grade 4 mathematics curriculum document 2016-2017 · 1st 6-weeks addition breaking each number into...

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Grade 4 Mathematics Curriculum Document

2016-2017

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Table of Contents Cover Page Pg. 1 Table of Contents Pg. 2 Trouble Shooting Guide Pg. 3 Best Practices in the Math Classroom Pg. 4 Problem Solving 4-Square Model Pg. 6 Problem Solving with Pictorial Modeling/ Strip Diagrams Pg. 7 Number Sense/ Number Talks Pg. 8 Year at a Glance Pg. 17 TEKS Spiraling Checklist Pg. 18 Mathematics Process Standards Pg. 21 Math Instructional Resources Pg. 22 Bundle 1: Representing and Comparing Whole Numbers and Decimals Pg. 24 Bundle 2: Whole Number and Decimal Operations (Addition and Subtraction) Pg. 34 Bundle 3: Whole Number Operations (Multiplication and Division) Pg. 40 Bundle 4: Algebraic Reasoning Pg. 49 Bundle 5: Fractions Pg. 54 Bundle 6: Geometry and Measurement Pg. 62 Bundle 7: Data Analysis Pg. 79 Bundle 8: Personal Financial Literacy Pg. 84 Bundle 9: STAAR Review and Testing Pg. 88 Bundle 10: Extended Learning Pg. 88

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Trouble Shooting Guide

• The 2015-2016 Mathematics Curriculum Document for Grade 4 includes the following features:

• The NISD Curriculum Document is a TEKS-Based Curriculum.

• Year at a Glance Indicating Bundle Titles and Number of Days for Instruction

• Color Coding: Green- Readiness Standards, Yellow- Supporting Standards, Blue- Process Standards,

Purple- ELPS, Strike-Out- Portion of TEKS not Taught in Current Bundle

• NISD Math Instructional Focus Information

• The expectation is that teachers will share additional effective resources with their campus Curriculum &

Instructional Coach for inclusion in the document.

• The NISD Curriculum Document is a working document. Additional resources and information will be

added as they become available.

• **Theresourcesincludedhereprovideteachingexamplesand/ormeaningfullearningexperiencestoaddresstheDistrictCurriculum.InordertoaddresstheTEKStotheproperdepthandcomplexity,teachersareencouragedtouseresourcestothedegreethattheyarecongruentwiththeTEKSandresearch-basedbestpractices.Teachingusingonlythesuggestedresourcesdoesnotguaranteestudentmasteryofallstandards.Teachersmustuseprofessionaljudgmenttoselectamongtheseand/orotherresourcestoteachthedistrictcurriculum.

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NISD Math Focus

Best Practices in the Math Classroom • Teaching for Conceptual Understanding: Math instruction should focus on developing a true understanding of the math concepts being

presented in the classroom. Teachers should avoid teaching “quick tricks” for finding the right answers and instead focus on developing student understanding of the “why” behind the math. Math is not a list of arbitrary steps that need to be memorized and performed, but is, rather, a logical system full of deep connections. When students see math as a set of disconnected steps to follow they tend to hold many misconceptions, make common mistakes, and do not retain what they have learned. However, when students understand the connections they have fewer misconceptions, make less errors, and tend to retain what they have learned.

• Developing Student Understanding through the Concrete-Pictorial-Abstract Approach: When learning a new math concept, students should be taken through a 3-step process of concept development. This process is known as the Concrete-Pictorial-Abstract approach. During the concrete phase, students should participate in hands-on activities using manipulatives to develop an understanding of the concept. During the pictorial phase, students should use pictorial representations to demonstrate the math concepts. This phase often overlaps with the concrete phase as students draw a representation of what they are doing with the manipulatives. During the abstract phase, students use symbols and/or numbers to represent the math concepts. This phase often overlaps with the pictorial phase as students explain their thinking in pictures, numbers, and words. If math concepts are only taught in the abstract level, students attain a very limited understanding of the concepts. However, when students go through the 3-step process of concept development they achieve a much deeper level of understanding.

• Developing Problem Solving Skills through Quality Problem Solving Opportunities: Students should be given opportunities to develop their problem solving skills on a daily basis. One effective approach to problem solving is the think-pair-share approach. Students should first think about and work on the problem independently. Next, students should be given the opportunity to discuss the problem with a partner or small group of other students. Finally, students should be able to share their thinking with the whole group. The teacher can choose students with different approaches to the problem to put their work under a document camera and allow them to talk through their thinking with the class. The focus of daily problem solving should always be Quality over Quantity. It is more important to spend time digging deep into one problem than to only touch the surface of multiple problems.

• Developing Problem Solving Skills through Pictorial Modeling: One of the most important components of students’ problem solving development is the ability to visualize the problem. Students should always draw a pictorial representation of the problem they are trying to solve. A pictorial model helps students to better visualize the problem in order to choose the correct actions needed to solve it. Pictorial modeling in math can be done with pictures as simple as sticks, circles, and boxes. There is no need for detailed artistic representations. One of the most effective forms of pictorial modeling is the strip diagram (or part-part-whole model in lower grades). This type of model allows students to see the relationships between the numbers in the problem in order to choose the proper operations.

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• Developing Students’ Number Sense: The development of number sense is a critical part of a student’s learning in the mathematics classroom. The ability to reason about numbers and their relationships allows students the opportunity to think instead of just following a rote set of procedures. The standard algorithms for computation may provide students with a quick answer, but they do not allow for development of student thinking and reasoning. The standard algorithms should not be abandoned completely, but should be used as one of many ways of approaching a computation problem. It is, however, very important that students have the opportunity to develop their number sense through alternative computation strategies before learning the standard algorithm in order to prevent students from having a limited view of number relationships.

• Creating an Environment of Student Engagement: The most effective math classrooms are places in which students have chances to interact with their teacher, their classmates, and the math content. Students should be given plenty of opportunities to explore and investigate new math concepts through higher-order, rigorous, and hands-on activities. Cooperative learning opportunities are critical in order for students to talk through what they are learning. The goal should be for the student to work harder than the teacher and for the student to do more of the talking.

• Higher Level Questioning: The key to developing student thinking is in the types of questions teachers ask their students. Teachers should strive to ask questions from the top three levels of Bloom’s Taxonomy to probe student thinking.

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NISD Math Focus

Developing Problem Solving through a 4-Square Model Approach • The 4-square problem solving model should be used to help guide students through the problem solving process. It is important that

students complete step 2 (pictorial modeling) before attempting to solve the problem abstractly (with computation). When students create a visual model for the problem they are better able to recognize the appropriate operation(s) for solving the problem.

Dragon Problem Solving 1. What is the problem asking?

2. This is how I see the problem. (pictorial/ strip diagram)

3. This is how I solve the problem. (computation)

4. I know my work is correct because... (justify)

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NISD Math Focus

Developing Problem Solving through Pictorial Modeling/ Strip Diagrams Part-Part-Whole: Addition and Subtraction

Whole Unknown Alan had some marbles. He lost 12 of them. Then he had 32 left. How many did he have to begin with?

? 12 32

Part Unknown Steven had 122 peanuts. He ate 71 of them. How many peanuts are left?

122 71 ?

Another Part Unknown Some adults and 12 children are on a bus. There are 31 people on the bus. How many adults are on the bus?

31 ? 12

Comparison: Addition and Subtraction

How Much More Unknown Alex has 47 toy cars. Keisha has 12 toy cars. How many more cars does Alex have than Keisha?

47 12 ?

Smaller Quantity Unknown Fran spent $84 which was $26 more than Alice spent. How much did Alice spend?

84 ? 26

Larger Quantity Unknown Barney has 23 old coins. Steve has 16 more old coins than Barney. How many old coins does Steve have?

? 23 16

Equal Parts: Multiplication and Division

Whole Unknown Kim has 4 photo albums. Each album has 85 pictures. How many photos are in her 4 albums?

? 85 85 85 85

Amount for Each Part Unknown Pam put the same number of apples in each of 4 bags. She ended up with 52 apples in bags. How many apples did she put in each bag?

52 ? ? ? ?

Number of Equal Parts Unknown Fred bought some books. Each book cost $16. He spent $48 on books. How many books did he buy?

48 16

Comparison: Multiplication and Division

Larger Quantity Unknown Raul has 17 markers. Linda has 3 times as many. How many markers does Linda have?

? 17 17 17 17

Smaller Quantity Unknown Maria has 24 cards. She has 3 times as many cards as Jamal has. How many cards does Jamal have?

24 ? ? ? ?

Number of Times as Many Unknown Ann’s teacher is 39 years old. Ann is 13 years old. Ann’s teacher is how many times as old as Ann?

39 13 13

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NISD Math Focus

Developing Number Sense through Number Talks

What is a Number Talk? A Number Talk is a short, ongoing daily routine that provides students with meaningful ongoing practice with computation. A Number Talk is a powerful tool for helping students develop computational fluency because the expectation is that they will use number relationships and the structures of numbers to add, subtract, multiply and divide.

Number Talks should be structured as short sessions alongside (but not necessarily directly related to) the ongoing math curriculum. It is important to keep Number Talks short, as they are not intended to replace current curriculum or take up the majority of the time spent on mathematics. In fact, teachers need to spend only 5 to 15 minutes on Number Talks. Number Talks are most effective when done every day.

A Rationale for Number Talks

http://www.mathsolutions.com/documents/9781935099116_ch1.pdf

Number Talks 6-Weeks Focus

Operation Strategies 1st 6-Weeks Addition Breaking Each Number Into its Place Value (Decomposing Each Number), Making Landmark

or Friendly Numbers, Doubles/ Near-Doubles, Making Tens, Compensation, Adding Up in Chunks (Decomposing One Number)

2nd 6-Weeks Subtraction Adding Up, Removal or Counting Back, Place Value and Negative Numbers, Keeping a Constant Difference, Adjusting One Number to Create and Easier Problem

3rd 6-Weeks Multiplication Repeated Addition or Skip Counting, Making Landmark or Friendly Numbers, Partial Products, Doubling and Halving, Breaking Factors into Smaller Factors

4th 6-Weeks Multiplication Repeated Addition or Skip Counting, Making Landmark or Friendly Numbers, Partial Products, Doubling and Halving, Breaking Factors into Smaller Factors

5th 6-Weeks Division Repeated Subtraction or Sharing/ Dealing Out, Multiplying Up, Partial Quotients, Proportional Reasoning

6th 6-Weeks Division Repeated Subtraction or Sharing/ Dealing Out, Multiplying Up, Partial Quotients, Proportional Reasoning

* Highlighted strategies must be introduced. Other strategies may be used with teacher discretion.

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Number Talks

Addition Strategy Examples Strategy Example(s)

Breaking Each Number Into its Place Value (Decomposing Each Number)

116 + 118 (100 + 10 + 6) + (100 + 10 + 8) 100 + 100 = 200 10 + 10 = 20 6 + 8 = 14 200 + 20 + 14 = 234

Making Landmark or Friendly Numbers

116 + 118 +2 116 + 120 = 236 236 – 2 = 234

Doubles/ Near-Doubles 116 + 118 -2 -3 115 + 115 = 230 230 + 4 = 234

Making Tens 116 + 118 (110 + 4 + 2) + (110 + 8) 110 + 110 + (2 + 8) + 4 110 + 110 + 10 + 4 230 + 4 = 234

116 + 118 (110 + 6) + (110 + 4 + 4) 110 + 110 + (6 + 4) + 4 110 + 110 + 10 + 4 = 234

Compensation 116 + 118 - 2 +2 114 + 120 = 234

116 + 118 +4 -4 120 + 114 = 234

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Adding Up in Chunks (Decomposing One Number)

116 + 118 116 + (100 + 10 + 4 + 4) 116 + 100 = 216 216 + 10 = 226 226 + 4 = 230 230 + 4 = 234

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Number Talks

Subtraction Strategy Examples Strategy Example(s)

Adding Up 123 – 59 59 + 1 = 60 60 + 40 = 100 100 + 23 = 123 1 + 40 + 23 = 64

Removal or Counting Back

123 – 59 123 – (10+10+10+10+10+3+6)

123- 59 123 – (20+30+3+6) 123 -20 = 103 103 – 30 = 73 73 – 3 = 70 70 – 6 = 64

123 – 59 123 – (50 + 9) 123 – 50 = 73 73 – 9 = 64

Place Value and Negative Numbers

123 – 59

100 20 3 - 50 9

100 -30 -6

100 – 30 – 6 100 – 30 = 70 70 – 6 = 64

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Keeping a Constant Difference

123 – 59 123 + 1 = 124 - 59 + 1 = - 60 64

300 – 78 300 – 1 = 299 - 78 – 1 = - 77 222

Adjusting One Number to Create and Easier Problem

123 – 59 123 123 -59 + 1 = - 60 63 + 1 = 64

123 – 59 123 123 - 59 – 6 = - 53 70 – 6 = 64

123 – 59 123 + 6 = 129 - 59 -59 70 – 6 = 64

123 – 59 123 – 4 = 119 -59 -59 60 + 4 = 64

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Number Talks

Multiplication Strategy Examples Strategy Example(s)

Repeated Addition or Skip Counting

6 x 15 15 + 15 + 15 + 15 + 15 + 15 15 + 15 = 30 30 + 15 = 45 45 + 15 = 60 60 + 15 = 75 75 + 15 = 90

6 x 15 15 + 15 + 15 + 15 + 15 + 15 2 x 15 = 30 2 x 15 = 30 90 2 x 15 = 30

6 x 15 15 + 15 + 15 + 15 + 15 + 15 6 x 10 = 60 6 x 5 = 30 90

Making Landmark or Friendly Numbers

9 x 15 9 groups of 15 + 1 (group of 15) 10 x 15 = 150 150 – 15 = 135

Partial Products 12 x 9 (10 + 2) x 9 10 x 9 = 90 2 x 9 = 18 90 + 18 = 108

12 x 9 (6 + 6) x 9 6 x 9 = 54 6 x 9 = 54 54 + 54 = 108

12 x 9 12 x (5 + 4) 12 x 5 = 60 12 x 4 = 48 60 + 48 = 108

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Doubling and Halving

Breaking Factors into Smaller Factors

8 x 25 (4 x 25) + (4 x 25) 100 + 100 = 200

8 x 25 (2 x 4) x 25 2 x (4 x 25) 2 x 100 = 200

8 x 25 8 x (5 x 5) (8 x 5) x 5 40 x 5 = 200

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Number Talks

Division Strategy Examples Strategy Example(s)

