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RemediationMath

2010

Ninth GradeRemediation Math

Table of Contents

Unit 1: Number and Operation........................................................................................1

Unit 2: Data Analysis.......................................................................................................13

Unit 3: Measurement.......................................................................................................24

Unit 4: Geometry..............................................................................................................39

Unit 5: Probability...........................................................................................................54

Unit 6: Algebra.................................................................................................................68

Unit 7: Algebra Connections...........................................................................................86

Louisiana Comprehensive CurriculumRemediation Math

Course Introduction

The Louisiana Department of Education issued the Comprehensive Curriculum in 2005. The curriculum was revised based on teacher feedback, an external review by a team of content experts from outside the state, and input from course writers in 2008. Remediation Math was added in 2010. It is to be taken by students who scored Unsatisfactory on the 8th Grade LEAP in mathematics. This course must be completed prior to enrolling in a mathematics course for Carnegie credit.

District Implementation GuidelinesLocal districts are responsible for implementation and monitoring of the Louisiana Comprehensive Curriculum and are delegated the responsibility to decide if

units are to be taught in the order presented substitutions of equivalent activities are allowed GLES can be adequately addressed using fewer activities than presented permitted changes are to be made at the district, school, or teacher level

Districts are requested to inform teachers of decisions made.

Implementation of Activities in the ClassroomIncorporation of activities into lesson plans is critical to the successful implementation of the Louisiana Comprehensive Curriculum. Lesson plans should be designed to introduce students to one or more of the activities, to provide background information and follow-up, and to prepare students for success in mastering the Grade-Level Expectations associated with the activities. Lesson plans should address individual needs of students and should include processes for re-teaching concepts or skills for students who need additional instruction. The total time frame for the provided units is 28 weeks to allow for such re-teaching. Appropriate accommodations must be made for students with disabilities.

FeaturesContent Area Literacy Strategies are an integral part of approximately one-third of the activities. Strategy names are italicized. The link (view literacy strategy descriptions) opens a document containing detailed descriptions and examples of the literacy strategies. This document can also be accessed directly at http://www.louisianaschools.net/lde/uploads/11056.doc.

A Materials List is provided for each activity and Blackline Masters (BLMs) are included to assist in the delivery of activities or to assess student learning. A separate Blackline Master document is included in the course folder posted at http://www.louisianaschools.net/lde/saa/2108.html.

GLEs addressed are listed in the title line for each activity in Career Diploma Courses. Underlined GLE numbers indicate the major emphases of the activity.

The Access Guide to the Comprehensive Curriculum is an online database of suggested strategies, accommodations, assistive technology, and assessment options that may provide greater access to the curriculum activities. Click on the Access Guide icon found on the first page of each unit or by going directly to the url http://sda.doe.louisiana.gov/AccessGuide.

http://sda.doe.louisiana.gov/AccessGuide

http://www.louisianaschools.net/lde/saa/2108.html

http://www.louisianaschools.net/lde/uploads/11056.doc


http://sda.doe.louisiana.gov/AccessGuide%00%E5%A1%B9%EF%92%81%E1%B4%BB%E4%A1%BF%E2%B2%AF%E5%B6%82%E8%97%84%E6%8C%A7%00%00%EA%AE%A5

2010 Comprehensive Curriculum


Unit 1: Number and Operations

Time Frame: Approximately four weeks

Unit Description

This unit focuses on operations with rational numbers including integers, fractions, decimals, ratios and proportions. A problem solving context is used that connects the mathematical operations to real life applications. Multiple ways of representing numbers will also be explored.

Student Understandings

Students will compare rational numbers in problem solving contexts and represent numbers in multiple ways. They will also develop a fundamental understanding of proportional reasoning and use these skills to solve real-life problems.

Guiding Questions

1. Can students compare rational numbers using symbols and position on the number line?

2. Can students simplify expressions using the order of operation rules and evaluate errors in the use of the rules?

3. Can students read and write large numbers in scientific notation? 4. Can students use proportional reasoning to solve real-life problems? 5. Can students find unit cost rates? 6. Can students communicate the meaning and application of key vocabulary terms from

this unit?

Unit 1 Grade-Level Expectations (GLEs)

GLE # GLE Text and BenchmarksGrade 8Number and Number Relations1. Compare rational numbers using symbols (i.e., <, , =, , >) and position

on a number line (N-1-M) (N-2-M)2. Use whole number exponents (0-3) in problem-solving contexts (N-1-M)

(N-5-M) 3. Estimate the answer to an operation involving rational numbers based on

the original numbers (N-2-M) (N-6-M)4. Read and write numbers in scientific notation with positive exponents (N-3-

M)5. Simplify expressions involving operations on integers, grouping symbols,

and whole number exponents using order of operations (N-4-M)

Remediation Math Unit 1Number and Operations 1



GLE # GLE Text and Benchmarks6. Identify missing information or suggest a strategy for solving a real-life,

rational-number problem (N-5-M)7. Use proportional reasoning to model and solve real-life problems (N-8-M)8. Solve real-life problems involving percentages, including percentages less

than 1 or greater than 100 (N-8-M) (N-5-M)9. Find unit/cost rates and apply them in real-life problems (N-8-M) (N-5-M)

(A-5-M)

Sample Activities

For this unit, have students maintain a learning log (view literacy strategy descriptions). A learning log is a notebook, binder or some other repository that students maintain to record ideas, questions, reactions, observations, thoughts, reflections, critical skill summaries, vocabulary self-awareness charts, vocabulary cards, etc. Documenting ideas in a log about the content being studied forces students to “put into words” what they know or do not know. Learning logs can become the place for virtually any kind of content-focused writing.

Activity 1: Comparing Rational Numbers (GLE: 1)

Materials List: number line (approximately 5 feet long, numbered from -5 to 5 in increments of ¼ ), Post-It® pad, Vocabulary Self-Awareness Chart BLM, Fractions, Decimals, Percents BLM, Comparing Rational Numbers BLM

Begin the unit the activity with a vocabulary self-awareness chart (view literacy strategy descriptions). Because students bring a range of word understandings to the learning of new topics in the content areas, it is important to assess students’ vocabulary knowledge before reading or doing other tasks involving text. This awareness is valuable for students because it highlights their understanding of what they know as well as what they still need to learn to fully comprehend the content.

Words are introduced at the beginning of the unit, and students complete a self-assessment of their knowledge of the words. Identify target vocabulary for the unit and provide students with a list of terms as shown in the Vocabulary Self-Awareness Chart BLM. Students may add terms to the list as the unit progresses. Each vocabulary word is rated according to the student’s understanding. A plus sign indicates a high degree of comfort and knowledge, a check mark indicates uncertainty, and a minus sign indicates the word is brand new to them. An example and a definition of the words should also be included. Over the course of the unit, students add new information to the chart. The goal is to bring all students to a comfortable level with the unit’s content vocabulary. Because students continually revisit their vocabulary charts to revise their entries, they have multiple opportunities to practice and extend their growing understanding of important content terms.

Add terms, if needed, and then distribute the Vocabulary Self-Awareness Chart BLM. Have students rate their understanding of each word. Ask students to keep this chart handy throughout






the unit and to add information as new words are learned. Students should complete the Vocabulary Self-Awareness Chart BLM in pencil and be prepared to revisit the Vocabulary Self-Awareness Chart BLM daily during the unit to make revisions and additions. When the unit is complete, the Vocabulary Self-Awareness Chart BLM should be complete and accurate. Students should place their completed chart in their math learning log (view literacy strategy descriptions).This will give them the opportunity to review it in the future, such as prior to testing.

Tell the students that comparing numbers is easier when the numbers are written in the same form. For example, when comparing two numbers, one written in fraction form and the other in decimal form, the comparison will be easier if the number written in fraction form is changed to its decimal representation. Review the process of converting between fractions, decimals, and percents with the students. Allow students to practice this skill using the Fractions, Decimals, Percents BLM.

Part A: Place the number line so that it is accessible to the students. Give each student in the class a Post-It® note with a number written on it. Be sure to include numbers written in different forms, pi, absolute values, square roots, fractions, decimals, negative numbers and positive numbers. After distributing the numbers, give the students about one minute to determine where on the number line their number should be placed (without discussion). Allow the students to place their numbers on the number line. Once the numbers have been placed on the number line, allow the students to discuss placement of the numbers and make any necessary changes. Prompt students about any incorrect placements if they don’t recognize them. After the numbers have been correctly placed, have the students draw the number line in their notebooks and place the numbers in the proper position. These notes will help them understand comparing these same numbers with symbols in Part B of this activity.

Part B: Have students write the definition of the following symbols and give an example of each: <, , =, , > Using the Post-It® notes from the number line above, randomly choose two of the numbers and place them on the board with space between them. Have a student volunteer to come to the board to write the appropriate symbol describing the relationship between the two numbers. Other students provide feedback. In addition to the symbol comparison, students should also compare the symbol representation to the number line placement from Part A of this activity. Tell students to note that the small number always lies to the left of the larger number on the number line. Have students use this reasoning as they explain why one number is less than or greater than another. Students record the correct answers in their notebooks. Students will use these notes as support when comparing individual sets of numbers using the Comparing Rational Numbers BLM. Give the students the Comparing Rational Numbers BLM and instruct them to use their notes from Parts A and B to complete the activity.





Activity 2: Order Matters! (GLE: 5)

Materials List: small (12 x 12) individual dry erase boards (one per group of four students), dry erase markers, erasers, Order Matters! BLM

In this activity, students will practice using the order of operations to evaluate expressions. Remind students of the mnemonic PEMDAS, which stands for

Please Excuse My Dear Aunt Sally| | | | | |Parentheses Exponents Multiplication Division Addition Subtraction

After reminding students of the mnemonic, share the meaning in terms of the following rules:

Rule 1: Simplify all operations inside parentheses (grouping symbols)Rule 2: Simplify all exponents, working from left to right.Rule 3: Perform all multiplication and division, working from left to right.Rule 4: Perform all addition and subtraction, working from left to right.

Give students the following problem to solve in their notebooks. 12 ÷ 3(7 – 4) + 23 – 5 Tell students to solve the problem using the rules above. Review the steps to solving the problem with the students.

White Board Activity (Explain to students that they should not talk or help each other during the process of solving the problems.)

1. Place students in groups of 4. 2. Provide each group with a dry erase board, marker and eraser.3. Give each group an order of operations problem on an index card. Be sure the problems

include integers, grouping symbols, and whole number exponents. 4. The first person in the group copies the problem on the white board and completes the

first step in solving the problem based on the rules above. 5. The first person then passes the board to the person to the right. That person completes

the next step. 6. This process continues until a final answer is determined. The person holding the white

board raises his/her white board and the teacher checks to verify that the answer is correct.

7. At this point, students in the group discuss the problem. Any incorrect answers must be reviewed and corrected by the group (with assistance from the teacher if necessary).

8. Repeat the process with two additional order of operations problems. (These can be the same problems completed by other groups in the class.)

This activity can be made even more interesting by giving each group the same problem and allowing groups to work competitively, without talking, to determine which group can solve the problem in the least amount of time.



Now that students have had practice and discussion in solving order of operations problems, have each student independently complete the Order Matters! BLM.

Note: Small dry erase boards are available through several companies that sell math manipulatives and/or school supplies. They can also be inexpensively made by going to a home improvement store (e.g., Home Depot, Lowe’s) and purchasing bathroom tile board. The store will cut them into the appropriate size for a nominal fee.

Part II

Challenge the students to create an order of operations problem using a modified story chain (view literacy strategy descriptions). Story chains are especially useful in teaching math concepts, while at the same time promoting writing and reading. The process involves a small group of students writing a story problem using the math concepts being learned and then solving the problem. Writing out the problem in a story provides students a reflection of their understanding. This is reinforced as students attempt to answer the story problem.

After a new math concept is learned, groups of students should be formed. The group size will vary depending upon the nature of the math concept/computation. The first student initiates the story. The next adds a second line; the next, a third line, etc. The last student is expected to solve the problem. All group members should be prepared to revise the story based on the last student’s input as to whether it was clear or not. Students can be creative and use information and characters from their everyday interests and media.

In this activity, students will develop and solve order of operations problems. Divide students into groups of four and assign each group an identifying number or name. Tell students that each group will use the following process to write order of operation problems that will eventually be given to the other groups to solve:

Student 1 – writes one expression and an operation on a sheet of paper – passes the sheet to Student 2

Student 2 – adds another expression and operation – passes the sheet to Student 3Student 3 – adds another expression and operation – passes the sheet to Student 4Student 4 – solves the problem and records the answerThe group determines if the answer is correct.

Example:Student 1 writes 42

Student 2 writes + (6 - 3)Student 3 writes − 8Student four solves and gets 11Written on the sheet of paper when the process is complete should be 42 + (6 - 3) – 8 = 11.

Repeat this procedure three times, with Student 2 starting a problem, then Student 3 starting another problem, and then Student 4 starting so that a total of 4 problems are created and solved.




Each group will transfer all the problems it created (but not answers) to a sheet of blank paper. Make sure each group also writes its group number or name on the paper. Collect the work and distribute to the other groups to solve. When all groups are finished, the sheets are returned to the groups that created the problems. If there are discrepancies in solutions between the groups’ answers, there should be discussion between the groups to determine what the correct solutions should be.

At the end of this activity, have each student write a reflection on the process for solving problems using the order of operations in their own words. Also, have students list the most common mistake made by them and the members of their group. This reflection should be placed in their math learning log (view literacy strategy descriptions). Documenting ideas in a log about content current being studied forces students to “put into words” what they know or do not know. This process offers a reflection of understanding that can lead to further study and alternative learning paths.

Activity 3: Who’s Right and Who’s Wrong? (GLE: 5)

Materials List: Who’s Right and Who’s Wrong BLM

Provide the students with the Who’s Right and Who’s Wrong BLM. This BLM provides students with order of operations problems and 3 answers to each. Only one of the three answers will be correct. The students will determine which answer is correct and determine the order of operation rule that was broken that led to the two incorrect answers.

Activity 4: How Big is the Number? (GLE: 4)

Materials List: Internet access, Scientific Notation and Rounding BLM, Research Summary BLM

Explain to students that scientific notation is a simplified way to express very large (or very small) numbers with many zeros. For example, the number 125,000,000 is written in standard notation. This number written in scientific notation is 1.25 × 108. Numbers written in scientific notation have one nonzero digit to the left of the decimal, followed by any significant digits written to the right of the decimal point in the number (the zeros at the end are dropped, but embedded zeros are kept). This decimal is then multiplied by a power of 10 which would produce a number equal to the original number. Demonstrate to students that 10,500 would be written as 1.05 x 10 4. The two zeros at the end are dropped, but the 0 embedded between 1 and 5 is kept.

Ask students if they remember how to easily find the power of ten that is required for the scientific notation. They should know that there is a relationship between the power and the number of places that the decimal is moved.




Write 125,000,000 on the board and review the actual movement of the decimal with the students. Give students the numbers 56,000,000 and 5,900,000,000 to practice writing in scientific notation. Provide another number with embedded zeros for students to try.

Also explain to students that in some situations numbers are rounded before writing in scientific notation. To round a number to a given place value, first locate the digit in the place value you are rounding to. Look to the right of that digit. If the number to the right is 5 or greater, add one to the digit and drop the remaining digits. If the number to the right is 4 or less, keep the digit as is and drop the remaining digits. Ask students to round 34,609 to the nearest thousand (as indicated by the arrow) and to round 1,749,000 to the nearest hundred thousand (as indicated by the arrow). 34609 1749000

Be prepared to review place value and its relation to the rounding algorithm, especially if students struggle with the process.

Provide students with the Scientific Notation and Rounding BLM for additional practice.

Once students have completed the practice activities with scientific notation and rounding, give each student a copy of the Research Summary BLM. Students will need access to the Internet to find the information for the activities. Students may work individually or in pairs to complete the Research Summary BLM. In the first part of the BLM, the students will find the distance, in miles, of each planet from the sun and write in standard and scientific notation. In the second part of the BLM, the students will find the populations of 10 Louisiana cities of their choice, write the population in standard notation, round the population to the nearest ten thousand, and write the rounded population in scientific notation.

Activity 5: Estimation is the Key (GLE: 5)

Materials List: Estimation BLM

Let students know that an exact answer is not always necessary. In many real life situations, an estimate can be very useful. One of the best approaches in determining estimates is to round the numbers in a problem prior to calculating. Distribute the Estimation BLM to the students and have each student complete it individually. Review the directions to the activity and complete the problem below as an example.

1. Determine an estimated answer for each problem without a calculator. The goal is to estimate the answer within 10% of the exact answer. Show work and answer in the second column for each problem.

2. Once the second column for each problem is complete for all problems, determine the exact calculation.

3. Determine the percent difference by using the following formula:



[(Estimate – Exact) ÷ Exact] × 100. If the percent difference is ± 10%, use a different strategy, place in part B and determine the percent difference. If your estimated answer is within ± 10% of the exact answer, move to the next problem.

403 × 2.09a. _____ × _____ = ______

b. _____ × _____ = ______403 × 2.09 = _____

a. ______

b. ______

Note: Encourage students to try to determine the exact answer without the use of a calculator.

Activity 6: A Good Estimate May Be All You Need (GLE: 3)

Materials List: Estimation Situations BLM, paper, pencil

Remind students about their previous work in Activity 5. Explain to students that in the current activity they will continue their estimation work with real-world scenarios. Ask the students for real-world situations where just an estimate may be appropriate. Some of the most common student suggestions may be dealing with making purchases, buying on sale, traveling, earning money from a part-time job, etc. Inform the students that these are the kinds of situations they will be dealing with in this activity. Since in many of these situations the student may not have a calculator available, do not allow students to use a calculator for this exercise. Distribute the Estimation Situations BLM to the students. Review the first scenario with the students. Tell the students to try to envision themselves in these situations as they do their work. At the conclusion of this activity, have student volunteers present a problem including the estimation strategy used, the answer, and why the answer is a reasonable estimate.

Activity 7: Proportional Thinking (GLEs: 7, 8, 9)

Materials List: Vocabulary Card BLM, pencils, index cards, paper

This activity will incorporate the use of a modified vocabulary card strategy (view literacy strategy descriptions). When students create vocabulary cards, they see connections between words, examples of the word, and the critical attributes associated with the word. This vocabulary strategy also helps students with their understandings of word meanings and key concepts by relating what they do not know with familiar concepts. Vocabulary cards require students to pay attention to words over time, thus improving their memory of the words. In addition, vocabulary cards can become an easily accessible reference for students as they prepare for tests, quizzes, and other activities with the words.

Proportional reasoning is one of the “Big Ideas” of math that is the capstone of the elementary math curriculum and the cornerstone of algebra and beyond. Therefore, it is critical that students develop a deep and meaningful understanding of this concept. One of the best ways to ensure that students develop that type of understanding is to begin with a meaningful understanding of the vocabulary terms and how they are applied. Vocabulary cards require students to think through the meaning of the word and provide an example of its use in context.





Begin the study of proportional reasoning with a class discussion in a questioning format that helps students to remember prior learning and make connections between previously learned material. Examples of questions to ask are listed below. These questions are intended to get students to think about their prior learning experiences. Allow students to openly discuss their thoughts.

1) What is a ratio? How can it be written?2) What is a proportion? How does it relate to a ratio?3) Is a fraction a ratio? What is an example of an equivalent fraction?4) What is a real life example of a rate? What is a unit rate?5) How have you seen proportions used in geometry?6) Is pi a ratio?

Following the discussion, let students know that they will now begin the process of developing a deeper understanding of these foundational concepts that will help them not only in algebra, but in solving real-life problems. Provide students with a list of vocabulary words that they need to understand. Suggested words are ratio, proportion, rate, unit rate, percent, fraction, and equivalent fraction. Other words should be added as necessary. Hand out the Vocabulary Card BLM. Review the general Vocabulary Card BLM with the students and explain the different parts. Then share the completed Vocabulary card for the word ‘Ratio’ with the students. Address any questions students may have. Now pass out the index cards and allow students to work in pairs to complete a vocabulary card for each of the words. Provide resource materials (textbook or handouts) for the students to use to assist them in completion of the cards. A good resource for teachers can be found at the following website: http://wps.ablongman.com/ab_vandewalle_math_5/12/3114/797191.cw/index.html

Observe and support students as they complete the cards. Check the cards as the students work through the development to ensure that the cards are accurate. One option is to have a general discussion of the terms before the cards are completed. The wording can be developed by the class with the teacher’s input to ensure correct information is recorded. In closing this lesson, let students know that they need to study their cards and be prepared to use them as they complete the other activities in this unit.

Activity 8: Rates, Ratios and Proportions (GLEs: 7, 9)

Materials List: It’s All in the Thinking BLM, pencil, paper, calculator

Have students review their vocabulary cards from Activity 7. Discuss the questions from Activity 7 to help students make connections with what they have learned from that activity.

Review and practice writing ratios with students using the examples provided below. Students should write the examples in their notebooks for future reference. Emphasize the importance of the order of the wording in a problem for writing ratios. Use examples like the following for the practice. Write the following ratios:


http://wps.ablongman.com/ab_vandewalle_math_5/12/3114/797191.cw/index.html


1) 5 dogs to 6 cats2) 18 CDs to 8 DVDs3) The ratio of boys to girls in this class

Now have students review the definition of proportion from their vocabulary cards. Discuss the definition and connect it to ratio as “two ratios equivalent in value to each other.” Show how equivalent fractions have the same value and that ratios can be fractions. Provide examples.

