Lead K – 9 IST and Supplemental Lesson Materials by Great Minds

Lead

Facilitating Successful Implementation of Eureka Math2TM K–9 A Story of Units® Story of Ratios® A Story of Functions® Supplemental Materials

Supplemental Materials

Lead: Facilitating Successful Implementation of Eureka Math2, K–9

Contents Exploring the Strategic Curriculum Design ................................................................................................... 3 Module and Topic: Grade 4 Module 3 Topic A ......................................................................................... 3 Lesson: Grade 4 Module 3 Lesson 1........................................................................................................ 17 Supporting Teacher Preparation ................................................................................................................. 32 Lesson: Grade 7 Module 3 Lesson 7........................................................................................................ 32 Topic: Grade 7 Module 3 Topic B ............................................................................................................ 50 Module: Grade 7 Module 3 ..................................................................................................................... 52 Credits ......................................................................................................................................................... 56 Works Cited ................................................................................................................................................. 56 Appendix A .................................................................................................................................................. 57 Implementation Support Tool (IST) ........................................................................................................ 57 Appendix B .................................................................................................................................................. 75 Scenarios ................................................................................................................................................. 75 Appendix C .................................................................................................................................................. 77 Implementation Benchmarks.................................................................................................................. 77

A Story of Units®

Fractional Units ▸ 4 TEACH

Module

1 2 3 4 5 6

Place Value Concepts for Addition and Subtraction

Place Value Concepts for Multiplication and Division

Multiplication and Division of Multi-Digit Numbers

Foundations for Fraction Operations

Place Value Concepts for Decimal Fractions

Angle Measurements and Plane Figures

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Before This Module

Overview

Grade 4 Module 2

Multiplication and Division of Multi-Digit Numbers

In module 2, students multiply two-digit numbers by one-digit numbers by using the distributive property and divide two- and three-digit numbers by one-digit numbers by using the break apart and distribute strategy. They rename the larger factor or total as tens and ones and then multiply or divide each part, as applicable. They use a place value chart, area model, and equations to represent the multiplication and division. Students express single customary units and mixed customary units of length in terms of smaller units. They show the conversions by using tape diagrams, number lines, and conversion tables and add or subtract to find the sum or difference of measurements. Students also identify factors and multiples of numbers within 100.

Topic A Multiplication and Division of Multiples of Tens, Hundreds, and Thousands Students multiply and divide multiples of 10, 100, and 1000 by focusing on place value units. They use place value disks and write equations in unit form to help them recognize that they can use familiar multiplication and division facts 2100 ÷ 3 = 21 hundreds ÷ 3 to find products and quotients. Application of the associative property helps students to rewrite two-factor multiplication = 7 hundreds expressions as three-factor expressions so, again, they can = 700 multiply by using familiar facts. To help prepare for multiplication of two-digit numbers by two-digit numbers, an area model is used to show that multiplying a multiple of 10 by a multiple of 10 results in a number with the unit of hundreds.

Topic B Division of Thousands, Hundreds, Tens, and Ones Students divide numbers of up to four digits by one-digit numbers. They draw an area model, represent the divisor as one side length, and compose the unknown side length by building up to the total. Students also represent the division on a place value chart. They decompose the totals into place value units, divide each unit, and record long division in vertical form alongside the place value chart to reinforce conceptual understanding. They recognize that, although the value of the unit is different, the process of dividing each unit remains the same. 4

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568 ÷ 4 = hundreds

tens

ones

1 4 5 - 4 1 - 1 -

4 0 6 0 6 6

2 0 0 8 0 8 0 8 8 0

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EUREKA MATH2

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Topic C Multiplication of up to Four-Digit Numbers by One-Digit Numbers Students apply the distributive property to multiply numbers of up to four digits by one-digit numbers. They break apart the larger factor by place value and multiply the number of each unit by the one-digit factor. They represent the multiplication by using place value charts, area models, and vertical form. Students record partial products in vertical form by recording each partial product separately and by recording them together on one line.