Sharing/ Dealing Out 30 ÷ 5

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

5 x 6 = 30

30 ÷ 5

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

5 x 6 = 30 First Deal: 5 x 2 = 10

Second Deal: 5 x 2 = 10 Third Deal: 5 x 2 = 10

Multiplying Up 72 ÷ 8 8 x 5 = 40 8 x 4 = 32 8 x 9 = 72 72 ÷ 8 = 9 72 ÷ 8

8 x 5 = 40

8 x 4 = 32

76 ÷ 4 4 x 10 = 40 4 x 5 = 20 4 x 4 = 16 4 x 19 = 76 76 ÷ 4 = 19

4 x 10 = 40

4 x 5 = 20 4 x 4 = 16

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Partial Quotients 56 ÷ 4 40 ÷ 4 = 10 16 ÷ 4 = 4 56 ÷ 4 = 14

88 ÷ 4 40 ÷ 4 = 10 40 ÷ 4 = 10 8 ÷ 4 = 2 88 ÷ 4 = 22

Proportional Reasoning 88 ÷ 4 88 ÷ 4 = 44 ÷ 2 = 22 ÷ 1 = 22

88 44 22 ---- = ---- = ----- = 22 4 2 1

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Year at a Glance

First Semester Second Semester 1st 6-Weeks 4th 6-Weeks

• Bundle #1- Representing and Comparing Whole Numbers and Decimals (24 days)

• Bundle #2- Whole Number and Decimal Operations (Addition

and Subtraction (5 days)

• Bundle #5 (cont.)- Fractions (19 days)

• Bundle #6- Geometry and Measurement (9 days)

2nd 6-Weeks 5th 6-Weeks • Bundle #2 (cont.)- Whole Number and Decimal Operations

(Addition and Subtraction (9 days)

• Bundle #3- Whole Number Operations (Multiplication and Division) (20 days)

• Bundle #6 (cont.)- Geometry and Measurement (25 days)

• Bundle #7- Data Analysis (8 days)

3rd 6-Weeks 6th 6-Weeks • Bundle #3 (cont.)- Whole Number Operations (Multiplication and

Division) (5 days)

• Bundle #4- Algebraic Reasoning (15 days)

• Bundle #5- Fractions (9 days)

• Bundle #8- Personal Financial Literacy (5 days)

• Bundle # 9- STAAR Review and Testing (6 days)

• Bundle #10- Extended Learning (17 days)

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TEKS Spiraling Checklist

*The following checklist is intended to assist with planning for spiraled review of previously taught material. The lavender shaded boxes indicate the 6-weeks in which the TEKS are introduced/taught. For each 6-weeks following initial instruction, there are six boxes in which to indicate when the TEKS were reviewed. For documentation purposes, the date the TEKS were reviewed can be written in the provided boxes. TEKS can be reviewed through center activities, problem of the day, entry/exit tickets, mini-lessons, etc.

1st 6-Weeks 2nd 6-Weeks 3rd 6-Weeks 4th 6-Weeks 5th 6-Weeks 6th 6-Weeks 4.2A

4.2B

4.2C

4.2D

4.2E

4.2F

4.2G

4.2H

4.3A

4.3B

4.3C

4.3D

4.3E

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4.3F

4.3G

4.4A

4.4B

4.4C

4.4D

4.4E

4.4F

4.4G

4.4H

4.5A

4.5B

4.5C

4.5D

4.6A

4.6B

4.6C

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4.6D

4.7A

4.7B

4.7C

4.7D

4.7E

4.8A

4.8B

4.8C

4.9A

4.9B

4.10A

4.10B

4.10C

4.10D

4.10E

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Mathematical Process Standards

• Process standards MUST be integrated within EACH bundle to ensure the success of students.

4.1A 4.1B 4.1C 4.1D 4.1E 4.1F 4.1G apply mathematics to problems arising in everyday life, society, and the workplace

use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution

select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems

communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate

create and use representations to organize, record, and communicate mathematical ideas

analyze mathematical relationships to connect and communicate mathematical ideas

display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication

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Math Instructional Resources

Resource Print/Online Description EnVision Both Textbook Adoption https://www.pearsontexas.com/#/ Motivation Math Both Supplemental Curriculum https://www.mentoringminds.com/customer/account/login/ Engaging Mathematics Print Collection of Mini-Lessons for All TEKS from Region IV http://www.region4store.com/catalog.aspx?catid=1171582 Reasoning Mind (STAAR Readiness) Online Online STAAR Supplemental Curriculum http://www.rmcity.org/ Think Through Math Online Online Supplemental Curriculum https://lms.thinkthroughmath.com/users/sign_in\ Thinking Blocks Online Online Problem Solving Practice with Strip Diagrams http://www.mathplayground.com/thinkingblocks.html 4th Grade Math Games Print Collection of Engaging and Low-Prep Math Games for Skill

Practice http://maccss.ncdpi.wikispaces.net/file/view/4thgrade_GAMES_8.22.14.pdf/519547634/4thgrade_GAMES_8.22.14.pdf Epic! For Educators Online Search for Literature Connections for Math Content https://www.getepic.com/educators Number Talks (Sherry Parrish) Print Develop Number Sense Through a Daily Number Talk

Routine TEKSing Toward STAAR Print STAAR Based Supplemental Curriculum Lessons for Learning (North Carolina) Print Collection of Engaging and Rigorous Math Lessons http://maccss.ncdpi.wikispaces.net/file/view/CCSSMathTasks-Grade4-2014new.pdf/521000852/CCSSMathTasks-Grade4-2014new.pdf Math Learning Center (Bridges) Print Collection of Engaging and Rigorous Math Lessons http://catalog.mathlearningcenter.org/catalog/supplemental-materials-elementary/lessons-activities-grade-4-free NCTM Illuminations Online Search for Engaging and Rigorous Math Lessons by Grade

and Topic http://illuminations.nctm.org/ Math Coach’s Corner Online Math Blog from a Master Texas Math Teacher, Coach, and

Consultant http://www.mathcoachscorner.com/ Promethean Planet Online Tools and Lessons for Interactive Whiteboard http://www.prometheanplanet.com/en-us/ Interactive Math Glossary Online TEA Interactive Math Glossary http://www.texasgateway.org/resource/interactive-math-glossary?field_resource_keywords_tid=math%20teks&sort_by=title&sort_order=ASC&items_per_page=5

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TEKS Information for Teachers TEA STAAR Resources Online TEA Information Regarding STAAR Math http://tea.texas.gov/student.assessment/staar/math/ TEA Math Resources Online TEA Supporting Information for Math TEKS http://tea.texas.gov/Curriculum_and_Instructional_Programs/Subject_Areas/Mathematics/Resources_for_the_Revised_Mathematics_TEKS/ Lead4Ward Resources Online Math TEKS Instructional Resources and Supporting

Information http://lead4ward.com/resources/ TEKS Resource System Online Math TEKS Instructional Resources and Supporting

Information http://www.teksresourcesystem.net/module/profile/Account/LogOn

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Course: Grade 4 Math Bundle 1: Representing and Comparing Whole Numbers and Decimals

Dates: August 22nd - September 23rd (24 days)

TEKS 4.2A: interpret the value of each place-value position as 10 times the position to the right and as one-tenth of the value of the place to its left 4.2B: represent the value of the digit in whole numbers through 1,000,000,000 and decimals to the hundredths using expanded notation and numerals 4.2C: compare and order whole numbers to 1,000,000,000 and represent comparisons using the symbols >, <, or = 4.2D: round whole numbers to a given place value through the hundred thousands place 4.2E: represent decimals, including tenths and hundredths, using concrete and visual models and money 4.2F: compare and order decimals using concrete and visual models to the hundredths 4.2G: relate decimals to fractions that name tenths and hundredths 4.2H: determine the corresponding decimal to the tenths or hundredths place of a specified point on a number line 4.3G: represent fractions and decimals to the tenths or hundredths as distances from zero on a number line

ELPS Learning Strategies 1A: use prior knowledge and experiences to understand meanings in English 1F: use accessible language and learn new and essential language in the process Listening 2C: learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions 2D: monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed Speaking 3D: speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency Reading 4C: develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials 4F: use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language

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Vocabulary Unit Vocabulary Billions Dollar Hundredths Ones Round Cent Equal to (=) Least to Greatest One-tenth Tens Consecutive multiples Expanded Notation Less than (<) Part of the whole Tenths Decimals Fractions Millions Penny Ten-times Digit Greater than (>) Money denominations Place value Thousands Dime Greatest to Least Number line Position Whole Distance Hundreds Numerals

Cognitive Complexity Verbs: interpret, represent, use, compare, order, relate, determine Academic Vocabulary by Standard: 4.2A: place value, position, one-tenth, ten-times 4.2B: place value (billions, millions, thousands, hundreds, tens, ones, tenths, hundredths), decimals, digit, expanded notation, numerals 4.2C: place value (billions, millions, thousands, hundreds, tens, ones, tenths, hundredths), digit, equal to (=), greater than (>), greatest to least, least to greatest, less than(<) 4.2D: consecutive multiples, hundreds, tens, ones, number line, place value, round 4.2E: decimals, hundredths, money denominations (penny, cent, dime, dollar), number line, part of the whole, place value, tenths, whole 4.2F: decimals, digit, equal to (=), greater than (>), greatest to least, least to greatest, less than(<), hundredths, place value, tenths 4.2G: decimal, expanded notation, fractions, hundredths, parts of a whole, place value, tenths, whole 4.2H: decimal, hundredths, number line, parts of a whole, tenths, whole 4.3G: decimals, distance, fractions, hundredths, number line, tenths

Suggested Math Manipulatives

Base 10 Blocks Money Place Value Charts Place Value Disks 10 x 10 Grid Paper

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Bundle 1: Vertical Alignment

3.2B describe the mathematical relationships found in the base-10 place value system through the hundred thousands place

4.2A: interpret the value of each place-value position as 10 times the position to the right and as one-tenth of the value of the place to its left

5.2A represent the value of the digit in decimals through the thousandths using expanded notation and numerals

2.2B use standard, word, and expanded forms to represent numbers up to 1,200 2.2A use concrete and pictorial models to compose and decompose numbers up to 1,200 in more than one way as a sum of so many thousands, hundreds, tens, and ones 3.2A compose and decompose numbers up to 100,000 as a sum of so many ten thousands, so many thousands, so many hundreds, so many tens, and so many ones using objects, pictorial models, and numbers, including expanded notation as appropriate

4.2B: represent the value of the digit in whole numbers through 1,000,000,000 and decimals to the hundredths using expanded notation and numerals


2.2D use place value to compare and order whole numbers up to 1,200 using comparative language, numbers, and symbols (>,<,=) 3.2D compare and order whole numbers up to 100,000 and represent comparisons using the symbols >,<,=

4.2C: compare and order whole numbers to 1,000,000,000 and represent comparisons using the symbols >, <, or =

5.2B compare and order two decimals to thousandths and represent comparisons using the symbols >,<,= 6.2D order a set of rational numbers arising from mathematical and real-world contexts

3.3H compare two fractions having the same numerator or denominator in problems by reasoning about their sizes and justifying the conclusion using symbols, words, objects, and pictorial models

4.2D: round whole numbers to a given place value through the hundred thousands place

6.5C use equivalent fractions, decimals, and percents to show equal parts of the same whole

4.2E: represent decimals, including tenths and hundredths, using concrete and visual models and money


4.2F: compare and order decimals using concrete and visual models to the hundredths

5.2B compare and order two decimals to thousandths and represent comparisons using the symbols >,<,= 6.2D order a set of rational numbers arising from mathematical and real-world contexts

2.2F name the whole number that corresponds to a specific point on a number line 3.3A represent fractions greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 using concrete objects and pictorial models, including strip diagrams and number lines 3.3B determine the corresponding fraction greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 given a specified point on a number line

4.2G: relate decimals to fractions that name tenths and hundredths

6.2B identify a number, its opposite, and its absolute value 6.2C locate, compare, and order integers and rational numbers using a number line

2.2E locate the position of a given whole number on an open number line 3.2C represent a number on a number line as being between two consecutive multiples of 10; 100; 1,000; or 10,000 and use words to describe relative size of numbers in order to round whole numbers

4.2H: determine the corresponding decimal to the tenths or hundredths place of a specified point on a number line

4.3G: represent fractions and decimals to the tenths or hundredths as distances from zero on a number line

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Bundle 1: Teacher Notes

TEKS/Student Expectations

Instructional Implications Distractor Factors Supporting Readiness Standards

TEA Supporting Information

4.2A: interpret the value of each place-value position as 10 times the position to the right and as one-tenth of the value of the place to its left

As students represent numbers utilizing base ten blocks, instruction should relate how the values of each place value position become ten times bigger as we move left on a place value chart (i.e. a ten rod is ten times larger than the ones units; the hundred flat is ten times larger than the ten rods). Students should also recognize how the values of each place value position become ten times smaller as we move right on a place value chart (i.e. one unit is one-tenth the size of a ten rod; a ten rod is ten times smaller than a hundred flat). Relating the value and representation of money may support students with understanding the relationship of related place value (i.e. if you had $1.00 and you wanted to make ten times more, you would have $10.00; if you have $10.00 and wanted to make ten times more, you would have $100.00; if you have $100.00 and you wanted to have ten times less, you would have $10.00; if you have $10.00 and wanted to make one-tenth as much, you would have $1.00).

This standard describes the mathematical relationship found in the base-10 place value system; this understanding will support students in identifying the value of each digit in a number in order to represent numbers in expanded notation and to effectively compare/order numbers. 4.2B: represent the value of the digit in whole numbers through 1,000,000,000 and decimals to the hundredths using expanded notation and numerals

The place-value positions address whole numbers through (less than or equal to) 1,000,000,000 and decimals to the hundredths (greater than or equal to 0.01).