Demonstrate that when two fractions are equivalent, their cross-products are equal. Example:

2 10 = 4 5

20 = 20

The property of equivalent fractions can be used to find an unknown denominator or numerator in one of the fractions. Model this for the students using an example similar to the one below.

Step 1: =

Step 2: 3(x) = (6)(2)

Step 3: 3x = 12

Step 4: =

Step 5: x = 4

Therefore, =

As a word problem, this could have been the following:

In Ms. Jones’ class the ratio of boys to girls was 2 to 3. If the same ratio is in Mr. Smith’s class and he has 6 girls in his class, how many boys are in his class?

Let the students know that they will now practice solving proportion problems. Distribute the It’s All in the Thinking BLM. Encourage students to use calculators only when necessary.



Activity 9: Working with Percents (GLE: 8)

Materials List: Working with Percents BLM, pencil, paper, calculator

In this activity, students will solve real-life problems dealing with percentages. Have students review their percent vocabulary card from Activity 7. Remind students that percent means part of 100 and that percents can also be written as fractions and decimals. Remind students that they practiced this in Activity 1. Review if necessary.

Review the following examples with students to remind them of how to find the percent of a number and use this in real-life problems.

Example 1: Sixty percent of Mr. Jones’ class passed his English test. There are 30 students in his class. How many students passed the English test?

Example 2: Best Buy has a sale on all electronic equipment. David wants to purchase a camera whose regular price is $149.99. If the camera is on sale for 25% off what is the amount of the discount and what is the sale price of the camera not including tax?

As these examples are reviewed with students, remind them how to change a percent to a decimal and how to use strategies for rounding learned in Activity 1. Distribute the Working with Percents BLM to students.

Activity 10: 4-Step Problem Solving (GLEs: 2, 6)

Materials List: 4-Step Problem Solving BLM, pencil, paper, calculator

Ask students about some of the problem solving strategies they have used to solve word problems. Some of the examples they may give are draw a picture, guess and check, work backwards or make a table. Remind students of the 4-Step Problem Solving Method: (1) understand, (2) plan, (3) solve and (4) look back. Discuss each step with a focus on the following:Understand: Read and re-read the problem to get the facts and understand the question(s). Is enough information given? Plan: Based on the information given, determine a strategy that can be used to solve the problem.Solve: Use the selected strategy to solve the problem.Look Back: Determine if the answer makes sense. Check for errors.

Give the students the following problem and ask them to solve it using the 4-Step method.

Sarah’s mother works part time at the local Wal-Mart where she earns $8.75 per hour. She usually works 20 hours per week. She plans to buy a new High Definition television that costs $1189. She will pay 9% tax on the television. Will she earn enough money



working her normal hours over the next one-and-a-half months to purchase the television?

Ask for volunteers to share their processes and answers with the class, and be sure that students identify information for each step of the strategy. Give students the 4-Step Problem Solving BLM and have them work in pairs to solve each problem. They should also be prepared to identify any problem in which there is not enough information to solve.

Sample Assessments

General Assessments

Have students work in pairs to develop five real-life word problems involving percents, proportions or unit rates. Provide students with examples such as shopping, investments, commissions, traveling, sports, etc. Ask each pair to choose one of its problems to present to the class.

Have students review their learning log entries and write summary reflections on what they have learned during this unit. This reflection should be placed in the learning log.

Activity-Specific Assessments

Activity 2 : Assess the Order Matters! BLM using the answer sheet. Provide feedback to students. Have them determine and write about any mistakes they have made and then correct their work.

Activity 4: Assess the Scientific Notation and Rounding BLM using the answer sheet. Provide feedback to the students.

Activity 7 : Have students use the Quiz-Quiz-Trade strategy to review their vocabulary

cards. Give each student a card or slip of paper with the question (or vocabulary word) on one side and the answer (or definition) on the other side.

Have all students stand and pair up. One student reads the question to the other student. The other student answers. The first student corrects, if necessary, then gives a praise statement about something positive that was said (whether the answer is right or wrong). The students switch roles. When both questions have been asked and answered, the students trade questions and then raise their hands. They look around for other students with their hands raised, and pair up with new partners. Allow this process to continue for about 10 minutes or 5 rotations.




Unit 2: Data Analysis

Time Frame: Approximately four weeks

Unit Description

This unit focuses on representations of data using appropriate graphs and displays and analyzing and making predictions from data. Central tendency and how factors in a data set affect these measures are also explored.


Students distinguish between the characteristics of the different types of graphs including line graphs, circle graphs, histograms, bar graphs, scatter plots and stem-and-leaf plots. Based on understanding the characteristics of graphs, students determine the best data display for a given situation. Students organize and display data using circle graphs and the box-and-whisker plot. Students develop a deep understanding of central tendency and are able to explain how specific data affects measures of central tendency and which measure of central tendency is most appropriate for a given situation.

Guiding Questions

1. Can students determine the most appropriate data display for a given situation? 2. Can students develop and analyze a box-and-whisker plot by using its range and

quartiles? 3. Can students accurately determine how specific data factors affect the measures of

central tendency of a given set of data? 4. Can students choose the most appropriate measure of central tendency for a given

situation? 5. Can students match a graph to a described situation using the characteristics of the

graph? 6. Can students construct a circle graph?


GLE # GLE Text and BenchmarksGrade 8Data Analysis, Probability, and Discrete Math34. Determine what kind of data display is appropriate for a given situation (D-

1-M)35. Match a data set or graph to a desired situation, and vice versa (D-1-M)36. Organize and display data using circle graphs (D-1-M)37. Collect and organize data using box-and-whisker plots and use the plots to

interpret quartiles and range (D-1-M) (D-2-M)

Remediation Math Unit 2Data Analysis 13



GLE # GLE Text and Benchmarks38. Sketch and interpret a trend line (i.e., line of best fit) on a scatter plot (D-2-

M) (A-4-M) (A-5-M)39. Analyze and make predictions from discovered patterns. (D-2-M)40. Explain factors in a data set that would affect measures of central tendency

(e.g. impact of extreme values) and discuss which measure is most appropriate for a given situation (D-2-M)

Sample Activities

Activity 1: Understanding Graphs (GLE: 34)

Materials List: sources of data for graphs (magazines, newspapers, etc.), colored paper, construction paper or posters, scissors, glue sticks, markers, Vocabulary Self-Awareness Chart BLM, Understanding Graphs BLM, Comparing and Contrasting Graphs Word Grid BLM, learning log

In Part 1 of this activity, the students will review important vocabulary terms using vocabulary self-awareness (view literacy strategy descriptions). Distribute the Vocabulary Self-Awareness Chart BLM to the students and have them complete it. In completing the chart, the students have an opportunity to reactivate their prior knowledge about graphs and highlight their understanding of what they know, as well as what they still need to learn to fully comprehend the content. The students will revisit the chart during the course of the unit, revise their entries, and have multiple opportunities to practice and extend their understanding of important content terms.

In Part II of this activity, the students continue to develop their knowledge of graphs by reacquainting themselves with information studied in previous classes. Prior to beginning this part, give the students a comprehensive overview of graphs to help develop a foundation for this activity. This overview should be done in a discussion format with a questioning approach to help students continue to connect prior learning experiences. Use the Understanding Graphs BLM to support this overview discussion. Be sure to review and discuss which type of graph is best for a specific situation and when it is inappropriate to use a particular type of graph.

Place students in groups of three and provide them with data sources from which to make the following types of graphs: bar graph, histogram, stem-and-leaf plot, line graph, circle graph, and scatter plot. Each group should be identified by a name or number. These should be creative projects such as foldables, posters, brochures, etc.

The projects should include(1) name of graph(2) example of graph (drawn or from magazine or newspaper)(3) the type of data displayed by this graph(4) example of when this type of graph may be used




(5) Reflection question: What should be considered when choosing an appropriate graph for a given set of data? (Answers from this question will be used in subsequent lessons.)

The completed projects should be presented to the class and displayed in the classroom for reference during the study of the unit.

To review and deepen the understanding of graphs, a word grid (view literacy strategy descriptions) will be used to compare and contrast graphs. To take full advantage of word grids, they should be co-constructed with students, so as to maximize participation in the learning process. Place a simple word grid on the wall that will serve as an example for explaining how it’s constructed and used. After analyzing a demonstration word grid, students will be much better prepared to create and study from one with actual disciplinary content.

A sample word grid is provided below.

Sample word grid for fruitTree grown Edible skin Citrus Has seeds

Orange 2 1 2 2Apple 2 2 0 2Strawberry 0 2 0 2Grapes 0 2 0 1Banana 2 0 0 0 2 – always 1 – sometimes 0 – never

Provide students with a blank word grid. A large version of the grid could be put on poster paper and attached to the wall or one could be projected from an overhead or computer. As critical related terms and definitions are encountered, students should write them into the grid. Invite students to suggest key terms and features, too.

Distribute the Comparing and Contrasting Graphs Word Grid BLM. Begin the discussion of similarities and differences between graphs by referring to the information from the discussion using the Understanding Graphs BLM. Pause to allow students time to digest this discussion, and invite the students to suggest entries for the word grid. As each entry is added to the word grid, place the proper number in the appropriate row. See the example of the partially completed Comparing and Contrasting Graphs Word Grid BLM below.

Comparing and Contrasting Graphs Word GridHas x and y

axisCategorical

dataNumerical data only

Individual data points

Circle graph 0 2 1 0Line graph 2 0 2 2 2 – always 1 – sometimes 0 – never

Once the grid is complete, allow students to review with a partner the information in the grid before quizzing them about the similarities and differences between the graphs. In this way,





students will make a connection between the effort they put into completing and studying the grid and the positive outcome on assessments and other tasks that require them to distinguish between the characteristics of the different graphs. Students should place the completed grid in their learning log (view literacy strategy descriptions). Explain to the students that the learning log will be kept in the classroom as a repository that they maintain in order to record ideas, questions, reflections and to summarize newly learned content. Keeping this information in the learning log will provide students with easy access to this information for review for tests and quizzes and also serves as reference materials for upcoming units.

Activity 2: Data, Data, Data (GLEs: 34, 36)

Materials List: graph paper; pencils; Data, Data, Data BLM

In this activity, students continue in the same groups as in Activity 1. Give each group a set of data from Part I of the Data, Data, Data BLM. The same data should be used with more than one group for comparison and discussion between groups. Based on the data, the students choose an appropriate graph for displaying the data. Each individual student in the group then displays the data graphically using the chosen graph. This gives each student an opportunity to practice his/her graphing skills with the support of his/her group members.

As the groups complete their graphs, ask students to choose one of their group member’s graphs to share with another group along with the data set they were given. The other group completes an analysis of the graph using the template found in Part II of the BLM. As students work on this activity, the teacher should use questioning strategies to help the students extend their reasoning in choosing the appropriate graph to further develop their understanding of the process.

Activity 3: Data Everywhere! (GLEs: 38, 39)

Materials List: Data Everywhere! BLM, graph paper, tape measure, pencils

Begin the discussion for this activity with an overview of scatterplots. Students need to understand that when graphing real life data, a straight line does not always emerge. Investigating scatterplots will give students an opportunity to explore different data sets that demonstrate different kinds of relationships or no relationship at all. Explain to students that if two variables are related linearly, the scatterplot of their data will approximate a straight line. When this exists, this line is called a line of best fit because it describes the general trend of the data. Share graphical depictions of scatterplots with positive, negative and no trends.

Provide students with the Data Everywhere! BLM and graph paper. Instruct the students to graph the data in Table 1 and sketch a trend line to determine the relationship between the data. For Table 2, have the students work in groups of three to measure each students’ arm span and height in centimeters using the measuring instruments provided. Model the proper technique for performing the measurements. Have students place their measurements in Table 2, and display




them on the overhead or a large chart. Ask students to record the data on the BLM, graph the data, and sketch a trend line. Have the class reflect on the relationships between the values in the tables. Make sure students made the right connections.

Activity 4: Circle Up (GLE: 36)

Materials List: compasses or discs for drawing circles, protractors, pencils, Circle Up BLM

Remind students that circle graphs are usually used to quickly display information about the relationship of parts to a whole. One example is the use of percentages to represent data.

Review the following example with the students.

Six hundred people were surveyed using the following question: Which one of the following four teams, New Orleans Saints, Indianapolis Colts, Dallas Cowboys, or New York Giants, do you think will win the Super Bowl? The responses were as follows:

NFL TeamNo. of

ResponsesNew Orleans

Saints 300Indianapolis

Colts 150

Dallas Cowboys 90NewYork

Giants 60

Present the data to the students. Ask the students to define the whole and the parts. The whole is the total number of responses and the parts are the numbers of people who voted for the different NFL teams. Now tell the students to determine the percentage of responses for each team. Support students as they work through this process. Once complete, they should have the following information.

NFL TeamNo. of

ResponsesPercent of Responses

New Orleans Saints 300 50%

Indianapolis Colts 150 25%

Dallas Cowboys 90 15%New York

Giants 60 10%



Ask students if there is anything they notice about the percentages. Things they might notice are twice as many voted for the Saints as the Colts, and the sum of those voting for the Cowboys and the Giants is the same as the number who voted for the Colts. If students don’t notice or remember that the sum of all percentages should be 100%, provide leading questions which will allow them to remember this and why it is true. To ensure that students will be able to accurately make the sectors of a circle graph, review the proper use of a protractor. Draw an angle on an overhead or on the board and model using a protractor to measure the degrees in the angle. Give the students a protractor and several angles to measure. Monitor students as they measure the angles to ensure proper use of the protractor.

Tell the students that they are now ready to construct the circle graph for the NFL data above. Draw a circle on the overhead or board and model how to use the protractor to draw various sectors of the circle based on the percentages given. Have the students focus on the following steps:

1. Convert each percent into degrees by finding the % of 360°. Place this data in the table by adding a column.

2. Draw a line from the center of the circle to the edge (a radius).3. Use the protractor to measure the first angle using the center point as 0°.4. Draw a line from that point to the edge of the circle (another radius).5. Using the last radius as the 0° mark, measure the number of degrees for the next

angle.6. Continue this process to get the degrees for each segment.

Suggest to the students that they may want to shade the sections of their circles with different colors or fill them with different designs to help distinguish between them. Now have students title and label their circle graphs.

Ask the students if they see any interesting relationships between their segments. For example, 50% for New Orleans represents half of the circle and the 25% for the Indianapolis Colts is half of a half or a fourth of the circle. Note: Because the sectors’ sizes for this graph are easy to draw, the teacher may want to point out that a protractor here is not really needed; however, knowing how to find and use the degree measures for the sectors will be critical when completing the BLMs.

Distribute the Circle Up BLM. For the first part of this activity, students are presented with car sales data from a local dealership and will graphically display the distribution of models sold by constructing a circle graph. Students should use their protractors to determine the sector sizes.

For the second part of this activity, students will brainstorm about the items that contribute to the cost of a vacation trip. They will include things like travel, lodging, meals, entertainment, etc. They will determine costs for each area, calculate the percentages, and construct a circle graph that graphically shows the cost distribution for their vacation.

At the end of this activity, have the students write a reflection on the process for constructing a circle graph. Ask students to place their completed graphs from the BLM and the reflection in



their learning logs (view literacy strategy descriptions). Explain to the students that the learning log will be kept in the classroom as a repository that they maintain to record ideas, questions, reflections and to summarize newly learned content. Keeping this information in the learning log will provide students with easy access to this information for review for tests and quizzes and also serves as reference materials for upcoming units.

Activity 5: A Picture is Worth a Thousand Words (GLEs: 35, 39)

Materials List: A Picture is Worth a Thousand Words BLM, paper pencil

This activity helps students think about change. What is changing? Over what time period is it changing? What is the rate? Is there a pattern? As an introduction to this activity, provide students with a graph that shows something changing over time with numbers included. Have the students explain the graph. Explaining the graph will help the students to verbalize the relationship between the two parameters. Then explain to them what meaning the graph would have if there were no numbers. Show the same graph with no numbers. The discussion should focus on how the change occurs over time.

Now share with students a new graph that does not have numbers, such as the one below. Have the students brainstorm what the graph could represent. One example might be, as the number of dogs at the animal adoption center increases, the pounds of dog food consumed increases. Ask students for other examples.

Now present the A Picture is Worth a Thousand Words BLM to the students and have them work in pairs. Instruct them to follow the directions in Part I by writing a short story to explain what could be happening in each graph. In Part II, they should sketch a graph to represent the given information. Each pair will present one of its stories from Part I or a sketch of one of the graphs from Part II.




Activity 6: Center Stage! (GLE: 40)

Materials List: calculator, paper, pencil, Center Stage BLM

To interpret and analyze measures of central tendency, students must first understand the concept of central tendency and how to calculate the different measures. Explain to students that measures of central tendency are ways to describe where the center of a set of data lies. Typical measures of central tendency are mean, median and mode, with the mean being the most common. Also share that another name for the mean is the average. Have the students refer to the Vocabulary Self-Awareness Chart BLM completed in Activity 1 to review and update these terms. Use the following set of data to allow the students to practice calculating each of these measures:

14, 18, 19, 16, 14, 20, 12, 18, 14Mean – The sum of the data items, divided by the number of data items.Median – The middle number or the average of the two middle numbers when the data items are listed in order.Mode – The most frequently occurring item, or items, in a data set. There may be more than one mode or no mode.Range – The difference between the largest and the smallest numbers in a set of data.

For this data set, Mean = 16.1, Median = 16 and Mode = 14, Range = 8

Emphasize how close the mean and median are to each other.

Have the students complete a SPAWN (view literacy strategy descriptions) writing assignment. SPAWN is an acronym that stands for five categories of writing options (Special Powers, Problem Solving, Alternative Viewpoints, What If? and Next). Begin by targeting the kind of thinking students should be exhibiting. If students are to anticipate the content to be presented or reflect on what has just been learned, then certain prompts work best.

Next, select a category of SPAWN that best accommodates the kind of thinking about the content that students are to exhibit. For example, if the teacher wants students to regard recently learned material in unique and critical ways, the Alternative Viewpoints category prompts writing of this nature. If, on the other hand, the teacher desires students think in advance about an issue and brainstorm their own resolutions, the Next and Problem Solving prompts may work best.

Present the SPAWN prompt to students. This can be done by simply writing it on the board or projecting it from the overhead or computer. If an anticipatory prompt, students will need to see it and begin writing before the new material is presented. If a reflective prompt, it should be revealed after new content has been covered. Allow students to write their responses within a reasonable period of time. In most cases, prompts should be constructed in such a way that adequate responses can be made within 10 minutes. Students should be asked to copy the prompt in their notebooks before writing responses and to record the date.

Since this is not formal writing, SPAWN writing should not be graded as such. Instead of a thorough assessment of students’ SPAWN writing, most teachers who use this strategy give




simple grades such as points for completing responses. SPAWN writing should be viewed as a tool students can use to reflect on and to increase their developing disciplinary knowledge and critical thinking.

For this assignment the student will write from a W or, What If? perspective. This is an individual writing task. Give the students the following prompt:

What would happen to the mean, median and mode if one of the values of 14 was replaced with a 35 and then with a 4?

Have students predict what will happen before performing the actual calculations.

Give students about 10 minutes to complete their written response. When all students are finished, ask students to share their ideas with the class and discuss the responses. Then have students determine the effects through calculations.

Allow this discussion to lead to a discussion about the meaning of the different measures from a real life perspective. For example, what if students owned a shoe store and needed to buy more shoes. Would they want to know the mean shoe size or the mode shoe size? Let this lead to a discussion of when it is appropriate to use mean, median or mode.

Give the students the Center Stage BLM and ask them to complete it individually. Once the activity is complete, as a class, discuss numbers 6 through 10 asking students to share their reasoning for their choice.

Activity 7: Box-and-Whisker Plots (GLE: 37)

Materials List: paper, pencil, Box-and-Whisker Plots BLM

Begin the class discussion with a review of the parts of a box-and-whisker plot and an explanation of each part. Display a box-and-whisker plot like the one below and refer to the different parts during the review.

The first quartile is the middle (the median) of the lower half of the data. One-fourth of the data lies below the first quartile and three-fourths lies above. The first quartile also identifies the 25th percentile, indicating that 25% of the values in the data set lie below this value.

The second quartile is another name for the median of the entire set of data. The median is the 50th percentile indicating that half of the data falls below and half above this value.

The third quartile is the middle (the median) of the upper half of the data. Three-fourths of the data lies below the third quartile and one-fourth lies above. The third quartile then identifies the 75th percentile for the data set.

The interquartile range is the difference between the third and first quartiles. The lower extreme is the lowest value in the data set. The upper extreme is the highest value in the data set.



Review the construction of a box and whisker plot with the following data:

80, 75, 90, 95, 65, 65, 80, 85, 70, 100

Median: 80 Minimum: 65Lower quartile: 70 Maximum: 100Upper quartile: 90 Interquartile range: 20

This should be modeled as a whole class activity by using a large sheet of bulletin board or butcher paper. Students are given the numbers on post-it notes and allowed to complete the process by placing the numbers in order, finding the median, lower quartile, upper quartile, etc. The teacher supports this through questioning such as, “How do you find the median of this data set?” Once the students have identified the specific data, students then plot the appropriate points and draw the box-and-whisker plot. The teacher should ask interpretive questions such as, “What’s happening in the lower quartile?”

