unit overview grade/course building mathematical practices ...€¦ · should progress from...

LCI Math 07/12/13 Math6/Building Mathematical Practices Unit 1

Unit Overview

Unit Title Building Mathematical Practices Grade/Course Level 6

Subject/Topic An introduction to the mathematical practice standards through exploring logic, word problems, operations on whole numbers and operations on decimals.

Unit Length 10 days Sequence Placement First two weeks of the school year

Big Idea The Standards for Mathematical Practice build mathematical reasoning

Essential Question

What does it mean to persevere in problem solving? What does it look and sound like to construct a viable argument and critique the reasoning of others? Why is it important to have multiple strategies to manipulate numbers?

Unit Resources

Holt, Mathematics Course 1; Springboard: Middle School 1; Core Connections: Course 1; Mathematics for Elementary School 5A (Japanese curriculum); http://www.svmimac.org

Unit Materials Graph paper, rulers, poster paper, coloring materials, and a projector

Additional Activities and

Tasks (Available on

the OUSD Core

Curriculum website)

Number Talk Series #1 “Whole Number Operations”, “Go Figure” PowerPoint presentations, Mathematical Practices Daily Reflection Journal


Unit Storyline

Entry Task “The BART Problem”: Students use prior knowledge to figure out whether or not they will have a window seat on the BART train, given that there are 4 seats in a row with an aisle in between. This task is designed to understand how students attack a problem, and to give students their first chance to see the value of learning from each other’s strategies

Lesson Series 1

Make sense of complex problems and persevere in solving them, while constructing viable arguments and critique the reasoning of others: This lesson series offers the teacher a menu of logic activities, through which students explore multiple strategies for approaching a problem, learn from their classmates, and can identify an exemplar mathematical explanation.

Formative Task

“The Bookstore”: Students solve questions about the prices of items for sale at a bookstore. Students compare purchase prices, determine amount of change, and reason whether or not repeated addition or multiplication is more efficient. This task uncovers students’ prior knowledge of addition and subtraction of multi-digit decimal numbers, and gives students an opportunity to work collaboratively in a team. Students get an opportunity to practice critiquing and building on their teammates’ thinking.

Lesson Series 2

Adding and subtracting multi-digit decimals à Multiplying and dividing multi-digit decimals: Increasing their understanding of concepts they explored in the formative task, students deepen their understanding of addition and subtraction of multi-digit decimals. Students extend that knowledge to multiplication and division of multi-digit decimals. This lesson series offers the teacher a menu of activities where students can create and use visual representations to model multiplication and division of multi-digit decimals. Throughout the lesson series, re-engagement activities may be needed to help students deepen their understanding of place value when adding and subtracting of multi-digit decimals.

Summative Task “Sensible Division”: Students find sensible answers to 5 scenarios that can be answered by dividing 100 by 6. Students determine when an exact answer is necessary, or when it is appropriate to round up or round down. Students demonstrate on an individual poster how their thinking has shifted around how to create an exemplar mathematical explanation.

Please Note: There are more non-routine tasks, activities, and lessons in this unit than days. Please consider the following as you plan:

Required Core content - 5 Days Max. (Ensure students have these learning experiences):

Elective Core content – 5 Days Min. (Select accordingly from these lesson series activities and lessons):

Entry Task – “The BART Problem” Adapted from Japan Lesson Series 1 – “How does it grow?” CPM 1.1.3 Formative Task – “Bookstore” Adapted from SpringBoard Lesson Series 1 – “Go Figure” Power Point

Summative Task – “Sensible Division” Adapted from MARS Lesson Series 1 – “Party Time” POM Lesson Series 1 – “Pennies” Adapted from CPM Lesson Series 2 – “How Much will it Cost?” Adapted from Japan Lesson Series 2 – “How Much did we Pay?” Adapted from Japan


Unit Overview

Common Core State Standards Essential Mathematics Supporting Mathematics Standards for Mathematical Practices

• Fluently divide multi-digit numbers using the standard algorithm. (6.NS.2)

• Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. (6.NS.3)

• Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. (5.OA.3)

• Make sense of complex problems and persevere in solving them (MP1)

• Reason abstractly and quantitatively (MP2) • Construct viable arguments and critique the

reasoning of others (MP3) • Model with mathematics (MP4) • Use appropriate tools strategically (MP5) • Attend to precision (MP6) • Look for and make use of structure (MP7) • Look for an express regularity in repeated

reasoning (MP8)


Unit Overview

Academic Vocabulary

Tier II (words that are cross-curricular) Tier III (words that are math-specific) • Sensible • Justify • Reason • Viable • Argument • Persevere • Complex • Precision • Structure • Regularity • Repeated

• Dot diagram • Figure • Array • Bar model • Decimal • Relationship • Pattern • Representation • Model


Unit Entry Task

Learning Goal Learning Experience Framing Student Learning Engaging Every Learner

Uncover how students make sense of a math problem. Mathematical Practice of the Day: Make sense of complex problems and persevere in solving them (MP1)

The Bart Problem (1 day) Adapted from a Japanese Research Lesson

Materials: Prepare index cards labeled with numbers between 20 and 60. Post the Mathematical Practices in class to remind yourself and the students. Into: Conduct a discussion about the Mathematical Practices and introduce the idea of perseverance through a brief whole class brainstorm. Begin with a class discussion by asking students: • “When you ride the bus or BART, where do you like

to sit?” Frame the scenario that the class is going on a field trip and will be taking BART. Each student wants to know if they will have a window seat. Provide students with the seating chart for the first 3 rows of the BART train. Tell students that the seats right by the door that face each other will be reserved for the adult chaperones. Therefore students will only sit in the two rows that run alongside the windows. Through: Pass out cards to each student and tell them that their card has their seat number on it. Each student has a different number. Students should work individually but they can talk with their group and compare strategies. Beyond: Select students to present their strategies during the whole-class discussion. Presentations should progress from counting strategies (like extending the seating chart), finding patterns of adding four, finding a pattern with the right window seat and multiples of four, to finding a pattern with the remainder when dividing by four. At the end of class, ask students to find out where seat number 157 (or other large number). Ask students which strategy they used and why. Close the lesson with a journal reflection that connects this task to perseverance using the reflection form.

Pass out cards strategically so that students who like to be challenged get a higher number that is not divisible by four. You may also want to put a lower number on one side and a higher number on the other side so that students can choose the number they want to work with, or you can assign the number for them. Encourage student to student discussion by passing out cards so that each group of 4 has a left window seat, left aisle seat, right aisle seat and right window seat. Develop perseverance by pulling students who are stuck getting started to the side and prompting them with questions about the seating chart of the first 3 rows, like: • What seat is this? • How do you know where the window seats are? • What is this number? • Can you keep going and find your number? Move thinking forward while students are working by asking clarifying questions, like: • What if you do the same here just for the window

seats? • How do you know this? • So who would sit by the window? • Can you decide on this just on the tens place? • Did you understand what we were working on

before? • Do you need to count it? • Do you really need to write down the whole thing?

Notes to Self:

Building Mathematical Practices Unit – Entry Task Adapted from a Japanese Research Lesson 1

1

Math 6 Name: Date: Period:

The BART Problem

Question:

Is your seat a window seat or not? Use the seating

chart below to help you answer this question.