4.2B: represent the value of the digit in whole numbers through 1,000,000,000 and decimals to the hundredths using expanded notation and numerals

In accordance with the standard, students must identify the place value position and value of each digit in a given whole number (i.e. 452,631; the digit 5 is in the ten thousands position and it has a value of 50,000). Instruction should include representing whole numbers in expanded notation (i.e. 452,631; 400,000 + 50,000 + 2,000 + 600 +

* Students may want to identify the hundredths place value as the oneths. * Students may not understand the difference between the digit in a given place value and the value of the digit. * Students may confuse the place

The SE reflects the representing of whole numbers through (less than or equal to) 1,000,000,000 and decimals to the hundredths (greater than or equal to 0.01). For the number 3.94, the 3 in the ones place is 3; the 9 in

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30 + 1). In conjunction with 4.2E/F, as students begin representing decimals with base ten blocks, instruction should relate how to represent those values in expanded notation (i.e. if a hundred flat represents one whole then three ten rods represent three-tenths and five unit cubes represent five-hundredths; 1 + 0.3 + 0.05 = 1.35).

value position for a given digit and the value of the digit. * Students may confuse the value of the base ten blocks as they have been used to represent both whole numbers and fractional parts of a whole.

the tenths place is 0.9; and the 4 in the hundredths place is 0.04; and 3.94 is the sum of 3 ones, 9 tenths, and 4 hundredths. The expanded notation for 3.94 may be represented as: * 3.94 = 3 x 1 + 9 x 0.1 + 4 x 0.01; * 3.94 = 3 x 1 + 9 x 1/10 + 4 x 1/100; * 3.94 = (3 x 1) + (9 x 0.1) + (4 x 0.01); or * 3.94 = (3 x 1) + (9 x 1/10) + (4 x 1/100)

4.2C: compare and order whole numbers to 1,000,000,000 and represent comparisons using the symbols >, <, or =

As students compare the value of numbers, they need to be able to relate their understanding of place value (i.e. the number 5,342 is greater than 3,226 because the digit 5 in 5,342 means there are 5 thousands which is a value of 5,000; however, the digit 3 in 3,226 means there are only 3 thousands which is a value of 3,000). Students will compare two numbers using the correct academic vocabulary (i.e. 5,342 is greater than 3,226). It is important for students to recognize the inverse comparison statement as well (i.e. 3,226 is less than 5,342). Instruction should connect the comparative language to the symbols (>, <, =). It is critical that the students understand how to correctly read each of the symbols not as a trick to remember directionality of the symbols (i.e. the alligator’s mouth eats the bigger number). Encourage students to write and articulate two comparison statements during activities (i.e. 5,342 > 3,226 and 3,226 < 5,342). The standard also has students

In Grade 2, students compared values through 1,200; grade 3, through 120,000; grade 4 through 1,000,000,000. This standard will continue to support students’ knowledge of comparing and ordering of numbers which will extend to thousandths place value in grade 5. 5.2B compare and order two decimals to thousandths and represent comparison using the symbols >, <, or =

Specificity regarding notation is included with the inclusion of the symbols >, <, or =.

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ordering three or more numbers from least to greatest or greatest to least. The use of an open number line will allow students to order more efficiently as numbers increase from left to right. Students can associate this with ordering from least to greatest; numbers decrease from right to left on a number line can be associated to ordering from greatest to least.


Instruction should begin with skip counting by tens, hundreds, thousands, ten-thousands, and hundred-thousand for students to understand consecutive multiples. Students should represent these benchmark values through the use of an open number line. When students are given a specific number to locate on an open number line, teachers will begin to assess students’ understanding of place value (i.e. students place the number 125,387 between 100,000 and 200,000), the relative position of numbers (i.e. the number 100,000 would be indicated first and the number 200,000 would be indicated second on the open number line), and the magnitude of numbers (i.e. students physically place the number 125,387 closer to 100,000 than 200,000). Students will apply this understanding to the rounding of whole numbers to the nearest 10, 100, 1,000, 10,000, and 100,000.

As students will be asked to solve problems using all four operations, it will be important to estimate solutions prior to solving. Rounding is one way of estimating values. Students will be able to evaluate if their solutions are reasonable. 4.4A: add and subtract whole numbers and decimals to the hundredths place using the standard algorithm 4.4H: solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders

The phrase “to a given place value through the hundred thousands place” is more precise than “to the nearest ten, hundred, or thousand.”

4.2E: represent decimals, including tenths and hundredths, using concrete and visual models and money

Through the use of base ten blocks, students will represent values of tenths and hundredths. It is imperative that instruction begin with defining the value of the whole (i.e. the hundred flat will represent one whole, the tens rod will represent tenths, and the unit cube will represent the hundredths). As

As students will be asked to relate decimals to fractions, it will be critical for them to have a visual representation for the value of a decimal. Relating decimals to money (i.e. a dime represent one-tenth) will also support their understanding of the part to whole relationship in fractions (i.e. a dime represents

The SE separates the representations of decimals, including tenths and hundredths, from other skills with decimals. Students are not expected to represent decimals smaller than hundredths.

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base ten blocks can represent both whole number and fractional parts of the whole, it is important to vary the whole (i.e. if the hundred flat represents a whole, then the ten rod represents tenths and the unit cube represents hundredths; if the ten rod represents a whole, then the hundred flat represents ten and a unit cube represents a tenth, etc.). In conjunction with 4.2A, connecting the value of each place value position as being ten times the position to the right and as one-tenth the value to the left. Connecting values of coins to the whole dollar may also support students with the representation of decimals and place value relationships (i.e. a dime $0.10 is ten times more than a penny $0.01; a dollar $1.00 is ten times more than a dime $0.10; a dime $0.10 is one-tenth the size of a dollar $1.00; a penny $0.01 is one-tenth the value of a dime $0.10).

10/100 or 1/10 of a dollar). 4.2G: relate decimals to fractions that name tenths and hundredths

4.2F: compare and order decimals using concrete and visual models to the hundredths

In adherence to the standard, comparing and ordering of decimals is through the use of concrete objects and/or pictorial models. In conjunction with 4.2B, as students compare the value of decimals, they need to be able to relate their understanding of place value (i.e. 2.42 is less than 2.75 because the digit 4 in 2.42 means there are four tenths which is a value of 0.40; however, the digit 7 in 2.75 means there are seven tenths which is a value of 0.70). Students will compare two decimal values using the correct academic vocabulary (i.e. 2.42 is less than 2.75). It is important for students to recognize the inverse comparison statement as well (i.e. 2.75 is greater than 2.42).

Developing the conceptual understanding of the value of decimals through the lens of concrete/pictorial models in grade 4 will be critical for students to be able to compare and order decimal values abstractly in grade 5. 5.2B compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =

The SE separates the comparing and ordering of decimals to the hundredths using concrete and visual models from other skills with decimals. Students can but are not expected to use the symbols >, <, or = with these comparisons.

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Instruction should connect the comparative language to the symbols (>, <, =). It is critical that students understand how to correctly read each of the symbols not as a trick to remember directionality of the symbols (i.e. the alligator’s mouth eats the bigger number). Encourage students to write and articulate two comparison statements during activities (i.e. 2.42 < 2.75 and 2.75 > 2.42). The standard also has students ordering three or more numbers from least to greatest or greatest to least. In conjunction with 4.2H, the use of number lines will allow students to order more efficiently. Numbers increase from left to right on a number line can be associated to ordering from least to greatest. Numbers decrease from right to left on a number line and can be associated to ordering from greatest to least.

4.2G: relate decimals to fractions that name tenths and hundredths

In conjunction with 4.2E, as students represent decimals using base ten blocks and money, instruction can relate those visuals to fractions (i.e. if a hundred flat represents one whole, then a ten rod is one-tenth its value; if a hundred flat represents one whole, then a unit cube is one-hundredth its value; if it takes 10 dimes to make a dollar, then a dime is one-tenth the value of the dollar; if it takes 100 pennies to make a dollar, then a penny is one-hundredth the value of a dollar). Because base ten blocks can represent both whole numbers and fractional parts of the whole, it is important to vary the whole (i.e. if the hundred flat represents a whole, then the ten rod represents tenths and the unit cube represents

* Students may view the part to whole relationship as different values for decimals vs. fractional representations. * Students may think the more digits in a decimal, the greater the value. For example, some students will think 0.05 is greater than 0.5.

Students may relate decimals to fractions that name tenths and hundredths using concrete and pictorial models.

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hundredths. If the ten rod represents a whole then a hundred flat represents ten and a unit cube represents a tenth, etc.). In conjunction with 4.3G, number lines can also be used to relate decimals to fractions (i.e. 0.10 and 1/10 are the same distance away from zero on a number line). Instruction should relate how the proper articulation of decimal representation yields the fractional representation (i.e. 0.7; “seven tenths”; 7/10).


Students will utilize number lines containing intervals that are divided equally between two whole numbers. Students will be asked to identify the decimal value of a point represented on a given number line. Students will need to determine the number of parts that make up the whole (the total number of intervals between two given whole numbers) to identify tenths or hundredths appropriately.

Locating decimal values as a specific point on a number line will support students in comparing/ordering decimals in grade 5. In conjunction with 4.3G, as students identify decimal and fractional values on a number line they will begin to relate the equivalent values of tenths (0.10; 1/10) and hundredths (0.01; 1/100) as they fall on the same point on the number line. 4.2G: relate decimals to fractions that name tenths and hundredths 5.2B compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =

When paired with 4.3C, decimals can be developed from fractions that have equivalent fractions with denominators of ten or one hundred, such as 0.25 from ¼ or 0.4 from 2/5. (This will be addressed in the fraction bundle.)


In adherence to the standard, instruction is limited to representing decimals to the tenths and hundredths on a number line. Students will utilize number lines containing intervals that are divided equally between zero and one whole. Students will be asked to represent a given decimal/ fraction value on a number line.

Locating decimal values as a specific point on a number line will support students in comparing/ordering decimals in grade 5. When students identify decimal and fractional values on a number line they will begin to relate the equivalent values of tenths (0.10; 1/10) and hundredths (0.01; 1/100) as they fall on the same point on the number line. 4.2G: relate decimals to fractions

The fractions that may be represented are of the form a/b, where a and b are whole numbers and b>0, including when a>b. This aligns to 4.3A. When used in conjunction with 4.3C, decimals can be developed from fractions that have equivalent fractions with denominators of ten or one hundred such as ¼=0.25 or 2/5=0.4.

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that name tenths and hundredths 5.2B compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =

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Course: Grade 4 Math Bundle 2: Whole Number and Decimal Operations (Addition and Subtraction)

Dates: September 26th-October 14th (14 days)

TEKS 4.2D: round whole numbers to a given place value through the hundred thousands place 4.4A: add and subtract whole numbers and decimals to the hundredths place using the standard algorithm 4.4G: round to the nearest 10, 100, or 1,000 or use compatible numbers to estimate solutions involving whole numbers

ELPS Learning Strategies 1A: use prior knowledge and experiences to understand meanings in English 1F: use accessible language and learn new and essential language in the process Listening 2C: learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions 2I: demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs Speaking 3B: expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication 3D: speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency 3E: share information in cooperative learning interactions Reading 4C: develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials 4D: use pre-reading supports such as graphic organizers, illustrations, and pre-taught topic-related vocabulary and other pre-reading activities to enhance comprehension of written text 4F: use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language

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Vocabulary Unit Vocabulary Add Difference Hundreds Place value Sum Compatible numbers Digit Nearest (10, 100, 1000) Round Tens Consecutive multiples Estimate Number line Solutions Tenths Decimals Expanded notation Ones Subtract Whole

Cognitive Complexity Verbs: add, subtract, use, round, estimate Academic Vocabulary by Standard: 4.2D: consecutive multiples, hundreds, tens, ones, number line, place value, round 4.4A: add, decimals, difference, digit, expanded notation, place value, subtract, sum, tenths, whole 4.4G: compatible numbers, estimate, nearest (10; 100; 1,000), round, solutions


Base 10 Blocks Money Place Value Charts Place Value Disks

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2.4A recall basic facts to add and subtract within 20 with automaticity 2.4B add up to four two-digit numbers and subtract two-digit numbers using mental strategies and algorithms based on knowledge of place value and properties of operations 2.4C solve one-step and multistep word problems involving addition and subtraction within 1,000 using a variety of strategies based on place value, including algorithms 2.4D generate and solve problem situations for a given mathematical number sentence involving addition and subtraction of whole numbers within 1,000 3.4A solve with fluency one-step and two-step problems involving addition and subtraction within 1,000 using strategies based on place value, properties of operations, and the relationship between addition and subtraction

4.4A: add and subtract whole numbers and decimals to the hundredths place using the standard algorithm

5.3A estimate to determine solutions to mathematical and real-world problems involving addition, subtraction, multiplication, or division

3.4B round to the nearest 10 or 100 or use compatible numbers to estimate solutions to addition and subtraction problems

4.4G: round to the nearest 10, 100, or 1,000 or use compatible numbers to estimate solutions involving whole numbers

5.2C round decimals to tenths or hundredths

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The phrase “to a given place value through the hundred thousand place” is more precise than “to the nearest ten, hundred, or thousand.”

4.4A: add and subtract whole numbers and decimals to the hundredths place using the standard algorithm

In conjunction with 4.2B/E, students will represent decimal values through the hundredths with concrete objects/ pictorial models, apply their understanding of place value to identify the digit in each position, and employ the use of expanded notation to develop strategies when solving addition and subtraction decimal problems. These actions should parallel the introduction of the

* Students may not align the appropriate place values when adding/ subtracting decimals. * Students may not understand that adding and subtracting decimals is joining and separating parts of the same whole.

The SE includes the addition and subtraction of decimals to the hundredths place using the standard algorithm. Problems may include both whole numbers and decimal values. Students may be expected to add decimals to the hundredths using concrete and pictorial models with

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standard algorithm in order to move students from the concrete to the abstract learning. Be sure to vary the context of problems. Joining: Sarah had $2.50. Juan gave her $1.95. How much money does Sarah have now? Sarah had $2.50. Juan gave her some money. Now, Sarah has $4.43. How much money did Juan give Sarah? Sarah had some money. Juan gave her $1.95. Now Sarah has $4.43. How much money did Sarah have to begin with? Separating: Sarah had $4.43. She lent Juan $2.50. How much money does Sarah have now? Sarah had $4.43. She lent some money to Juan. Now Sarah has $1.95. How much money did she lend to Juan? Sarah had some money. She lent Juan $2.50 and now has $1.95. How much money did Sarah have before lending Juan money? Comparing: Juan has $1.95 and Sarah has $4.43. How much more money does Juan have than Sarah? If Juan has $2.50 less than Sarah, how much does Juan have if Sarah has $4.43?

connections to the standard algorithm.