Support students in working through these calculations and explaining to them what the measures mean. For instance, the median is the middle value of all the data if data is placed in numerical order and splits the data set in half; the first quartile is the median or middle of the lower half of the data, etc.

Instruct students to copy the example in their notes and ask any remaining questions. Now present students with the Box-and-Whisker Plot BLM to allow them to continue to practice.

Sample Assessments

General Assessments

Provide the student with a mean, median and mode of a set of data. Have students create a set of data that will result in the given mean, median and mode.

Have the students estimate the sectors of a circle graph for the following set of data: 25%, 50%, 12% and 13%.

Respond to the following prompt: If the outlier in a data set is extremely low, how does it impact the mean of the data set? If the outlier is removed, how is the mean affected?




Activity 1 : Have the students select two graphs from various magazines, newspapers, etc. and explain why they believe the author used this type of graph.

Activity 4 : Have students survey the class on their favorite music artist (given 5 to choose from). Students then use this data to construct a circle graph representing the data.

Activity 5 : Have students sketch a graph of the following situations: (1) water level in a bathtub as the water is draining, (2) the temperature over a 24 hour period on a July day in Baton Rouge (or a city near their homes).

Activity 6 : Have each student write a reflection on the use of mean, median and mode in real-life situations.




Unit 3: Measurement

Time Frame: Approximately 4 weeks.

Unit Description

This unit explores basic concepts in measurement (metric and customary) in a real-world problem solving context with an emphasis on appropriate units and conversions between units within the same system. Basic 2- and 3-dimensional shapes, their surface areas, and volumes are also explored.


Students develop an understanding of various units of measure in the U.S. and metric systems and practice converting between units within the same systems. In doing so, they deepen their understanding of calculating the area and volume of rectangular prisms and cylinders. They also recognize the relationships between changes in dimensions and the resulting changes in volume of rectangular prisms. Students use proportional relationships and reasoning to make conversions and draw representations to visualize the connections between units.

Guiding Questions

1. Can students accurately determine the surface area and volume of prisms and cylinders?

2. Can students convert between U.S. and metric units within the same system?3. Can students choose the appropriate measurement units in real world applications?4. Can students describe the relationship between a change in linear dimensions and the

change in volume of a rectangular prism and its volume?


GLE # GLE Text and BenchmarksGrade 8Number and Number Relations7. Use proportional reasoning to model and solve real-life problems (N-8-M)9. Find unit/cost rates and apply them in real-life problems (N-8-M) (N-5-M) (A-5-M)Algebra16. Explain and formulate generalizations about how a change in one variable results in

a change in another variable (A-4-M)Measurement17. Determine the volume and surface area of prisms and cylinders (M-1-M) (G-7-M)18. Apply rate of change in real-life problems, including density, velocity, and

international monetary conversions (M-1-M) (N-8-M) (M-6-M)

Remediation Math Unit 3Measurement 24



GLE # GLE Text and Benchmarks19. Demonstrate an intuitive sense of the relative sizes of common units of volume in

relation to real-life applications and use this sense when estimating (M-2-M) (G-1-M)

20. Identify and select appropriate units for measuring volume (M-3-M)21. Compare and estimate measurements of volume and capacity within and between

the U.S. and metric systems (M-4-M) (G-1-M)22. Convert units of volume/capacity within systems for U.S. and metric units (M-5-M)Geometry27. Construct polyhedra using 2-dimensional patterns (nets) (G-4-M)32. Model and explain the relationship between the dimensions of a rectangular prism

and its volume (i.e., how scale change in linear dimension(s) affects volume) (G-5-M)

Patterns, Relations, and Functions48. Illustrate patterns of change in dimension(s) and corresponding changes in volumes

of rectangular solids (P-3-M)

Sample Activities

For this unit, have students maintain a learning log (view literacy strategy descriptions). A learning log is a notebook, binder or some other repository that students maintain in order to record ideas, questions, reactions, observations, thoughts, reflections, critical skill summaries, vocabulary self-awareness charts, vocabulary cards, etc. Documenting ideas in a log about the content being studied forces students to “put into words” what they know or do not know. Learning logs can become the place for virtually any kind of content-focused writing.

Activity 1: Understanding Surface Area (GLEs: 17, 27)

Materials List: rulers, Post-it® notes, markers, boxes (rectangular and cylindrical), calculators, one inch square tiles, Vocabulary Self-Awareness Chart BLM, Surface Area BLM

This activity uses a modified vocabulary self-awareness (view literacy strategy descriptions) strategy that has been modified by adding a section for students to include common units for measuring volume and surface area. Although this activity addresses surface area only, it is important to assess student’s awareness of surface area and volume at the beginning of this unit. This awareness is valuable for students because it highlights their understanding of what they know, as well as what they still need to learn, to fully comprehend the concepts.

Add terms to the chart as needed, and then distribute the Vocabulary Self-Awareness Chart BLM. Have students rate their understanding of each word. Ask students to keep this chart handy throughout the unit and to add information as new words are learned and as they get a better understanding of the common units used for measuring surface area and volume. Students should complete the Vocabulary Self-Awareness Chart BLM in pencil and be prepared to revisit it daily during the unit to make revisions and additions. When the unit is complete, the





Vocabulary Self-Awareness Chart BLM should be complete and accurate. Students should place their completed chart in their math learning log ((view literacy strategy descriptions). This will give them the opportunity to review it in the future, such as prior to testing.

Review the process, with examples, for determining the area of a rectangle. Place emphasis on using the proper units. Then move into a discussion of surface area using a box to model surface area. Ask students for real-life applications of surface area. Expect answers such as to determine how much paper needed to wrap the box or how much paint is needed to paint the box. Have students update their vocabulary self-awareness chart with real-life examples.

Remind students of their work with calculating surface area in previous grades. Let them know that this activity will help them to fully understand the formula used for calculating the surface area of a rectangular prism or box. Place students in cooperative learning groups of three. Distribute the following materials to each group: a rectangular prism (box), ruler, square inch tiles, calculator, and the Surface Area BLM. Instruct students to complete the activity as indicated on the BLM which requires them to create two-dimensional nets for their three-dimensional boxes. Monitor the groups as they work to ensure that the students are following directions and measuring the surface areas properly. Some boxes will have extensions used to connect the sides of the boxes. For example, there may be a flap that is used to insert the top so that it closes securely. Make sure students measure only the part of the boxes that is considered exposed surface area. As the groups finish the rectangular prism activity, allow groups to work on the extension activity using the cylindrical box. All groups do not have to complete the extension activity. Each group will have an opportunity to see the work of the other groups in the next part of this activity.

The professor know-it-all (view literacy strategy descriptions) strategy will be used to conclude this activity. Once coverage of content has been completed, the professor know-it-all strategy can be enacted. The strategy is appropriate to review material learned for the purpose of internalizing the content. In this activity, the students will use the strategy to show their understanding of surface area.

Begin by forming groups of three or four students. Give students time to review the content just covered. Indicate to students that groups will be called on randomly to come to the front of the room and provide “expert” answers to questions from their peers about the process they used for determining the surface areas of their boxes, how to make accurate measurements, how they handled fractional measurements, etc. Tell the groups that they are to generate 3-5 questions they might anticipate being asked about what they just experienced and learned from measuring and calculating surface area. They should also write questions that they can ask other “experts.”

To add a level of novelty to the strategy, some teachers keep ties, graduation caps and gowns, lab coats, clip boards, or other symbols of professional expertise for students to don when playing the role of the “professors.”

Call a group to the front of the room and ask them to face the class, standing shoulder to shoulder and to bring their box and measurement work with them. The “professors” invite questions from





the other groups. Students in the audience should ask their prepared questions first, then add others if more information is desired.

When the strategy is first employed, demonstrate with the class how the “professors” should respond to their peers’ questions. Typically, students are asked to huddle after receiving a question, discuss briefly how to answer it, and then have a “professor” spokesperson give the answer.

Remind students asking the questions to think carefully about the answers received and to challenge or correct the professor know-it-alls if answers are not correct or need elaboration or amending. After 5 minutes or so, call a new group of professor know-it-alls to the front of the class and continue the process of students questioning students.

Initially, it may be necessary and helpful to model the various types of questions expected from students about the content. For example, students should ask the “professors” both factual and higher-level questions.

At the conclusion of this activity, have the students write a reflection discussing at least five things they learned about determining surface area. This reflection should be placed in their math learning log (view literacy strategy descriptions). This will give them the opportunity to review it in the future, such as prior to testing.

Activity 2: Understanding Surface Area and Volume (GLEs: 17, 19, 20)

Materials List: centimeter cubes, small boxes, one inch square tiles, rulers (metric and US markings), masking tape, Understanding Surface Area and Volume BLM, LEAP Reference Sheet BLM, marbles and sand

Have students refer to their vocabulary self-awareness chart (view literacy strategy descriptions) developed in Activity 1 and review their definitions and examples for volume and surface area. Based on this information, ask the students to compare and contrast volume and surface area. Ensure that the students understand that volume is the space contained in a figure such as inside a box, jar, etc. To make this clear to the students, use a box similar to one of the boxes used in Activity 1. Model the difference between surface area of the box and the volume by indicating that the surface area refers to the area of the faces of the box and the volume is a measurement of the inside of the box (how much it will hold). Have students openly discuss their thoughts on these concepts. Be sure that one point of discussion is that for each measurement, the unit of measure must be known.

Give each student a square tile and a ruler. Ask him/her to describe the tile by giving its proper name and dimensions of each side. The students should conclude that the tiles are square shaped and each side measures one inch in length. Model how to measure surface area in square inches using the tiles by placing tiles on the overhead projector and explaining to students what the covered area measures in square inches. For example, place 9 square inch tiles on the overhead in a 3 x 3 format with the sides touching and tell students that the area covered by the





tiles measures 9 square inches (9 in2). Have pairs of students work together to estimate the area of a desk in square inches using the square inch tiles.

Ask the students to draw a square centimeter using their rulers and compare it to a square inch. Ask students what a square yard would look like. Have students model this on an area of the floor using masking tape. Ask students to compare cm2, in2, and yd2. Monitor the class discussion of the comparisons to ensure that students are making the correct comparisons. At the end of the discussion, have students reflect on the discussion, write a reflection on the comparisons, and place this reflection in their math learning log (view literacy strategy descriptions). This will give them the opportunity to review their thoughts and reasoning in the future.

To reinforce the proper concept of cubic units in measuring volume, remind the students that surface area is measured in square units because the objects being measured are two dimensional (length and width), but volume is measured in cubic units because the objects being measured are three dimensional (length, width and height (or depth). Engage the students in a discussion about whether the squares could be used to determine the volume of a three dimensional figure and have them explain their thoughts. Show students a cube and explain that a cube has six square-shaped faces. (Remind them the lengths of all sides of a square are equal.) Now show them a centimeter cube and explain that each side of the cube measures one centimeter in length. Hold up a small box and ask the students how they could use the centimeter cubes to determine the volume of the box. Use an open discussion format to help students understand that the volume of the box is the number of centimeter cubes that fit into the box. Continue this discussion and modeling to help students understand the concept. To conclude this part of the lesson, ask students why they believe particular shapes such as the square and cube are used as standard measurement units for surface area and volume instead of other shapes. If students don’t see the benefit of using these shapes, prod them by asking what would happen if circles were chosen to measure area or spheres were used to measure volume. Demonstrate the disadvantage of using spheres by filling a small box with marbles and then pouring sand over the marbles. Ask students where the sand went if the box was full of marbles to emphasize the importance of using units that have no gaps when measuring volume.

Ask students to describe a cubic inch and a cubic yard (as they think back on their work with the square units). Give the students about a few minutes to write a reflection about how they would determine which unit (cm3, in3, ft3, m3or yd3) would be the most appropriate to use in a given situation. When reflections are completed, have volunteers discuss what they have written. Address any misconceptions or questions. Have students place this reflection in their math learning log (view literacy strategy descriptions). This will give them the opportunity to review their thoughts and reasoning in the future.

Distribute the Understanding Surface Area and Volume and LEAP Reference Sheet BLMs. Provide students with practice problems for calculating the area and volume of a rectangular solid. Use this practice as an opportunity to review the formulas with the students. Also remind students of the formula for calculating the area and volume of a cylinder. After reviewing the formulas, allow students to work in pairs to complete the Understanding Surface Area and






Volume BLM. Once the activity is complete, allow the students to review and discuss their answers as a class.

Activity 3: Exploring Volume Changes (GLEs: 16, 17, 32, 48)

Materials List: learning log, pencil, calculator, LEAP Reference Sheet BLM

Present the students with a picture of a rectangular prism labeled with the following dimensions: length = 5 inches, width = 3 inches and height = 8 inches. The LEAP Reference Sheet BLM can be used for this as well as reviewing the formula for volume of the prism. Have students calculate the volume of the prism. Ask students to predict how the volume would change if the length increased by 2 inches. Students should understand that the volume would increase. Have students calculate the new volume and compare it to the initial volume. Tell students that the purpose of the activity is to find a way to calculate the volume of a box when given information about changes made to the dimensions of a box.

Have the students copy the following table into their learning log and complete the first row using the prism dimensions above.

Original volume Volume with length doubled

Volume with length and width doubled

Volume with length, width and height

doubled

The answers should be as follows:

Original volume Volume with length doubled

Volume with length and width doubled

Volume with length, width and height

doubled120 in3 240 in3 480in3 960in3

Ask the students if they see a relationship between the numbers. Have several students discuss the relationship they see.

Have students complete the other two rows using prisms with the following dimensions:Prism 1: length = 2 in, width = 4 in, height = 10 inPrism 2: length = 8 cm, width = 3 cm, height = 6 cm

Once students complete the table, have a general class discussion about how the changes in dimensions affect the volume of each prism. The idea is for students to conclude that when one dimension doubles, the volume doubles or is two times greater. When two dimensions double, the volume is four times greater. When three dimensions are doubled, the volume is eight times greater.



Give other examples with all of the dimensions changed by different factors. Ensure that students can generalize the situation by stating that the new volume can be found by multiplying the original volume by all of the factors used. For example, if the original volume is 12 cubic units and each side is doubled, the new volume is (2)(2)(2)(12) or 96 cubic units. If one dimension is tripled, and another divided by three, the new volume would be (3)(1/3)(12) or 12 cubic units. Since (3) and 1/3 are reciprocals, the two factors have no effect on the volume.

Now that students have an idea of how a change in a dimension affects the volume, a modified directed reading-thinking activity (DR-TA) (view literacy strategy descriptions) will be used to engage students in predicting how the volume will change when the dimensions are changed. DR-TA is a process in which students make and check predictions about a text as they read. In this case, students will be making and verifying predictions about a new problem related to what they have just learned about volume.

For this activity, students should recall what they have learned about how the volume changes with changes in dimensions. Proceed through the following sequence of events:

1. Introduce the background knowledge as shared in the examples above. Write on the board, “Given the following dimensions: length = 2 in, width = 4 in, height = 10 in, how would the volume change if the dimensions are tripled.”

2. Ask students to recall what they have learned and make predictions about this situation. Have students write their predictions in their math learning log (view literacy strategy descriptions). Ask students to share their predictions with a partner and allow them to revise their predictions if necessary.

3. After students have written their predictions and shared them with a partner, have them calculate the volume changes when the dimensions are tripled.

4. Discuss the calculations and poll students to share their predictions and compare to the calculated value.

After a class discussion about the comparison between predicted and calculated volumes of the rectangular solid when the dimensions are tripled, have students use the same process to predict the volume change for each of the following scenarios:

1) one dimension is doubled and one dimension is tripled, and2) one dimension is tripled and another dimension is divided by three.

The students should be allowed to complete scenario 1 by going through steps 1 through 4 above, before being given scenario 2. Monitor the discussion when the comparisons between predicted and calculated volumes are being discussed to ensure that students are drawing appropriate conclusions.

Provide additional practice problems asking students to use generalizations to find the new volumes rather than reworking the problem with new box dimensions. Address any misconceptions as students work.






Activity 4: Conversions and Comparisons (GLEs: 7, 21, 22)

Materials List: Conversions and Comparisons BLM, LEAP Reference Sheet BLM, pencils, calculators

Introduce this activity by reminding students of the two systems of measurement, customary and metric. Explain the difference between volume and capacity. - Volume is how much space a given material occupies. - Capacity is the maximum volume a container can hold. Review the common units for measuring volume and capacity using the table below as a guide. Have a discussion around this information and ask students for real-life applications of these units of measure.

Common Units for Measuring Volume/CapacityCustomary Metricounce millilitercup literpintquartgallon

Present models of the units. This can be accomplished by using purchased models from any school supply store or by asking students to bring in items that have the stated volumes. It is important that students are familiar with what the units look like as they develop their sense of measurement. This will also help them to understand the concept of converting between units in the same system.

Introduce the words convert and conversion in a measurement context by explaining that to convert means to represent a quantity in a different unit. For example, use the LEAP Reference Sheet BLM to remind students that 2 cups can also be represented as 1 pint. Two cups or one pint represent the same amount of liquid. To model this, take one pint of water and show students that the water will fill 2 cups. Using this reference point, ask the students to convert 8 cups to pints, or 8 cups = ___ pints. Give students one minute to think about the answer, and then ask for volunteers to give their answers and an explanation. Review the conversion process with the students. The method of establishing a proportion in which the units in the ratios are the same is shown below.



Metric system conversions can be made using the same process. For example, model the conversion of 2 L = __ mL.

1000 mL x mL ---------- = ------ 1 L 2 L

x = (2)(1000)

x = 2000 mL

Students may also be familiar with the conversion process based on movement of the decimal point when using metric measures. In this process, the conversion of 2 L to __ mL would be based on multiplying by 1000 or moving the decimal point 3 places to the right. Allow students to use the method that allows them to be successful. Be sure that students understand that the use of powers and moving the decimal point applies only to the metric system and why this is true.

Present students with extra problems for guided practice.

Tell students that they will not make exact conversions between the metric and customary systems. Instead, they will compare and estimate measurements within the same system. As an example, a liter is a little more than a quart. Other benchmarks that are used are the thickness (height) of dime is about a millimeter and the width of finger is about a cm (have students measure to see which of their fingers can be used for such an estimate) Use models to help students understand these comparisons. Ask students for real-life examples of where these units are used.

Distribute the Conversions and Comparisons BLM. Monitor students as they complete the BLM and offer assistance where necessary. Collect student work for review and discussion.

Activity 5: Area and Volume Conversions (GLEs: 7, 22)

Materials List: Area and Volume Conversions BLM, pencils, calculators

In this activity, students will practice converting between area measurements (in2, ft2, and yd2) and volume measurements (in3, ft3 and yd3). To begin this activity, have the students refer to



their Activity 2 reflections in their math learning logs (view literacy strategy descriptions) where they compared square units for measurement and cubic units for volume. Ask students to share short summaries of their reflections on the unit comparisons for area measurements and volume measurements. Correct any misconceptions noted.

Focus the students on area measurements and tell them that since they know that there are different units for representing area measurements, they will now review conversions between those units. Review with students the meaning of “square yard” and draw a model of a square yard on the board and label each side. Ask the students how many square feet are in a square yard. Most students are likely to say that there are 3 ft2 in 1 yd2 because there are 3 feet in 1 yard. If students give this initial response, continue to ask questions leading them to think more deeply about the question. Suggest that they use their knowledge of feet and yards to draw the square feet within the square yard. After a couple of minutes, begin the process of drawing in the square feet and allow the students to finish. Present the completed drawing as represented below.

1 yd2 9 ft2 1 yd2

Represented as a ratio, ----------. In other words there are 32 or 9 ft2 in 1 yd2. 9 ft2

Use this ratio to find how many square feet are in 6 square yards. Ask students to set up a proportion and then solve it based on what they previously learned about establishing proportions to solve conversion problems.

1 yd2 6 yd2

--------- = --------- 9 ft2 x ft2

Solving the proportion produces a result of 54 ft2. Review the proper placement of the exponent when writing these measures. For example, some students have the misconception that three square feet is written 32 ft instead of 3 ft2. Discuss the fact that 32 feet actually means 9 feet and that in 3 ft2 the exponent is used to describe the unit of measure.

Remediation Math Unit 3Measurement

1 ft2 1 ft2 1 ft2

1 ft2 1 ft2 1 ft2

1 ft2 1 ft2 1 ft2

33



Have students use their reasoning skills to determine how many square inches are in one square foot 1 ft2 = ___ in 2 . After a few minutes help students conclude that 1 ft2 = 144 in2 (or 122). Address any questions about the relationships between units of area measurement.

Change the focus to volume conversions. Ask students to share the basic difference between area measurements and volume measurements. Students should remember that area is two dimensional and is measured in square units while volume is three dimensional and measured in cubic units. Using similar reasoning as used with the area measurement, ask students how many cubic feet (ft3) are in one cubic yard (yd3), 1 yd3 = ___ ft3. Remind students that the units are cubed. Give students a couple of minutes to think about this, and then get responses from students. Help students think through the process and conclude that there are 33 or 27 ft3 in 1 yd3. Written as a ratio, 1 yd3

--------- 27 ft3

Ask students to determine how many cubic inches are in 3 cubic yards. Monitor students as they work, ensuring that they first determine how many cubic inches are in a cubic yard, and then use a proportion to find the answer to the problem.