First 3 rows of the BART train:

1 2

3 4 5 6 7 8

9 10 11 12

BART Problem Solving Task

Learning Target: I can generalize a pattern or write a mathematical/ verbal expression to determine which BART seat I will sit in Lesson Components Lesson Sequence Formative Assessment Initial Engagement DO NOW: Students have 5 minutes to complete 5-7 problems from previous

taught standards. Observations – walk the room checking to see what problems students are struggling with Shoulder Partners – Have students check answers and coach Equality Sticks – pull sticks to have students provide answers. If students have incorrect answers have them correct

LAUNCH 10 minutes

Introduce Learning Target – Read aloud to students Explain the situation. We are going to take BART on a school field trip. Show BART seating chart. 1 minute Think-Write-Round robin Share: Which is the best seat to sit in on BART? 2 minutes think and write, 2 minute round robin share and 2 minutes whole class share out Pass out individual Student bart tickets. Ask the class if they are sitting in their preferred seat? 3 min

Examples: Think –Pair Share One minute essay Dyad

EXPLORE 25 minutes 17 min

Individual Task: Given your specific seat number determine where you will be sitting i.e. left window, left aisle, right aisle or right window. Explain how you know. 10 minutes work time [ Teacher walks the classroom observing student work, provides individual or group coaching if necessary] Call on students to show and explain their work. Progressively document each student’s work on the board. [ have students copy the work you put up on the board into their notes] 15 minutes

Anticipated student response per seat Left window LA RA RW *Find the RW and add 1 *Find the RW and subtract 3 * Divide seat number by 4 with a remainder of 1 * 4x + 1 * 4x - 3

* Find the RW and subtract 2 * Find the LW and add 1 * Find the RA and subtract 1 * Divide seat number by 4 with a remainder of 2 * 4x + 2 * 4x - 2

* Find the RW and then subtract 1 * Find the LW and add 3 * Divide seat number by 4 with a remainder of 3 * 4x + 3 * 4x - 1

* Adding/skip counting by 4 * These seats are all of multiples of 4. * Divide your seat number by 4. *4x

Group Task: Make a poster showing where each person’s seat is on BART. Indicate row number and seat location. Explain using a generalized pattern or mathematical/verbal expressions how you know where your seat is located on BART 12 minutes poster work and 5 minute group share-out

Formative Assessment Exit Ticket. Ensure students have mastered the lesson. 3 minutes

13 30

27 48

14 49

55 20

72 23

29 46

76 69

66 83

14 31

28 50

15 26

33 41

47 51

58 60

67 81

19 29

30 35

38 54

61 64

74 77

1 2 3 4

5 6 7 8

9 10 11 12


Unit Lesson Series 1


Uncover multiple strategies for approaching simple math problems that go beyond procedural fluency. Use the mathematical concepts of logic, reasoning, and extending patterns. Mathematical Practice of the Day: Construct viable arguments and critique the reasoning of others (MP3)

How does it grow? (1-2 days) CPM CCR1 1.1.3 Core: 1-15 through 1-18

Into: Conduct a discussion about the Mathematical Practices and introduce the idea of constructing a viable argument. Write a whole number expression on the board from the Number Talks Series #1 document. If this is the first time your class will do a number talk, select a simple addition or subtraction problem like 49 + 51. Start with Team Roles and begin on 1-15 by making it clear that you will be listening for team questions and conversations. Students may struggle to be engaged at first so structure this task as a Participation Quiz. Encourage them to work with their team to find as many ways of seeing and describing the pattern. If a team struggles to see the 30th figure, you can call a team Huddle or a Swapmeet. Through: Have teams create posters based on 1-16, which has directions for poster making. Have posters placed strategically somewhere that does not cover-up the participation quiz scribing. Facilitate a whole classroom discussion where each team is asked questions about their poster. Support students’ understanding by making connections from one poster to another while asking questions like, “Why this is a good strategy?”, “How is this strategy the similar to the strategy on that poster?” Beyond: Have them work on 1-17 and repeat the process from before. For fast finishing groups, offer 1-18 as a challenge. Close the lesson by highlighting a positive behavior from each group, and then end with a journal reflection that connects this task to constructing viable arguments using the reflection form.

Develop mathematical reasoning by opening with a Number Talk, which helps students to articulate their cognitive process when using mental math in the classroom (see Instructional Toolkit for more Number Talks details). Crediting competency supports building student confidence and lessen status in the room by scribing students’ names next to the strategy they share out with the class. This also helps students to build off of each other’s thinking by referring to each other’s strategies by name. Supporting effective group work can be achieved with consistent use of Team Roles. Suggested roles for this task are a Resource Manager, Facilitator, Recorder/Reporter, and Task Manager. Develop structures for group work by using a Participation Quiz to help students know what type of mathematical discourse is relevant and appropriate for the task. It makes clear the criteria for success, not just with content but also with expectations for group work behaviors. (see Instructional Toolkit for more Participation Quiz details). Build a culture of success by providing students with a clear picture of what group work looks like, and how all work on a poster must be justified. Using color and arrows in writing expressions, tables, and diagrams helps students to connect the different parts in a representation, or this case each figure.

Number Talk Series #1

Grade Level: 6th Unit(s): Building Mathematical Practices & continued in Statistics Core Math Ideas: Whole Number Operations Notes about Number Talks:

A. Keep them short.

B. Encourage sharing and clarify students’ thinking. C. Choose related sequences of problems (see below).

D. Chart the students’ thinking so that it can be saved and referred to later.

E. Create a safe and supportive environment, i.e., accept answers without praise or criticism. F. Record the students’ thinking using correct notation on the board, on doc camera, or on chart

paper.

G. Give students lots of practice with the same kinds of problems. Frame for each Number Talk: Requires students to:

• Mentally do math • Explain your reasoning • Listen and think about classmates’

strategies

• Understand other classmates’ strategies • Ask your classmates questions • Revise your thinking • Explain someone else’s’ thinking

In this Number Talk, we are scribing how you: • Do Math mentally using a strategy • Explain your thinking verbally • Ask questions to understand other classmate’s methods • Reason how other classmate’s methods work

As you think, consider a way (strategy) or ways to answer the question. When you have your answer, think about how to say (articulate) your answer to the group. Be ready to share. Student Work Time Teacher presents the problem and waits as student’s think After about 10 seconds to 2 minutes of work time (depending on the grade level): Share solutions and strategies for the question. (6-8 min) Wrapping Up Number Talk: Highlight some of the specific strategies that were shared Number Talk Series #1: These may be used in an order in preparation for end of unit. They may also be continued in Statistics unit.