9 2

Topic D Multiplication of Two-Digit Numbers by Two-Digit Numbers Students apply the associative and distributive properties to multiply a two-digit number by a multiple of 10 and then progress to multiplying two-digit numbers by two-digit + numbers. Area models are used to represent the multiplication and to help students recognize how each factor is broken apart and multiplied. Students see that each part of one factor is multiplied by each part of the other factor. They record four partial products in the area model and in vertical form alongside the area model and then transition to recording two partial products in the same way. Students add the partial products to find the product.

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After This Module Grade 5 Modules 1 and 2 In grade 5 module 1, students multiply multi-digit whole numbers and develop fluency with the standard algorithm for multiplication. Students also divide with two-digit divisors and continue building conceptual understanding of multi-digit whole-number division. They find whole-number quotients and remainders. In module 2, students transition from finding whole-number quotients and remainders to fractional quotients.

Grade 5 Modules 1, 3, and 4 In modules 1, 3, and 4 of grade 5, students use multiplicative relationships to convert metric and customary units involving whole numbers, fractions, and decimals. In addition to expressing larger measurement units in terms of smaller units, they express smaller measurement units in terms of larger units.

Topic E Problem Solving with Measurement 40 lb – 14 lb 10 oz = 406 oz Students use multiplicative relationships to convert units of time and customary units of weight and 39 lb 16 oz 25 400 + 6 = 406 × 16 liquid volume to smaller units. They use conversion 16 oz – 10 oz = 6 oz 3 tables and number lines to express larger 120 39 lb – 14 lb = 25 lb measurement units in terms of smaller units and + 250 1 recognize that the smaller units are all multiples 400 of the same number. Students notice relationships in the conversion tables and use the tables to convert other amounts. Throughout the topic, students add and subtract mixed units by using different methods including the method of expressing larger units in terms of smaller units before adding or subtracting and the method of adding or subtracting like units, renaming as necessary.

Topic F Remainders, Estimating, and Problem Solving Students divide with numbers that result in whole-number quotients and remainders. They recognize the remainder as the amount remaining after finding a whole-number quotient, and they solve word problems that require interpretation of the whole-number quotient and remainder. Students estimate quotients by finding a multiple of the divisor that is close to the total and then dividing. They reason about the relationship between their estimate and the actual quotient and apply their thinking to assess the reasonableness of their answers to division word problems. Students use the four operations to solve multi-step word problems. They draw tape diagrams that represent the known and unknown information in the problem to help them find a solution path. After solving, students assess the reasonableness of their answers.

14 ÷ 3 14

Remainder of 2

14 = (3 × 4) + 2 6

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Contents Multiplication and Division of Multi-Digit Numbers Why . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Topic A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Multiplication and Division of Multiples of Tens, Hundreds, and Thousands Lesson 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Divide multiples of 100 and 1000.

Topic C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Multiplication of up to Four-Digit Numbers by One-Digit Numbers Lesson 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Apply place value strategies to multiply three-digit numbers by one-digit numbers.

Lesson 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Lesson 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Multiply by multiples of 100 and 1000.

Apply place value strategies to multiply four-digit numbers by one-digit numbers.

Lesson 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Multiply a two-digit multiple of 10 by a two-digit multiple of 10.

Represent multiplication by using partial products.

Topic B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Division of Thousands, Hundreds, Tens, and Ones Lesson 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Apply place value strategies to divide hundreds, tens, and ones.

Lesson 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Lesson 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Connect pictorial representations of division to long division.

Multiply by using various recording methods in vertical form.

Topic D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Multiplication of Two-Digit Numbers by Two-Digit Numbers Lesson 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Multiply two-digit numbers by two-digit multiples of 10. Apply place value strategies to multiply two-digit numbers by two-digit numbers.

Lesson 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 Multiply with four partial products.

Lesson 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Represent division by using partial quotients.

Lesson 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 Multiply with two partial products.