Ask for and address any questions students have about this conversion factor. Have the students reflect on the process for setting up the ratios or conversion factors when working with square and cubic units. Have several students share their reflections. The reflections should be placed in their math learning log (view literacy strategy descriptions) for future reference.

Distribute the Area and Volume Conversions BLM and have students work in pairs to complete. Review the answers with the students at the completion of the activity.

Activity 6: Understanding and Using Rates (GLEs: 7, 9, 18)

Materials List: Understanding and Using Rates BLM, pencils, calculators

Introduce this activity by telling students that they will practice making conversion for real-life applications involving density, speed, and money. Have students add the words velocity and density to their modified vocabulary self-awareness chart (view literacy strategy descriptions). Give the students about 5 minutes to complete the chart for velocity and density. Have volunteers share their thoughts with the class.

Give students a brief overview of velocity and density. Remind them that they either have or will have a more in-depth study of these two concepts in their science class. Some of the basic facts about density and velocity are as follows.

Density is the ratio of mass to volume. (Remind students that mass is the amount of matter in an object.)

Density is a measure of how tightly packed the particles are in a substance. Speed is distance traveled per unit time





Calculating speed is based on the formuladistance = rate ● time

Or distance = speed (how fast you are going) ● time

Give the students the following scenario and ask them to calculate the average speed.

It took 4 hours for a truck to travel 320 miles. What was the average speed of the truck?

Using the formula,

Distance = speed ● time; 320 miles = speed● 4 hours;

320 miles----------- = speed 4 hours

speed = 80 miles/hour

Discuss the scenario and address any misconceptions in the use of the formula. Now move to helping students understand density.

massDensity = ---------- volume

Give the students the following scenario and ask them to calculate density.

A piece of material has a mass of 125 grams and a volume of 5 cubic centimeters.

Mass 125 gramsDensity = -------------; Density = ---------------; Density = 25 grams/cm3

Volume 5cm3

Discuss the scenario and address any misconceptions in the use of the formula.

Explain to students that money is another measurement that can be converted to different units. Review some common monetary exchange rates with the students such as the ones below. Help students to understand the importance of exchange rates when traveling outside of the United States.



Type of MoneyValue of One Unit

(U.S. dollars), $U.S. Dollar 1.00Mexican Peso 12.52Euro 0.74British Pound 0.66Canadian Dollar 1.01

Give the students the following example for monetary exchanges.

If a cake costs $24.89 in Euros, how much does it cost in U.S. Dollars?

$1 x----- = --------0.74 Euro 24.89 Euro

(x)(0.74) = (1)(24.89)

0.74x = 24.89

24.89 x = ---------- 0.74 x = $33.64 U.S. Dollars

OR, 1.00 US24.89 Euro × ------------- = $33.64 (U.S. dollars) 0.74 Euro

Distribute the Understanding and Using Rates BLM. Students should complete this activity individually and then compare answers with a partner. Suggest students use proportions to solve the problems and show all work in order to explain the process for solving the problems to their partner.

Activity 7: Gummy Bear Lab (GLE: 18)

Materials List: gummy bears (1 per student), Gummy Bear Lab BLM (downloadable at http://sciencespot.net/Media/mmaniabearlab.pdf), 1 small (4 oz.) cup per student, ruler, scale or triple beam balance, calculator, water

This activity can be found at http://sciencespot.net/Media/mmaniabearlab.pdf. The BLM should be downloaded prior to starting the activity. Have students complete the lab as noted in the instructions on page 2. The following is summary of what students will do.


http://sciencespot.net/Media/mmaniabearlab.pdf

http://sciencespot.net/Media/mmaniabearlab.pdf


The students will perform a measurement experiment with gummy bears. They will measure the dimensions of a gummy bear, use a triple beam balance or scale to find its mass, and then calculate the bear’s density. The gummy bear will then be placed in water overnight. The next day, the students will measure the gummy bear again to determine if the dimensions, mass, and density have changed. After obtaining the new measurements, the students will answer a series of questions involving volume, mass and density.

Sample Assessments

General Assessments

Provide the students with several common boxes typically found in a household setting. Have the students measure the dimensions of the boxes in inches and centimeters and calculate the volume and surface area of each box.

Give the students a list of items that can hold water and have the students determine the best unit of measure for the containers.

Have the student write a letter to a middle school student explaining in general terms how the volume of a rectangular prism is affected by changes in the dimensions of the prism. Have students give specific dimensional and volume changes.


Activity 1 : Have students develop a presentation that proves and/or explains the formula for calculating the surface area of a rectangular prism. Students should use pictures or models to represent the different components of the formula.

Activity 2 : Provide students with common household containers that are rectangular or cylindrical. Allow students to choose one of the containers and estimate the volume including appropriate units. After estimating the volume, the students should measure the volume of the container and compare the measurement to their estimate.

Activity 4 : Have the students convert the following units of measure to fluid ounces: 3 gallons, 4 quarts, 10 pints, 16 cups.

Activity 5 : Have students complete a RAFT (view literacy strategy descriptions) writing

task. For this activity, the RAFT description is as follows:R – A square yardA – A middle school studentF – LetterT – Why I am equivalent to nine square feet

In other words, the student will take the role of a square yard and explain to the student why the square yard is equivalent to nine square feet. Ask students to place the completed RAFTs in their math learning log (view literacy strategy descriptions ).





Activity 6 : Have students plan a trip to another country and list the amount of money, in U.S. dollars, they need for food, lodging and travel. The students should then convert the U.S. dollars to the currency of the country they are visiting using the proper exchange rate.



Ninth GradeRemediation MathUnit 4: Geometry

Time Frame: Approximately 5 weeks

Unit Description

This unit focuses on angle relationships, transformations, the Pythagorean Theorem and similar triangles. The unit also extends the learning from Unit 3 by including an analysis of how changes in one or more dimensions of a shape affect the volume and surface area of that shape.


Students have a basic understanding of similarity, congruence and symmetry. Students are able to differentiate between the different types of transformations. Students are able to apply the concepts, properties and relationships of angles. Students can justify the Pythagorean theorem and apply it and its converse to real-life problems. Students continue to deepen their understanding of how changes in dimensions affect the volume and surface area of solids.

Guiding Questions

1. Can students demonstrate a conceptual understanding of the identified geometric terms?

2. Can students differentiate between the different types of transformations: reflections, rotations, translations, and dilations

3. Can students use proportional reasoning to determine the missing sides of similar triangles?

4. Can students demonstrate a conceptual understanding of angle relationships, concepts, and properties?

5. Can students justify the Pythagorean Theorem and apply the theorem and its converse in real-life situations?

6. Can students accurately predict how changes in dimension will affect the volume and surface area of solids?


GLE # GLE Text and BenchmarksGrade 8Number and Number Relations5. Simplify expressions involving operations on integers, grouping symbols, and

whole number exponents using order of operations. (N-4-M)7. Use proportional reasoning to model and solve real-life problems (N-8-M)Measurement17. Determine the volume and surface area of prisms and cylinders (M-1-M)(G-7-

M)

Remediation Math Unit 4Geometry 39



GLE # GLE Text and BenchmarksGeometry23. Define and apply the terms measure, distance, midpoint, bisect, bisector, and

perpendicular bisector (G-2-M)24. Demonstrate conceptual and practical understanding of symmetry, similarity,

and congruence and identify similar and congruent figures (G-2-M)25. Predict, draw, and discuss the resulting changes in lengths, orientation, angle

measures, and coordinates when figures are translated, reflected across horizontal or vertical lines, and rotated on a grid (G-3-M) (G-6-M)

26. Predict, draw, and discuss the resulting changes in lengths, orientation, and angle measures that occur in figures under a similarity transformation (dilation) (G-3-M) (G-6-M)

28. Apply concepts, properties, and relationships of adjacent, corresponding, vertical, alternate interior, complementary, and supplementary angles (G-5-M)

29. Solve problems involving lengths of sides of similar triangles (G-5-M) (A-5-M)

30. Construct, interpret, and use scale drawings in real-life situations (G-5-M) (M-6-M) (N-8-M)

31. Use area to justify the Pythagorean theorem and apply the Pythagorean theorem and its converse in real-life problems (G-5-M) (G-7-M)

32. Model and explain the relationship between the dimensions of a rectangular prism and its volume (i.e., how scale change in linear dimension(s) affects volume) (G-5-M)

Sample Activities

For this unit, have students maintain a learning log (view literacy strategy descriptions). A learning log is a notebook, binder or some other repository that students maintain in order to record ideas, questions, reactions, observations, thoughts, reflections, critical skill summaries, vocabulary self-awareness charts, vocabulary cards, etc. Documenting ideas in a log about the content being studied forces students to “put into words” what they know or do not know. Learning logs can become the place for virtually any kind of content-focused writing.

Activity 1: Bisecting Basics (GLEs: 5, 23)

Materials List: Vocabulary Self-Awareness Chart BLM, Vocabulary Card BLM, graph paper or the Coordinate Grid BLM for Activity 3, rulers

As this first part of this activity, the students will review important vocabulary terms using a vocabulary self-awareness chart (view literacy strategy descriptions). Distribute the Vocabulary Self-Awareness Chart BLM to the students and have them complete it. In completing the chart, the students have an opportunity to reactivate their prior geometric knowledge and highlight their understanding of what they know, as well as what they still need to learn, to fully comprehend the content. The students will revisit the chart during the course of the unit, revise their entries





and have multiple opportunities to practice and extend their understanding of important content terms.

In addition to the vocabulary self-awareness chart for a few key terms, the students will use a modified vocabulary card strategy (view literacy strategy descriptions). Give the students the Vocabulary Card BLM and have them complete it using the directions given. When students create vocabulary cards, they see connections between words, examples of the word, and the critical attributes associated with the word. This vocabulary strategy also helps students with their understandings of word meanings and key concepts by relating what they do not know with familiar concepts. Vocabulary cards require students to pay attention to words over time, thus improving their memories and use of the words. In addition, vocabulary cards can become an easily accessible reference for students as they prepare for tests, quizzes, and other activities with the words. Have students develop vocabulary cards for the following words: midpoint, bisector, perpendicular bisector, line segment and ordered pair.

Students will now apply the terms from the vocabulary cards to further deepen their understanding. Give each student a sheet of graph paper. Have students use the entire page to construct a coordinate system. Check to make sure the students have properly labeled the axes. Give the class the coordinate pairs for two endpoints of a horizontal line segment in Quadrant I. Ask the students to plot the points and connect the points to form a line segment. Check to see that the students have properly labeled the points and drawn the line segment. Ask the students to refer to their vocabulary cards and complete the following.

1. Find the midpoint of the line segment.2. Identify the coordinates of the midpoint.3. Draw a perpendicular bisector of the line segment.

Monitor the students as they complete their work. After about 5 minutes, have an open class discussion on the process the students used to complete the activity. During the discussion help students understand the process for finding the midpoint (average of the x-coordinates and average of the y-coordinates to find the x- and -y coordinates of the midpoint, respectively). Model this process with the given example. During the discussion of the perpendicular bisector, help students understand that while there are several ways to bisect a line, a perpendicular bisector intersects the line segment at right angles (90°) and passes through the segment. Show examples of other ways to bisect a segment with lines that are not perpendicular bisectors.

Provide students with coordinates for the following segments, and have them work through the process using the same coordinate grid.

1. A vertical line segment2. A horizontal line segment in Quadrant III3. A line segment with a positive or negative slope whose endpoints lie in two different

quadrants.




As a closing activity, have students respond to the following prompt in their math learning logs (view literacy strategy descriptions).

Prompt: Explain the process for drawing a perpendicular bisector of a line segment.

This will give them the opportunity to reflect on the work just completed and identify any misconceptions they still have. Have some of the students share their writing and discuss.

Activity 2: Am I Similar? (GLEs: 7, 24, 29)

Materials List: Vocabulary Self-Awareness Chart BLM from Activity 1, Similar Triangles BLM

Have students refer to their Vocabulary Self-Awareness Chart BLM that was completed in Activity 1. Discuss the following terms with the students: symmetry, similar / similarity, congruent. Prior to the discussion, have students fold a sheet of paper into thirds (lengthwise folds), open the paper, and draw a line down the paper marking the one-third line. As the discussion takes place, have the students take notes using split-page notetaking (view literacy strategy descriptions) to organize their notes during the discussion. One good way to begin teaching the split-page notetaking strategy is by showing students the difficulty of trying to study from poorly organized notes. Create an example of “disorganized” notes by looking through the material to be covered and writing out main points, key terms, and specific supporting information in a mixed-up way for a section of the content. Note that when students are trying to study and recall the material, it becomes a confusing process to sort out the important from the less important with a random notetaking scheme. An example of split-page notetaking can be given to students in a handout, presented as an overhead, or on a computer slide. Discuss the advantages of taking notes in this way. Show students how they can prompt recall by bending the sheet so that information in the right or left column is covered. The use of the note-taking guide created here will continue with other discussions within this unit and will serve as a good reference for unit assessments. See below an example of what the notetaking page might look like:

Split-page Notetaking Guide

NameDateBlock/Period

Topic: Geometry concepts

Symmetry

Similar

Congruent

An object is said to have line symmetry if when a line is drawn down the center, the images that are on each side of the line are identical

Two objects are similar if they are identical in shape, but not necessarily the same size.

Same shape, same size






After a short discussion, clear up any misconceptions and help students understand the terms and transfer this understanding to their notes. Conceptual examples of information to present are found in the table above. During the discussion, remind students how they may have used these terms in the past such as with angles, triangles, and other various shapes. This should lead to a discussion of similar triangles. Have students add this term to their notes. Ask students to give their thoughts on what similar triangles are based on the class discussion. Have students note that triangles are similar if they have the same shape but can be different sizes. Tell students that they are going to investigate the properties of similar triangles. Review how to write ratios, if needed. Give the students two similar triangles. Keep the ratios simple such as 1 to 2 or 1 to 3 and use whole number lengths. Lead a discussion about the term “corresponding sides.” Have students measure the corresponding side lengths, record them, and calculate the ratios. Then have the students measure the angles in each triangle and record those. A chart for recording measurements such as the one below can be provided or drawn on the board and copied by students. Corresponding sides and angles are listed by number, but if the triangles are labeled ABC and DEF, then corresponding segment and angle names should be used. Monitor students as they work.

Side Lengths Triangle 1 Side Length Triangle 2 Ratio of Side LengthsSide 1: Side 1:Side 2: Side 2:Side 3 Side 3:

Angle Measures Triangle 1 Angle Measures Triangle 2Angle 1: Angle 1:Angle 2: Angle 2:Angle 3: Angle 3

Lead a discussion in which students share their findings. They should note that corresponding sides have the same ratios and corresponding angles are congruent. It is important that students remember that statements showing equivalent ratios are called proportions. Lead them to understand that the terminology used to describe the relationship in similar figures is that corresponding sides are proportional. The students should be able to use the proper mathematical language when stating the relationships for corresponding sides and corresponding angles.

Review the basic concepts of proportionality and review the process of finding a missing term in a proportion. Present the students with two more similar triangles with one side length missing. Ask the students to find the length of the missing side. Give students about five minutes to complete this task. Have a student present and explain their work on the board. Then review with the students and encourage class discussion on the process. Provide one more class example if warranted. When the majority of the students understand the process, give the students the Similar Triangles BLM for additional practice. Monitor the students as they complete the BLM. Have students submit their completed work. Assess the work and discuss the work during one of the next classes.



Activity 3: Transformations – Focus on Translations (GLEs: 24, 25)

Materials List: Vocabulary Self-Awareness Chart BLM from Activity 1, Coordinate Grid BLM

Have students refer to their Vocabulary Self-Awareness Chart BLM that was completed in Activity 1. Discuss the following terms with the students: translation, reflection, and rotationDuring the discussion give students the simple, conceptual meanings to each of the words and have the students update their Vocabulary Self-Awareness Chart. The following examples should be considered.

Translation – a slide. Reflection – a flip over a line. Rotation – turning around a center.

As each simple meaning is discussed, model the transformations using two identical, non-symmetrical shapes on a coordinate grid overhead transparency. Tell the students that in this activity they will focus on understanding what happens during a translation of a figure on the coordinate plane.

Remind students that a translation is a slide. On the overhead grid, draw a scalene triangle with the vertices at intersecting points on the grid. Have students provide the coordinates for each vertex. Ask the students what would happen to the coordinates if the triangle was translated two units right and one unit up. Have the students predict and record what they think the coordinates of the vertices of the translated figure will be. Now have one of the students perform the translation on the projector. Get feedback from the students and provide support to insure the translation is correct. Ask the students to consider the size and shape of the two figures. They should note that the translated figure is the same shape and size as the original, resulting in congruent figures. Ask students how they could verify that this relationship is correct. Possible answers might be to measure the corresponding sides and angles or to place one figure on top of the other.

Distribute the Coordinate Grid BLM to students. Give the coordinates of Figure 1 below and have them plot the points and draw the figure. Next, direct the students to perform the indicated transformation. Monitor students as they work to ensure that they plot their points correctly and are making the correct movements with the translations.

Figure 1: A:(1,1), B:(4,1), C:(2, 2), D:(4, 2) Translate the figure 3 units left and 2 units up.

Once students have completed the example with Figure 1, give students the information below for Figure 2 and allow them to complete the translation.

Figure 2: A:(-3,-3), B: (4,-3), C:(-3, 3) Translate the figure 2 units to the right and 4 units up.

To close this activity, ask students what patterns or relationships they see between the number of units translated and the new coordinates. Probe the students to help them conclude that for each point, they are counting the number of spaces to move the point up or down on the y-axis and left



or right on the x-axis. Ask students to think about how to write a rule to find the coordinates of the translated figure using the coordinates of the original so they don’t have to count each time. For example, in Figure 1 above, students should see that they subtract 3 from the x-coordinate and add 2 to the y-coordinate to get the coordinates of the translated figure.

End by having the students individually write a reflection on what they have learned about translations in their math learning logs (view literacy strategy descriptions). Ask student volunteers to share their reflections. Address any misconceptions.

Activity 4: What’s Your Move? - Translations (GLEs: 24, 25)

Materials List: index cards, paper, pencils, masking tape, yarn

Before class, create a coordinate grid on the floor using masking tape. This is also a great opportunity to work with students in the gym or other area with open floor space if there is insufficient space in the classroom.

Ask a group of three or four students (depending on the shape to be modeled) to come to the front of the room. Give the group an index card with the coordinates of a figure written on it. For example, for a triangle the coordinates might be A (1, 1), B (2, 1), C (1, 3). Each student in the group is to represent a vertex on the figure and stand on the appropriate point on the grid. Once each student is standing on the appropriate grid point, use yarn to connect the vertices to form the figure.

Then call a second group of students to the front and give this group an index card indicating a translation of the first figure. Ask the second group to form the translated figure using the same process. The rest of the class should record the coordinates of the original figure and the coordinates of the translated figure and determine what translation was performed.

Continue this process with other groups and different figures.

Activity 5: Transformations – Focus on Reflections (GLEs: 23, 24, 25)

Materials List: patty paper, protractors, rulers, Vocabulary Self-Awareness Chart BLM from Activity 1

Begin this activity by asking students what a reflection is. Have them refer to their Vocabulary Self-Awareness Chart BLM that was completed in Activity 1. For a reflection, ask students to think of how things look when viewed in a mirror.

Give the students a right triangle drawn on a coordinate grid, and tell them that the y-axis is to be the line of reflection. Ask them to place a piece of patty paper on top of the grid and trace the figure and the line of reflection. Student will then fold the patty paper on the line of reflection and trace the image. Monitor students as they complete this activity. Also have a completed




model available for the student to see.

Once complete, ask the students a series of questions to help them deepen their understanding. Some questions to consider asking are: 1) What do they see? 2) Are the two figures congruent? Once the students indicate that the figures are congruent, challenge them to explain why. They should comment that the side lengths and angle measures are the same. Have them verify this by measuring the corresponding angles and corresponding side lengths. This can be done faster if some students measure angles and others measure sides. Have them note that a reflection produces an image that is the same size as the original, but the image is flipped over (reversed).

Lead the students through a discussion of how to find the location of the reflected vertices on the coordinate grid if the original coordinates are given. Encourage them to think about how this might be done before using the patty paper to help them see the connection. Ask them first to look at the locations of one vertex and its image on the grid and determine if they see anything visually that might help. Asking “How far from the line of reflection is each point?” is sometimes helpful if students need a place to start.

Students should notice that if they count the number of units from a vertex of the original triangle to the line of reflection and count the number of units from the line of reflection to the same vertex of the reflected triangle, those distances are the same. Have the students draw a line segment between these two vertices. Ask, “What is the relationship of the line segment to the line of reflection?” Help students to see that the line segment is perpendicular to the line of reflection and that the line of reflection bisects the line segment between the two vertices. Ask, “What name can be given to the line if it is perpendicular to the segment and cuts the segment in half?” Refer students to their learning log (view literacy strategy descriptions) responses from Activity 1. The students should conclude that the line of reflection is the perpendicular bisector of the segment that joins two corresponding vertices of reflected figures.