49 + 51 114 – 49 18 x 5 x 2 160 ÷ 8

25 + 78 12 x 8 4 x 50 x 5 19 ÷ 2

130 + 99 25 x 6 120 ÷ 6 13 ÷ 4

100 – 49 15 x 30 250 ÷ 50 21 ÷ 5

58 – 39 2 x 3 x 50 300 ÷ 15 96 ÷ 10




Uncover multiple strategies for approaching simple math problems that go beyond procedural fluency. Use the mathematical concepts of logic, reasoning, and extending patterns. Mathematical Practice of the Day: Construct viable arguments and critique the reasoning of others (MP3)

Go Figure Power Point (1-2 days)

Before the lesson: Select one of the PowerPoint activities in the “Go Figure” folder to get students thinking logically about patterns. Into: Write a whole number expression on the board from the Number Talks Series #1 document. If this is the first time your class will do a number talk, select a simple addition or subtraction problem like 49 + 51. Through: Pass out the “Figurative Patterns” handout to students. Allow students to work individually for about 10 minutes and then allow teams to work in groups of 4. Encourage students to share their strategies with each other. Beyond: Ask groups to choose one problem (Figurative Numbers #1, #2, or #3) and make a poster explaining their strategy for solving the problem. Their poster should include written explanations as well as diagrams. Then have students do a Gallery Walk and notice the different strategies presented. During the walk, include a focus for students where they must reference the work of another team. For example, “Find a poster with a different strategy than yours that you find interesting” and “What is it about the strategy that makes it interesting?”. Be ready to explain what makes it interesting and write down any questions you may have for that team.” The teacher should then lead the class in a whole-class discussion and guide students to comparing and contrasting different strategies. Close the lesson with a journal reflection that connects this task to the mathematical practices using the reflection form.

Develop mathematical reasoning by opening with a Number Talk, which helps students to articulate their cognitive process when using mental math in the classroom (see Instructional Toolkit for more Number Talk details). Build a culture of success by providing students with a clear picture of what group work looks like, and how all work on a poster must be justified. Using color and arrows in writing expressions, tables, and diagrams helps students to connect the different parts in a representation, or in this case each figure. Facilitate a whole-class discussion by helping students “see” where each strategy shows up on different posters (even if those posters show a totally different strategy). This will allow students to access the problem at their own level, but also move their learning forward by challenging them to recognize their own strategy in more sophisticated strategies. Build a culture of quality group work by using a Gallery Walk as an opportunity to have students view each other posters and look for good ideas to write down from them to then discuss.




Use the mathematical concepts of logic, reasoning, and counting methods. Mathematical Practice of the Day: Construct viable arguments and critique the reasoning of others (MP3)

Party Time (2 days) SVMI Problem of the Month

Into: Engage students in a discussion around whether or not you can predict how many people will show up when you host a party. A 3 Read Strategy can be used to read Level A and B for a better understanding of what they are being asked to do prior to getting started on the POM. Hopefully students will say that oftentimes the people that they invite also invite other people, so the amount of people who show up is often greater than the number of original invites. Have students do a quick think-pair-share to think about how they might use their knowledge about patterns to predict the number of people who might show up at a party. Through: Allow students to work individually for about 10 minutes and then allow teams to work in groups of 4. Students should share their strategies for the levels they completed, then work together to progress through the remaining levels. The goal is not to complete all the levels, but to develop mathematical reasoning skills, and learn from each other. Beyond: Ask groups to choose one level problem and make a poster explaining their strategy for solving the problem. Their poster should include written explanations as well as diagrams. Then have students do a Gallery Walk and notice the different strategies presented. The teacher should then lead the class in a whole-class discussion and guide students to comparing and contrasting different strategies. Close the lesson with a journal reflection that connects this task to the mathematical practices using the reflection form.

Supporting English Learners with academic vocabulary can help them to access the mathematics. Instead of handing out the task to the whole class, you may want to focus on the overall problem first and what the bold-faced words mean. Using the 3 Read Strategy would give students more access to the problem thus more feedback on their mathematical understanding rather than their language skills (see Instructional Toolkit for more 3 Read Strategy details). Understanding through multiple representations can provide access to the problem for students who are stuck. There are several different strategies that students can use to approach each level problem. Encourage teams to use strategies that they used on earlier levels to help them through a later level, if they get stuck. Attend to context by changing the costume options to ones that are familiar to your students, (i.e. French Maid), particularly on the Level C problem, where students may not know what some of the costumes look like. Facilitate a whole-class discussion by helping students “see” where each strategy shows up on different posters (even if those posters show a totally different strategy). This will allow students to access the problem at their own level, but also move their learning forward by challenging them to recognize their own strategy in more sophisticated strategies

Problem of the Month Party Time P 1 (c) Noyce Foundation 2007. To reproduce this document, permission must be granted by the Noyce Foundation: [email protected].

Level A Cindy had a party. She invited two guests. Her guests each invited four guests, and then those guests each invited three guests. How many people were at Cindy’s party? Explain how you determined your solution.

Problem of the Month Party Time


Level B At Leslie’s party ¼ of the people had long hair. One half of the people at the party were boys, ¼ of the girls had short blond hair. None of the boys had long hair. If there were 32 guests, what is the maximum number of girls who could have had short red hair? Show how you determined you answer and why you know you have a correct solution.


Level C Mia, Jake, Carol, Barbara, Ford and Jeff are all going to a costume party. Figure out which person is wearing what costume and when they arrived at the party.

• The person that arrived fourth was wearing bathing suit. • Barbara was the last to arrive. • Jake and Mia arrived and stayed together. • The first person was dressed as a French Maid. • Superman arrived right before Barbara. • The Potato Heads were always together at the party. • Ford was a Surfer Dude. • The French Maid was not Carol. • The Vampire arrived after Superman.


Level D Your Aunt is having a baby. You have created a party game for a baby shower. It is called pick the gender. You put pink and blue tiles into a bag. You ask two guests to pick one tile out of the bag without looking. You tell your guests that if they are the same color, player A wins and if they are two different colors, then player B wins. How many tiles of which colors did you put into the bag to make sure that both players have an equal chance of winning? Explain your solution and why it is fair.


Level E A man and his wife invite 5 other couples to a dinner party. As the guests arrive for drinks before dinner, they shake hands. Not everybody shakes everybody's hands, and of course no one shakes hands with his own spouse. Later, as they sit down to dinner, the host asks each other person, including his wife, “how many hands you shake?” He notices, to his surprise, that each respondent shook a different number of hands. How many did his wife shake? Explain your solution and justify your reasoning.


Primary Version Level A

Materials: Sets of counters.

Discussion on the rug: (Teacher asks the class.) “Who likes to have parties in our class? We are going to solve a problem that is about inviting friends to a party. Who would like to be our party host?” (Teacher invites a student to come forward). The teacher says to the host, “Let’s start by inviting three friends to the party.” (The student picks three friends to come forward.) “How many people are at the party?” the teacher asks the class. (Students share their answers and explain how they know.) The teacher says, “Suppose each of the friends phone two people to come to the party, how many will be at the party altogether?” (Students share their ideas and discuss solutions. Then they actually act it out and count the total number of people at the party).

In small groups: (Students have counters available.) Teacher says, “Cindy had a party. She invited two guests. Her guests each invited four guests, and then those guests each invited three guests. How many people were at Cindy’s party?” (Students work together to find a solution. After the students are done, the teacher asks students to share their answers and method.)

At the end of the investigation: (Students either discuss or dictate a response to this summary question.) “Explain and show how you know how many people are at the party.”

Problem of the Month Party Time

POM Teacher’s Notes Party Time Page 1 (c) Noyce Foundation 2007. To reproduce this document, permission must be granted by the Noyce Foundation: [email protected].