Lesson 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Choose and apply a method to divide multi-digit numbers.

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Lesson 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

Lesson 14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

Apply place value strategies to divide thousands, hundreds, tens, and ones.

Lesson 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Lesson 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 Apply the distributive property to multiply.

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Topic E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Problem Solving with Measurement

Resources

Lesson 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

Achievement Descriptors: Proficiency Indicators. . . . . . . . . . . . . . . . 460

Express units of time in terms of smaller units.

Lesson 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

Standards. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

Terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

Express customary measurements of weight in terms of smaller units.

Math Past. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

Lesson 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

Express customary measurements of liquid volume in terms of smaller units.

Topic F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 Remainders, Estimating, and Problem Solving

Works Cited. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 Credits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

Lesson 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Find whole-number quotients and remainders.

Lesson 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 Represent, estimate, and solve division word problems.

Lesson 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 Solve multi-step word problems and interpret remainders.

Lesson 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 Solve multi-step word problems and assess the reasonableness of solutions.

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Why Multiplication and Division of Multi-Digit Numbers Why does work with division come before multiplication in module 3? The work of module 3 builds on the work 4 304 300 of module 2. In module 2, students multiply 3 912 23 647 and divide with tens and ones. Division follows −900 × 62 × 3 multiplication and students relate the methods used 1 2 1 2 4 6 for division with those they use for multiplication 1 821 1 −12 + 1 280 (e.g., distributive property, area model). 1,9 4 1 1 0 1,4 2 6 Module 2 ends with a focus on factors and multiples. To continue the progression of learning from 912 ÷ 3 = 304 module 2 and provide an opportunity for application of the work with factors and multiples, division precedes multiplication in module 3. Students relate the methods used to divide with tens and ones to division of three- and four-digit numbers. Although the work focuses on division, students use their fluency with multiplication facts to help them divide. They then transition back to multiplication, first multiplying three- and four-digit numbers by one-digit numbers and then multiplying two-digit numbers by two-digit numbers.

Why do remainders come at the end of the module and not earlier in students’ work with division? Division resulting in whole-number quotients and remainders presents an added complexity to division, both conceptually and computationally. To provide ample time to refine students’ skills and conceptual understanding of both division and multiplication, remainders are introduced last in the module. Students find whole-number quotients and remainders and then immediately have opportunities to apply their new learning to situations and word problems in which the whole-number quotients and remainders are interpreted.

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1 40 9 37 5 - 360 15 - 9 6

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EUREKA MATH2

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What is the intent of using vertical form for multiplication and division? In grade 4, students multiply and divide by using strategies 708 3,5 6 8 based on place value and the properties of operations. This × 5 × 4 module uses the representations of place value charts, area 2 2 3 40 12 042 models, and vertical form because they are based on place +3 500 1 4,2 7 2 value understanding. The intent of vertical form is to provide 3,5 4 0 a written method to record the process of multiplication and division by using partial products and partial quotients. Vertical form can be more efficient than representing a problem with an area model or a place value chart. Because students mature in their understanding of multi-digit multiplication and division at different times, expect and accept a variety of representations. Fluency with the multiplication and division algorithm is not expected until grade 5 and grade 6, respectively.

Why is vertical form introduced alongside the place value chart for multiplication and division? Similar to what students experience 4,278 ÷ 3 = with addition and subtraction, thousands hundreds tens ones vertical form is introduced alongside the place value chart for multiplication and division to support conceptual understanding and the transition from a pictorial representation to a written representation. Each action represented in the place value chart (e.g., renaming units, adding or subtracting like units, distributing units, finding the total quantity of each unit) has a direct connection to a recording within vertical form. As students become proficient with recording in vertical form, they internalize the process and no longer require drawing on the place value chart to find the unknown or explain their work. Additionally, students not yet fluent with multiplication and division facts may find the place value chart helpful in keeping track of their calculations within vertical form. 10

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Achievement Descriptors: Overview Multiplication and Division of Multi-Digit Numbers Achievement Descriptors (ADs) are standards-aligned descriptions that detail what students should know and be able to do based on the instruction. ADs are written by using portions of various standards to form a clear, concise description of the work covered in each module. Each module has its own set of ADs, and the number of ADs varies by module. Taken together, the sets of module-level ADs describe what students should accomplish by the end of the year. ADs and their proficiency indicators support teachers with interpreting student work on • informal classroom observations, • data from other lesson-embedded formative assessments, • Exit Tickets, • Topic Quizzes, and • Module Assessments. This module contains the five ADs listed.