Next, have students create a table showing the coordinates of the original figure and the reflected figure. From this table, they should begin to see patterns emerge with the coordinates. For example, if the coordinates of a vertex of an original figure is (1, 2) and it is reflected over the y-axis, the corresponding vertex of the reflected figure will be (-1, 2). Help students make these connections for all of three points in the given triangle, noting that the x-coordinates are opposites and the y-coordinates are the same when points are reflected across the y-axis.

Give students a copy of the same right triangle drawn on a grid. They should now use the x-axis as the axis of reflection, draw the reflected image on the grid, and identify the vertices of the original and reflected images. Ask students to indicate how the coordinates of the reflected points can be determined from the original vertices. Monitor students as they work and address any misconceptions.




Activity 6: Transformations – Focus on Rotations (GLEs: 24, 25)

Materials List: protractors, rulers, Vocabulary Self-Awareness Chart BLM from Activity 1

Begin this activity by asking students what a rotation is. Have them refer to their Vocabulary Self-Awareness Chart BLM that was completed in Activity 1. Remind them that a rotation turns a figure about a point.

Give each student a coordinate grid. Have them draw two right triangles with the following coordinates:

Triangle 1: A (1, 1), B (2, 1), C (1, 3) Triangle 2: A’ (1, -1), B’ (1, -2), C’ (3, -1)

Model for students and check their work for correctness. Let them know that is read “A prime” and indicates that is the image of point A. Having such a name will be helpful in referring to a point and its image. Tell the students that Triangle 2 is the image of Triangle 1 rotated about the origin. Ask students to locate the origin and give its coordinates as way to review this term. Have the students draw a line from the origin through Point A of Triangle 1, and then have them draw a second line from the origin through Point of Triangle 2. Model this for students. Tell students to use their protractor to measure the angle formed by these two lines to determine the angle of rotation. Students should conclude that the angle of rotation is 90°. Ask them if the direction of rotation is clockwise or counterclockwise. They should note clockwise. It is also important to note that A and A’ are the same distance from the origin.

Ask as student to come to the front of the room. Have the student stand and then rotate 180 degrees clockwise. Have the student return to his original location and then rotate 180 degrees counter clockwise. This is to help students remember that a 180 degree rotation is an “about face” and that his original line of vision and his line of vision once rotated are along the same line. Draw a line on the board or overhead and ask students how to measure the number the number of degrees for a line. Model as needed to remember that the rotations just made and “measuring the number of degrees” are both 180 degrees.

Now give the students a coordinate grid. Have them draw a trapezoid with the following coordinates: A (1, 1), B (4, 1), C (2, 2), D (4, 2). Have them rotate the trapezoid 180° clockwise about the origin. This will be harder for students. Allow students time to work on this (in pairs if necessary). If students have trouble, prompt them to first draw a line from the origin through one of the points and then have think about the results of a 180 degree rotation and where the image point would be with respect to the origin and the original point. Encourage them to make predictions about location of the point’s image. They then should draw the image and check the angle measurement between the point and its image. Once the image of one point is completed, location of the images of other points will be easier. The coordinates of the rotated trapezoid are

(-1, -1), (-4, -1), (-2, -2), (-4, -2).



Close this activity by reviewing the trapezoid work with a projector and addressing any questions.

Note: Additional practice and reinforcement of each transformation will be addressed in the next activity.

Activity 7: Transformations: Putting It All Together (GLEs: 24, 25)

Materials List: protractors, rulers, Vocabulary Self-Awareness Chart BLM from Activity 1, Coordinate Grid BLM (see Activity 3 BLMs), Transformation BLM, computer access.

Have student volunteers briefly review each of the transformations. They may use their notes, learning logs, examples, etc.

Have students form pairs and give each student three copies of the Coordinate Grid BLM. Have them work to plot the following trapezoid on the grid and label it as original. A (2, 3), B (8, 3), C (4, 7), D (7, 7). Tell students to perform the following transformations on this figure. Have the students use a different grid for each transformation. In doing so, students should draw and label the new images and identify the coordinates of the new images. Although they are working in pairs, each student should do the work.

1. Translation: four left and five down2. Reflection: over the x-axis3. Rotation: 90° clockwise about the origin

For additional support, refer the students to the National Library of Virtual Manipulatives interactive website at http://nlvm.usu.edu/en/nav/category_g_3_t_3.html.

For additional practice, distribute the Transformations BLM and have the students work in pairs to complete it.

Activity 8: Dilation: Another Transformation (GLEs: 24, 26)

Materials List: Vocabulary Self-Awareness Chart BLM from Activity 1, Quadrant I Grid BLM

Begin this activity with a review of the transformations from Activity 3. Have students share what they remember from the activity and use this as an opportunity to update their Vocabulary Self-Awareness BLM. Explain to students that dilations are another type of transformation where a figure becomes larger or smaller by a specific factor.

Distribute the Quadrant I Grid BLM and ask students to label the x- and y-axes with numbers based on the points that need to be plotted. The coordinates are A (5,2), B (10, 1), C (10, 6), D (6, 6). After labeling the axes, ask students to plot and connect the points to create a four-sided shape. Have the students properly label each vertex. The students should then make a


http://nlvm.usu.edu/en/nav/category_g_3_t_3.html


new set of coordinates by multiplying each of the original coordinates by 2. They should then plot the new points and draw the resulting shape. Have the students measure the lengths of the sides of the original figure and the new figure formed from the dilation. Ask the students for their observations about the ratios of the sides. They should see that the ratio of the sides is 2 to 1 if comparing the new figure to the original. Ask them why they think this happened. Ask them to describe the relationship between the two figures. They should know the figures are similar with the new figure being twice as large as the original.

Now give the students another Quadrant I Grid BLM. Ask them to draw a four-sided polygon using the following points: A (6,18), B (33, 15), C (3,33), D (30,36) Label the coordinates and the vertices. Ask students to predict what would happen if each of the coordinates were

multiplied by . Give them time to record a prediction on their paper and then ask them to

perform the calculations. Students should make a set of the new coordinates, plot the points, and draw the resulting shape. Have students measure the lengths of the sides of the original figure and the new figure formed from the dilation and share their work with each other, discussing their observations and conclusions drawn. In this case, the figures are similar, but the new figure

is smaller and the size of the original.

Discuss the relationship between the assigned scale factors and the relationship between the two images in the two examples. Students should grasp that dilations create similar figures with the scale factor determining the ratios of the corresponding sides. Allow students to openly discuss their findings and remaining questions. Address any misconceptions.

Let students know that as a practical application, some of the computer animation techniques are based on transformation of figures. Encourage students to do some research in this area.

Activity 9: Effect of Dilation on the Volume of Solids (GLEs: 17, 32)

Materials List: paper, pencils, calculator, LEAP Reference Sheet BLM (from Unit 1, Activity 1)

In this activity, students will extend their learning of the how changes in the dimensions of a rectangular prism affect the volume of the prism and connect this learning with dilations.

Review Unit 3, Activity 3 by referring the students to the example used in that activity that can be found in the table in their learning logs. For reference, the original dimensions of the prism were length = 5 inches, width = 3 inches and height = 8 inches. Ask the students for the original volume and how the volume was determined. (This helps the students reflect on the formula for calculating volume of a prism.) Refer students to the LEAP Reference Sheet BLM if they are unable to recall the formula.

Now ask the students for the volume of that prism with length, width and height doubled. The students should respond with 960 in3. Have the students discuss their processes for determining this answer. As in Unit 3, Activity 3, help students generalize that the new volume can be found by multiplying the original volume by all of the factors used. In other words, since each



dimension was doubled, the new volume can be determined by multiplying 120 in3 by (2)(2)(2), or 120 ● 2 ● 2 ● 2. Ask the students how this connects to the concept of dilation. Students should conclude that the prism was dilated by a factor of 2. In general, when a solid is dilated by a scale factor, then the volume of the dilated solid is multiplied by the cube of the scale factor. In other words, for this example, the new volume is 120 in3 x (2)3, or 120 in3 x 8, or 960 in3. Have the students draw and label the original prism and the dilated prism. Check drawings to ensure that the students show that the dilated prism and volume are larger than the original.

Give students the following problems for additional practice. Students may be allowed to work in pairs to support each other, but each student should complete the practice.

Draw and label the following figures and their dilations according to the corresponding scale factors. Calculate the volume of each of the figures.

(1) Rectangular prism: length = 27 inches, width = 15 inches, height = 30 inches; dilate by a

scale factor of

(2) Cylinder: radius = 2 meters, height = 4 meters; dilate by a scale factor of 3.

Conclude this activity by having students write a reflection on what they have learned about how dilations affect the volume of a solid. This reflection should be shared with the class and placed in their math learning log. Clear up any misconceptions as students share their reflections.

Activity 10: The Pythagorean Theorem (GLE: 31)

Materials List: Centimeter Grid BLM or centimeter graph paper, Pythagorean Theorem BLM, calculators, pencils

Have students draw a right triangle in the center of the Centimeter Grid BLM with the two perpendicular sides having lengths of 3 and 4 units. Label the sides side a and side b and the hypotenuse or diagonal as side c. Model this entire process for the class using an overhead projector or some other form of technology.

Have students draw a square on each leg and on the hypotenuse and then find the areas of the three squares by counting the number of square units. The students will have to decide how to put partial squares together to form whole squares to determine the number of square units for the square on the hypotenuse. Students should record the area of each square and look for a relationship between the areas of the three squares. Work with the students to help them see that the sum of the areas of the two smaller squares is the same as the area of the square formed by the hypotenuse. For a more hands-on approach, students can cut out the squares and rearrange the smaller squares inside the large square or measure the length of the hypotenuse with a centimeter ruler and calculate the area of the square on the hypotenuse. Continue the discussion to help students understand the Pythagorean Theorem formula, a2 + b2 = c2 if a and b are legs and c is the hypotenuse of a right triangle.



Explain to students that the Pythagorean Theorem can be used to find the missing side of a right triangle when the other two sides are given. Allow students to use the formula to solve the following problems as a class.

(1) A right triangle with side lengths of 5 inches and 12 inches. Find the length of the hypotenuse.

(2) A right triangle with one side length of 7 inches and the hypotenuse length of 25 inches. Find the length of the other side.

Ask students to think of a way they can use the Pythagorean Theorem to determine if a triangle is a right triangle if the lengths of all sides are given. Allow students to discuss their thoughts. Then give students the following:

A triangle has side lengths of 5 inches and 7 inches and a hypotenuse of 12 inches. Is this triangle a right triangle?

Give students about 5 to 10 minutes to work through this problem using the Pythagorean Theorem. Have students share their answers and their reasoning while helping them to work through the problem. The triangle is not a right triangle because

52 + 72 ≠ 122 25 + 49 ≠ 144

74 ≠ 144

Distribute the Pythagorean Theorem BLM and allow students to practice solving problems using the Pythagorean Theorem. Review the student work and address any misconceptions.

Activity 11: Angle Relationships (GLEs: 23, 28)

Materials List: rulers, protractors, paper, pencils, Vocabulary Self-Awareness Chart BLM from Activity 1, Angle Relationships BLM

Have students refer to their Vocabulary Self-Awareness Chart that was started in Activity 1 and focus on the last six entries relating to angle relationships. Review the entries and have the students update their charts by providing them with the definitions of the following terms and having students draw an example of each in their chart: complementary angles, supplementary angles, alternate interior angles, alternate exterior angles, corresponding angles, adjacent angles and transversal. Model example drawings of each on the board or overhead to ensure students have the correct information in their charts.

Distribute the Angle Relationships BLM. Allow the students to work in pairs to determine the measures of all angles by using the information learned about angle relationships above. Monitor students as they work to ensure that the angle measurements and reasoning are correct.

Once all pairs have completed their work, review the answers and reasoning used to determine the measurements. Address any misconceptions or errors.



Place students in groups of four. Have the students draw two parallel lines cut by a transversal, label all angles, and measure one angle in each set of four angles where the transversal crosses the two parallel lines. Students should investigate the relationship between the alternate interior angles, corresponding angles, adjacent angles, and vertical angles.

Have students then draw on a poster or large piece of paper a model of this same labeled diagram indicating the measurement of the measured angles only. This will be presented to the class using the professor know-it-all (view literacy strategy descriptions) strategy. Tell the students that their groups will be called on randomly to come to the front of the room and provide “expert” answers to questions from their peers about identification of the different types of angles and their angle measures and reasoning without the use of notes. For example, the students may be asked to identify what type of angle that angle 5 is, its measure, and how they know. Also, the groups are asked to generate 3-5 questions about the content they might anticipate being asked and that they can ask other experts.

Close this activity by having the students complete a SPAWN writing (view literacy strategy descriptions) assignment, writing from a W or, What If? perspective. The students should complete this task individually. Give the students the following prompt: What would be the measure of the angles formed if the transversal that cuts two parallel lines is perpendicular to the parallel lines? What type(s) of angles would be formed? Draw a diagram showing this situation.

Give students about 10 minutes to complete their written responses. When all students are finished, ask students to share their ideas with the class and discuss the responses.

Activity 12: Scale Drawings (GLE: 30)

Materials List: rulers, paper, pencils, Scale Drawings BLM, poster board

Explain to students that a scale drawing is a drawing that is similar to a real object. The drawing is labeled to show the actual dimensions of the real object; however, the drawing is either a reduction or an enlargement of the real object. The value used to determine the reduction or enlargement is called the scale or scale factor.

Show students that they have already learned how to solve problems involving scale drawings when they learned to solve proportions. Remind them that they have used scale factors in dilations.

Work the following example with the class.A blueprint shows the master bedroom to be 5 inches long. The scale on the blueprint is 2 in: 8 ft. What is the scale? What is the actual length of the master bedroom?

The since 2 inches on the blueprint represents 8 feet in the room, the scale is 1 inch represents 4 feet. However, a proportion can be established to solve the problem without reducing the scale:

2 in 5 in






-------- = -------- 8 ft x ft

2x = 40

x = 20 ft; the length of the master bedroom is 20 feet.

Distribute the Scale Drawings BLM. Have students work in groups of three to work through the real-life situations. After the BLM has been completed, have each group discuss its scale drawing developed in problem 4. One person from each group will be chosen to be a part of the expert group in using a modification of the professor know-it-all (view literacy strategy descriptions) strategy. With this strategy, the teacher selects a group to become the “experts” on scale drawing required in the situation that is selected. The group should be able to justify its thinking as it explains its proportions or solution strategies to the class. Encourage questions from the other students related to scaling to challenge the know-it-alls. Also, tell the other students to listen carefully to the know-it-alls’ descriptions of scale drawing, and hold them accountable for accuracy. All students must prepare to be “experts” because they are not told prior to the beginning of the strategy which students will be chosen to be in the expert group.

Sample Assessments

General Assessments Have students respond to the following prompt in their learning log.

Write a real-life application of one of the concepts learned in this unit. Provide students with unlined paper and rulers to develop a scale drawing of their “dream

room” Provide the students with several right triangles that have a missing side measure. The

students will find the lengths of the missing sides.


Activity 2 : Provide students with similar triangles with missing side lengths, and have them find the missing sides using proportions. Students should show all of their work.

Activity 3: Provide students with a handout of a figure that has been rotated, translated, and reflected. Allow the students to match the transformation to the appropriate picture.

Activity 9: Give students the dimensions of a rectangular prism and ask them to determine the effect on volume and surface area of a dilation using scale factors of ½ and 3.

Activity 11: Have students work in pairs to develop two true statements and one false statement about angle relationships. (These statements should be verified by the teacher.) Each statement should be written on separate index cards. Have pairs of students challenge other pairs to determine which statements are true and which statement is false.





Ninth GradeRemediation MathUnit 5: Probability

Time Frame: Approximately three weeks

Unit Description

The unit explores the application of theoretical and experimental probability, sampling with and without replacement, permutations, and combinations and analyzing single and multiple event probability situations. A real-world problem solving approach is used to enhance conceptual understanding.


Students will deepen their understanding of determining the probability of simple, dependent, independent and mutually exclusive events. Based on this understanding, they will be able to make predictions of outcomes in various real-life situations. Students will also be able to determine what biased sampling is and how it impacts the decision-making process.

Guiding Questions

1. Can students determine the number of orderings (permutations) or combinations (groupings) that can occur under given conditions?

2. Can students calculate and interpret single- and multiple-event probabilities in a wide variety of situations, including independent, mutually exclusive, and dependent, non-mutually exclusive settings?

3. Can students use experimental data to make outcome predictions of independent events?

4. Can students explain the impact of sampling with and without replacement? 5. Can students use their knowledge of probability in real-world situations?


GLE # GLE Text and BenchmarksGrade 8Number and Number Relations8. Solve real-life problems involving percentages, including percentages less than

1 or greater than 100 (N-8-M) (N-5-M)Data Analysis, Probability, and Discrete Math41. Select random samples that are representative of the population, including

sampling with and without replacement, and explain the effect of sampling on bias (D-2-M) (D-4-M)

42. Use lists, tree diagrams, and tables to apply the concept of permutations to represent an ordering with and without replacement (D-4-M)

Remediation Math Unit 5Probability 54



GLE # GLE Text and Benchmarks43. Use lists and tables to apply the concept of combinations to represent the

number of possible ways a set of objects can be selected from a group (D-4-M)44. Use experimental data presented in tables and graphs to make outcome

predictions of independent events (D-5-M)45. Calculate, illustrate, and apply single- and multiple-event probabilities,

including mutually exclusive, independent events and non-mutually exclusive, dependent events (D-5-M)

Sample Activities

Activity 1: Counting it Up (GLEs: 42, 43)

Materials List: Permutations BLM, paper, pencils

Place the words Permutations and Combinations on the board. Ask students what they remember about these words and record key words from their responses. Allow students to think as you ask questions to stimulate the discussion. Examples of questions are (1) What do you remember about order? (2) What do you remember about tree diagrams? Try to stimulate the recall of prior knowledge. After a couple of minutes of discussion, give the students the definition of the two words and have them record the definitions in their math learning log (view literacy strategy descriptions). The students will refer to the definitions during this unit. The following definitions may be used.

Permutations - An arrangement or listing of items in which order is important (two listings of items in a different order are considered different listings – ABC is different than BAC) Combinations - An arrangement or listing of items in which order is not important (two listings of items in a different order are considered the same – ABC is the same as BAC).

Explain to students that in everyday language the term combination is used loosely without thinking of order. For example, a fruit salad may be considered a combination of apples, grapes and bananas and the order the fruit is in is not important. However, when a combination lock has the combination of 583, order does matter. The only way the lock will open is if the numbers are used in the correct order. In math, when the order of the arrangement matters, then those possibilities are called permutations.

Tell the students to use the definitions to answer the following question.

How many permutations are there of the letters M, N, and O?

Give students a couple of minutes to work on answering the questions. Then ask the students to volunteer their answers.

Explain to students that the answer to this question is six permutations. Show students that these permutations can be represented in several ways.





As a list: MNO, NMO, OMN, MON, NOM, and ONM. All of these are different because order matters with permutations, i.e., listing things in different a order is considered a different arrangement of the items.

In a tree diagram:Place the following tree diagram on the board or projector.

Use a questioning strategy to help the students develop the formula for determining permutations by multiplication. Consider asking the students the following questions: (1) In the tree diagram example, how many initial choices are available for the first letter? The students should note that there are three choices, M, N, or O. (2) After the first choice is made, how many choices remained for the second letter? (Remind students that the letters cannot repeat.) Students should note that there are two choices left. (3) After the first and second choices have been made, how many additional choices are available? Students should answer, “One remaining choice.” As students give their answers, refer to the tree diagram to point out the portion of the tree diagram that represents the numbers of choices they are giving.

Summarize their conclusions and write the number of choices on the board as the process is reviewed. Initially there were three choices. Write the number 3 on the board. After the first choice, there were two remaining choices. Write the number 2 to the right of the 3, ( 3 2) After the first two choices, there was one remaining choice. Write the number 1 to the right of the 2 ( 3 2 1). Remind the students that the number of possible permutations is 6. Ask students if they see any connection to the total number of permutations and the number of choices that exist at each step. The answer of 6 was produced with both the list and the tree diagram. Have students refer to the list and the tree diagram to make the connection. If students speculate that they can find the answer by multiplying the number of choices at each step, ask if they think they can do this in every case.

As another example, ask students to determine the number of permutations for the four colors red, blue, yellow and green. Have them first predict the number of permutations and then create a tree diagram to verify their predictions. Check the students’ work to ensure that they have the correct answers.



Ask the students to draw a tree diagram showing the number of possible permutations for 10 different letters: the letters A, B, C, D, E, F, G, H, I, and J. As the students begin this work, ask them to complete this in about two minutes. Students will quickly realize that it takes much longer to draw a tree diagram to represent the possible permutations and that the tree diagram would become very complicated. If students have previously proposed multiplying the number of choices found at each stage of developing the tree diagram, then begin a discussion about how many choices are available in the first step, the second step, etc. Ask students how you know when to stop such a process. If students have not yet made a connection, then lead them to see that the product of the number of remaining choices after each step will produce the correct answer.

Give students the following example for practice:

A store manager is designing a display for five types of cookies: Oreos, Nutter Butter, Chips Ahoy, Oatmeal Raisin, and Fig Newtons. How many different ways can the cookies be arranged for the display?