Teacher’s Notes Problem of the Month: Party Time

Overview: In the Problem of the Month, Party Time, students use mathematical concepts of logic, reasoning and counting methods. The mathematical topics that underlie this POM are knowledge of logic, deductive reasoning, counting principles/strategies, and a variety of mathematical representations such as tree diagrams, Venn diagrams, tables, charts, and matrices. In the first level of the POM, students determine the number of guests invited to a party through examination of set of invites, guests inviting other sets of guests. In part B, students are asked to determine the number of short red headed girls at a party given a number of logic clues. The students need to partition the whole using simple fraction (½ and ¼) amounts and logical reasoning of compound events. In level C, students are presented with a set of clues regarding the names, costumes and time of arrival at the party. The students are asked to sort each of the partygoers to their names, costumes and arrival times. In level D, the students are asked to determine when a game is fair for both players. Students justify their findings and explain when the game is fair. In the final level, students are asked to solve a complex logic puzzle. Students must defend their solution and explain how they solve the puzzle. Mathematical Concepts: Mathematical logic and deductive reasoning are key processes in doing all mathematics. Students can use problem solving strategies of working backward, making organized lists, creating simpler problems or using logical reasoning methods such as the process of elimination to solve logic problems. Mathematical representations such a diagrams, charts and tables help mathematicians systematically organize and record thinking and reasoning. Using deductive tools such as the process of elimination and conditionally linked tautologies lets mathematicians draw conclusions. All fields and strands of mathematics depend on reasoning and logic. Problem solving promotes this thinking. Representations support and illuminate logical reasoning. Logical Reasoning Mathematics requires logical reasoning. Using logic to justify one’s findings is central to doing mathematics. Justifications are built on known facts and the relationships of those facts. Logical reasoning is developed by using problem-solving strategies and verifying outcomes. Various logic techniques, tools, and charts are helpful in developing reasoning; these include using Venn diagrams, logic tables, understanding the precise language of logic statements. Use of Precise Language Logical arguments require the precise use of language. Logic statements are true or false, but never both true and false at the same time. A negation is the opposite of a statement that changes a true statement to a false one and a false statement to a true one. The sun will rise tomorrow is a logic statement. It is always going to be either true or false. Its negation is the sun will not rise tomorrow.


Conditional statements are sometimes referred to as implications. A conditional statement is made up of two logic statements in which the second statement is dependent on the first. Example: The Statement if it is Tuesday, then I will buy my wife flowers is a conditional statement. For the statement to be true, the outcome of buying flowers must occur every Tuesday. Of course, the true statement does not restrict me from buying flowers on Friday also. If for some reason Tuesday never occurs, then I would never need to buy flowers. It would be false, though, if Tuesday came and I forgot to buy flowers. A conjunction statement is two logic statements that are combined with the word and. I am going to the store and I am going to the movies is such a statement. It is true whenever both the first and second parts of the sentence are true. If I only went tot the movies and did not go to the store, then the statement would be false. A disjunction statement is two logic statements that are combined with the word or. I am going to the store or I am going to the movies is a disjunction statement. It is true whenever one of the two statements is true. If I only went to the movies and did not go to the store, then the statement would be true, because I only need to satisfy one of the conditions. If I satisfied both conditions by going to both the movies and the store, it would also be true. It would only be false if I did not go either to the store or the movies. A valid argument is a set of logic statements that leads to a logical conclusion. Deductive Proof Deductive reasoning involves ascertaining the necessary conditions and finding evidence to satisfy all conditions before drawing conclusions. A deductive proof is a set of conditional circumstances that imply other circumstances and known facts that support the original circumstances. The proof is verified by following a logical sequence of facts and conditional statements that eventually implies a conclusion. One method of deductive reasoning is a process of elimination. All possible conclusions must be considered to use this strategy. Each conclusion is considered and eliminated, as a possible occurrence until only a single conclusion is left that cannot be discounted. Using deductive reasoning, if all possible conclusions are false except for one, then the only possible conclusion must be the one remaining. Matrix Logic Tables Using a table or a matrix is an effective tool for solving some logic problems. A matrix logic table is a chart that helps in using the process of elimination. Examine all the logic statements. If the task is to find which one of several conditions is true, then a matrix logic table may be helpful. Draw a table, making sure all the possible conditions that are referred to in the logic statements are listed.


Using the logic statements determines whether certain conditions are true or false. If a condition is false, label the appropriate matrix box with an X. If a condition is true, label the matrix box with an O. Whenever you put an O in a box, you can mark all the other boxes in that row and column with an X. In that manner, you can eliminate possibilities to find other facts. If all the boxes in a single row or column are marked with an X, leaving only one possibility, then you can deduce that the open box can be marked O. Use the statements to complete the table by eliminating possibilities that are false or cannot occur, as well as finding conditions that you know are true. In each row and column, one and only one O should appear, indicating which condition is true and that the others are false. The logic problem is solved once the table is complete. Example: Four friends order four different things to eat: a hamburger, pizza, chicken, and fish.

• Rita likes Italian food. • Akami and Mike do not like to eat red meat. • The boys will not eat fish. • Raul works with Akami.

Hamburger Pizza Chicken Fish Rita X O X X Mike X X O X Akami X X X O Raul O X X X

Venn Diagrams A Venn diagram is a logic chart to help organize statements and determine valid conclusions. A Venn diagram is comprised of a universal set indicated by a large rectangle and a number of circles that may overlap to show subsets of the universal set. The circles are located inside the rectangle to indicate that all members of a set are also members of the universal set. The circles are labeled to indicate the sets they define. The overlap or intersection of two sets is a region where a member of both sets resides. A conjunction statement can be written about the members in the intersection regarding both sets. Example: Suppose one circle indicates students that are in the drama club and a second circle indicates students who play on the basketball team. Devon’s name appears in the intersection of those two circles. Then the conjunction statement Devon is in the drama club and Devon is on the basketball team is a true statement. A disjunction statement may be written about a member in one set who is not in the other set.


Example: If Marlo is in the drama club circle but not in the intersection of the two circles, then the statement, Either Marlo is in the drama club or Marlo plays on the basketball team is true. A circle inside a circle can show a conditional statement, because if you are a member of the smaller circle, then you must be a member of the larger circle also. Some problems involving Venn diagrams require that students determine the number of members in each region of the Venn diagram. Example: Suppose a Venn diagram was drawn to show people’s music preference from a survey that provided the choices of rock, country, or rap music. From the survey, the following statements were made.

• 25 people like only to listen to rock music. • 13 people made two choices and said they liked both country and rock music. • No one liked all three styles listed in the survey. • 5 people chose a style not mentioned in the survey. • There were 50 people surveyed. • The remaining people on the survey only chose rap.