In conjunction with 4.4A, students will estimate solutions to addition and subtraction problems prior to solving for the exact answer. Students will employ the understanding of representing a number on a number line between two consecutive multiples of 10, 100, 1000 (see 4.2D) as a means

As students will be asked to solve problems using all four operations, it will be important to estimate solutions prior to solving. Rounding and the use of compatible numbers will support students in estimating solutions and evaluating reasonableness of solutions.

This SE includes addition and subtraction as well as multiplication and division.

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of estimating sums and differences (i.e. 679 – 344 = ___; 700-300 = 400 or 680 – 340 = 340). In adherence to the standard, the intent of rounding is to estimate a solution; therefore, employing a specific rounding rule is not necessary. Compatible numbers is another means for estimating solutions (i.e. 679 – 344 = ___; 675 – 350 = 325).

4.4A: add and subtract whole numbers and decimals to the hundredths place using the standard algorithm 4.4H: solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders

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Course: Grade 4 Math Bundle 3: Whole Number Operations (Multiplication and Division)

Dates: October 17th- November 18th (25 days)

TEKS 4.2D: round whole numbers to a given place value through the hundred thousands place 4.4B: determine products of a number and 10 or 100 using properties of operations and place value understandings 4.4C: represent the product of 2 two-digit numbers using arrays, area models, or equations, including perfect squares through 15 by 15 4.4D: use strategies and algorithms, including the standard algorithm, to multiply up to a four-digit number by a one-digit number and to multiply a two-digit number by a two-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties 4.4E: represent the quotient of up to a four-digit whole number divided by a one-digit whole number using arrays, area models, or equations 4.4F: use strategies and algorithms, including the standard algorithm, to divide up to a four-digit dividend by a one-digit divisor 4.4G: round to the nearest 10, 100, or 1,000 or use compatible numbers to estimate solutions involving whole numbers 4.4H: solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders

ELPS Listening 2C: learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions 2E: use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language 2I: demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs Speaking 3C: speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired 3D: speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency 3F: ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments 3H: narrate, describe, and explain with increasing specificity and detail as more English is acquired Reading 4C: develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials 4F: use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language 4G: demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs

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Vocabulary

Unit Vocabulary Algorithms Distributive property Mental math Partial product Quotients Area model Dividend Multiplication Partial quotients Remainder Array Division Nearest (10, 100, 1000) Patterns Round Associative property Divisor Number line Perfect square Solutions Commutative property Equation Number sentence Place value Strategies Compatible numbers Estimate One-digit number Product Tens Consecutive multiples Factor Ones Properties of operations Two-digit number Digit Hundreds

Cognitive Complexity Verbs: determine, use, round, estimate, solve, represent, solve Academic Vocabulary by Standard: 4.2D: consecutive multiples, hundreds, tens, ones, number line, place value, round 4.4B: one-digit number, patterns, place value, product, properties of operations, two-digit number 4.4C: area model, array, equation/ number sentence, factor, multiplication, perfect square, partial product, product, two-digit number 4.4D: algorithms, digit, mental math, multiplication, partial products, place value, properties (commutative property, associative property, distributive property), strategies 4.4E: area model, array, division, equations, partial quotients, quotients 4.4F: algorithm, dividend, divisor, quotient, remainder, strategy 4.4G: compatible numbers, estimate, nearest (10, 100, 1000), round, solutions 4.4H: dividend, division, divisor, factor, multiplication, product, quotient, remainders


Base 10 Blocks, Grid Paper

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3.4E represent multiplication facts by using a variety of approaches such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line, and skip counting 3.4F recall facts to multiply up to 10 by 10 with automaticity and recall the corresponding division facts

4.4B: determine products of a number and 10 or 100 using properties of operations and place value understandings

5.3B multiply with fluency a three-digit number by a two-digit number using the standard algorithm 6.3E multiply and divide positive rational numbers fluently

3.4G use strategies and algorithms, including the standard algorithm, to multiply a two-digit number by a one-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties

4.4C: represent the product of 2 two-digit numbers using arrays, area models, or equations, including perfect squares through 15 by 15


3.4G use strategies and algorithms, including the standard algorithm, to multiply a two-digit number by a one-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties

4.4D: use strategies and algorithms, including the standard algorithm, to multiply up to a four-digit number by a one-digit number and to multiply a two-digit number by a two-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties


3.4K solve one-step and two-step problems involving multiplication and division within 100 using strategies based on objects; pictorial models, including arrays, area models, and equal groups; properties of operations; or recall of facts

4.4E: represent the quotient of up to a four-digit whole number divided by a one-digit whole number using arrays, area models, or equations

5.3C solve with proficiency for quotients of up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm 6.3E multiply and divide positive rational numbers fluently

4.4F: use strategies and algorithms, including the standard algorithm, to divide up to a four-digit dividend by a one-digit divisor

5.3C solve with proficiency for quotients of up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm 6.3E multiply and divide positive rational numbers fluently

3.4B round to the nearest 10 or 100 or use compatible numbers to estimate solutions to addition and subtraction problems


5.2C round decimals to tenths or hundredths

4.4H: solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainder

5.3B multiply with fluency a three-digit number by a two-digit number using the standard algorithm 5.3E solve for products of decimals to the hundredths, including situations involving money, using strategies based on place value understandings, properties of operations, and the relationship to the multiplication of whole numbers 6.3E multiply and divide positive rational numbers fluently

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The phrase “to a given place value through the hundred thousand place” is more precise than “to the nearest ten, hundred, or thousand.”

4.4B: determine products of a number and 10 or 100 using properties of operations and place value understandings

Through the use of technology (calculators), students can begin to see a pattern when multiplying a one-digit number by ten and/or hundred (i.e. 1 x 10 = 10; 2 x 10 = 20; 3 x 10 = 30, etc.). The partial product strategy can also be applied to determine products of two-digit numbers and 10 and/or 100 (i.e. 64 x 10 = ___; (60 + 4) x 10 = (60 x 10) + (4 x 10) = 600 + 40 = 640). Instruction should also relate to how the place value chart becomes ten times larger when moving one position to the left or one hundred times larger when moving two

Recognizing patterns in multiplying by 10 or 100 will support students in developing multiplication/division algorithms and being able to estimate products/quotients of multiplication/ division problems. 4.4H: solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders

Specificity has been added related to patterns. Students are to use properties of operations and place-value understandings to multiply a number by 10 or 100. Because place value in grade 4 extends from the hundredths to the one billions place, students may be expected to determine the product of a pair of numbers such as 4,000 and 10. The product is 10 times larger than 4,000, so the

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positions to the left. The automaticity of multiplying by 10 and 100 will be essential for the development of strategies/algorithms for multiplying multiple digit numbers (see 4.4D).

place value of the 4 increases from 1,000 to 10,000. When multiplying numbers such as 324 and 10, a student is expected to use the properties of operations and place value. Multiplying 324 by 10 means that each value from expanded notation is being multiplied by 10, so 10 x 324 = 10 x (300 + 20 + 4) = (10 x 300) + (10 x 20) + (10 x 4) = 3,000 + 200 + 40 = 3,240. Students are not expected to use the standard algorithm to determine the result of multiplying a number by 10 or 100.

4.4C: represent the product of 2 two-digit numbers using arrays, area models, or equations, including perfect squares through 15 by 15

The intent of this standard is to build the conceptual knowledge of two-digit multiplication (i.e. 12 x 13 = ___). Students will use a variety of methods to represent their understanding: Arrays:

100

10

10

10 1 1 10 1 1 10 1 1

Area Models:

It is critical for students to develop the conceptual understanding of multiplying two-digit numbers before moving to the abstract understanding of the standard algorithm. This supporting standard provides that developmental progression. 4.4H: solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders

This SE specifies that the factors may be 2 two-digit numbers and that perfect squares through 15 x 15 are included. This SE includes equations as representations for the product of 2 two-digit numbers. A student may be expected to describe an array or an area model with an equation that includes the factors as lengths and the product as the area. When paired with 4.1A, students solve problems. The intent of this SE is not a sole focus on computation. When paired with 4.1G, students may be expected to

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10 x 10 = 100

10 x 2 = 20

3 x 10 = 30

3 x 2 = 6

Equations:

Encourage students to demonstrate their understanding in more than one way. In adherence to the standard, instruction should include examples of perfect squares where the two factors are the same value (i.e. 12 x 12). Through the use of arrays and/or area models, students will begin to relate why such examples are identified as perfect squares as the side lengths are equal.

explain situations in word form. This SE complements the development of area formulas in 4.5C and 4.5D.

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4.4D: use strategies and algorithms, including the standard algorithm, to multiply up to a four-digit number by a one-digit number and to multiply a two-digit number by a two-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties

In accordance with the standard, students are to use strategies to develop their conceptual understanding of four-digit times one-digit multiplication and two-digit by two-digit multiplication. The use of mental math, partial products, and operational properties will allow students to build their flexibility in the use of numbers (i.e. 23 x 12 = (20 + 3) x (10 + 2) = (20 X 10) + (20 X 2) + (3 X 10) + (3 x 2) = 200 + 40 + 30 + 6 = 276). It is imperative to relate those actions to the steps found in the standard algorithm in order to move students from the concrete to abstract understanding (i.e. the partial products method multiplies the values of the digits; whereas, the traditional algorithms multiplies the digits within a given place value; students relate how multiplying 2 tens and 2 ones to yield a 4 in the tens place value of the traditional algorithm is the same thing as multiplying 20 x 2 to yield 40 in the partial products method). Encourage students to demonstrate their understanding in more than one way.

It is critical for students to develop their flexibility with multiplying multi-digit numbers b employing different strategies before moving to the abstract understanding of the standard algorithm. This supporting standard provides that development progression. 4.4H: solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders

The SE specifies “strategies” and “algorithms,” including the standard algorithm. Strategies may include mental math, partial products, and properties of operations.

4.4E: represent the quotient of up to a four-digit whole number divided by a one-digit whole number using arrays, area models, or equations

The intent of this standard is to build the conceptual knowledge of four-digit divided by one-digit whole numbers (i.e. 645 ÷ 5 = ___). In adherence to this standard, students represent the quotient using arrays and area models (see 4.4C for examples). Equations (i.e. partial quotient):

It is critical for students to develop the conceptual understanding of division of multi-digit numbers before moving to the abstract understanding of the standard algorithm. This supporting standard provides that development progression. 4.4H: solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders

The SE adds specificity to the picture forms as “arrays” and “area models” and the number form as “equations.” It also adds specificity to the division situations as they have “up to a four-digit whole number divided by a one-digit whole number.” When paired with 4.1A, students solve problems. The intent of this SE is not a sole focus on computation.

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In adherence to the standard, limit instruction to division of four-digit by one-digit.

When paired with 4.1G, students explain situations in word form.

4.4F: use strategies and algorithms, including the standard algorithm, to divide up to a four-digit dividend by a one-digit divisor

In adherence to the standard, instruction is limited to four-digit divided by a one-digit divisor. In conjunction with 4.4E, it is essential to relate the actions of the conceptual strategies to the standard algorithm. The partial quotient method uses the value of the numbers in the dividend; students determine the total number of equal groups that will divide into each value and then the total number of equal groups is added to determine the quotient. The traditional algorithm uses the digits in each place value to determine the total number of equal groups. The traditional algorithm moves through each place value one at a time subtracting out the total number of equal groupings created. Encourage students to demonstrate their understanding in more than one way.

It is critical for students to develop an efficient way to divide multi-digit numbers by a one-digit divisor. This learning will extend to interpreting the remainder in a division problem. 4.4H: solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders

This SE specifies “strategies” and “algorithms,” including the standard algorithm. Strategies may include mental math, partial products, and properties of operations.


In conjunction with 4.4A, students will estimate solutions to addition and subtraction problems prior to solving for the exact answer. Students will employ the understanding of representing a number on a number line between two consecutive multiples of 10, 100, 1000 (see 4.2D) as a means of

As students will be asked to solve problems using all four operations, it will be important to estimate solutions prior to solving. Rounding and the use of compatible numbers will support students in estimating solutions and evaluating reasonableness of solutions.

The SE includes addition and subtraction as well as multiplication and division.

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estimating sums and differences (i.e. 679 – 344 = ___; 700-300 = 400 or 680 – 340 = 340). In adherence to the standard, the intent of rounding is to estimate a solution; therefore, employing a specific rounding rule is not necessary. Compatible numbers is another means for estimating solutions (i.e. 679 – 344 = ___; 675 – 350 = 325).

4.4A: add and subtract whole numbers and decimals to the hundredths place using the standard algorithm 4.4H: solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders

4.4H: solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainder

In conjunction with 4.4D/4.4E, as students become more comfortable with the conceptual understanding of multiplication and division, instruction should move to a more abstract representation. In adherence to the standard, instruction should include one- and two-step problems (i.e. two steps with just multiplication; two steps with just division; one step with multiplication and one step with division). Students need to interpret and apply the use of the remainder of division problems in different ways (i.e. There were 225 cupcakes. If 8 cupcakes fit on a plate, how many plates will be filled with cupcakes? How many cupcakes will be leftover not on a plate? How many plates would be needed for all of the cupcakes to fit on a plate?).

* Students may try to apply “key words” to select the appropriate operation instead of understanding the context of the problem. * Students may not recognize a number sentence and its inverse as being equivalent (i.e. 42 ÷ 6 = ___ is the same thing as 6 x ___ = 42). * Students may not understand the context of a remainder in a real world division problem.

This SE includes the interpretation of remainders when dividing. “Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently” (National Research Council, 2001, p. 121).