Have the students determine the number of permutations. Once the pairs have finished their work, have them share their work with another pair of students. Then review the answer as a class. There are 120 different ways the store manager can arrange the cookie display.

Permutations can also be determined when only some members of the initial set are being used. For example, if the store manager could only display two of the five brands of cookies, how many possible display arrangements would be possible? Let students know that in this case the number of permutations is determined using the number of choices that are available to fill a position in the display, but the process of multiplying is the same.

5 cookie choices are possible for the first choice × 4 cookie choices possible for the second choice; therefore, there are 5 × 4 possible arrangements or 20 possible display arrangements. Have students show the work associated with this in a list or tree diagram.

Distribute the Permutations BLM and have the students work individually or in pairs to determine the number of possible arrangements for the situations given.

Activity 2: How Many are There? (GLEs: 42, 43)

Materials List: How Many are There? BLM, paper, pencils

Have the students refer to their math learning logs (view literacy strategy descriptions) and review the definitions of the words permutations and combinations. Ask students to state the basic difference between the two words. They should respond that with permutations, order matters (arranging items in different order is considered to be a different arrangement) and with combinations, order does not matter (arranging the same items in a different order is considered to be the same arrangement).




To review their work with permutations, ask students to determine the number of permutations of a play list of the following music: Jazz, Hip Hop, Pop and R&B. Have students complete a list, a tree diagram, and use the formula method. Students should be able to explain and show that there are twenty four permutations. Check student work to ensure they understand these concepts. Now, present the students with the questions below and give them a few minutes to determine the answers. Remind them to look at the difference between permutations and combinations.

1. How many combinations are there of the letters M, N, and O?2. How many combinations of 2 letters can be formed from the letters M, N, and O?

Ask students to share some of their answers. Explain to students that the answer to question one is one combination. This is true because order does not matter with combinations (a different ordering is considered the same list). Therefore, the combination MNO is the same as the combinations NMO, OMN, etc. The same items in any order are considered to be the same arrangement and are called a combination.

Now that the answer to the first question has been clarified, ask the students to recheck their answers to Question 2 and make any necessary changes. After a couple of minutes, ask students for their answers. Continue to question students about their answers and help them to come to the conclusion that the answer to Question 2 is three combinations, MN, MO, and NO. No other combination is possible because since order does not matter, any other 2-letter combination would be a duplication of one of these three.

Give the students the following problem for guided practice:

In forming a committee, two students are to be selected from a group of five students. How many different committees of two can be formed?

Begin this work by having the students write out all of the possible arrangements or permutations of two students. Allow students to work through this with minimal support. Monitor students as they work and offer support only when their work has errors. Once students have had an opportunity to list all of the possible arrangements, share the list below with them for clarity.

Identifying the 5 students as A, B, C, D, and E

PermutationsAB BA CA DA EAAC BC CB DB EBAD BD CD DC ECAE BE CE DE ED Now ask the students what must they do to determine the number of possible combinations and why. They should conclude that to determine the number of combinations, they must eliminate the duplications in the permutation list because order matters and therefore, for example, the



committee AB is the same as the committee BA. Once students have eliminated the duplication, their work should show 10 possible combinations as listed below.

CombinationsAB AC AD AE BCBD BE CD CE DE

Therefore, there are 10 possible combinations.

Distribute the How Many are There BLM? Have students work in pairs to complete. In this BLM, students will also determine if a situation involves a permutation or combination, or in other words, determine if order matters in the stated situation.

Activity 3: Experiments in Probability (GLEs: 44, 45)

Material List: Experiments in Probability BLM, coins, calculators, paper, pencils,

For this activity, students need to understand the difference between dependent events and independent events. Have students add the following definitions to their math learning log (view literacy strategy descriptions).

Dependent events - two events are dependent if the outcome or occurrence of the first affects the outcome or occurrence of the second.

Independent events - two events are independent if the outcome or occurrence of the first does not affect the outcome or occurrence of the second.

Ask students for examples of dependent and independent events.

Ask students if rolling a number cube and then rolling the cube again are dependent or independent events. The students should respond that these are independent events because the outcome of one roll does not affect the outcome of the next roll. Each time a person rolls, he/she can get one of 6 numbers that are on the number cube.

Continue this activity by asking what probability is. After receiving answers from students, ask them to write the following definition in their math learning log (view literacy strategy descriptions).

Probability – how certain it is that a particular event or outcome will occur. The ratio of successful outcomes to the total number of possible outcomes.

To further explain the concept of probability, work through determining the sample space for the possible outcomes for the sum obtained by rolling two number cubes as a class. Present the following table to the students:







Sums when Rolling Two Number Cubes1 2 3 4 5 6

123456

Help the students begin to complete the table. The numbers in the shaded areas represent the two number cubes. Allow the students to complete the table on their own. Then review their answers. The completed table of all possible outcomes, known as the sample space, is below. Have students add this term to their math learning log (view literacy strategy descriptions).

Sums when Rolling Two Number Cubes1 2 3 4 5 6 72 3 4 5 6 7 83 4 5 6 7 8 94 5 6 7 8 9 105 6 7 8 9 10 116 7 8 9 10 11 12

Number of successful outcomesBased on this example, Probability = --------------------------------------- Number of possible outcomes

Therefore, the probability of rolling a sum of 3 is: 2 1 P (3) = ---- = ---- because there are 2 sums of 3 out of a possible 36 sums. 36 18Note: Explain to students that P(3) represents the probability of rolling a sum of 3.

Ask the students to study the table and predict which sum has the greatest chance of being rolled. Students should deduct that since there are more sums of 7 than any other sum, seven is the sum with the greatest chance of being rolled. Ask the students to predict which sum has the least chance of being rolled? The students should deduct that the sums of 2 and 12 have the least chance of being rolled. Have students work in pairs to find the probabilities of the other sums, write the fractions in simplest form, then change each to a decimal and then to a percent probability.

The students should set up a table as the one below:




Probabilities of Rolling the Indicated Sums

Sum Probability(fraction)

Probability(simplified fraction)

Probability(decimal)

Probability (percent)

1

2

3

4

5

6

7

8

9

10

11

12

Upon completion of the activity, explain to students that the probabilities in the table represent the theoretical probability of rolling the various sums as it is based a formula to compare desired outcomes with possible outcomes. Experimental probability is based on the results of performing an experiment.

Distribute the Experiments in Probability BLM. Allow students to work in groups of three to determine the experimental probability of the events. Once each group completes its work, they will share their results with the class. Help students note the variability in the data.

Now compile the data from the individual groups to form one class set of data for each experiment. Have students recalculate the probabilities of the data from the whole class and compare it to their individual group probabilities. Students should note that as the number of



experiments increase, the experimental probabilities should move closer to the theoretical probability.Activity 4: Crossing the River (GLE: 44)

Material List: small objects such as cm cubes or counters to represent boats (12 for each student), Crossing the River BLM

In this activity, students play a game that helps them use their knowledge of probability, discover patterns, and develop mathematical strategies to win the game.

Review the game rules with the students and either choose pairs or allow students to choose their own opponent. Distribute the Crossing the River BLM and 12 “boats” to each student. Within each pair, each student’s boats should be of the same color, but of a different color than his/her partner. Although it should be obvious that the game is based on the probability of rolling sums with two number cubes, as in Activity 3, do not mention this to the students. Monitor the games as they play to see if students discover a game strategy based on the work done in Activity 3. As the class ends, review the game strategy and how it is based on probability.

Activity 5: Theoretical vs. Experimental Probabilities (GLE: 44)

Material List: brown paper bag for each student with 12 colored objects in each (4 blue, 5 green and 3 red), computer access.

In this activity, students will investigate theoretical and experimental probability. In advance, prepare a brown paper bag for each student with 12 colored objects of three different colors in each (for example, 4 blue, 5 green and 3 red). Objects should be light weight to prevent the bag from breaking during the shaking process. Centimeter cubes or small pieces of colored paper will work well.

First have students create on a sheet of paper a table like the one below for recording the results of their individual experiments.

Object color Tally of colors chosen

Totalchosen

Fraction chosen

ExperimentalProbability

1) Give each student a prepared bag. (Do not let the students know that their bags have the same objects in them.) Tell the students they will open the bag, but they must not look inside.



2) Have each student take out one object, record its color with a tally mark and then place the object back into the bag. Gently shake the bag to mix up the objects prior to the next pull.

3) Students should repeat this process 11 more times for a total of 12 times.4) Have each student complete the table including finding the experimental probability for

each color in their bag. 5) Have students close their bags, without looking in them and place them under their desks.6) Now place the students in groups of four. Have each group identify a person to create

another table like the one above and record the combined results of the individual members’ experiments.

7) Using the data from the combined results from the individual charts, determine the experimental probability for the group of four.

8) Provide a large chart for students to record the data for the class. Have each group place their group’s data on the chart and have the class determine the experimental probability for the class.

9) Then have each student pour out the contents of his/her bags and determine the actual number of each color they have in their bag and find the theoretical probability of each color.

10) In their groups of four, have the students address the following:- Compare the experimental and theoretical probabilities for their individual

probabilities, group probabilities and class probabilities, and write a statement or reflection about each comparison.

- Which probabilities were closer to each other? What is the reason for these probabilities being closer?

Monitor students as they complete these activities. Once each group has completed the activities, have a representative from each group share its group’s reflection statement with the class. Address any misconceptions or problems encountered. At the end of this activity, make sure students understand that as the number of experiments increase, the experimental probability approaches the theoretical probability.

For additional practice, have students visit the website below. This site provides a virtual spinner that allows students to spin multiple times and record the results of the spins. http://nlvm.usu.edu/en/nav/frames_asid_186_g_3_t_5.html?open=activities&from=search.html?qt=probability

First, have students look at the spinner and determine the theoretical probability. Then spin the spinner multiple times to see that as the number of spins increases, the experimental probability approaches the theoretical probability.

Activity 6: Special Probabilities (GLEs: 8, 45)

Material List: Special Probabilities BLM, calculators, paper, pencils, Rubric to Evaluate Presentations BLM (if specific-assessment is to be used)


http://nlvm.usu.edu/en/nav/frames_asid_186_g_3_t_5.html?open=activities&from=search.html?qt=probability

http://nlvm.usu.edu/en/nav/frames_asid_186_g_3_t_5.html?open=activities&from=search.html?qt=probability


Explain to students that two or more events that are influenced by each other are called dependent events. The probability of dependent events is found by multiplying the probability of the first event by the probability of the second event. Model the following example with the students.

A box contains 3 red pencils and 4 blue pencils. Two pencils are pulled from the box at random and placed on the table. Ask the students to explain why these events are considered dependent events. Help them conclude that they are dependent because when one pencil is removed from the box, the number of pencils of one color is reduced, thus affecting the probability of pulling the next color.

Ask students, “What is the probability of removing a red pencil on the first pull and a blue pencil on the second?”

P(red on the first pull) = , P(blue on the second pull) = or

To find the probability of removing a red and then a blue pencil multiply the probabilities; ×

= or or 28.57%

Provide additional examples if students need additional support understanding this concept.

Now introduce the concept of mutually exclusive events. If two or more events cannot happen at the same time, the events are said to be mutually exclusive. The probability of occurrence of two or more mutually exclusive events is the sum of the probabilities of the individual events. Share an example of determining the probability of rolling a 3 or 4 when rolling one number cube. When the cube is rolled one time, a 3 or a 4 can be rolled, but not both. The probability of rolling a 3 or 4 is the sum of the probability of rolling a 3 and the probability of rolling a 4.

P(3 or 4) = P(3) + P(4) = + = = or 33.33%

Distribute the Special Probabilities BLM and allow students to work in pairs to complete it. Closely monitor students as they complete the activity. Make sure the students can explain the reasoning behind their work.

The professor know-it-all (view literacy strategy descriptions) strategy will be used to conclude the activities on probabilities. Through this activity, students will be given opportunities to demonstrate what they have learned about the different types of events and how to determine their probabilities.

Place students in groups of four. Have students review their work on independent events, dependent events and mutually exclusive events. They should understand the difference between the events, be able to provide an example of each, and know how to determine their probabilities of occurrence. Explain that they will be called on randomly to come to the front of the class to




provide “expert” answers to questions from their peers about whether events are dependent, independent or mutually exclusive and how to calculate the probabilities for a given situation. Also, the groups are asked to generate 3-5 questions about the different events and the determination of probabilities that they anticipate being asked and that they can ask other experts.

See the Activity-Specific Assessments at the end of this unit for evaluating presentations.

Activity 7: Sampling: Is It Biased or Random? (GLEs: 41, 44)

Materials List: Opinionnaire BLM, paper, pencils

Begin this activity with a discussion about surveys. For example, ask the students to share some examples of surveys that can be or have been conducted. Some examples to stimulate student thinking are 1) sports surveys to determine the player of the week, 2) surveys to determine the most popular cell phone, or 3) a survey to determine the most popular brand of tennis shoe. Tell students that due to the sampling process, the survey results can be misleading depending on who is surveyed and how the questions are structured. Present the students with the terms biased sampling vs. random sampling. Explain to students that random sampling occurs when every individual in the population has an equal chance of being selected. In biased sampling, a sampling method is used that does not give every individual in the population an equal chance of being selected. Based on the sports example above, if the survey for the player of the week were taken during a game between only two schools, the survey would be biased toward the players at those schools, limiting the input of individuals that would favor players from other schools.

In this activity the students will use an opinionnaire (view literacy strategy descriptions) to determine whether a survey process represents biased sampling or random sampling. Opinionnaires are highly beneficial in promoting deep and meaningful understandings of content area topics by activating and building relevant prior knowledge, and by building interest and motivation to learn more about particular topics. Opinionnaires also promote self-examination, value youths’ points of view, and provide a vehicle for influencing others with their ideas. .Distribute the Opinionnaire BLM and tell students to respond individually to the statements and be prepared to explain their responses by explaining the reason for their choice. Give the students about ten minutes to complete the BLM. Monitor students as they work to make sure they are giving a reason for each choice.

Next, put students in pairs and have them compare and discuss their responses to the opinionnaire. Emphasize that there is no correct answer at this stage and students should discuss freely. Open the discussion to the whole class so many points of view about the accuracy of the information can be expressed.

Transition from the discussion to reviewing each of the situations and help the students to understand the correct answer and the basis for the answer. During this process, address any




misconceptions about random vs. biased sampling and have students make corrections to their original choices.



Activity 8: Sampling With and Without Replacement (GLEs: 41, 44)

Materials List: paper, pencils, a brown paper bag with ten objects of different colors: five of one color, three of another color, one of a third color and one of a fourth color

In this activity, students will predict the number of objects of each color in a bag using the random sampling process. In advance, prepare a brown paper bag with 10 objects of different colors: five of one color, three of another color, one of a third color and one of a fourth color. Let the students know that there are ten objects in the bag. Allow one student to pull one object from the bag without looking and place the object on his/her desk. The class should record the color of the object removed. Repeat this process four more times. Tell students that this type of sampling is sampling without replacement, because the objects removed were not returned to the bag. Based on this information, have students work in pairs to predict the number of each color object in the bag. Have the students discuss their results as a class and their reasoning for their predictions.

Now, put all of the objects back in the bag. Have one student pull one object from the bag without looking. The class should record the color of the object removed. The student is then allowed to put the object back into the bag. Repeat this process four more times. Tell students that this type of sampling is sampling with replacement, because the objects removed were returned to the bag. Based on the information recorded, have students work in the same pairs to predict the number of each color object in the bag. Have the students discuss their results as a class and their reasoning for their predictions.

Remove the objects from the bag so that students can see how many of each color are in the bag. As a class, compare the results between the data generated by sampling with and without replacement. Ask the pairs of students to write a reflection about their predictions as compared with the actual colors of the objects in the bag. Their reflections should be placed in their math learning log (view literacy strategy descriptions).

Sample Assessments

General Assessments

Have students play a game of chance and analyze the probabilities of winning. Have the students modify the rolling two number cubes activity to determine the products

instead of the sum. Have students create their own activity with brown paper bags and different colored

objects, and challenge their peers to make the appropriate predictions.





Activity 2 : Have students compare and contrast permutations and combinations and provide an example of each.

Activity 3 : Assess the table of probabilities and their corresponding representations as simplified fractions, decimals and percents to determine if the work performed by the students was correct. If deficiencies in these skills are noted, more practice work should be provided.

Activity 6 : Use the Rubric to Evaluate Presentations BLM or a similar rubric to evaluate the professor-know-it-all presentations.

Activity 7 : Have students create a sampling scenario and poll their peers to determine if the scenario represents random or biased sampling.




Unit 6: Algebra

Time Frame: Approximately five weeks

Unit Description

This unit focuses on developing a basic understanding of equations, expressions, inequalities and graphing. Making connections between verbal, numeric, graphic, and symbolic representations of relationships is also explored. Problem solving in meaningful contexts is used to further develop conceptual understanding.


Students deepen their understanding of representing situations graphically by accurately plotting points and creating graphs of real-life situations. Students are able to look at these graphs and determine if they are linear or non-linear and share what this means in regard to the data. Students are also able to switch between various representations of data. Students begin to see and extend patterns in data using pictures and numbers. Students begin to master the process for solving equations and inequalities without technology to ensure that they have a strong knowledge of integer operations.

Guiding Questions

1. Can students solve one and two-step equations with and without technology? 2. Can students switch between the various representations of data: verbal, graphical,

tabular and equations? 3. Can students graph multi-step linear equations and inequalities? 4. Can students explain and form generalizations about rates of change in real-life

applications? 5. Can students construct and use a table of values from a given equation and graph it on

the coordinate plane? 6. Can students distinguish between linear and non-linear real-life situations?


GLE # GLE Text and BenchmarksGrade 729. Plot points on a coordinate grid in all 4 quadrants and locate the coordinates of a

missing vertex in a parallelogram (G-6-M) (A-5-M)Grade 8Number and Number Relations5. Simplify expressions involving operations on integers, grouping symbols, and whole

number exponents using order of operations (N-4-M)

Remediation Math Unit 6Algebra 69



GLE # GLE Text and Benchmarks8. Solve real-life problems involving percentages, including percentages less than 1 or

greater than 100 (N-8-M) (N-5-M)Algebra10. Write real-life meanings of expressions and equations involving rational numbers

and variables (A-1-M) (A-5-M)11. Translate real-life situations that can be modeled by linear or exponential

relationships to algebraic expressions, equations, and inequalities (A-1-M) (A-4-M) (A-5-M)

12. Solve and graph solutions of multi-step linear equations and inequalities (A-2-M)13. Switch between functions represented as tables, equations, graphs, and verbal

representations, with and without technology (A-3-M) (P-2-M) (A-4-M)14. Construct a table of x- and y-values satisfying a linear equation and construct a graph

of the line on the coordinate plane (A-3-M) (A-2-M)15. Describe and compare situations with constant or varying rates of change (A-4-M)16. Explain and formulate generalizations about how a change in one variable results in

a change in another variable (A-4-M)Geometry33. Graph solutions to real-life problems on the coordinate plane (G-6-M)Patterns, Relations, and Functions46. Distinguish between and explain when real-life numerical patterns are

linear/arithmetic (i.e., grows by addition) or exponential/geometric (i.e., grows by multiplication) (P-1-M)

47. Represent the nth term in a pattern as a formula and test the representation (P-1-M), (P-2-M), (P-3-M), (A-5-M)

48. Illustrate patterns of change in dimension(s) and corresponding changes in volumes of rectangular solids(P-3-M)

Sample Activities

Activity 1: Operations with Integers (GLE: 5)

Materials List: Operations with Integers BLM, pencils, paper

For success in algebra, it is very important that students have a strong foundation in operations with integers. To that end, in this activity, students will review and practice operations with integers.

Introduce the activity by introducing situations that represent negative and positive integers to help students with connections between integers and real-life situations. Examples of such situations are as follows and should be presented as questions to the students.

1) A loss of 5 yards in a football game - 52) A deposit of $25 to a savings account + 253) A 7 degree temperature decrease - 74) Three feet below sea level - 3



Ask students for other real-life examples representing negative and positive integers. Correct any problems or misconceptions.

Now prepare students to move into operations with integers. Remind students of their previous studies with integer operations and the rules associated with them acknowledging that they may have heard the rules expressed in different ways. It is important to tie the development of the rules to prior knowledge.

Adding integers:

Use two-colored counters to model addition of integers. For examples, red counters can represent negative integers and blue counters positive integers. For example, -3 + 5 can be modeled by giving each student 3 red counters and 5 blue counters. When making matched pairs of red and blue, there are 2 blue counters remaining. Therefore,-3 + 5 = 2. Model problems of various types including finding the sum of integers with the same sign. Let students provide the rules, and assist them in coming to class consensus on the wording of the rules and examples.

Like Signs: Add numbers and keep the sign. Example: 3 + 5 = 8; -3 + -5 = -8Unlike Signs: Find the difference of the two numbers (disregarding the signs). The answer takes the sign of the number with the largest absolute value. Examples: -3 + 5 = 2; 3 + -5 = -2

Subtracting integers:

One method of developing the concept of “adding the opposite” to perform subtraction is through the use of patterns that rely on prior knowledge. Ask students to quickly write the problems below and write the answers:

5 – 4 =5 – 3 =5 – 2 =5 – 1 =5 – 0 =

Solicit and record students’ responses as shown below.