Using the statements, there must be a rectangle with three overlapping circles: one labeled rock, one country, and one rap. The sum of the numbers in the diagram must be 50. The region outside the circle must be 5 for those who chose a different style of music. The region that only likes rock music has 25 people. The region overlapping country and rock, but not including the rap area has 13 people. Seven people are in the rap-only region. All other regions are empty. Reading this Venn diagram, one can tell that 38 people like rock music. Fairness: The term fairness in probability means that there is an equal likelihood of events occurring. Fairness can be used to describe an object being used to generate chance, such as a number cube or a coin. If you roll the number cube, do all sides have an equal chance of appearing, if yes the number cube is considered fair. Fairness can also be used to describe a contest. Does each

5

rock 25 13 country

7 rap


contestant have an equal chance of winning, if the answer is yes then the contest may be considered fair. Game Reasoning/Strategy: Reasoning strategies are important for students. In NIM games, students are often successful by making the problem simpler. Suppose the goal of a NIM game is to pick up the last stick, and one can pick up either 1 or 2 sticks on a turn. By simplifying the game to one stick, then two sticks, then three sticks, etc., a pattern emerges. Once a pattern is found, a strategy to utilize the pattern is created. That is a basic principle of game theory. There are many other strategies used in game theory. A common strategy used by people planning games is to engage in a process of elimination. As a game player one might have a large number of possible moves or options. Many of these options or moves lead to ultimate failure. If one eliminates possible failing moves, then the game becomes simpler and the probability of success greater. Many puzzles and games become easier if you backtrack from the ending. If there are decision forks (picture a maze), it is easier to identify the best option by working in a reverse direction. These are just some strategies useful in winning games and identifying best game playing strategies. Sample Space: To calculate the probability of an event, it is necessary to determine the number of possible outcomes produced by that event. A sample space is used for this purpose. A sample space is the list of all possible outcomes. From this list, a subset of successful occurrences of the event can be determined and counted. The probability can be calculated by dividing the number of occurrences by the total number of possible outcomes. Representations: Representations of an event or sample space may be determined through lists, charts, diagrams or area models. Matrix tables, Venn Diagrams, and tree diagrams are representations of sample spaces and are often used to determine all possible outcomes of an event and its probability. An area model is a geometric display of all possible outcomes for a given situation. The model is partitioned into regions depicting the different possible outcomes. The size of the regions corresponds to the number of outcomes in a particular event. These representations are helpful in visualizing probability and other measures of chance. Students may need to create sample space charts to make sense of problems. These may be tables, Venn diagrams, tree diagrams, or area models. Below are examples of sample spaces used in the context of solving probability problems. In problems that involve rolling of two number cubes, a table may be used to show the sample space of all possible sums.

1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12


In problems that involve using a deck of cards finding the probability that one would draw a Jack or a Heart, a Venn diagram may be used. In a problem that involves rolling one number cube and flipping one coin, the tree diagram shows all possible outcomes:

Start One Two Three Four Five Six Heads Tails Heads Tails Heads Tails Heads Tails Heads Tails Heads Tails In a problem that involves selecting marbles from a bag with 2 blue, 6 yellow and 4 red marbles, an area model might be used to explore probabilities. Often students present their solutions through the use of sample spaces. This may clarify thinking and help communicate understanding to others. Throughout the POM, the students may refer to sample charts to understand and solve probability problems. Counting Principles Counting principles are important tools in determining sample space size and numbers of successful occurrences of an event. There are two basic counting functions, permutation and combination. A permutation is the number of ways one can order r items from a set of n objects. For example, how many different batting orders (1…9) can one makes from a team of 12 players? The order matters because the permutation specifies the order as well as the size of the set.

12 Hearts

3 Jacks 1J♥

36 other cards

2 blue

4 red 6 yellow


Therefore there are 12 ways to pick the first player, times 11 ways to pick the second player, times 10 ways to pick the third play and so forth down to 4 ways to pick the ninth and final player. Therefore the permutation of 9 players in order out of 12 is 12•11•10•9•8•7•6•5•4 = 79,833,600 ways. A combination is a selection of r items without order from a set of n objects. An example in a similar context is how can you choose 9 players to play from a group of 12 players. The answer is 220 unique teams but position or order does not matter; it is just the different make up of the team. Factorials Factorial is an operation that is indicated by a natural number followed by the exclamation sign (!). A factorial is the product that results from multiplying a natural number by each of its subsequent natural numbers down to one. For example, 5! is 5•4•3•2•1 or 120. Factorials are useful in calculating permutations and combinations. As shown in the example of a permutation, the permutation expression of 12•11•10•9•8•7•6•5•4 could have been written in factorial notations such as 12!/3!


Solutions Level A Cindy had a party. She invited two guests. Her guests each invited four guests, and then those guests each invited three guests. How many people were at Cindy’s party? Explain how you determined your solution. The are several ways that students might solve this problem. One method is a tree diagram. The method shows 2 guests then 8 guests then 24 guests invited to the party plus Cindy, for a total of 35 people. This can also be calculated using multiplication: 2 first invited guests plus 2 times 4 second invited guests plus 2 times 4 times 3 third invited guest plus Cindy.

Cindy’s Party

First guests Invited

First guests Invited

Second guests Invited








Third guests invited

























Level B At Leslie’s party ¼ of the people had long hair. One half of the people at the party were boys, ¼ of the girls had short blond hair. None of the boys had long hair. If there were 32 guests, what is the maximum number of girls who could have had short red hair? At the most, 4 girls could have had short red hair. Show how you determined you answer and why you know you have a correct solution. Students may use diagrams to help sort the clues.

32 people

To start, suppose the large square includes all 32 people at the party. Half were boys, so half are girls

16 boys

16 girls

16 boys

8 girls long hair

8 girls short hair

One fourth of the people had long hair but none were boys, so 8 girls had long hair.

4 blond short hair

16 boys

8 girls long hair

One fourth of the girls had short blond hair, so 4 girls had short blond hair

That leaves just one-fourth of the girls or 1/8 of all party goers to have short red hair

4 blond short hair

16 boys

8 girls long hair So at most, only 4 people at the

party could have had short red hair.


Level C Mia, Jake, Carol, Barbara, Ford and Jeff are all going to a costume party. Figure out which person is wearing what costume and when they arrived at the party.

• The person that arrived fourth was wearing bathing suit. • Barbara was the last to arrive. • Jake and Mia arrived and stayed together. • The first person was dressed as a French Maid. • Superman arrived right before Barbara. • The Potato Heads were always together at the party. • Ford was a Surfer Dude. • The French Maid was not Carol. • The Vampire arrived after Superman.

One way to solve a logic problem such as this is to design a table. Using the clues, you can identify matches. When a match is made you can x out all the rest of the possibilities in the row and column of the match. This is a systematic way to use the process of elimination.

Party Guests Vampire

Potato Head

Potato Head

French Maid

Super man

Surfer Dude 1st 2nd 3rd 4th 5th 6th

Mia x x 0 x x x x 0 x x x x Jake x 0 x x x x x x 0 x x x Carol x x x x 0 x x x x x 0 x Barbara 0 x x x x x x x x x x 0 Ford x x x x x 0 x x x 0 x x Jeff x x x 0 x x 0 x x x x x

From the table we see that: Mia was a Potato Head and arrived with Jake, 2nd and 3rd at the party. Jake was the other Potato Head. Carol was Superman and arrived 5th. Barbara was a vampire and arrived 6th. Ford was the surfer dude and arrived 4th. Jeff was the French maid and arrived 1st.