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Course: Grade 4 Math Bundle 4: Algebraic Reasoning

Dates: November 28th-December 16th (15 days)

TEKS 4.5A: represent multi-step problems involving the four operations with whole numbers using strip diagrams and equations with a letter standing for the unknown quantity 4.5B: represent problems using an input-output table and numerical expressions to generate a number pattern that follows a given rule representing the relationship of the values in the resulting sequence and their position in the sequence

ELPS Learning Strategies 1B: monitor oral and written language production and employ self-corrective techniques or other resources Listening 2I: demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs Speaking 3G: express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics 3H: narrate, describe, and explain with increasing specificity and detail as more English is acquired Reading 4C: develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials 4F: use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language

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Vocabulary

Unit Vocabulary Addend Equation Number pattern Rule Subtraction Addition Factor Numerical expressions Sequence Sum Difference Input-output table Position Strip diagram Variable Division Multiplication Product

Cognitive Complexity Verbs: represent, generate Academic Vocabulary by Standard: 4.5A: addend, addition, difference, division, equation, factor, multiplication, product, strip diagram, subtraction, sum, variable (letter standing for an unknown quantity) 4.5B: input-output table, number pattern, numerical expressions, position, rule, sequence


Cuisenaire Rods (as strip diagrams), dominoes

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2.7C represent and solve addition and subtraction word problems where unknowns may be any one of the terms in the problem 3.5A represent one- and two-step problems involving addition and subtraction of whole numbers to 1,000 using pictorial models, number lines, and equations 3.5B represent and solve one- and two-step multiplication and division problems within 100 using arrays, strip diagrams, and equations 3.5D determine the unknown whole number in a multiplication or division equation relating three whole numbers when the unknown is either a missing factor or product

4.5A: represent multi-step problems involving the four operations with whole numbers using strip diagrams and equations with a letter standing for the unknown quantity

5.4B represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity 6.7B distinguish between expressions and equations verbally, numerically, and algebraically 6.10A model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts 6.10B determine if the given value(s) make(s) one-variable, one-step equations or inequalities true

3.5E represent real-world relationships using number pairs in a table and verbal descriptions

4.5B: represent problems using an input-output table and numerical expressions to generate a number pattern that follows a given rule representing the relationship of the values in the resulting sequence and their position in the sequence

5.4C generate a numerical pattern when given a rule in the form y = ax or y = x + a and graph 5.4D recognize the difference between additive and multiplicative numerical patterns given in a table or graph 6.4A compare two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships

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Instructional Implications Distractor Factors TEA Supporting Information

4.5A: represent multi-step problems involving the four operations with whole numbers using strip diagrams and equations with a letter standing for the unknown quantity

In conjunction with 4.4A/D/F/H, students will represent +/-/x/÷ problems with an equation in which the unknown quantity is represented with a variable (letter). In adherence to the standard, the equation must be a multi-step problem involving a mixture of the four operations. The use of strip diagrams (part-part-whole mat) may support the understanding of how to represent such equations (i.e. The Wildcats basketball team scored 75 points. Michael scored 35 of the points by himself and the remaining points were made by Damon and Rayshawn. If Rayshawn and Damon scored the same number of points, how many points did each boy score?

Total Number of Points Scored by Wildcat’s Team (whole)

75 Michael’s

Points (part)

Damon’s Points (part)

Rayshawn’s Points (part)

35 P P 75 = 35 + P + P where P = points). Encourage students to write more than one equation for every problem (i.e. 75 = 35 + P + P; 75 = 35 + 2P; (75-35) ÷ 2 = P).

* Students who do not have an understanding of the context of the problem may incorrectly represent the expression/equation (i.e. there are two times as many girls as boys in the classroom; 2 x B = G not 2 x G = B). * Students may try to apply “key words” to select the appropriate operation instead of understanding the context of the problem.

Strip diagrams may include a letter representing the unknown quantity. When paired with 4.1G, students may explain situations in word form.

4.5B: represent problems using an input-output table and numerical expressions to generate a number pattern that follows a given rule representing the relationship of the values in the resulting sequence and their

In accordance to the standard, students should be given a real world situation (i.e. number of wheels on a tricycle) and asked to represent the number pattern in an input-output table (see below). Table representations should be both vertical and horizontal. Students also have to

* Students may identify a pattern by comparing input to input values and/or output to output values instead of input to output values. * Students may confuse a

The “input” is the position in the sequence. The “output” is the value in the position. For example, the sequence 4, 8, 12, 16 …. can be listed in table format where the “input” is the position of the value in the sequence. The first position has a value of four, the second position has a value of 8, etc.

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position in the sequence represent the pattern in a numerical expression (i.e. w = 3x t). A process column (a.k.a. what’s my rule) may be added to the input-output table to help support students with that understanding. Students should state two numerical expressions for every pattern (i.e. w = t x 3; t = w ÷ 3).

Number of

Tricycles

Process Number of Wheels

0 0 x 3 0 1 1 x 3 3 2 2 x 3 6 3 3 x 3 9 4 4 x 3 12 5 5 x 3 15 t t x 3 w

multiplicative pattern for an additive pattern as they view multiplication as repeated addition. * Students may not recognize the equivalency of a numerical expression and its inverse (i.e. w = t x 3 is the same expression as t = w ÷ 3).

Input, Position Numerical Expression

Output, Value

1 4 x 1 4 2 4 x 2 8 3 4 x 3 12 4 4 x 4 16

The output is four times the input. The value of the position is four times the position number. A problem might ask for the value of the 23rd position for this relationship. Students would be expected to determine that the input is 23, the numerical expression is 4 x 23, and the output is 92. Students may be expected to work with problems that reflect the addition and subtraction of a whole number to the input or the multiplication or division of the input by a whole number. The values used with multiplication and division should align to 4.4D and 4.4F. The “rule” or the description of the operation that is applied to the input could be given to the students. Alternately, students could determine the rule. Students are not expected to represent relationships that use more than one operation.

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Course: Grade 4 Math Bundle 5: Fractions Dates: January 2nd – February 10th (28 days)

TEKS 4.2H: determine the corresponding decimal to the tenths or hundredths place of a specified point on a number line 4.3A: represent a fraction a/b as a sum of fractions 1/b, where a and b are whole numbers and b > 0, including when a > b 4.3B: decompose a fraction in more than one way into a sum of fractions with the same denominator using concrete and pictorial models and recording results with symbolic representations 4.3C: determine if two given fractions are equivalent using a variety of methods 4.3D: compare two fractions with different numerators and different denominators and represent the comparison using the symbols >, =, or < 4.3E: represent and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations 4.3F: evaluate the reasonableness of sums and differences of fractions using benchmark fractions 0, 1/4, 1/2, 3/4, and 1, referring to the same whole 4.3G: represent fractions and decimals to the tenths or hundredths as distances from zero on a number line

ELPS Learning Strategies 1A: use prior knowledge and experiences to understand meanings in English 1B: monitor oral and written language production and employ self-corrective techniques or other resources Listening 2C: learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions 2I: demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs Speaking 3B: expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication 3D: speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency 3G: express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics 3H: narrate, describe, and explain with increasing specificity and detail as more English is acquired

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Reading 4F: use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language 4G: demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs

Vocabulary

Unit Vocabulary Addition Difference Equivalent fractions Number line Sum Benchmark fraction Distance Fraction Numerator Sum of fractions Common denominator Equal denominators Greater than (>) Parts of a whole Tenths Comparison symbol Equal parts Hundredths Properties of operations Unit fractions Decimal Equal to (=) Identity property Reasonableness Whole Denominator Equations/ number sentences Less than (<) Subtraction

Cognitive Complexity Verbs: represent, decompose, use, determine, compare, evaluate, solve Academic Vocabulary by Standard: 4.2H: decimal, hundredths, number line, parts of a whole, tenths, whole 4.3A: denominator, equal parts, fraction, numerator, sum of fractions, unit fractions, whole 4.3B: denominator, equations/ number sentences, fraction, numerator, sum of fractions 4.3C: distance, equal parts of a whole, equivalent fractions, identity property, number line, whole 4.3D: common denominator, comparison symbol, denominator, equal to (=), fractions, greater than (>), less than (<), numerator 4.3E: addition, difference, equal denominators, fractions, number line, properties of operations, subtraction, sum, whole 4.3F: benchmark fraction, difference, parts of a whole, reasonableness, sum, whole 4.3G: decimals, distance, fractions, hundredths, number line, tenths


Cuisenaire Rods Fraction Bars Fraction Circles Pattern Blocks Snap Cubes 2-Color Counters

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2.2E locate the position of a given whole number on an open number line 3.2C represent a number on a number line as being between two consecutive multiples of 10; 100; 1,000; or 10,000 and use words to describe relative size of numbers in order to round whole numbers


3.3D compose and decompose a fraction a/b with a numerator greater than zero and less than or equal to b as a sum of parts 1/b

4.3A: represent a fraction a/b as a sum of fractions 1/b, where a and b are whole numbers and b > 0, including when a > b

6.2E extend representations for division to include fraction notation such as a/b represents the same number as a ÷ b where b ≠ 0

3.3D compose and decompose a fraction a/b with a numerator greater than zero and less than or equal to b as a sum of parts 1/b

4.3B: decompose a fraction in more than one way into a sum of fractions with the same denominator using concrete and pictorial models and recording results with symbolic representations

3.3F represent equivalent fractions with denominators of 2, 3, 4, 6, and 8 using a variety of objects and pictorial models, including number lines 3.3G explain that two fractions are equivalent if and only if they are both represented by the same point on the number line or represent the same portion of a same size whole for an area model

4.3C: determine if two given fractions are equivalent using a variety of methods



4.3D: compare two fractions with different numerators and different denominators and represent the comparison using the symbols >, =, or <


4.3E: represent and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations

5.3H represent and solve addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations 5.3K add and subtract positive rational numbers fluently

4.3F: evaluate the reasonableness of sums and differences of fractions using benchmark fractions 0, 1/4, 1/2, 3/4, and 1, referring to the same whole

5.3H represent and solve addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations 5.3K add and subtract positive rational numbers fluently


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Students will utilize number lines containing intervals that are divided equally between two whole numbers. Students will be asked to identify the decimal value of a point represented on a given number line. Students will need to determine the number of parts that make up the whole (the total number of intervals between two given whole numbers) to identify tenths or hundredths appropriately.

Locating decimal values as a specific point on a number line will support students in comparing/ordering decimals in grade 5. In conjunction with 4.3G, as students identify decimal and fractional values on a number line they will begin to relate the equivalent values of tenths (0.10; 1/10) and hundredths (0.01; 1/100) as they fall on the same point on the number line. 4.2G: relate decimals to fractions that name tenths and hundredths 5.2B compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =

When paired with 4.3C, decimals can be developed from fractions that have equivalent fractions with denominators of ten or one hundred, such as 0.25 from ¼ or 0.4 from 2/5.

4.3A: represent a fraction a/b as a sum of fractions 1/b, where a and b are whole numbers and b > 0, including when a > b

Instruction should begin with students identifying the whole (i.e. a square has been divided into four equal parts; therefore, we will be working with denominators of four). In adherence to the standard, instruction should extend to fractions greater than one whole. Students should label each part of the whole with its appropriate unit fraction (i.e. if a square represents one whole, then each part of the whole represents one-fourth).

As students begin counting each portion of a given fraction (i.e. one-fourth, two-fourths, three-fourths, etc.), instruction can then relate a number sentence to those actions (i.e. ¼ + ¼ + ¼ + ¼ + ¼ = 5/4). Students should understand that each part of

When paired with 4.1D, students may represent a/b as a sum of fractions 1/b using concrete and pictorial models, which includes improper fractions when a>b. The sum of fractions, such as 1/5 + 1/5 + 1/5, may be referenced as an expression.

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the whole represents a unit fraction (i.e. ¼) and as they count each unit fraction, they are orally stating its cumulative sum. Students will extend the sums of fractions in more than one way (see 4.3B).

4.3B: decompose a fraction in more than one way into a sum of fractions with the same denominator using concrete and pictorial models and recording results with symbolic representations

In conjunction with 4.3A, as students represent improper fractions as a sum of fractions (i.e. 5/4 = ¼ + ¼ + ¼ + ¼ + ¼), instruction should extend the representation of those number sentences to in more than one way (i.e. 5/4 = 2/4 + 2/4 + 1/4; 5/4 = 3/4 + 2/4; 5/4 = 4/4 + 1/4 etc.). It is imperative that instruction relate each component of the number sentence to the pictorial model as a means of moving students from the concrete to the abstract understanding (i.e. 5/4 = 2/4 + 2/4 + 1/4 ; students shade each portion of the area model a different color to represent the different addends of the equation).

In alignment with 4.3E, equations are limited to fractions with like denominators only. Instruction should include examples with whole numbers.

As students decompose fractions as the sum of fractions with like denominators in more than one way (i.e. given five out of four parts of a whole shaded, students represent shaded portion as ¼ + ¼ + ¼ + ¼ + ¼ = 5/4; 4/4 + 1/4 = 5/4; 1 + 1/4 = 5/4; 2/4 + 2/4 + 1/4 = 1 ¼ ), they will extend that understanding to adding fractions with like denominators. This will include the use of whole numbers in addition of fractions. 4.3E: represent and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations

This SE expects students to describe fractions as a sum of unit fractions, such as 5/2= ½ + ½ + ½ + ½ + ½. In this SE, students are also expected to express 5/2= 3/2+2/2; 5/2=1/2+4/2; 5/2=2/2+2/2+1/2; and 2 ½ =1 + 1 + ½. Students are expected to use concrete models such as fraction strips or fraction bars and pictorial models such as strip diagrams and to record the appropriate number sentences.

4.3C: determine if two given fractions are equivalent using a variety of methods

Instruction should include a variety of methods to determine if two fractions are equivalent. Begin with the use of manipulatives (i.e. if a hexagon pattern block represents one whole, then a trapezoid pattern block represent ½, and a triangle pattern block represents 1/6; one trapezoid = 3 triangles, therefore ½ = 3/6), extend to the pictorial models (i.e. a square divided into two, four, and eight equal parts reflect ½ being the same amount of area as 2/4 and 4/8), relate to number lines (i.e. ½, 2/4, 3/6, and 4/8 are all the same distances away from zero on a number line). It is imperative for

Determining if two fractions are equivalent will support students in being able to compare and order fractions with different denominators. 4.3D: compare two fractions with different numerators and different denominators and represent the comparison using the symbols >, =, or <

Methods may include concrete models, such as fraction strips and fraction bars, and pictorial models, such as strip diagrams. Methods also include numeric approaches. Students are expected to use methods that prove that the multiplicative relationship between the numerators is the same as the multiplicative relationship between the

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instruction to progress from the concrete object to pictorial, before introducing the multiplicative identity property (a number multiplied by one will not change the value of the number). Instruction should relate the actions in the identity property to the visual (i.e. ½ x 1 = ___; ½ x 2/2 = 2/4 should be demonstrated through the use of a strip of paper divided into two equal parts representing ½; folding that paper in half again does not change the length of the piece of paper but now divides the whole into four equal parts where 2/4 represents the same as ½).

denominators or prove that the multiplicative relationship between the numerator and denominator in each fraction is the same. Simplifying each fraction would be another approach. For example, ¾ and 9/12 are equivalent because each fourth is separated into 3 equal parts to make 9/12, and 6/8 and 9/12 are equivalent because they both simplify to ¾. Methods may also include multiplying by a fraction equivalent to one.