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EUREKA MATH2

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4.Mod3.AD1

4.Mod3.AD2

4.Mod3.AD3

4.Mod3.AD4

Solve multi-step word problems by using the four operations, including problems that require interpreting remainders in context, represent these problems by using equations, and assess reasonableness of answers.

Multiply whole numbers of up to four digits by one-digit whole numbers, and multiply 2 two-digit whole numbers.

Divide whole numbers of up to four digits by one-digit whole numbers.

Express larger units of time, customary units of weight, and liquid volumes in terms of a smaller unit by using tables.

4.OA.A.3

4.NBT.B.5

4.NBT.B.6

4.MD.A.1

4.Mod3.AD5

Solve word problems that require expressing measurements of larger units of time, customary units of weight, and liquid volumes, in terms of a smaller unit. 4.MD.A.2

The first page of each lesson identifies the ADs aligned with that lesson. Each AD may have up to three indicators, each aligned to a proficiency category (i.e., Partially Proficient, Proficient, Highly Proficient). While every AD has an indicator to describe Proficient performance, only select ADs have an indicator for Partially Proficient and/or Highly Proficient performance. An example of one of these ADs, along with its proficiency indicators, is shown here for reference. The complete set of this module’s ADs with proficiency indicators can be found in the Achievement Descriptors: Proficiency Indicators resource.

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ADs have the following parts: • AD Code: The code indicates the grade level and the module number and then lists the ADs in no particular order. For example, the first AD for grade 4 module 1 is coded as 4.Mod1.AD1. • AD Language: The language is crafted from standards and concisely describes what will be assessed. • AD Indicators: The indicators describe the precise expectations of the AD for the given proficiency category. • Related Standard: This identifies the standard or parts of standards from the Common Core State Standards that the AD addresses.

AD Code: Grade.Module.AD#

AD Language

EUREKA MATH2

4 ▸ M3

4.Mod3.AD2 Multiply whole numbers of up to four digits by one-digit whole numbers, and multiply 2 two-digit whole numbers.

Related Standard

RELATED CCSSM

4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Partially Proficient

Proficient

Highly Proficient

Complete or identify a representation of a multiplication calculation for whole numbers of up to four digits by one-digit whole numbers and 2 two-digit whole numbers.

Multiply whole numbers of up to four digits by one-digit whole numbers, and multiply 2 two-digit whole numbers.

Identify and explain an error in a multiplication calculation for whole numbers of up to four digits by one-digit whole numbers and 2 two-digit whole numbers.

Which model shows 3 × 1,078? A.

3 B.

3 C.

3 D.

1,000

3,000

210

1,000

700

3,000

2,100

1,000

300

210

Multiply.

Casey found 3 × 1,078. Her work is shown.

3 × 1,078 =

1,000

700

3,000

2,100

AD Indicators

3,000 + 2,100 + 24 = 5,124 Casey made a mistake. Explain the mistake and what she should do to correct it.