Level D Your Aunt is having a baby. You have created a party game for a baby shower. It is called pick the gender. You put pink and blue tiles into a bag. You ask two guests to pick one tile out of the bag without looking. You tell your guests that if they are the same color, player A wins and if they are two different colors then player B wins. How many tiles of which colors did you put into the bag to make sure that both players have an equal chance of winning? Explain your solution and why it is fair. Solution: For considering a mathematical argument for this problem let’s define T as the total number of tiles in the bag and there will be exactly two colors pink and blue, where P stands for the number of pink tiles and B stands for the number of blue tiles and although it can be reconsidered for other colors in the spirit of generalities P > B. So T = P+B. Approach 1: Through trial and error one can find that if you have four tiles, only one combination of tiles provides a fair chance. That combination would be three pink and one blue. There are 12 ways to select the four tiles from the bag. This systematic list shows all possible permutations, noting that half (6) are matches and half (6) are non-matches. P1 P2 match P1 P3 match P1 B1 non-match P2 P1 match P2 P3 match P2 B1 non-match P3 P1 match P3 P2 match P3 B1 non-match B1 P1 non-match B1 P2 non-match B1 P3 non-match Making lists for other possible combinations show that equal matches and non-matches are not possible for four. Also other totals do not seem to produce matches. It is not until 9 that one can find another combination of pinks and blues that produce equal matches and non-matches. In that case, it is 6 pinks and 3 blues. Continue exploration provide other cases. The following table shows the total and combinations that create equal matches and non-matches.


Total Pink Blue 4 3 1 9 6 3 16 10 6 25 15 10 36 21 15 49 28 21 64 36 28

Looking at the table, the totals are always perfect squares and the pink and blue values are the triangular numbers. There is a solution for every perfect square greater than 1 tile. The pink and blue tiles can be found using the following formulas: Pink = ((n+2)(n+1))/2, where n is a natural number Blue =(n (n+1))/2, where n is a natural number Total = ((n+2)(n+1))/2 + (n (n+1))/2, where n is a natural number Simplifying the right side of this equation shows that T = (n+1)2 . Therefore, every perfect square greater than 1 is equal to the sum of two consecutive triangular numbers. So Approach 2 The total ways you can select two tiles with T tiles in the bag is T P 2 = T • (T-1). There needs to be an equal number of matching colors as non-matches. Therefore the combinations of Pinks and Blues will be the same as Blues and Pinks. So, the number of Pink • Blue + the number of Blue • Pink must be half the amount of tiles in the bag. Or 2 times Pink times Blue is equal to half of T(T-1) in order to have equal chances. Writing that relationship as an equality one gets 4•Pink • Blue = (Pink+Blue) • (Pink+Blue – 1) If you consider only a positive domain for pinks and blues, then this relationship can also be written as: Pink – Blue = √ (Pink +Blue) This illustrates that for Pink and Blue to be integers, the total T must be a perfect square. It also shows that the difference between pink and blue must be the square root of T. Finally we can create functions that find the amount of pinks and blues given any T, where T is a perfect square. P = (T + √T)/2 and B = (T - √T)/2


Below is a spread table that compares all pairs of natural numbers and uses the relationship Pink – Blue = √ (Pink +Blue). The resulting matrix shows with the value TRUE when there is a case of equal matches and non-matches.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 2 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 3 TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 4 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 5 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 6 FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 7 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 8 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 9 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE

10 FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 11 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 12 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 13 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 14 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 15 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE 16 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 17 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 18 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 19 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 20 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 21 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE 22 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 23 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 24 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 25 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 26 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 27 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 28 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 29 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 30 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 31 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 32 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 33 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 34 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE


Level E A man and his wife invite 5 other couples to a dinner party. As the guests arrive for drinks before dinner, they shake hands. Not everybody shakes everybody's hands, and of course no one shakes hands with his own spouse. Later, as they sit down to dinner, the host asks each other person, including his wife, how many hands he or she shook. He notices, to his surprise, that each respondent shook a different number of hands. How many did his wife shake? I started with inviting one couple. In that case, you have four people and at most a person can have two handshakes (can’t shake hands with self or spouse). Therefore, two other have only 1 handshake and the fourth has no handshake. That means that the host has to have the duplicate number (1 handshake) because everyone else needs to be different. The person with the most handshakes needs to be married to the one with no handshakes. That leaves the other person with one handshake being the spouse. Following that logic and inviting two couples, you have six people (see diagram below). The most handshakes by a single person must be 4 (A). That person’s spouse (F) must have 0 handshakes. Then you need a person with 3 (B). That leaves two people (C, D) with 2, and a person (E) with 1. The host has to be one of the two’s because everyone else need a unique total. Since the B hasn’t shaken with E, they must be married. That leaves the other person who shook 2 hands to be the spouse. So I believe there is a pattern.

A

B

C

F

D E


I created a table of the 5 couples plus the two hosts. I started with the first couple and had one partner shake hands with everyone possible making 10 handshakes. The spouse then must shake hands with no one. I went to the second couple and had one shake hands with 9 people and 1 for the spouse. I continued this pattern. Therefore the host and hostess each ended up with 5 using the logic above. 1A 1B 2A 2B 3A 3B 4A 4B 5A 5B H1 H2 Total 1A 0 0 1 1 1 1 1 1 1 1 1 1 10 1B 0 0 0 2A 1 0 0 1 1 1 1 1 1 1 1 9 2B 1 0 0 1 3A 1 1 0 0 1 1 1 1 1 1 8 3B 1 1 0 0 2 4A 1 1 1 0 0 1 1 1 1 7 4B 1 1 1 0 0 3 5A 1 1 1 1 0 0 1 1 6 5B 1 1 1 1 0 0 4 H1 1 1 1 1 1 0 0 5 H2 1 1 1 1 1 0 0 5 Total 10 0 9 1 8 2 7 3 6 4 5 5

So following the logic above, if my wife and I invite n couples and handshake sequence of event followings the rules of the problem, then I will shake n hands and so will my wife.




Decompose quantities into sums of multiple parts and represent them with words, symbols and diagrams. Mathematical Practice of the Day: Make sense of complex problems and persevere in solving them (MP1)

“Pennies” Adapted from “How can I represent it?” (1 day) CPM CCR1 1.2.1 Core: 1-41 through 1-44

Materials: Have enough pennies for students to work together. Into: Write or create a whole number expression on the board from the Number Talks Series #1 document that continues to build on previous talks. After reading the lesson introduction, show students some pennies and ask them to quickly determine how many pennies you have shown them. Display eleven pennies that have been randomly placed on the overhead for a brief period of time and then remove them from sight. Ask students to volunteer how many pennies they saw and how they know. Through: Pass out the student handout and ask teams to begin. Circulate as students work and ask questions that encourage them to think in new ways. Students often decide to form “towers” of 10 pennies. After about 10 minutes, or when most teams have finished an arrangement, lead a brief (approximately 5 minutes) gallery walk in which students have a chance to view the arrangements created by other teams. Direct students to draw diagrams for particularly interesting arrangements that they see as they walk. When teams return to their seats, direct them to questions 3 and 4 on their handout in which they will look for a clearer organization strategy. Beyond: Distribute materials for poster. The goal is to show as many representations (diagrams, expressions, words) as they can of their arrangement as well as to highlight connections among those representations. Lead another gallery walk so that students get to see the posters created by other teams. Ask students to take note of the connections other teams have found among their multiple representations. Close the lesson with a journal reflection that connects this task to the mathematical practices using the reflection form.