5 – 4 = 15 – 3 = 25 – 2 = 35 – 1 = 45 – 0 = 5

Ask students what patterns they see. They should notice that as the number being subtracted decreases, the answer (difference) increases.

Ask students to write the next problem in the pattern. They should write



5 – (-1) = ___. Ask them what the difference should be based on the pattern. It maybe necessary to review the pattern found in the differences again. The differences of 1, 2, 3, 4, 5 are increasing, so 6 has to be the next difference; therefore, 5 – (-1) = 6.

5 – 4 = 15 – 3 = 25 – 2 = 35 – 1 = 45 – 0 = 55 – (-1) = 6

Ask students for another way to get 6 when starting with 5. They should see note that 5 + 1 = 6. Thus, 5 – (- 1) = 6 means the same as 5 + 1 = 6. Lead a class discussion about how to read and interpret the subtraction sign as meaning “add the opposite of.” Tell students that they have already applied this rule in the past and may not have known it. Connect “adding the opposite” to a simple problem such as 7 – 2 is 5, noting that 7 + (-2) is also 5. Practice with several examples, and then let the class come to consensus on the rule for subtracting integers and use of examples.

Add the opposite of the number to be subtracted.

Example: 6 – 2 becomes 6 + (-2). Follow the rule for adding integers with unlike signs 6 + (-2) = 4

Example: -6 – 2 becomes -6 + (-2). Follow the rules for adding integers with like signs -6 + (-2) = -8

Multiplying and dividing integers:

Patterns also work well to develop the rules for determining signs in multiplication and division of integers. This pattern can be built on students’ prior knowledge of a positive number times a positive number, and a positive number times zero.

5 * 3 = 155 * 2 = 105 * 1 = 55 * 0 = 5

Following the above pattern,5 * (-1) = -5 This could also be reinforced by knowing the meaning of multiplication (adding -1 + -1 + -1 + -1 + -1 = -5). Therefore, a positive times a negative results in a negative. If needed, have students do another pattern, replacing 5 with another positive number.



The rule for multiplying two negative numbers can be based on a pattern similar to the one below:

3 * -1 = -3 2 * -1 = -2 1 * -1 = -1 0 * -1 = 0-1 * -1 = 1 (as the factor in first column decreases, the product increases)-2 * -1 = 2

After examining the signs in the pattern, students should see that a negative number times a negative number results in a positive number.

Since multiplication and division are inverse operations, point out that -2 * -1 = 2 also means 2 -1 must equal -2. Therefore, a positive number divided by a negative number is a negative. The

same process can be used with any of the other multiplication rules to show that the rules for signs are the same in division as in multiplication.

Continue with pattern development and practice as needed, and then have the class come to consensus on the rules for multiplying and dividing integers. Review the terms product and quotient.

Multiply or divide the numbers. To determine the sign of the answer for two numbers, like signs produce a positive product or quotient; unlike signs produce a negative product or quotient.

Examples: 3 • 5 = 15; -3 • (-5) = 15; -3 • 5 = -15 15 ÷ 3 = 5; - 15 ÷ (-3) = 5; - 15 ÷ 3 = -5

Because the result is a time saving device, it may be beneficial to have students multiply and/or divide several sets of one digit integers with varying signs. After students work the problems, have them compare the number of negative numbers to the sign of the answers. They should see the following pattern:

If more than two numbers are being multiplied or divided, an even number of negative signs will produce a positive answer and an odd number of negative signs will produce a negative answer.

Once the rules have been established, give students an opportunity to practice operations with integers using the guided practice instructional format. Address any misconceptions.

Distribute the Operations with Integers BLM. Allow students to work in pairs to answer the problems. Monitor students as they work and check answers for correctness. In closing this activity, remind students to review and practice their rules and commit them to memory.



Activity 2: Integer War Game (GLE: 5)

Materials List: one deck of cards for each pair of students, paper, pencils, calculator

This game allows students to practice their skills with the integer operations of addition, subtraction, and multiplication.

Before starting the game, place students in pairs. Have each pair write each of the rules for operations with integers and give an example of each without using their notes. Then allow the students to use their notes to check and correct their rules. Instruct students to review the rules in preparation for the Integer War game where they will be competing with each other.

Explain to the students the cards are given signs in Integer War. The red cards represent negative numbers and the black cards represent positive numbers. The number on the card will represent the card’s value. A face card has a value of 10 and an ace has a value of 1. Remove the jokers from the decks. Four different rounds will be played.

Distribute a deck of cards and a calculator to each pair of students. The cards should be split evenly between the two students. The calculator will only be used to verify disputed answers if the teacher is not available.

Round 1: Addition with integers

Model the game for the whole class by working with one student. Each player flips over a card. The first one to add the numbers correctly and say the answer wins the cards. This round will be played for 10 minutes. The person with the most cards at the end of the 10 minutes will win the round. 2-minute reflection: Each student writes the rule for adding integers.

Round 2: Subtraction with integers

Each player flips over a card. The number on the second card down is subtracted from the first card down. As an example, if the first card is Red 8 and second card is Black 2, the problem is -8 – 2. The first one to subtract the numbers correctly and say the answer wins the cards. This round will also be played for 10 minutes. The person with the most cards at the end of the 10 minutes will win the round.

2-minute reflection: Each student writes the rule for subtracting integers.

Round 3: Multiplication with integers

Each player flips over a card. The first one to multiply the cards correctly and say the answer wins the cards. This round will also be played for 10 minutes. The person with the most cards at the end of the 10 minutes will win the round.



2-minute reflection: Each student writes the rule for multiplying integers and also connects the rule for dividing integers.

Note: Expectations for behavior must be established by the teacher and communicated to the students before the game begins. Consider having rewards for those students with proper decorum to be distributed as good behavior is observed during the game. Another suggestion is to have integer operation worksheets available as an alternate activity for those students who have trouble meeting classroom expectations.

Activity 3: Back to the Basics (GLE: Grade 7: 29 )

Materials List: Coordinate Graphing BLM, graph paper, pencils

In this activity, students practice and show their mastery of the basics of graphing on a coordinate grid learned in Grade 7 (GLE 29).

Open with a discussion of basic graphing. Distribute the Coordinate Graphing BLM. Have students (1) label the x- and y-axes, (2) label the origin, (3) identify the four quadrants and (4) plot and label the following points (3,7), (8,1, (-5, 2), (3,-4), -6,-2), (7,-10), (-1,9), and (-8,-1). Monitor students as they individually complete the BLM. After the students have completed the work, show a transparency of the correct graph to them. Ask the students to review the completed BLM and identify any errors in their work. Ask several students to explain where they had errors and use this discussion as an opportunity to clear up any misconceptions.

For additional practice, have the students plot and label the following points on the same graph. The points to plot are as follows: (-8,1), (-4,7), (0,5), (5,0), (-4,-8), (8,-2), (-9,4), (-2,-2) Monitor students as they work and assist those that have problems properly graphing the point. Update the transparency with the first set of points by adding (or allowing students to add) the last group of points.

To end the activity, have the students work in pairs to complete a RAFT (view literacy strategy descriptions) writing task. For this activity, the RAFT description is as follows:

R – 9th grade studentA – 6th grade studentF – LetterT – How to plot points on a coordinate graph and avoid mistakes when doing so.

Be sure to use the proper math terminology.

In other words, the student will take the role of a 9th grade student writing a letter to a 6th grade student explaining how to plot points on a coordinate graph and listing several mistakes to be careful to avoid. The students should use proper math terminology, such as x-coordinate, y-coordinate, origin, etc.

Once the RAFTS are completed, students should share them. As the students read their RAFTed assignments, other students should listen for accuracy. Listening to students’ RAFTs will allow





the teacher to evaluate whether students adequately understood the material and if further teaching or independent practice is needed. Ask students to place the RAFTs in their math learning log (view literacy strategy descriptions) for later review.

Activity 4: Expressions: Verbal to algebraic (GLEs: 10, 14)

Materials List: graph paper, pencils

In this activity, students will review and practice the process of translating verbal phrases into algebraic expressions. Remind students that algebraic expressions have at least one number, one variable and one operation. An example of an algebraic expression is n + 8. Ask students what the n, +, and 8 represent. They should respond variable, operation, and constant, respectively. Give the students the following algebraic expressions and ask them what the variable, operation,

and constant are: 3x and .

Now that students have a clear understanding of what an algebraic expression is, have them translate an algebraic expression to a verbal statement. Ask the students to translate n + 8 to a verbal phrase. Example responses are 1) a number increased by 8; 2) a number plus 8; or 3) the sum of a number and 8. Give students several more examples where they translate from an algebraic expression to a verbal phrase and vice versa.

For additional, individual practice, have students complete the following table.

Verbal phrase Algebraic expressionx - 12

ten more than a number9 - x

twice a number

Review students’ responses as a class and have students make corrections where necessary.

In the next part of this activity, students will connect their knowledge of translating verbal to algebraic expressions with plotting points and creating a graph of an algebraic equation. Give students the following situation:

A local automated car wash charges $8 for its “Works” car wash.

Have students create a table of x and y-coordinates showing the amount of money in dollars (y) generated when 1, 4, 6 and 10 cars (x) are washed. Assist students with setting up the table if necessary. The completed table is represented below.




x y# cars dollars

1 84 326 4810 80

Have students graph this data on a sheet of graph paper. Monitor the students to ensure they set the graph up correctly with the proper labels, scales, points, etc. For students that make mistakes plotting the points, have them refer to the RAFT letter in their math learning log (view literacy strategy descriptions).

As an introduction to the next activity, ask students to explain the relationship they see between the number of cars and the money generated. As the discussion progresses, help the students to understand that the value of x is multiplied by 8 to determine the amount of money. In other words, help students to conclude that the equation representing the car wash situation is y = 8x.

Activity 5: Words to Equations to Tables to Graphs (GLEs: 12, 13, 14)

Materials List: graph paper, pencils, Words to Equations to Tables to Graphs BLM

Explain to students that data can be represented in several ways. When given data in one form, it can be used to generate different representations. For example, write the following verbal description on the board: y is x increased by 2

Ask students to write an equation to represent this information. Ask several students to volunteer their answers, and discuss the answers to help students understand the correct translation to an equation of y = x + 2. Now have students use the equation to develop a table of x- and y-coordinates, and use these coordinates to construct a graph. Monitor students as they work. Once most students have completed their work, present the table, equation, and graph as four different representations of the same information. Have students share a real-life application that can be represented by this equation.

Give students the following coordinates: (-2, -4), (-1, -2), (0, 0), (1, 2), (2, 4)

Have students make a table, create a graph, write an equation, and write a verbal description of the information. Monitor students as they work and when complete, have several students present their work.

Provide students with the Words to Equations to Tables to Graphs BLM. Allow the students to work in pairs to complete the activity. Although students are working in pairs, each student should complete the activity.





Activity 6: Solving Equations (GLEs: 5, 10, 12)

Materials List: Solving Equations BLM, paper, pencils, calculators

Present students with the following scenario:

Sarah works as a babysitter and charges $7 per hour plus a fee of $10 for each job. Sarah earned $38 from one of her babysitting jobs. Using the equation, 7x + 10 = 38, where x is the number of hours worked, how many hours did Sarah work?

Help students to connect the equation to the situation by discussing how the parts of the equation represent portions of the scenario. For example, ask students what each part of the equation represents, and tie this discussion back to the discussion on translating verbal phrases.

Review the process of solving equations with the students starting with the concepts of isolating the variable by using the reverse of the order of operations. In using an example such as the one below, ask students questions during each step of the process such as the following: What should be done first and why? What should be done next? Why should we subtract instead of add?

7x + 10 = 38 7x + 10 – 10 = 38 – 10 (subtract 10 from both sides) 7x = 28 (performed subtraction)

= (divide both sides by 7)

x = 4 (Sarah worked 4 hours.)

Remind the students to check their answer by substituting the value of x in to the equation

7x + 10 = 38 7(4) + 10 = 38 28 + 10 = 38 38 = 38

Since both sides of the equation are equal to each other, the answer of 4 is correct. Address any questions students may have about this problem.

Give students the following scenario:

Larry’s cell phone provider charges a monthly fee of $25 plus 10¢ per minute of usage. If Larry’s cell phone bill last month was $85, how many minutes did Larry use his cell phone?

Have students set up the equation and solve it. Monitor students as they work. After about five minutes, ask for a volunteer to present his/her work to the class. Ensure that each step is shown, a



rationale for each step is given, and the correct answer of 600 minutes is given. Then have another student check the work to verify that the answer is correct.

Give students the following problems to complete for guided practice.

1) = 48 2) + 7 = 11 3) -3 + 4x = 13 4) 8x – 3.5 = 36.5

Monitor students as they work to ensure that they show their work for all of the problems.

Distribute the Solving Equations BLM and have students complete it individually by solving each problem and checking all of their answers. Encourage students to ask clarifying questions as they work.

Activity 7: Building with Toothpicks (GLEs: 13, 47)

Materials List: flat toothpicks, paper, pencils

Place the students in pairs and provide each pair about 75 to 100 toothpicks. Have the students make each of the shapes below using their toothpicks.

Figure 1 Figure 2 Figure 3 Figure 4

Ask students to complete the table below by recording the number of toothpicks needed to form the perimeter of each shape. Instruct students to look for and discuss with their partner any patterns they see.

Figurenumber

Perimeter (in

toothpicks)

1

2

3

4



After students have shown that they understand the concept of perimeter and have the correct perimeter for each of the figures, have them complete the following tasks:

1. Use a pattern from the figures to determine the perimeter of Figure 5. Show in pictures or explain in writing how you arrived at your answer.

2. What would the perimeter of figure 20 be? Explain your answer.

3. Write a formula that you could use to find the perimeter of any shape n. Explain, in writing, how you found your formula.

To close this activity, students will use brainstorming (view literacy strategy descriptions). Brainstorming involves students working together to generate ideas quickly without stopping to judge their worth. In brainstorming, students in pairs or groups freely exchange ideas and lists in response to an open-ended question, statement, problem, or other prompt. Students try to generate as many ideas as possible, often building on a comment or idea from another participant. This supports creativity and leads to expanded possibilities. The process activates students’ relevant prior knowledge, allows them to benefit from the knowledge and experience of others, and creates an anticipatory mental set for new learning.

Allow students to work in groups of four to brainstorm ideas for five minutes to identify how using patterns to find perimeter would be beneficial in real-life situations. Then ask each group to share its ideas with the class. Record the group ideas on the board. Once the ideas have been recorded, continue a discussion about the answer to help students to understand the value of seeing and using patterns in real-life situations.

Note: This activity was adapted from the NCTM Navigating through Algebra series, 2005.

Activity 8: Fall Sports Banquet (GLEs: 11, 13, 14, 16, 47)

Materials List: square tiles (approximately 4 per group of three), Fall Sports Banquet BLM

Place the students in groups of three and give them the Fall Sports Banquet BLM and a set of square tiles. Explain to students that they will determine the number of people that can sit at tables with two different arrangements. In the first arrangement, the square tables are separate, no sides touching. In the second arrangement, the tables touch on one side. See the diagrams on the BLM for specific details. Stress that in completing the activity, students show all of their work either in written statements or diagrams and explain how they determined the rule.

Have students write a reflection explaining how they used the tiles to help them “see” the pattern and how they translated what they saw in the pattern into an algebraic expression. Students should place the reflection in their math learning log (view literacy strategy descriptions) for future reference when switching between functions as tables and algebraic expressions.





After the students have completed the BLMs and their reflections, open a discussion with the students by asking them to compare the patterns created in the x/y tables that were based on the two different arrangements of dining tables. Ask, “As x changes by 1 in each of the table arrangements, how does y change?” Help students to see that in the first arrangement (tables separated), as the number of tables increases by 1, the number of people that can be seated increases by 4. Have students compare these changes to the changes found in the data when using the second arrangement (table sides touching). They should be able to state that when number of tables increases by 1, the number of people that can be seated increases by 2. Ask students to graph the two data sets, labeling each with the equations of the lines formed. Lead a discussion of which line is steeper and why. The discussion should also include the fact that both sets of data have a constant rate of change with the situation of having separate tables showing a faster rate of change than when tables are placed next to one another.

Activity 9: Linear or Non-linear? (GLEs: 8, 11, 14, 33, 46)

Materials List: graph paper, pencils, calculators, Quadrant I Grid BLM

Present the students with the following scenario and have them work in pairs to determine if the situation is linear or non-linear.

Michael has a car washing business from which he earns $60 each month. Each month Michael saves $30 from the money he earned. Consider these two situations:a) Each month Michael saves $30 in a shoe box under his bed. At the end of a year,

how much money has Michael saved? orb) Each month, Michael gives the $30 to his mother, who pays him 5% interest each

month on the accumulated balance, which includes all interest earned to date. At the end of a year, how much money has Michael saved?

Have students make tables and graphs to represent each situation. The graphs should be developed on the same Quadrant I Grid BLM. Examine both graphs. How are they different? One situation may be characterized as being linear. Which situation is linear? How can you tell?

After the students complete the activity, explain that Situation B involves compound interest. Compound interest involves paying interest upon the interest. Have students explain why the graph representing a compound interest situation would generate a nonlinear graph. Students should place their explanation in their math learning log (view literacy strategy descriptions) for future reference. Be sure to address any misconceptions as the pairs share their explanations with the class.




Activity 10: Not Necessarily Equal (GLE: 12)

Materials List: pencils, paper, Not Necessarily Equal BLM

Present examples to the students to help them understand the difference between equations and equalities. In explaining inequalities, explain and give examples of the following terms and their corresponding symbols: less than, greater than, less than or equal to, greater than or equal to.

Ask students what they remember about the differences in the processes for solving equations and solving inequalities. Tell them that the same rules that they used to solve equations apply to solving inequalities with one exception: multiplying or dividing by a negative number reverses the direction of the inequality. To review this, give students the following inequality and ask students why it is true.

-10 > -15

Tell students that they are to divide each side by 5 and ask what the resulting inequality will be. Students should get -2 > -3. Prompt students to notice that they have divided both sides of the inequality by a positive number and the resulting inequality is true.

Now ask the students to divide each side by -5 and provide the results. Students should notice that the results of division produce 2 and 3; however, writing 2 > 3 is not true. To have a valid inequality, the inequality symbol must change from “is greater than” to “is less than” resulting in 2 < 3. In this case, the division was by a negative number which requires that inequality symbol be changed.

Have students solve and review the following as an application of this rule:

-2x + 5 ≤ 15

After solving this example, have students graph the answer on a number line. Lead a discussion about the fact that there are many solutions to this inequality, specifically any number greater than or equal to -5. Remind students that this does not apply just to integers, but to fractions and decimals as well (e.g., -1.56, ½ )

Distribute the Not Necessarily Equal BLM to the students. In completing this BLM, students will practice solving inequalities. Some of the inequalities will require the students to take a real-life problem and set up the inequality before solving it.

Activity 11: Change, Is It Constant or Not? (GLEs: 10, 15, 16, 33, 46)

Materials List: graph paper, pencils,

Write “Constant Change” and “Varying Change” on the board. Ask student volunteers to explain these terms. During the discussion, help students understand that for situations with constant change, the change in a value over time is the same, and with situations with varying



change, the change in a value over time is not constant. Have students use a What If? prompt of SPAWN writing (view literacy strategy descriptions) to identify real-life situations with constant and varying change. For example, have the students write the following prompt in their math learning log (view literacy strategy descriptions) and respond: “ What if your mom recently started a weight loss program and lost 6 pounds the first week, 4 pounds the second week, and 2 pounds the third week. Explain the type of change this represents.” Give students about five minutes to write their responses after writing the prompt. After the five minutes, allow the students to share their writing with a partner to stimulate discussion. If they don’t see it, use questioning techniques to guide students to an understanding that is a varying rate of change.

Present students with a second What If? prompt to write in their math learning log (view literacy strategy descriptions) and write a response. “What if you earn $9 per hour? Explain the type of change this represents in earnings over the course of an eight hour work day?” After about five minutes, allow the students to share their responses with a partner. If students struggle, suggest that they create a table of hours worked and money earned. After the partner discussion, allow a class discussion of the examples and the students’ thoughts on constant and varying change. Students should see that the second situation is a constant rate of change as the amount of money earned increased by $9 each hour. Address any misconceptions about constant and varying change.

Give students the following and have them work in pairs. They should describe and compare the two situations by developing graphs and writing an explanation comparing the shapes of the two graphs. The graphs should be done on separate sheets of graph paper.

Mr. Jones and his family decided to drive from Baton Rouge to New Orleans on Interstate 10. The data in Table A represents their trip to New Orleans down Interstate 10 where x represents time in minutes and y represents miles driven.

Table A

x(time in minutes)

y(distance in miles)

10 1020 2030 3040 4050 5060 60

On the family’s return trip to Baton Rouge, there was construction work being done in some of the areas. The data in Table B represents its trip back to Baton Rouge.







Table B

Monitor the students as they work to ensure that they complete their graphs correctly. In writing their descriptions of the graphs, tell students to indicate which graph represents constant change and why. Once the graphs and written explanations are complete, allow the students to discuss their findings.