463

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Topic A Multiplication and Division of Multiples of Tens, Hundreds, and Thousands In topic A, students use familiar representations, strategies, and methods to multiply and divide multiples of 10, 100, and 1000 by one-digit numbers. Students also multiply a multiple of 10 by a multiple of 10, resulting in hundreds. The work of this topic extends the work of module 2 topic A, where students multiply and divide multiples of 10 by one-digit numbers. The topic begins with division. Students divide multiples of 10, 100, and 1000 by using place value disks and unit form to represent the total. They use familiar facts to divide and then name the quotient in standard form. By focusing on the place value of the units being divided, students recognize that although the units change, the familiar division fact remains the same. For example, 2100 ÷ 3 = 21 hundreds ÷ 3 = 7 hundreds = 700. Similarly, students then multiply multiples of 10, 100, and 1000 by a one-digit number by focusing on place value and by using place value disks and unit form. The associative property helps students connect multiplying in unit form to multiplying in standard form. Students break apart and regroup factors to find familiar facts. For example, by using the associative property, 4 × 6 hundreds = 24 hundreds is represented in standard form as 4 × 600 = 4 × (6 × 100) = (4 × 6) × 100 = 24 × 100 = 2400. Four-digit numbers are written without a comma to help students think of a number such as 2400 as 24 hundreds. The topic concludes with multiplication of a multiple of 10 by a multiple of 10. Arrays are drawn to show that multiplying tens by tens results in a different unit, hundreds. This is new learning for students and helps to prepare them for multiplying a two-digit number by a two-digit number in topic D. Equations are initially written in unit form. Students see that although each factor is named by its unit, they can use a familiar multiplication fact to help them multiply. For example, 60 × 50 = 6 tens × 5 tens = 30 hundreds = 3000. Equations are then written in standard form. In topic B, students apply their knowledge of dividing multiples of 10, 100, and 1000 to divide numbers of up to four digits by one-digit numbers.

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EUREKA MATH2

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Progression of Lessons Lesson 1

Lesson 2

Lesson 3

Divide multiples of 100 and 1000.

Multiply by multiples of 100 and 1000.

Multiply a two-digit multiple of 10 by a two-digit multiple of 10.

100

4 × 5 ones = 20 ones 6 ones ÷ 2 = 3 ones

4 × 5 tens = 20 tens 4 × 5 hundreds = 20 hundreds 4 × 5 thousands = 20 thousands

6 tens ÷ 2 = 3 tens

6 hundreds ÷ 2 = 3 hundreds

1,000 1,000 1,000

6 thousands ÷ 2 = 3 thousands

Representing multiplication with place value disks or multiplying by using unit form helps me see that if I know 2 × 6 = 12, then I know 2 × 6 hundreds = 12 hundreds. I can use the associative property to help me multiply when the equations are written in standard form. For example,

Arrays, area models, and unit form equations help me see that multiplying a multiple of 10 by a multiple of 10 results in a multiple of 100. I can multiply the number of tens in each factor by using a familiar multiplication fact to find the number of hundreds in the total.

4 × 500 = 4 × (5 × 100) = (4 × 5) × 100 = 2000.

1,000 1,000 1,000

Representing division with place value disks or dividing by using unit form helps me see that if I know 6 ÷ 2 = 3, then I know 6 hundreds ÷ 2 = 3 hundreds.

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LESSON 1

Divide multiples of 100 and 1000.

EUREKA MATH2

4 ▸ M3 ▸ TA ▸ Lesson 1

Name

Date

Divide. Use unit form to help you. a. 800 ÷ 2 =

hundreds ÷

hundreds

400

b. 1200 ÷ 4 =

300

= 12 hundreds ÷ 4

Lesson at a Glance Students divide multiples of 100 and 1000 by using place value disks and unit form and making connections to division facts. They select a strategy and solve a multiplicative comparison problem with division.

Key Question

= 3 hundreds

• How can thinking about place value units help us to divide multiples of 100 and 1000?

= 300

Achievement Descriptor 4.Mod3.AD3 Divide whole numbers of up to four digits by one-digit

whole numbers. (4.NBT.B.6)

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EUREKA MATH2

4 ▸ M3 ▸ TA ▸ Lesson 1

Agenda

Materials

Lesson Preparation

Fluency

Teacher

Gather at least 12 thousands disks, 12 hundreds disks, 12 tens disks, and 12 ones disks for each student and the teacher.