4.3D: compare two fractions with different numerators and different denominators and represent the comparison using the symbols >, =, or <

Students will compare fractions with unlike denominators in a variety of ways (i.e. concrete objects, pictorially, number line, and the identity property). In conjunction with 4.3C, as students begin to understand how to employ the use of the multiplicative identity property to determine if two fractions are equivalent, instruction can extend its use to comparing fractions (i.e. 2/3 > 1/4 because 2/3 x 4/4 = 8/12 and 1/4 x 3/3 = 3/12 and 8/12 > 3/12). It is imperative that the pictorial representation relates to use of the identity property in order to move students from the concrete to abstract understanding. In adherence to the standards, students will only compare two fractions; they do not have to order three or more. Encourage students to state two comparison statements to ensure understanding (i.e. 3/8 < 3/6 and 3/6 > 3/8).

* Students may not understand that the compared fractions relate to the same whole. * Students may not understand that larger denominators yield smaller parts of a whole; the smaller denominators yield larger parts of a whole. * Students may not understand that creating an equivalent fraction does not change the fractional value. * Students may not view the comparison statement 3/8 < 3/6 is the same as 3/6 > 3/8. * Students that rely on a trick to determine directionality (i.e. the alligators mouth eats the bigger number), may not be able to read comparison symbols

The SE includes specificity for the number of fractions a student compares. When paired with 4.1D, students may compare fractions using concrete and pictorial models. The SE builds on 3.3H, where students compare two fractions having the same numerator or denominator.

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correctly. It is important that students read the comparison statement from left to right.

4.3E: represent and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations

In adherence to the standard, instruction is limited to the addition and subtraction of fractions with like denominators. In conjunction with 4.3B, students continue to use manipulatives and pictorial models to demonstrate their concrete understanding of addition/subtraction of fractions with like denominators. Instruction will move to the use of number lines (i.e. 5/4 + ¼ can be represented beginning with a point on 5/4 then modeling the movement of one-fourth to the right, landing on 6/4 or 1 ½ on the number line) and properties of operations (i.e. 5/4 + 1/4 = (4/4 + 1/4) + 1/4 = 4/4 + (1/4 + 1/4) = 1 + 2/4 = 1 2/4 or 1 ½).

* Students may want to add the numerators and denominators. * Students may not understand the adding and subtracting of fractions as joining and separating parts of the same whole.

Objects that build to the number line include fraction strips and fraction bars and other linear fraction models. Pictorial models include sketches of the linear fraction models and strip diagrams. Properties of operations with the addition and subtraction of fractions with equal denominators connects to the decomposing of fractions included in 4.B. Improper fractions and mixed numbers may be included. For example, 7/8 + 11/8 could be thought of as 7/8 + (1/8 + 10/8) = (7/8 + 1/8) + (8/8 + 2/8) = 8/8 + 8/8 + 2/8= 2 2/8= 2 ¼..

4.3F: evaluate the reasonableness of sums and differences of fractions using benchmark fractions 0, 1/4, 1/2, 3/4, and 1, referring to the same whole

Although 4.3E limits the “solving” of addition and subtraction of fractions to that of like denominators, this standard addresses the ability to use benchmark fractions in “evaluating” the reasonableness of sums/difference of fractions that may or may not have like denominators. Instruction should include evaluating whether a given answer to unlike denominator problem is reasonable (i.e. If Marco has a water bottle filled with 16 ounces of water and he drank 4 1/6 ounces before lunch and 6 7/8 ounces after lunch, would it be reasonable to say he had about 2 ounces of water left in the bottle at the end of the day?). Students

Through the use of benchmarking of fractions, students will be able to estimate prior to solving problems and determine if a solution is reasonable. 4.3E: represent and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations 5.3K add and subtract positive rational numbers fluently

For example, when estimating 7/8 + 2/8, one might estimate 7/8 as 1 and 2/8 as ¼. The estimated sum would be 1 ¼.

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employ fractional benchmarks to determine reasonableness (i.e. When referencing 4 1/6 ounces of water, 1/6 is less than half which means Marco drank about 4 ounces in the morning. When referencing 6 7/8 ounces of water, 7/8 is more than half which means Marco drank about 7 ounces after lunch; Marco drank a total of 11 ounces of water which means there is about 5 ounces of water left in the bottle at the end of the day). Instruction should begin with students visually representing the fractional amounts with concrete objects, pictorial models, or the use of the number line to observe the positioning being closer to 0, ¼. ½, ¾, or 1. To support students with benchmarking to the nearest half and/or fourth, have them identify the total number of parts that make up a whole (i.e. in the fraction 3/8, eight is the denominator), determine half that amount of the denominator (i.e. half of 8 is 4; 3/8 is less than half) and determine a fourth by taking half of half (i.e. half of 8 is 4 and half of 4 is 2; 3/8 is more than ¼).


In adherence to the standard, instruction is limited to representing decimals to the tenths and hundredths on a number line. Students will utilize number lines containing intervals that are divided equally between zero and one whole. Students will be asked to represent a given decimal/fractional value on a number line.

Locating fractional and decimal values as a specific point on a number line will support students in comparing/ordering decimals in grade 5. When students identify decimal and fractional values on a number line, they will begin to relate the equivalent values of tenths (0.10; 1/10) and hundredths (0.01; 1/100) as they fall on the same point on the number line. 4.2G: relate decimals to fractions that name tenths and hundredths 5.2B compare and order two decimals to thousandths and represent comparisons using symbols >, <, or =

The fractions that may be represented are of the form a/b, where a and b are whole numbers and b>0, including when a>b. This aligns to 4.3A. When used in conjunction with 4.3C, decimals can be developed from fractions that have equivalent fractions with denominators of ten or one hundred such as ¼=0.25 or 2/5=0.4.

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Course: Grade 4 Math Bundle 6: Geometry and Measurement Dates: February 13th-April 7th (34 days) TEKS

4.5C: use models to determine the formulas for the perimeter of a rectangle (l + w + l + w or 2l + 2w), including the special form for perimeter of a square (4s) and the area of a rectangle (l x w) 4.5D: solve problems related to perimeter and area of rectangles where dimensions are whole numbers 4.8A: identify relative sizes of measurement units within the customary and metric systems 4.8B: convert measurements within the same measurement system, customary or metric, from a smaller unit into a larger unit or a larger unit into a smaller unit when given other equivalent measures represented in a table 4.8C: solve problems that deal with measurements of length, intervals of time, liquid volumes, mass, and money using addition, subtraction, multiplication, or division as appropriate 4.6A: identify points, lines, line segments, rays, angles, and perpendicular and parallel lines 4.6B: identify and draw one or more lines of symmetry, if they exist, for a two-dimensional figure 4.6C: apply knowledge of right angles to identify acute, right, and obtuse triangles 4.6D: classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size 4.7A: illustrate the measure of an angle as the part of a circle whose center is at the vertex of the angle that is "cut out" by the rays of the angle. Angle measures are limited to whole numbers 4.7B: illustrate degrees as the units used to measure an angle, where 1/360 of any circle is one degree and an angle that "cuts" n/360 out of any circle whose center is at the angle's vertex has a measure of n degrees. Angle measures are limited to whole numbers 4.7C: determine the approximate measures of angles in degrees to the nearest whole number using a protractor 4.7D: draw an angle with a given measure 4.7E: determine the measure of an unknown angle formed by two non-overlapping adjacent angles given one or both angle measures

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ELPS Learning Strategies 1A: use prior knowledge and experiences to understand meanings in English Listening 2C: learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions 2I: demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs Speaking 3D: speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency 3E: share information in cooperative learning interactions 3F: ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments 3G: express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics Reading 4C: develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials 4D: use pre-reading supports such as graphic organizers, illustrations, and pre-taught topic-related vocabulary and other pre-reading activities to enhance comprehension of written text 4F: use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language

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Vocabulary

Unit Vocabulary 1/360 of a circle Division Mass Part of a circle Sides Acute angle Equiangular Measurement units Perimeter Square Acute triangle Equilateral Measurements Perpendicular lines Square units Addition Equivalent measures Metric Points Subtraction Adjacent angles Formulas Money Polygon Symmetry Angle measure Geometric attributes Multiplication Protractor Table Angles Inner scale Non-overlapping Quadrilateral Time Area Isosceles Number sentence/ equation Rays Trapezoid Center Length Obtuse angle Rectangle Two-dimensional Center mark Line Obtuse triangle Relative size Units Congruent Line of symmetry Outer scale Rhombus Vertex/vertices Customary Line segment Parallel lines Right angle Width Degree Liquid volume Parallelogram Right triangle Zero edge Dimensions

Cognitive Complexity Verbs: represent, use, determine, solve, identify, convert, apply, illustrate, classify, draw Academic Vocabulary by Standard: 4.5C: area, dimensions, formulas, length, perimeter, rectangle, square, width 4.5D: area, dimensions, length, perimeter, rectangles, square units, width, units 4.6A: angles, geometric attributes, lines, line segments, parallel lines, perpendicular lines, points, rays 4.6B: congruent, lines of symmetry, symmetry, two-dimensional 4.6C: acute angles, acute triangles, obtuse angles, obtuse triangles, right angles, right triangles 4.6D: angles, attributes, parallel lines, perpendicular lines, polygons, rhombus, parallelogram, trapezoid, rectangle, square, quadrilateral, sides, two-dimensional, vertex/vertices, isosceles, equilateral, equiangular, acute, right, obtuse 4.7A: center, measure of an angle, part of a circle, rays, vertex of an angle 4.7B: angle, vertex of an angle, degrees, 1/360 of a circle, one degree 4.7C: angle vertex, angle, center mark, degrees, inner scale, line, outer scale, protractor, ray, zero edge 4.7D: angle, angle vertex, center mark, degrees, inner scale, outer scale, protractor, zero edge 4.7E: angle, adjacent angles, degrees, non-overlapping 4.8A: customary, length, liquid volume, mass, measurement units, metric, relative size 4.8B: customary, equivalent measures, measurements, metric, table 4.8C: addition, division, length, liquid volume, mass, measurement, money, multiplication, number sentence/ equation, perimeter, subtraction, time

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AngLegs Pattern Blocks Power Polygons Protractors Geoboards Measuring Tape Yard Sticks/ Meter Sticks Rulers Centimeter Cubes Color Tiles Clocks Grid Paper Coins/Money

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3.7B determine the perimeter of a polygon or a missing length when given perimeter and remaining side lengths in problems

4.5C: use models to determine the formulas for the perimeter of a rectangle (l + w + l + w or 2l + 2w), including the special form for perimeter of a square (4s) and the area of a rectangle (l x w)

5.4G use concrete objects and pictorial models to develop the formulas for the volume of a rectangular prism, including the special form for a cube (V = l x w x h, V = s x s x s, and V = Bah) 6.8B model area formulas for parallelograms, trapezoids, and triangles by decomposing and rearranging parts of these shapes

3.7B determine the perimeter of a polygon or a missing length when given perimeter and remaining side lengths in problems

4.5D: solve problems related to perimeter and area of rectangles where dimensions are whole numbers

5.4H represent and solve problems related to perimeter and/or area and related to volume 6.8C write equations that represent problems related to the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers 6.8D determine solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers

4.8A: identify relative sizes of measurement units within the customary and metric systems

5.7A solve problems by calculating conversions within a measurement system, customary or metric 6.4H convert units within a measurement system, including the use of proportions and unit rates

4.8B: convert measurements within the same measurement system, customary or metric, from a smaller unit into a larger unit or a larger unit into a smaller unit when given other equivalent measures represented in a table


2.9G read and write time to the nearest one-minute increment using analog and digital clocks and distinguish between a.m. and p.m. 2.9D determine the length of an object to the nearest marked unit using rulers, yardsticks, meter sticks, or measuring tapes 2.9A find the length of objects using concrete models for standard units of length 2.9B describe the inverse relationship between the size of the unit and the number of units needed to equal the length of an object 2.9E determine a solution to a problem involving

4.8C: solve problems that deal with measurements of length, intervals of time, liquid volumes, mass, and money using addition, subtraction, multiplication, or division as appropriate


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length, including estimating lengths 3.7B determine the perimeter of a polygon or a missing length when given perimeter and remaining side lengths in problems 3.7C determine the solutions to problems involving addition and subtraction of time intervals in minutes using pictorial models or tools such as a 15-minute event plus a 30-minute event equals 45 minutes 3.7D determine when it is appropriate to use measurements of liquid volume (capacity) or weight 3.7E determine liquid volume (capacity) or weight using appropriate units and tools 2.8A create two-dimensional shapes based on given attributes, including number of sides and vertices 3.6B use attributes to recognize rhombuses, parallelograms, trapezoids, rectangles, and squares as examples of quadrilaterals and draw examples of quadrilaterals that do not belong to any of these subcategories

4.6A: identify points, lines, line segments, rays, angles, and perpendicular and parallel lines

5.5A classify two-dimensional figures in a hierarchy of sets and subsets using graphic organizers based on their attributes and properties

2.8A create two-dimensional shapes based on given attributes, including number of sides and vertices 3.6B use attributes to recognize rhombuses, parallelograms, trapezoids, rectangles, and squares as examples of quadrilaterals and draw examples of quadrilaterals that do not belong to any of these subcategories

4.6B: identify and draw one or more lines of symmetry, if they exist, for a two-dimensional figure

2.8A create two-dimensional shapes based on given attributes, including number of sides and vertices 3.6B use attributes to recognize rhombuses, parallelograms, trapezoids, rectangles, and squares as examples of quadrilaterals and draw examples of quadrilaterals that do not belong to any of these subcategories

4.6C: apply knowledge of right angles to identify acute, right, and obtuse triangles


2.8C classify and sort polygons with 12 or fewer sides according to attributes, including identifying the number of sides and number of vertices 3.6A classify and sort two- and three-dimensional figures, including cones, cylinders, spheres, triangular and rectangular prisms, and cubes, based on attributes using formal geometric language

4.6D: classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size


4.7A: illustrate the measure of an angle as the part of a circle whose center is at the vertex of the angle that is "cut out" by the rays of the angle. Angle measures are limited to whole numbers

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4.7B: illustrate degrees as the units used to measure an angle, where 1/360 of any circle is one degree and an angle that "cuts" n/360 out of any circle whose center is at the angle's vertex has a measure of n degrees. Angle measures are limited to whole numbers

4.7C: determine the approximate measures of angles in degrees to the nearest whole number using a protractor

4.7D: draw an angle with a given measure 4.7E: determine the measure of an unknown angle

formed by two non-overlapping adjacent angles given one or both angle measures

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4.5C: use models to determine the formulas for the perimeter of a rectangle (l + w + l + w or 2l + 2w), including the special form for perimeter of a square (4s) and the area of a rectangle (l x w)

In adherence to the standard, students should manipulate concrete objects and associate those actions to their respective formulas (i.e. building a 4 by 6 array with color tiles, two side lengths of 4 and two side lengths of 6 yield a perimeter of 4 + 4 + 6 + 6 = 20 or (4 x 2) + (6 x 2) = 8 + 12 = 20). Instruction should connect how a square is also a rectangle and the formula for a rectangle can be used to solve for the perimeter of a square (i.e. building a 6 by 6 array with color tiles yields a perimeter of 6 + 6 + 6 + 6 = 24 or (6 x 2) + (6 x 2) = 24). As a square is a special type of rectangle where all four sides are equal in length, one could multiply its side length by four; hence, P=4s (i.e. building a 6 by 6 array yields a perimeter of 6 x 4 = 24).