Supporting effective group work can be achieved with team roles. Support teams in learning how to work effectively together by communicating expectations at the beginning of the lesson, model the kinds of questions you hope to hear students ask each other, and reinforce construct team behaviors throughout the lesson. Attend to misconceptions when teams are working, asking questions such as, • “How many pennies are in your tower? • “How can I tell that there are not 9 or 11?” § This can help students to recognize that this is

not a very effective strategy. Similarly, a 10 by 10 array of pennies is not immediately distinguishable from a 9 by 9 or an 11 by 11 array. You might ask students to consider what number they can arrange in such a way that they are easily recognized.

§ In problem 3b, it might be difficult for some students to draw a diagram without drawing each one of the individual pennies. You can ask questions such as, “How could you represent that group of pennies without drawing each one? Could you represent parts of your arrangement separately?”

§ On problem 4, encourage students to find many ways to represent their arrangements using words, numbers and symbols.

Access prior knowledge by helping students recall that multiplication can be represented with repeated addition by asking students for different ways to determine the number of pennies in a rectangular array.

Notes to Self:

Building Mathematical Practices Unit – Lesson Series 1 Adapted from Core Connection: Course 1, Lesson 1.2.1 1

1


Pennies Today you will focus on finding several ways to represent a quantity (an amount of something) in different ways. As you work, ask your teammates the following questions:

How can we represent the quantity with numbers and symbols? How can we represent it with words? How can we represent it with diagrams?

1. How can you figure out how many objects are in a pile without counting each one? Are there some ways objects can be arranged so that it is easy to see how many there are? Your task: Your teacher will bring your team a handful of pennies. As a team, organize the pennies so that anyone who looks at your arrangement can easily see how many pennies your team has. Keep working until all members of your team agree that your arrangement is the clearest and easiest to interpret. (Note: Someone looking at your pennies should know how many there are without having to believe what you tell them, so arranging your pennies into the shapes of the numerals of your number does not count!)

2. Are some arrangements easier to interpret than others? Your teacher will direct you to participate in a gallery walk so that you can see how other teams have arranged their pennies. You will walk to the desks or table of each of the other teams in the class to see how they have arranged their pennies.

As you walk, notice how easy or difficult it is for you to see how many total pennies each team has. When you see an arrangement that helps you know quickly and easily what the number of pennies is, consider what makes that particular arrangement easy to total.

Building Mathematical Practices Unit – Lesson Series 1 Adapted from Core Connection: Course 1, Lesson 1.2.1 2

2

3. How could you make your arrangement even clearer?

a. Work with your team to rearrange the pennies to improve how well others can understand the quantity represented. Use what you noticed on the gallery walk to help you do this.

b. On your own paper, draw a diagram that represents your new

arrangement without drawing all of the pennies themselves.

c. Compare your diagram with those made by your teammates. Are some diagrams clearer matches to the arrangement than others?

As a team, decide on the best way to represent your arrangement in a diagram. Consider using ideas from multiple drawings. When all team members have agreed on the best diagram, copy it onto your paper.

4. Work with your team to represent your arrangement of pennies using words, numbers and symbols. Write at least three different expressions that represent your quantity. Some number and symbol representations may match certain diagrams more closely than others. Identify which expressions most closely match your team’s chosen arrangement.

5. As a team, create a poster that shows your team’s best arrangement of your pennies along with the diagrams and number expressions that represent it. Show as many connections as you can among the pennies, diagrams, and number expressions.


Unit Lesson Formative Task


Work collaboratively to add, subtract, and multiply decimals. Mathematical Practice of the Day: Construct viable arguments and critique the reasoning of others (MP3)

Bookstore (1-2 days) Adapted from SpringBoard: Middle School 1, Activity 2.6

Into: Write or create a whole number expression on the board from the Number Talks Series #1 document that continues to build on previous talks and whole number operational sense. Have students think about the chart and work independently to fill it in. After individual think time, have a whole class discussion on how to fill-in the chart. This should be kept relatively short with conversations about how to represent different units like cents and dollars. Through: Students work with a group of 4 to solve questions 2 and 3 on their handout. Encourage teams to reach consensus on the answer and their explanation (but each student can use their own strategy) before moving onto the next question. Stop the groups to facilitate a whole class discussion on the errors that Juan and Isabel have made. Scribe each group’s reasoning as part of their quiz. Provide feedback on participation by highlighting some of the positive behaviors and mathematical reasoning being observed. Have groups complete the remainder of the paper. Beyond: Ask groups to come to the document camera and present their findings to the whole group. Select and sequence groups based on the work done during group time. Close the lesson with a journal reflection that connects this task to the mathematical practices using the reflection form.

Allowing for think time provides many students benefit from working on a task by themselves for 10 minutes before working with a partner. This allows students to elicit their own understanding of the task and take ownership of their own learning. Providing group roles like establishing a Resource Manager, Facilitator, Recorder/Reporter, and Task Manager can support developing effective student engagement during group work. Structuring group work (option 1) using a Participation Quiz helps students know what type of mathematical discourse is relevant and appropriate for the task. It makes clear the criteria for success, not just with content but also with expectations for group work behaviors. Structuring group work (option 2) using talking chips help to ensure equity of voice during group work. Give students 2-colored chips (e.g. red on one side, green on the other side). When working in small groups, if a student would like to speak, they flip their chip over to the green side. This helps students monitor their airtime. Providing sentence starters like, “I understand your way but…” or “I wonder if our ways are similar because I noticed…”, students can understand how to communicate with their teammates using academic language. Encourage student to student discussion by having this time to practice their questioning strategies, like “How do you know that?”, “Can you explain to me your strategy?”, etc.

Notes to Self:

Building Mathematical Practices Unit – Formative Task Adapted from SpringBoard, Middle School 1, Activity 2.6 1

1


Bookstore Mrs. Grace’s middle school class operates a bookstore before school and during lunch. Juan and Isabel are in charge of ordering supplies and selling items. The table below lists the items they sell and the price per item. 1. Mrs. Grace asked Juan and Isabel to rewrite the price chart so that each price

is written in cents and dollars. Complete the chart for them.

Item Price Price in Cents Price in Dollars

Pencil 25¢

Pack of Notebook Paper $1.32

Ruler $0.52

Ink Pen 43¢

Pack of Markers $2.35

Eraser 19¢

2. Juan fills in his chart for pencils like this…

Item Price Price in Cents Price in Dollars

Pencils 25¢

25¢

0.25¢

Is he right or wrong? Why? 3. Isabel sells a pack of markers and an eraser to her friend, Cierra. She adds like this… $2.35 +19¢ $21.35 Cierra says, “Really? That seems expensive!” Who is right? Explain why.


2

4. Kalen buys two pencils, an ink pen, and a box of markers. What will they

cost? 5. Elan purchases a pack of notebook paper and an eraser. What will they cost?

6. Who spent more on their purchases, Kalen or Elan? How much more?

Explain how you arrived at your answer. 7. Alena has $5.00 to spend at the bookstore and would like to buy a box of

markers, 2 erasers, 1 pencil, and a pack of notebook paper. Does she have enough money? Show and explain how you know.

8. DJ wants to buy 20 pencils but he may not have enough money. How can he

find out how much they will cost? Explain.