Launch Learn

10 min 5 min

35 min

• Represent Division with Place Value Disks • Use Unit Form to Divide

• Place value disks set • Teacher computer or device* • Projection device* • Teach book*

• Divide Multiples of 100

Students

• Problem Set

• Place value disks set

Land

10 min

• Dry-erase marker* • Learn book* • Pencil* • Personal whiteboard* • Personal whiteboard eraser* * These materials are only listed in lesson 1. Ready these materials for every lesson in this module.

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EUREKA MATH2

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Fluency

Happy Counting by Hundreds Students visualize a number line while counting aloud to develop familiarity with Happy Counting.

Teacher Note

Invite students to participate in Happy Counting. When I give this signal, count up. (Demonstrate.) When I give this signal, count down. (Demonstrate.) When I give this signal, stop. (Demonstrate.) Let’s count by hundreds. The first number you say is 0. Ready? Signal up, down, or stop accordingly.

100 200 300 400

500 600 700

600 500

Choose signals that you are comfortable with, such as thumbs-up, thumbs-down, and an open hand. Show your signal and gesture accordingly for each count. The goal is to be clear and crisp so that students count in unison. Avoid saying the numbers with the class; instead, listen for errors and hesitation.

600 700 800

Continue counting by hundreds within 2,500. Change directions occasionally, emphasizing crossing over multiples of 1,000 and where students hesitate or count inaccurately.

Whiteboard Exchange: Divide in Unit and Standard Form Students divide ones or tens in unit form and write the equation in standard form to prepare for dividing multiples of 10, 100, and 1,000. Display 6 ones ÷ 2 =

ones.

What is 6 ones ÷ 2? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

3 ones

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EUREKA MATH2

4 ▸ M3 ▸ TA ▸ Lesson 1

Display the answer. 6 ones ÷ 2 =

Write the equation in standard form.

ones

tens

6÷2=3

Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

6 tens ÷ 2 = 60 ÷ 2 = 30

Display the equation.

tens, displaying the answer after each question

Continue with 6 tens ÷ 2 = or prompt. What is 6 tens ÷ 2?

Write the equation in standard form. Repeat the process with the following sequence:

15 ones ÷ 3 =

ones

15 ÷ 3 = 5 15 tens ÷ 3 = 150 ÷ 3 = 50

EM2_0403TE_A_L01.indd 19

24 ones ÷ 4 =

ones

tens

24 ÷ 4 = 6 5

tens

24 tens ÷ 4 = 240 ÷ 4 = 60

4/13/2021 10:29:35 AM

EUREKA MATH2

4 ▸ M3 ▸ TA ▸ Lesson 1

Choral Response: Rename Place Value Units Students say a number in standard form and then rename the number by using tens or hundreds to prepare for dividing by multiples of 10, 100, and 1,000. After asking each question, wait until most students raise their hands, and then signal for students to respond. Raise your hand when you know the answer to each question. Wait for my signal to say the answer. Display 1 hundred 5 tens. What is 1 hundred 5 tens in standard form?

150 Display the answer. On my signal, rename 1 hundred 5 tens by using only tens.

15 tens Display the answer. Continue with 1 thousand 5 hundreds, displaying the answer after each question or prompt.

1 hundred 5 tens = 150 = 15 tens 1 thousand 5 hundreds = 1,500 = 15 hundreds = 150 tens

What is 1 thousand 5 hundreds in standard form?

1,500 On my signal, rename 1 thousand 5 hundreds by using only hundreds.

15 hundreds Now rename 1 thousand 5 hundreds by using only tens.

150 tens Repeat the process with the following sequence: 2 hundreds 7 tens

4 hundreds 8 tens

2 thousands 7 hundreds

4 thousands 8 hundreds

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3/25/2021 10:07:51 AM

EUREKA MATH2

Launch

4 ▸ M3 ▸ TA ▸ Lesson 1

Students relate dividing one-dollar bills, ten-dollar bills, and hundred-dollar bills.