As students use models to develop the conceptual understanding of perimeter, this supporting standard builds on a student’s flexibility with the various formula representations. 4.5D: solve problems related to perimeter and area of rectangles where dimensions are whole numbers

Models may include manipulatives and grids.

4.5D: solve problems related to perimeter and area of rectangles where dimensions are whole numbers

Instruction should merge the concepts of area and perimeter to ensure understanding of the concepts (i.e. perimeter is measuring the length around an object; area measuring the space inside an object). A variety of examples for students to understand the difference between area and perimeter should be applied such as the amount of paper needed to cover the bulletin board vs. the amount of border needed to decorate around the bulletin board. Be sure to vary the context of the problems. Given the dimensions of the rectangle, determine the area and perimeter; given the perimeter of a square, determine the area; given the area of a square, determine the perimeter; given the length of one side and the perimeter, determine the area; given the length of one side and the area,

* Through the lens of a real world problem, students confuse when to apply the use of area vs. perimeter. * Students may not apply the correct unit of measure when reflecting area (i.e. square inches) vs. perimeter (i.e. inches).

Problems may include situations where perimeter or area is known and the student is asked to determine a missing dimension.

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determine the perimeter. It is important for students to understand how linear measurements like perimeter are recorded in units and area is reflected in square units.

4.8A: identify relative sizes of measurement units within the customary and metric systems

In measurement, students need a point of reference to identify the relative size of measurement units. Benchmarks need to be provided for both customary and metric (i.e. and inch is about the length of the bend of my index finger, a centimeter is the length of a unit cube in the base 10 blocks, a foot is the length of a ruler, a decimeter is the length of a ten rod in base ten blocks). Relative size needs to be applied to length (i.e. an inch is about the length of the bend of my index finger), liquid volume (i.e. a gallon can relate to a jug of milk), and mass (one M&M candy is approximately one gram). This understanding will allow students to better estimate and/or apply reasonableness to solutions.

Measurement conversions may be a component of solving problems of length, liquid volume, and mass. Therefore, identifying the relative size of various customary and metric measures will support estimation and reasonableness of solutions. 4.8C: solve problems that deal with measurements of length, intervals of time, liquid volumes, mass, and money using addition, subtraction, multiplication, or division as appropriate

When paired with 4.1C, students may use tools such as rulers to solve problems related to identifying relative sizes of measurement units. In this SE, measurement units include units of length, capacity, and weight within the customary system. In this SE, measurement units include units of length, capacity, and weight within the metric system. Mass and weight are related, but do not have the same measurements. Mass is the quantity of matter. Weight is the amount of force exerted by gravity for a given amount of mass.

4.8B: convert measurements within the same measurement system, customary or metric, from a smaller unit into a larger unit or a larger unit into a smaller unit when given other equivalent measures represented in a table

In accordance to the standard, students will apply measurement conversions through the use of a table. In conjunction with 4.5B, students should employ the use of the process column (a.k.a. what’s my rule) to better understand how units convert. Instruction should include both conversions from a smaller to a larger unit and a larger to a smaller unit.

Meters Process Kilometers 1 1 x 1,000 1,000 2 2 x 1,000 2,000

Measurement conversions may be a component in solving problems of length, liquid volume, and mass. Through the use of tables, students will build their flexibility in moving between units. 4.8C: solve problems that deal with measurements of length, intervals of time, liquid volumes, mass, and money using addition, subtraction,

Within this SE, the conversions will be “one-step” conversions from a smaller unit to a larger unit or from a larger unit to a smaller unit when students are provided other equivalent measures represented in a table as shown in the example below. Equivalent measures are shown in the table below.

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3 3 x 1,000 3,000 10 10 x

1,000 10,000

21 21 x 1,000

21,000

Ounces Process Pounds

16 16 ÷ 16 1 32 32 ÷ 16 2 64 64 ÷ 16 4 96 96 ÷ 16 6

160 160 ÷ 16 10 Instruction should include the conversion of customary (i.e. inches to feet; ounces to pounds) and metric (i.e. centimeters to meters; kilograms to grams). In conjunction with 4.8C, examples of measurement conversions should apply to length, liquid volume, and mass.

multiplication, or division as appropriate 4.5B: represent problems using an input-output table and numerical expressions to generate a number pattern that follows a given rule representing the relationship of the values in the resulting sequence and their position in the sequence

Feet Inches 2 24 3 36 4 48 6 72

Based on the information in the table, how many inches are in 7 feet?

4.8C: solve problems that deal with measurements of length, intervals of time, liquid volumes, mass, and money using addition, subtraction, multiplication, or division as appropriate

In conjunction with 4.5A, students will not only be asked to represent multi-step word problems in a number sentence/equation, but they will be asked to solve the problems. In accordance to the standard, examples should include measurement of length (i.e. perimeter), intervals of time (i.e. elapsed time), liquid volumes (i.e. filling/dispensing of containers), mass (i.e. comparison of weight), and money (i.e. comparison of money). Such problems may include the addition/ subtraction of fractions (see 4.3E/F), addition/ subtraction of decimals (see 4.4A), multiplication of four-digit by one-digit whole numbers (see 4.4D), and division of four-digit whole numbers divided by a one-digit whole number (see 4.4F). Instruction should include examples from both metric and customary units of measure.

* Students may not understand that calculating time is based on 60 minutes in an hour unlike our base-ten system when adding/ subtracting whole numbers. * Students may rely on “key words” to determine the operation instead of understanding the context of the problem.

When paired with 4.1C, students may use tools such as a clock with gears or a stopwatch to solve problems related to intervals of time. Students are expected to solve problems involving intervals of time, or elapsed time, without tools. This SE includes problems that include intervals of time and money. Specificity has been added for capacity and measures of liquid volume. Mass is typically taught with the metric system. The customary unit for mass is the “stone,” although this is rarely taught. Weight is not part of this SE.

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Students may be asked to solve problems to combined or separate measures of mass. Students are not expected to calculate mass. The calculation of mass is a grade 8 science skill. Multiplication or division problems that deal with money are limited to amounts expressed as cents or dollars with no decimal values as decimal multiplication and division are in the TEKS for grade 5.

4.6A: identify points, lines, line segments, rays, angles, and perpendicular and parallel lines

In conjunction with 4.6D, students will need to understand the geometric attributes outlined in this standard in order to classify two-dimensional figures. In accordance to the TEKS, students need to able to not only identify, but analyze the attributes of the various geometric terms. (See table below.)

Students are expected to identify points, lines, line segments, rays, angles, and parallel and perpendicular lines. Students may indicate that two lines are perpendicular with the symbol for a right angle.

Students may indicate that two lines are parallel using chevrons or arrows. If two sets of lines are present, students may differentiate based on the number of chevrons or arrows.

When paired with 4.1D, students describe these figures using concrete and pictorial models.

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4.6B: identify and draw one or more lines of symmetry, if they exist, for a two-dimensional figure

In adherence to the standard, students are to not only identify, but physically draw lines of symmetry. The standard limits the figures to two-dimensional; however, instruction should include regular and irregular two-dimensional shapes. Encourage students to identify/ draw more than one line of symmetry should more exist. As outlines in the TEKS, students need to develop generalizations about lines of symmetry (i.e. lines of symmetry divide a figure into two congruent parts; corresponding points on each half are equidistant from the line of symmetry).

To verify symmetry with a reflection, one must first identify the line(s) of symmetry. The revised SE addresses the identification of the lines of symmetry. When paired with 4.1E and 4.1F, students use the line(s) of symmetry as line(s) of reflection to verify the placement of the line of symmetry.

4.6C: apply knowledge of right angles to identify acute, right, and obtuse triangles

In accordance with 4.6A, students identified perpendicular lines and described how the angles formed are 90 degrees, or right angles. As students begin measuring angles (see 4.7C/D), they can begin to classify angles into three categories: acute (less than 90 degrees), right (exactly 90 degrees), and obtuse (greater than 90 degrees). This knowledge of angle classification can then define various triangles: acute triangle (all angles measure less than 90 degrees), right triangle (one angle measures 90 degrees), and obtuse triangle (one angle measures more than 90 degrees).

Students are expected to use a right, or 90 degree angle as a benchmark to identify acute, right, and obtuse angles. Triangles are named according to their largest angle.

4.6D: classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size

Students will compare/ contrast the attributes (i.e. number of vertices, sides, angles) and absence and presence of various properties (i.e. parallel lines, perpendicular lines, angle sizes) of various two-dimensional figures. In adherence to the TEKS, students will be able to better classify various two-dimensional figures by developing generalizations about their properties such as all parallelograms can be called a quadrilateral because they have four sides and four angles, but cannot all be called a rectangle because not all parallelograms have four right angles; a trapezoid can be

Classification is part of defining the essential attributes of a geometric figure. Students are expected to classify all 4-sided figures with opposite sides parallel and four right angles from adjacent sides that are perpendicular as rectangles. Isosceles is an adjective that indicates that two sides of a polygon have the same measure. Examples include isosceles triangles and isosceles trapezoids.

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called a quadrilateral because it has four sides and four angles, but cannot be called a parallelogram because it only has one set of parallel sides, etc.

Right is an adjective that indicates that a polygon has a ninety degree angle. Examples include right triangles and right trapezoids. Equilateral is an adjective that indicates that indicates that all sides of a polygon have the same length. Equiangular is an adjective that indicates that all angles in a polygon have the same measure. Students may indicate that two lines are perpendicular with the symbol for a right angle.

Line segment is perpendicular to line segment

Students may indicate that two lines are parallel using chevrons or arrows. If two sets of lines are present, students may differentiate based on the number of chevrons or arrows.

4.7A: illustrate the measure of an angle as the part of a circle whose center is at the vertex of the angle that is "cut out" by the rays of the angle. Angle measures are limited

In conjunction with 4.6A, students should identify the point in the middle of the circle that would be equal distance from that point to any point on the circle. This point would be identified as the center of

This SE lays the foundation for work with central angles in G12 and radian measures in P4.

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to whole numbers the circle. The center of the circle will serve as the vertex of two rays. The center of the circle will serve as the common endpoint of two rays (or as the vertex of an angle). The part of the circle that is “cut out” by the rays is identified as the angle. In adherence to the standard, students should be able to illustrate such understanding.

4.7B: illustrate degrees as the units used to measure an angle, where 1/360 of any circle is one degree and an angle that "cuts" n/360 out of any circle whose center is at the angle's vertex has a measure of n degrees. Angle measures are limited to whole numbers

Just like length is measured in linear units, time is measured in seconds/minutes/ hours, weight can be measured in pounds/ ounces; angles have their own measurement units called degrees. Instruction could use the example of a 360 on a skateboard related to 360 degrees in a circle (i.e. a skateboarder turns a complete circle in air around on the skateboard to perform a 360; the turn all the way around a circle measures 360 degrees). The vertex of the angle is the center of a circle (i.e. the point on which the skateboarder begins his rotation). Relating how far a rider turns his skateboard may support students with the understanding. In adherence to the standard, students must illustrate how each degree represents 1/360 of a circle; therefore, a 180 on a skateboard would actually represent 180/360 or ½ a turn of a circle or 180 degrees. Should a skateboarder only be able to turn his board a 90 degree angle means he/she completed a 90/360 turn or ¼ of a complete circle.

This SE leads to the development of radian measures in P4B.

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4.7C: determine the approximate measures of angles in degrees to the nearest whole number using a protractor

Just like a ruler can measure length in inches or a clock can measure time in minutes/ hours or a scale can measure weight in pounds/ ounces, a protractor can measure angles in degrees. Instruction should identify the different components of a protractor (i.e. inner scale, outer scale, center mark, zero edge). In conjunction with 4.6C, students should first identify a given angle as acute, right, or obtuse and then estimate an angle measurement prior to actually measuring. Instruction should model how to place the center of the protractor on the vertex of the angle and align the zero edge of the protractor with one of the rays of the angle. Where the other ray crosses the protractor determines the measurement of the angle. Depending if the given angle was previously identified as an acute, right, or obtuse angle will determine whether to record the inner or outer angle measure reflected on the protractor. Students should notice how the protractor represents a half of a circle or 180 degrees. Just like a ruler will only measure lengths that are 12 inches, a protractor will only allow you to measure angles of 180 degrees or less. The composing (i.e. an image of a 260 degree angle would have to be decomposed into two separate angles with each angle measurement added together) or decomposing (i.e. the measure of the inner angle could be subtracted from 360 degrees) of angles would have to be applied to determine the measurement of an angle measuring more than 180 degrees. Instruction should also include the use of images of a protractor and a given angle to determine the measure of the angle. Examples may or may not align to the zero edge of the protractor (i.e. an image of a protractor reflecting one ray aligning

The angle measurements do not necessarily have to begin with 0 degrees. For example:

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with the 50 degree unit of measure and the other ray aligning with the 130 degree unit of measure; 130 degrees – 50 degrees = 80 degrees; angle measures 80 degrees).