3




Uncover multiple strategies for approaching simple math problems that go beyond procedural fluency. Mathematical Practice of the Day: Make sense of complex problems and persevere in solving them (MP1)

How much will it cost? (1-2 days) Adapted from Mathematics for Elementary School 5A (a Japanese textbook), Chapter 3

Into: Write a whole number expression on the board from the Number Talks Series #1 document that is based in whole number multiplication (i.e. 60 x 3). Introduce a pair-share where students solve a multiplication problem using a number line model. Through: Give students the handout “How much will it cost?” Allow students individual think time and then ask students to work in their small group to solve the problem. Once the small group has reached consensus, the group creates a collaborative poster showing and explaining their thinking with words, diagrams, and math. Provide checklists or rubrics so that students understand the components of an exemplar poster. Beyond: Conduct poster presentations where students have a chance explain their poster and comment on others; noticing good explanations and explanations that need improvement. Have a whole-class discussion where students make connections between the different strategies that each group used, as well as identify components of an exemplar poster. Groups revise their posters. Close the lesson with a journal reflection that connects this task to the mathematical practices using the reflection form.

Understanding through multiple representations can be made accessible for students with a pair-share in the beginning of class. Allow students to use diagrams and manipulatives to get started if they are struggling with the number line model. Develop perseverance by pulling students who are stuck getting started to the side and prompting them with questions about the seating chart of the first 3 rows, like: • How did you solve our first multiplication

problem? • What are we trying to figure out? • What do we already know? • What can we figure out? Throughout the lesson, call student attention to the perseverance by verbally recognizing students who demonstrate it. Build a culture of quality group work by using a Gallery Walk as an opportunity to have students view each other posters and look for good ideas to write down from them to then discuss.

Building Mathematical Practices Unit – Lesson Series 2 Adapted from Mathematics for Elementary School 5A, Chapter 3 1

1


How much will it cost? Your school wants to find out how much it will cost to rent a jumper obstacle course for a field day. Your cousin’s school rented one for 3.4 hours and paid $180 for each hour. If your school rents from the same company, for the same amount of time, how much will it cost in total to rent the jumper? Use the diagram below to show how you would figure out your answer.

0 180 ? (dollars)

0 1 2 3 3.4 4 (hours)




Create and use visual representations to divide decimals. Mathematical Practice of the Day: Make sense of complex problems and persevere in solving them (MP1)

How much did we pay? (1-2 days) Adapted from Mathematics for Elementary School 5A (a Japanese textbook), Chapter 4

Materials: Graph paper and colored pencils. Into: Write a whole number expression on the board from the Number Talks Series #1 document that continues to build on previous talks. Try division at this point if possible, and division with remainders by Day 2. Ask students to give a recap of their learning from the “How much it will cost?” lesson. Pose today’s question using the 3 Read Strategy with the whole class this will provide a classroom discussion to make sure that all students understand the process before releasing students to work individually. Through: Give students the handout “How much did we pay?” Allow students individual think time and then ask students to work in their small group to solve the problem. Once the small group has reached consensus, the group creates a collaborative poster showing and explaining their thinking with words, diagrams, and math. Provide checklists or rubrics so that students understand the components of an exemplar poster. Beyond: Conduct a gallery walk where students have a chance to comment on each group’s poster, noticing good explanations and explanations that need improvement. Have a whole-class discussion where students identify components of an exemplar poster. Groups revise their posters. Close the lesson with a journal reflection that connects this task to perseverance using the reflection form.

Supporting English Learners with academic vocabulary can help them to access the mathematics. Instead of handing out the task to the whole class, you may want to focus on the overall problem first and what the bold-faced words mean. Using the 3 Read Strategy would give students more access to the problem thus more feedback on their mathematical understanding rather than their language skills (see Instructional Toolkit for more 3 Read Strategy details). Using color and arrows in writing expressions, tables, and diagrams helps students to connect the different parts in a representation, or this case each figure. Some students may benefit by using different colored pencils to organize their thinking. Structuring group work by using talking chips help to ensure equity of voice during group work. Give students 2-colored chips (e.g. red on one side, green on the other side). When working in small groups, if a student would like to speak, they flip their chip over to the green side. This helps students monitor their airtime. Build a culture of quality group work by using a Gallery Walk as an opportunity to have students view each other posters and look for good ideas to write down from them to then discuss.

Notes to Self:

Building Mathematical Practices Unit – Lesson Series 2 Adapted from Mathematics for Elementary School 5A, Chapter 4 1

1


How much did we pay? Your school decides to rent a jumper obstacle course from a different company than your cousin’s school. Your school ended up only using the jump for 2.8 hours and the total price was $420. How much did you school pay per hour? Who got a better deal: your school or your cousin’s school? Use words and the diagram below to explain your reasoning.

0 ? 420 (dollars)

0 1 2 2.8 3 (hours)


Unit Summative Task


Relate a given division calculation to appropriate practical situations. Determine when an exact answer is necessary, or when it is appropriate to round up or round down. Mathematical Practices of the Unit: Make sense of complex problems and persevere in solving them (MP1) Construct viable arguments and critique the reasoning of others (MP3)

Sensible Division (1-2 days) Adapted from MARS, Grade 6 2007

Materials: Construction paper or halve chart paper and calculators. Into: Write or create a whole number expression on the board from the Number Talks Series #1 document that continues to build on division with remainders. Ask students to solve the division problem 100 ÷ 6 individually in the box. Have them partner share after individual think time. Students may use a calculator for this assessment. Explain that students will be using this answer today in their task. Through: Allow students to work in partner groups to find sensible answers to the 5 scenarios on their worksheet. Encourage students to explain their thinking to their partner and build off of each other’s reasoning. Sentence frames and questioning stems can be provided to support discourse, like “What did we do with the remainder?” “Why did we round or why didn’t we?” Beyond: Have students choose two questions to make individual posters that explain how they arrived at their answers. Their posters should include words, pictures and math. Include on posters an explanation as to why these questions have different answers when the same math is used to solve them. Have a gallery walk and have students identify posters that they feel are exemplar posters and why. Have students write on the Mathematical Practices Reflection form.

Attend to context by having calculators available for students to check their division and not get stuck in calculations. Reviewing expectations for a final product will provide students with a clear picture of what an exemplar poster is using checklists, rubrics, etc. Providing sentence starters like, “I understand your way but…” or “I wonder if our ways are similar because I noticed…”, students can understand how to communicate with their teammates using academic language. Encourage student to student discussion by practicing their questioning strategies, like “How do you know that?”, “Can you explain to me why you chose that answer?”, etc.

Notes to Self:

Building Mathematical Practices Unit – Summative Task Adapted from MARS Grade 6 2007 1

1


Sensible Division In the box to the right, calculate 100 ÷ 6 Write the result here: This result can be used to give a sensible answer to the following questions. With your group, discuss what would be a sensible answer to each question and justify your thinking. If there is a question where the division problem 100 ÷ 6 cannot be used to answer the question, talk about why not and talk about how you could change the problem so that 100 ÷ 6 could be used. 1. How much does each person pay when 6 people share the cost of a meal

costing $100? 2. 100 students each need a pencil. Pencils are sold in packs of 6. How many

packs are needed? 3. What is the cost per gram of shampoo costing $6 for 100 grams? 4. How many movies costing $6 each can be downloaded for $100? 5. What is the average distance per day traveled by a hiker on the Appalachian

Trail, who covers 100 miles in 6 days?

unit overview grade/course building mathematical practices ...€¦ · should progress from...

Documents