Present the students with the graph produced by the data from Table A. Through a questioning process, help the students to see how to move from one point to the next on the graph. For example, to move from the point (10, 10) to the point (20, 20), one moves up 10 units on the y-axis and over 10 points on the x-axis. Allow students to describe the move to the next point. As students practice this process, introduce the concept of slope (rise/run) to them. Let students know that since the change from one point to the other is the same, it shows that the rate of change is constant for this situation. Have the students look at the graph from the data in Table B to see that the change in moving from one point to the other is not the same for all the points in this graph. This shows that non-linear graphs do not have a constant rate of change.

Have students create a table of values (at least 5 points) and a graph for the equation y = 3x. Is the graph linear or non-linear? As x changes by 1, by what value does y change? What real-life situation could be represented by the equation? Have student volunteers discuss their graphs, explain their reasoning behind whether or not the graph is linear or non-linear, and share their real-life situation that could be represented by the equation. Address any errors or misconceptions.

Activity 12: A Flower Garden (GLEs: 11, 13, 14)

Materials List: graph paper, pencils, calculator, colored pencils (3 for each student)

Provide students with a handout containing the information below. Allow students to work in pairs to determine which local store has the best pricing for Mrs. Johnson’s flower garden.

Mrs. Johnson is planning to prepare her flower bed for spring flowers. To do so, she must purchase top soil to help build up her bed. She wants to spend the least money possible on the top soil so she will be able to buy as many flowers as possible. Therefore,

Remediation Math Unit 6Algebra

x(time in minutes)

y(distance in miles)

10 1020 2030 2540 2850 3360 43

84


she has been careful to check the local advertisements for sales on top soil. The advertisements show the pricing information below:

Local store Sale priceWal-Mart $10 per bagTarget Buy 2 for $14 each, get one freeHome Depot Buy 1 for $16, buy all additional at ¼ off

Complete the table below to determine where Mrs. Johnson should purchase her top soil in order to save the most money.

No. of bags 1 2 3 4 5 6 7 8 9 10Wal-MartTargetHome Depot

Graph the costs for the top soil on the same coordinate graph. Use different colors to represent each store’s prices. Which store(s) produces graphs that are linear? Which store(s) produces graphs that are non-linear? Explain your reasoning.

If Mrs. Johnson needs to buy 3 bags of top soil, from which local store should she make her purchase? What is the basis for your decision?

If Mrs. Johnson needs to buy 8 bags of top soil, from which local store should she make her purchase? What is the basis for your decision?

Note: This activity was adapted from the NCTM “Let’s Go Home” activity.

Sample Assessments

General Assessments

Have students create verbal descriptions of algebraic equations and vice versa. Provide students with algebraic equations and have them solve the equations and

explain each step. These equations should be solved without the use of a calculator in order to allow students to also practice integer operations.

Provide students with a pictorial representation of a pattern and have students determine what the nth pattern would be.




Activity 6: Give students a linear graph and have them create a table of values and write an equation to represent the situation.

Activity 9: Give students two real-life scenarios, one linear and one non-linear. Based on the information, have the students predict which situation is linear and which is non-linear and give their reasoning. Then have students create graphs from the situation to evaluate their choice.

Activity 10: Have students write a verbal description of the following inequality: y > x – 3.

Activity 11: Have students think of a real-life situation with a constant rate of change and one with a varying rate of change and explain the differences in their graphs.




Unit 7: Algebra Connections

Time Frame: Approximately three weeks

Unit Description

The focus of this unit is to help students connect mathematics to real-life experiences. The activities provide the opportunity to practice concepts and skills critical for success in algebra.


Students will use proportional reasoning to solve real-life problems and connect the affects of the sampling process. They will also explore how the dimensions of a shape affect its volume. Students will also deepen their understanding of algebra applications by translating between real-life situations and expressions, equations and inequalities.

Guiding Questions

1. Can students use proportional reasoning to solve real-life problems? 2. Can students translate real-life situations to expressions, equations and inequalities? 3. Can students find the nth term in a pattern? 4. Can students determine the volume of a cylinder and describe how changes in the

dimension affect the volume? 5. Can students set up, solve and graph multi-step linear equations and inequalities

based on real-life situations?


GLE # GLE Text and BenchmarksGrade 8Number and Number Relations7. Use proportional reasoning to model and solve real-life problems (N-8-M)Algebra10. Write real-life meanings of expressions and equations involving rational

numbers and variables (A-1-M) (A-5-M)11. Translate real-life situations that can be modeled by linear or exponential

relationships to algebraic expressions, equations, and inequalities (A-1-M) (A-4-M) (A-5-M)

12. Solve and graph solutions of multi-step linear equations and inequalities (A-2-M)

13. Switch between functions represented as tables, equations, graphs, and verbal representations, with and without technology (A-3-M) (P-2-M) (A-4-M)

Remediation Math Unit 7Algebra Connections 87



GLE # GLE Text and Benchmarks14. Construct a table of x- and y-values satisfying a linear equation and

construct a graph of the line on the coordinate plane (A-3-M) (A-2-M)Measurement17. Determine the volume and surface area of prisms and cylinders (M-1-M)

(G-7-M)Geometry33. Graph solutions to real-life problems on the coordinate plane (G-6-M)Data Analysis, Probability, and Discrete Math41. Select random samples that are representative of the population, including

sampling with and without replacement, and explain the effect of sampling on bias. (D-2-M) (D-4-M)

Patterns, Relations, and Functions46. Distinguish between and explain when real-life numerical patterns are

linear/arithmetic (i.e., grows by addition) or exponential/geometric (i.e., grows by multiplication) (P-1-M) (P-4-M)

47. Represent the nth term in a pattern as a formula and test the representation (P-1-M) (P-2-M) (P-3-M) (A-5-M)

Sample Activities

Activity 1: Experimenting with Volume (GLE: 17)

Materials List: tape, 2 - 8 ½ x 11 transparencies for each class (or sheets of paper), ruler, fill material (rice, packing peanuts, cereal, etc.), calculators

In this activity, students will experiment with the volume of cylinders made from the same size paper. Open this activity by asking students to compare and contrast volume and surface area. As the students discuss the comparison, hold up a cylindrical shape and have the students continue the discussion referring to the cylindrical shape shown. Ask students to give the formula for calculating the volume of a cylinder. Students may refer to any of their reference materials. If necessary, help students to remember the formula for the volume of a cylinder, V = πr2h, where V is volume, π is 3.14, r is the radius of the cylinder and h is the height of the cylinder. Write this formula on the board and ask the students to write it down.

Show the students two transparency sheets and note that the sheets are the same size. Get six student volunteers (three on each side of the desk) to come to the front of the class to make cylinders out of the transparency sheets. Instruct the other students to observe as the cylinders are made. Have one group of students use a transparency sheet and tape to form a “bottom-less” cylinder such that 11” is the height of the cylinder. The edges should meet exactly with no overlap or gaps. The other group of students will make the other cylinder such that 8.5” is the height. Again, there should be no overlap or gaps. The edges should meet exactly. Have the students stand the two cylinders on the desk in front of the class. The tall cylinder will be



referred to Cylinder A and the short cylinder will be referred to as Cylinder B. Label the cylinders accordingly. Have another model of the two cylinders available for reference.

An anticipation guide (view literacy strategy descriptions) will be used to guide the students’ response to several statements about the volumes of the cylinders formed. Anticipation guides are best used with information that is verifiable as is the case with the volumes of the cylinders.

1. Give each student a handout with the following statements and ask the students to respond “yes” or “no” to the first three questions and write the reason for their answers in the space provided.

Anticipation Guide Statements

a. The cylinders have the same volume. Yes ___ No ___

b. The short cylinder, cylinder B, has the greater volume. Yes ___ No ___

c. The tall cylinder, cylinder A, has the greater volume. Yes ___ No ___

Explain the reasons for your answers.

2. Put the students in pairs and have them compare and discuss their responses to the statements. Encourage students to freely discuss their answers and reasons for their choices.

3. After about 5 minutes, open the discussion to the whole class so many different opinions, beliefs, points of view and hunches about the accuracy of the statements are expressed.

4. Transition from the discussion into the experiment to determine the correct answer. Have four other volunteers come to the front of the class. Two students will work with Cylinder A and two students will work with Cylinder B. Each pair of students will place its base-less cylinder in a small box and make sure that the bottom of the cylinder is held firmly against the box. Fill the cylinders with the available fill material, and measure the amounts of fill material to determine which cylinder holds the most. Allow the students options for determining which amount of fill material is greater using the available equipment. For instance, they can weigh the material if a scale is available, or they can transfer the material to other labeled containers such as measuring bottles, beakers, etc. and compare the amounts.

After confirming that Cylinder B holds the greater amount, have the students work in pairs to share their initial responses to the anticipation guide statements, and then explain why Cylinder B holds the greater amount. Walk around to monitor students as they work, and correct any initial incorrect responses to the guide statements. After a few minutes, remind the class that it may be able to use the formula to help it find the answer. When monitoring students, help them to conclude that Cylinder B has a larger radius than Cylinder A, and since radius is squared in




the formula, it has a greater impact than the height. Have the students determine the radius and height of each cylinder and use the formula to calculate the volume of each cylinder. From this work, lead the class in a discussion of how the dimensions of the cylinder affected the volume of the cylinder although the transparency paper was the same size in both cases.

Ask the students to make a third cylinder by folding a new transparency sheet lengthwise and cutting it in half. Each piece measures 4.25 inches by 11 inches. Tape these two pieces together to form a rectangle that is 4.25 inches by 22 inches. Have the students predict whether or not this new cylinder will hold more than the other two cylinders based on the knowledge they just gained from the previous experiment. After students make their predictions and discuss their reasoning, conduct the experiment and compare the volumes. Initiate a discussion to help the students see that the taller and narrower cylinders hold less than the shorter, stouter cylinder. Since the radius is squared, it has more of an effect on the volume than the height. Therefore, starting with rectangles of the same surface areas, the cylinder with the largest radius will have the greatest volume.

Activity 2: Spreading Rumors (GLEs: 33, 46, 47)

Materials List: paper, pencil, calculator, graph paper

Give students the following table of values:

n 1 2 3 4 5 203 6 9 12 15 ?

Ask the students to explain any patterns they see in how the numbers in the top row can be used to generate the numbers in the bottom row. Ask them to write in words what is happening with the numbers. Students should see that the bottom numbers are three times the top numbers. Another way of stating this is 3n, where n stands for the number in the top row. In other words the rule here would be 3n. (Remind students that 3n means 3 times n). So the value for n = 20 would be 3 x 20 or 60. Have students graph this data to see that the graph is linear. Ask students to discuss why the graph is linear and express the type of change occurring in the bottom row of numbers. Share additional examples of rules if necessary.

Begin a short, general discussion about how quickly rumors spread. Tell the students that today they will analyze a situation and predict how fast a rumor can spread by determining the rule for the pattern in the situation.

Give students the following scenario and allow them to work in pairs:

In homeroom, a student decided to start a rumor that school would be closed Friday due to the expectation of an unusually high snow fall.

The student tells one other student, so at the end of the first class, 2 students know the rumor.



After the second class, each of the first 2 people tells one other student, so now 4 students know.

After the third class, each of the 4 students tells one other student, so now 8 students know the rumor.

If the pattern continues through seven class periods, how many total people know the rumor? Create a table showing the data. What if the rumor continues at the same rate the following day? Determine a rule for finding the number of students who know the rumor after the nth class.

The table generated by students should look similar to the one below.

Have students write in words what they see happening with the number of people who now know the rumor. The students should see that the numbers keep doubling, or multiplying by two. Ask the students how many 2’s are multiplied together to determine the number of students for the first through third classes. Have the students write this out. For example, after class period 1, there’s only one 2. After class period 2, there are two twos multiplied together to get 4, 2 × 2. After class period 3, there are three twos multiplied together to get 8, 2 × 2 × 2. Ask students if there is another way to write these numbers. Students should eventually see that another way to write the numbers is 21, 22 and 23 for the first three classes, or 2n. Have students use this rule to determine how many students will know the rumor after the 2nd day or 14th class.

Give students a sheet of graph paper and have each of the students create a graph of the data in the table for the first 7 classes. Remind students to use proper scaling, labels, etc. After creating the graph, have students compare the shape of this graph to the graph developed above. They should note that the second graph is not linear. Extend this discussion and share with the students that this graph is exponential. Help students to see that the rate of change is not constant with this graph as with the linear graph created above.

Activity 3: Real-Life Algebra (GLEs: 10, 14, 33 )

Materials List: graph paper, pencils, Real-Life Algebra BLM

Remind students of the work they did in Unit 6 on translating verbal phrases to algebraic expressions and expressions. Present the following examples to students and have them translate them into verbal phrases.

Example 1: x + 3 Example 2: 7x + 10 = 31

Remediation Math Unit 7Algebra Connections

Class Period # of students1 22 43 84 ?5 ?6 ?7 ?

91


Students should provide answers such as (1) a number added to three and (2) the sum of seven times a number and ten is thirty-one. Praise students on their ability to translate the equation and expression to verbal phrases.

To transition into real-life applications, use student questions for purposeful learning (SQPL) (view literacy strategy descriptions) SQPL promotes purposeful reading and learning by prompting students to ask and answer their own questions about content. Purposeful learning is associated with higher levels of engagement and achievement. Therefore, using SQPL encourages students to be more engaged in the learning process.

Put the following statement on the board or overhead for students to read:

Algebraic expressions and equations can be used to represent or model real-life situations.

Have students work in pairs to brainstorm two to three questions they would like to have answered to support their understanding of this concept. A typical question might be, “How is translating an expression or equation to a verbal phrase, different from or the same as translating to a real-life situation?” Or, “Why is it important to know how to translate real-life situations to expressions or equations?” Give students about five minutes to develop their questions. As a class, have each pair of students present one of its questions and write this question on chart paper or the board. Give the class time to read each of the questions presented and select the ones to use. Tell students that they will revisit these questions as they continue to study how algebraic expressions and equations can be used to represent real-life situations.

Use questioning to help the students extend their ability to translate expressions to real-life situations by asking them how they would take the expression from above, x + 3, and put it into a real-life situation. Give students a minute or two to think about this and ask questions. If the class struggles, help it get started with the translation. For example,

Assume x + 3 represents the amount of money Larry has and the x represents the amount of money Molly has. The expression would translate to Larry has three dollars more than Molly.

Have students discuss this situation with their partners and refer back to the questions listed on the board or chart paper. Does this example answer any of their questions? Can they see how algebraic expressions can represent real-life situations? Ask several pairs to share their thoughts. Address any misconceptions or gaps in understandings.

Refer students to the second example, 7x + 10 = 31. Give them a few minutes to think of a real-life situation that could be represented by this equation. Walk around the class to support students as they work. After a few minutes, ask several students to share their situations. Provide feedback as students share their work. Be prepared to help students through the process if necessary. For instance, an example of a situation for the equation is:




The equation can represent the amount of money Sarah earned for babysitting Mr. and Mrs. Johnson’s son last Saturday night. The variable x represents the number of hours worked. The expression would translate as follows: Last Saturday night Sarah earned thirty-one dollars by babysitting for the Johnsons for x hours at a rate of seven dollars per hour plus a ten dollar fee for purchasing gas to get there.

As a class, discuss the example to check for understanding. After the discussion, have students create a table of x- and y- values and construct a graph for the situation above. Remind students to properly label their graphs. Have the students classify the graph as linear or non-linear and write an explanation of their choice. After all graphs are complete, students should compare their graphs with those of a partner, note any differences, and make corrections where necessary.

Have students refer to the original questions listed on the board or chart paper and note how what they have learned has helped them to answer those questions.

For additional practice, distribute the Real-Life Algebra BLM and allow students to work in pairs to complete. Monitor students as they work.

Activity 4: Real-Life Algebra, Too (GLEs: 11, 12)

Materials List: paper, pencils, graph paper, Real-Life Algebra, Too BLM

This activity is an extension of Activity 3. Students will be given real-life situations to translate into algebraic expressions and equations.

Initiate a discussion on the learning from Activity 3. Have the students refer to the questions they generated in Activity 3 and what they learned in that activity.Ask students if they have any questions about the process of translating algebraic expressions and equations into real-life situations. Address any questions. If there are no questions, ask students questions such as 1) Why is it important to focus on the specific wording in the situation? 2) How can you check to ensure that the expression or equation matches the situation? Continue to ask students questions to get them to think through the process of switching between real-life situations and algebraic equations and expressions.

Tell students that they will now do the opposite by translating a real-life situation to an algebraic expression or equation. Give students the following real-life situation and have them write an algebraic expression or equation that represents it.

Jeffrey had some money in his pocket. He gave his little brother 3 dollars to go to the store. How much money did Jeffrey have then?

Ask students to write down their responses and then call for volunteers to share. An example of an expression for the situation above to represent how much Jeffrey had after giving his brother three dollars is j – 3. Address any questions students may have about the expression. Ask students why only an expression, not an equation, should be written to represent this situation.



Be sure that students state what the variable represents. In this case, j is how much money Jeffrey had initially.

Give students the following real-life situation and have them write an equation or expression to represent it.

Mia wants to buy a video game that costs $73.48 (including tax). She has already saved $45.00 and wants to know how much more money she needs to buy the game.

Ask students to write down their responses. After about a minute, have the students share their responses with a partner. Call on volunteers to share their responses and give feedback as the students share. An example of an equation to represent the situation above is m + 45 = 73.48, where m represents the amount of money that Mia needs. Ask students why an equation and not an expression should be used to represent this situation. Have students solve the equation to determine how much more money Mia needs to save.

Distribute the Real-Life Algebra, Too BLM and allow students to work in pairs to complete it. As a class, review each of the situations to ensure that students have the correct equations and solutions to the equations.

To close the activity, have the students write a reflection comparing the processes for translating from algebraic equations and expressions to real-life situations and vice versa. Have the students place their reflection in their math learning log (view literacy strategy descriptions).

Activity 5: How Much Will I Earn? (GLEs: 11, 13, 14)

Materials List: graph paper, paper, pencils

This activity is designed for students to show mastery of writing and solving equations, creating data tables, and graphing the data from a table. Allow students to complete this activity individually. Provide the following scenario and questions to students.

You are planning to start a lawn service. You have decided to charge your clients $35.00 for an average size lawn. Based on this fee, you want to determine how much you will earn for mowing l number of lawns.1. Write an equation to represent the situation above.2. Create a table of values and a graph showing the amount you would earn for mowing up

to seven lawns in one day.

Have students answer the following questions:1. Is the graph linear or non-linear? Explain your answer.2. Based on the graph, how much will you earn for mowing 5 lawns? Explain how you

used the graph to find this answer.3. How many lawns will you have to mow to earn at least $450.00? Write an inequality to




represent this situation.4. What is the rate of change of the graph?

Monitor students as they work and pay particular attention to how they set up their equation and graph. Also make sure students provide explanations for their answers. Provide support when necessary. As a class, discuss this activity and ask students to focus on the most difficult parts for them. Address any misconceptions or gaps in knowledge. Once the class discussion is complete, have each student reflect on his/her learning from this activity by writing a reflection in his/her math learning log (view literacy strategy descriptions).

Activity 6: Going Fishing (GLEs: 7, 41)

Materials List: paper lunch bags, beans (or other counters), 5 oz. cups, paper plates, paper, pencils, calculators, Going Fishing BLM

The capture-recapture method is a statistical method used to estimate the size of a population. For example, a scientist could use this process to estimate the number of fish in a pond.

In this activity, students will take on the role of scientists in estimating the number of fish in a pond. Divide students into groups of three. Provide each group with a paper lunch bag

containing dried beans (enough to fill the bag about full), a 5 oz cup, a permanent marker, a

paper plate, and the Going Fishing BLM. The paper bag serves as the pond, the beans represent the fish, and the cup will be used as a fishing net.

Once each group has its materials, review the BLM with students to ensure they have read and understood all of the instructions. Work through the first part with the students. Walk around to monitor students as they complete the activity, and address any problems or misunderstandings.

Have the student groups present their work and discuss the variations in their population estimate and actual populations. Have students use the What IF? Prompt in SPAWN writing (view literacy strategy descriptions) to respond to the following prompt in their math learning logs (view literacy strategy descriptions).

How might the population estimate been affected if during the sampling process the bag was not shaken?

After about five minutes, have each group share their responses. Then ask for other factors that might impact the accuracy of the estimate.

Note: This lesson was adapted from the Florida Center for Instructional Technology







Sample Assessments

General Assessments

Provide students with a table of values. Have the students graph the data and note if the graph is linear or non-linear.

Have students develop their own real-life situations with corresponding representations as expressions, equations or inequalities.

Have students develop a brochure on situations where the capture-recapture method is used to determine population sizes.

Provide students with additional pattern problems and allow them to determine the value of the nth term.


Activity 1 : Give students various cylinders with different dimensions and have the students place the cylinders in order from least to greatest by predicting the volume of each. After the predictions, have the students calculate the volume of each cylinder and determine if their order prediction were correct.

Activity 3 : Give students other expressions, equations, and inequalities and have the students write real-life situations for them.

Activity 4 : Give students other real-life situations and have them translate them to expressions, equations, and inequalities.

Activity 5 : Have students create their own real-life situation, write an equation, create a table of values, and graph the data.

Activity 6 : Allow students to combine their beans into one container and perform the sampling procedure as a class. Once the sampling is complete, students will determine an estimate based on the sampling process and compare it to the actual population which has been determined by combining the data from each of the groups.


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