4.7D: draw an angle with a given measure

In conjunction with 4.7C, instruction should extend from the measuring of angles using a protractor to drawing the angle. Instruction should begin with students identifying whether the given angle measurement is acute or obtuse. They should move on to drawing a ray, aligning the center of the protractor to what will become the vertex of the angle, aligning the zero edge of the protractor to the drawn edge, marking the given angle measurement (reminding students to refer to the appropriate unit marks for acute/ obtuse angles), and drawing a ray from the center of the angle to the indicated mark.

When paired with 4.1C, students may use tolls such as protractors to solve problems related to the measurement of angles.

4.7E: determine the measure of an unknown angle formed by two non-overlapping adjacent angles given one or both angle measures

Instruction should define and provide examples and non-examples of non-overlapping adjacent angles. Adjacent angles are to be defined as two angles that do not overlap that share a common ray and a common point (vertex).Students should begin to visualize how two adjacent angles will yield a larger angle.

Non-overlapping adjacent angles can include more than two angles.

As students begin to determine the measure of an unknown angle, be sure to

The two non-overlapping angles may include complementary and supplementary angles.

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vary the context of the problems (i.e. two angle measurements given for the two adjacent angles, determine the measurement of the larger angle; given the measurement of the larger angle and one of the adjacent angles determine the measurement of the other adjacent angle. Given that the larger angle is a right angle and the measure of one of the adjacent angles, determine the measure of the other adjacent angle. Given the larger angle represents a half of a circle and the measures of one of the adjacent angles, determine the measure of the other adjacent angle). This understanding will begin to lay the foundation for supplementary and complementary angle measurements.

Term Example Geometric Properties

Point Specifies location; does not have length, width, or depth Line Has only one dimension of length; does not have width or depth; connects two points and continues on in

both directions Line Segment Has only one dimension of length; does not have width or depth; two points (called endpoints) and all

points between Ray Contains a point (endpoint) and extends one direction Angle

A figure formed by two rays that share the same endpoint (vertex)

Perpendicular Lines

Two lines that intersect forming a 90 (right) angle

Parallel Lines

Two lines that never intersect

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Course: Grade 4 Math Bundle 7: Data Analysis

Dates: April 10th- April 21st (8 days)

TEKS 4.9A: represent data on a frequency table, dot plot, or stem-and-leaf plot marked with whole numbers and fractions 4.9B: solve one- and two-step problems using data in whole number, decimal, and fraction form in a frequency table, dot plot, or stem-and-leaf plot

ELPS Listening 2E: use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language 2I: demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs Speaking 3G: express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics Reading 4D: use pre-reading supports such as graphic organizers, illustrations, and pre-taught topic-related vocabulary and other pre-reading activities to enhance comprehension of written text 4F: use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language

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Vocabulary

Unit Vocabulary Categorical data Decimal Frequency table Labels Stem-and-leaf plot Categories Dot plot Graph titles Scaled intervals Whole number Data Fraction

Cognitive Complexity Verbs: represent, solve Academic Vocabulary by Standard: 4.9A: categories, data, dot plot, frequency table, graph titles, labels, scaled intervals, stem-and-leaf plot 4.9B: categorical data, decimal, dot plot, fraction, frequency table, stem-and-leaf plot, whole number

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2.10B organize a collection of data with up to four categories using pictographs and bar graphs with intervals of one or more 3.8A summarize a data set with multiple categories using a frequency table, dot plot, pictograph, or bar graph with scaled intervals

4.9A: represent data on a frequency table, dot plot, or stem-and-leaf plot marked with whole numbers and fractions

5.9A represent categorical data with bar graphs or frequency tables and numerical data, including data sets of measurements in fractions or decimals, with dot plots or stem-and-leaf plots 6.12A represent numeric data graphically, including dot plots, stem-and-leaf plots, histograms, and box plots

2.10C write and solve one-step word problems involving addition or subtraction using data represented within pictographs and bar graphs with intervals of one 3.8B solve one- and two-step problems using categorical data represented with a frequency table, dot plot, pictograph, or bar graph with scaled intervals

4.9B: solve one- and two-step problems using data in whole number, decimal, and fraction form in a frequency table, dot plot, or stem-and-leaf plot

5.9C solve one- and two-step problems using data from a frequency table, dot plot, bar graph, stem-and-leaf plot, or scatterplot 6.13A interpret numeric data summarized in dot plots, stem-and-leaf plots, histograms, and box plots

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Instructional Implications

Distractor Factors Supporting Readiness Standards


4.9A: represent data on a frequency table, dot plot, or stem-and-leaf plot marked with whole numbers and fractions

According to the TEKS, students need to collect, organize, and display their own data. Personalizing such activities will allow students to understand and make more sense of the data and summarize more appropriately. Instruction needs to include multiple categories (i.e. extend the survey question, “Do you like cats or dogs?” to “What is your favorite animal?”). In accordance with the standard, data should be represented on a frequency table, dot plot, or stem-and -leaf plot. Instruction should begin with frequency tables and dot plots, as students will have had previous experience with those representations (see 3.8A). In adherence to the standard, those representations need to extend to include the use of fractional units. Students will need more support with the development of stem-and-leaf plots. This type of graph will distribute the data by separating one place value from the other or the parts from the whole when working with fractions.

stem leaf 9 0 4 8 3 4 5 7 9 7 0 2 2 9 6 5 8 9 5 3

* In using the stem-and-leaf plot, students may have difficulty determining what values represent the stem and leaf. * In using the stem-and-leaf plot, students may not represent repeated values (i.e. math grades: 84, 56, 92, 84, 87, 91; students may only reflect 84 one time on the graph when the value 84 needs to be recorded twice). * When representing the same set of data on all three types of graphs, students may interpret the data as different because of the difference in the visual representations. * When representing the same set of data vertically and horizontally, students may interpret the data as different because of the difference in visual representations.

A dot plot may be used to represent frequencies. A number line may be used for counts related to numbers. A line labeled with categories may be used as well if the context requires. Dots are recorded vertically above the number line to indicate frequencies. Dots may represent one count or multiple counts if so noted. Students begin work with dot plots in grade 3. A stem-and-leaf plot organizes data in numerical order according to place value. The stem represents the place values preceding the last digit(s). The leaves provide the frequency counts for the range of numbers included in that row of the stem-and-leaf plot.

The use of stem-and-leaf plots begins in grade 4. Data may be in the form of whole numbers and/or fractions. Students begin work with stem-and-leaf plots in grade 4. The fractions 1, 1 ¼, 1 ¼, 1 ½ could be represented 1I 0/4 ¼ ¼ ½ Stem values should be consecutive values. It is possible to have stem values without associated leaf values.

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8 l 4 = 84 The larger place value or the whole of a fraction is identified as the stem; the smaller place value or the part of the whole is called the leaf. Instruction should include representing the same set of data on all three types of graphs.

4.9B: solve one- and two-step problems using data in whole number, decimal, and fraction form in a frequency table, dot plot, or stem-and-leaf plot

Instruction should vary the context of the problems being asked of the students (i.e. joining, separating, comparing, and distance). In adherence to the standard, whole number/ decimal/ and fractional data should be represented. In conjunction with 4.9A, as students have graphed the same data set on three different types of graphs, students could then solve the same problem using the different graph representations to model their flexibility in moving among the different types of graphs.

This supporting standard merges the use of whole numbers, decimals, and fractions with various graph representations. Through the interpretation of data on a graph, students should be able to apply their ability to solve addition/ subtraction of fractions and decimals and multiplication/ division of whole numbers. 4.9A: represent data on a frequency table, dot plot, or stem-and-leaf plot marked with whole numbers and fractions

A frequency table shows how often an item, a number, or a range of numbers occurs. Tallies and counts are used to record frequencies. Students begin work with frequency tables in grade 3. Students begin work with dot plots in grade 3. Students begin work with stem-and-leaf plots in grade 4. Arrows are only required when a number line is used for a dot plot.

A dot plot of categorical data is called a “Cleveland” plot.

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Course: Grade 4 Math Bundle 8: Personal Financial Literacy

Dates: April 24th – April 28th (5 days)

TEKS 4.10A: distinguish between fixed and variable expenses 4.10B: calculate profit in a given situation 4.10C: compare the advantages and disadvantages of various savings options 4.10D: describe how to allocate a weekly allowance among spending; saving, including for college; and sharing 4.10E: describe the basic purpose of financial institutions, including keeping money safe, borrowing money, and lending

ELPS Listening 2C: learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions 2D: monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed 2I: demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs Speaking 3B: expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication 3D: speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency

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Vocabulary

Unit Vocabulary Advantages Cost Lending Saving Sharing Allowance Disadvantages Money market funds Savings accounts Spending Borrowing Financial institutions Profit Savings bonds Variable expenses Certificates of deposit Fixed expenses Revenue Savings options Weekly College savings

Cognitive Complexity Verbs: distinguish, calculate, compare, describe Academic Vocabulary by Standard: 4.10A: fixed expenses, variable expenses 4.10B: profit, revenue, cost 4.10C: advantages, disadvantages, savings options, savings accounts, certificates of deposit, money market funds, savings bonds 4.10D: allowance, college savings, saving, sharing, spending, weekly 4.10E: financial institutions, borrowing, lending

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3.9A explain the connection between human capital/labor and income

4.10A: distinguish between fixed and variable expenses 5.10A define income tax, payroll tax, sales tax, and property tax

2.11F differentiate between producers and consumers and calculate the cost to produce a simple item 3.9B describe the relationship between the availability or scarcity of resources and how that impacts cost

4.10B: calculate profit in a given situation

2.11A calculate how money saved can accumulate into a larger amount over time 3.9E list reasons to save and explain the benefit of a savings plan, including for college

4.10C: compare the advantages and disadvantages of various savings options

2.11C distinguish between a deposit and a withdrawal 4.10D: describe how to allocate a weekly allowance among spending; saving, including for college; and sharing

5.10D develop a system for keeping and using financial records 6.14A compare the features and costs of a checking account and a debit card offered by different local financial institutions 6.14C balance a check register that includes deposits, withdrawals, and transfers

2.11D identify examples of borrowing and distinguish between responsible and irresponsible borrowing 3.9D explain that credit is used when wants or needs exceed the ability to pay and that it is the borrower's responsibility to pay it back to the lender, usually with interest

4.10E: describe the basic purpose of financial institutions, including keeping money safe, borrowing money, and lending

5.10C identify the advantages and disadvantages of different methods of payment, including check, credit card, debit card, and electronic payments 6.14B distinguish between debit cards and credit card

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TEKS/Student Expectations Instructional Implications Supporting Readiness Standards TEA Supporting Information 4.10A: distinguish between fixed and variable expenses

Fixed expenses can be defined as those costs that do not change frequently, are consistent over time; costs that you do not have the ability to manipulate (i.e. rent, taxes). Variable expenses are those costs that change more frequently; costs that can be manipulated to be more or less (i.e. amount of money spent on gas, eating out, going to the movies). Instruction can merge content standards with the understanding of fixed and variable expenses.

Distinguishing between fixed and variable expenses will support one’s ability to manage their financial resources more effectively for a lifetime of financial security.

Fixed and variable expenses depend upon the situation. (See Interactive Math Glossary.) Fixed expenses are a set amount consistent from month to month. Variable expenses vary in cost from month to month.

4.10B: calculate profit in a given situation Instruction should define profit as the total amount earned from the sale of goods/services minus the amount spent (i.e. Natalie sold 5 charm bracelets for $20 each. She spent $25.50 on materials to make the bracelets. Natalie would have earned a profit of 474.50). Instruction can merge content standards with the understanding of profit in order to solve problems.

Calculating profit may support one’s ability to manage their financial resources more effectively for a lifetime of financial security.

Profit = revenue-cost

4.10C: compare the advantages and disadvantages of various savings options

Students need to understand that there are different types of savings options available (i.e. regular savings accounts, certificates of deposit, money market funds, and savings bonds). In adherence to the standard, students should compare the advantages and disadvantages to the different types of savings plans (i.e. regular savings accounts do not require a lot of money to invest and you can withdraw funds at any time; however, the interest earned is less than a cd or money market account).

Comparing various savings plan options will support one’s ability to manage their financial resources more effectively for a lifetime of financial security.

Saving options may include a piggy bank, and interest-bearing account, certificates of deposits, and bonds.

4.10D: describe how to allocate a weekly allowance among spending; saving, including for college; and sharing

In accordance with the standard, students need to understand how to manage their personal finances through the lens of a weekly allowance. Instruction should include a fair amount of spending, saving (including for college) and sharing.

Describing how to allocate various funds will support one’s ability to manage their financial resources more effectively for a lifetime of financial security.

When paired with 4.1G, the expectation is that students explain the value of the allocation.

4.10E: describe the basic purpose of financial institutions, including keeping money safe, borrowing money, and lending.

When paired with 4.1G, the expectation is that students explain the nature of financial institutions.

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Course: Grade 4 Math Bundle 12: STAAR Review and Testing

Dates: May 1st- May 8th (6 days)

TEKS -Review previously taught skills/TEKS -Address STAAR questions STEMS at varied DOK levels

Course: Grade 4 Math Bundle 13: Extended Learning

Dates: May 9th – June 1st (17 days)

TEKS Continued Problem Solving 4.4H: solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders 4.5A: represent multi-step problems involving the four operations with whole numbers using strip diagrams and equations with a letter standing for the unknown quantity Continued Number Sense 4.3E: represent and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations 4.3F: evaluate the reasonableness of sums and differences of fractions using benchmark fractions 0, 1/4, 1/2, 3/4, and 1, referring to the same whole 4.4A: add and subtract whole numbers and decimals to the hundredths place using the standard algorithm 4.4D: use strategies and algorithms, including the standard algorithm, to multiply up to a four-digit number by a one-digit number and to multiply a two-digit number by a two-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties 4.4F: use strategies and algorithms, including the standard algorithm, to divide up to a four-digit dividend by a one-digit divisor Project Based Learning -Introduce project-based learning activities with strong integration of TEKS that need extended exposure for mastery.

grade 4 mathematics curriculum document 2016-2017 · 1st 6-weeks addition breaking each number into...

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