Language Support

Present the situation: Liz plays a board game with 2 friends. She has 6 one-dollar bills, 6 ten-dollar bills, and 6 hundred-dollar bills to divide equally among the 3 players.

Consider using strategic, flexible grouping throughout the module. • Pair students who have different levels of mathematical proficiency.

Display the picture of the bills. Invite students to think–pair–share about how Liz can divide the one-dollar bills.

• Pair students who have different levels of English language proficiency.

She can give 2 one-dollar bills to each player because 6 ÷ 3 = 2. Invite students to think–pair–share about how Liz can divide the ten-dollar bills.

• Join pairs to form small groups of four. As applicable, complement any of these groupings by pairing students who speak the same native language.

She can give 2 ten-dollar bills to each player. What equation represents sharing the 6 ten-dollar bills among 3 players?

6÷3=2 Repeat the process with the 6 hundred-dollar bills. What if there were 6 thousand-dollar bills to share among 3 players? How many bills would they get? They would each get 2 bills. The values of the bills are different. How can the answer be 2 each time? Even though the values are different, it’s the same number of bills each time. We start with 6 and divide by 3. Invite students to turn and talk about whether they think familiar division facts could help them divide other units, such as tens, hundreds, or thousands. Transition to the next segment by framing the work. Today, we will divide multiples of 10, 100, and 1000.

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EUREKA MATH2

4 ▸ M3 ▸ TA ▸ Lesson 1

Learn

Represent Division with Place Value Disks Materials—T/S: Disks

Students represent a division expression in unit form with place value disks. Write the expression 6 ones ÷ 2. Invite students to use place value disks to represent 6 ones divided into 2 equal groups and to write an equation in unit form to match. How did you represent the expression with place value disks? I divided 6 ones disks into 2 equal groups. Each group has 3 ones. What is 6 ones ÷ 2?

3 ones Give partners 1 minute to use place value disks to represent the following expressions and to write a division equation in unit form for each situation.

6 tens ÷ 2

6 ones ÷ 2 = 3 ones

6 tens ÷ 2 = 3 tens 10

100

6 hundreds ÷ 2 6 thousands ÷ 2 Display the picture of the completed representations.

6 hundreds ÷ 2 = 3 hundreds 100

100

How are the solutions similar and different? Each problem is represented by 6 disks divided into 2 equal groups with 3 disks in each group.

1,000 1,000 1,000

6 thousands ÷ 2 = 3 thousands 1,000 1,000 1,000

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EUREKA MATH2

4 ▸ M3 ▸ TA ▸ Lesson 1

The total in each problem is 6 of something, but the units are different. They are either 6 ones, 6 tens, 6 hundreds, or 6 thousands. Each quotient is 3 of something. It’s a different unit each time: 3 ones, 3 tens, 3 hundreds, and 3 thousands. Repeat the process, inviting students to represent the following expressions with place value disks and to write a division equation in unit form.

12 ones ÷ 3 12 tens ÷ 3 12 hundreds ÷ 3 12 thousands ÷ 3 Display the picture of the three representations. Invite students to write 12 hundreds in standard form. Why is thinking of this number as 12 hundreds instead of 1 thousand 2 hundreds helpful when dividing? It is helpful because then we could think about 12 ÷ 3 = 4. That helps us to know 12 hundreds ÷ 3 = 4 hundreds. Renaming 1 thousand 2 hundreds as 12 hundreds is useful because then I can use a division fact I know, 12 ÷ 3 = 4. Invite students to turn and talk about how naming the total in each expression by using one place value unit helped them divide multiples of 10, 100, and 1000.

EM2_0403TE_A_L01.indd 23

100

12 ones ÷ 3 = 4 ones

Teacher Note

12 tens ÷ 3 = 4 tens

Throughout the lesson, four-digit numbers are represented without commas to support students in thinking of the numbers in unit form. For example, 1,200 is written as 1200 to support students in thinking of the number as 12 hundreds.

12 hundreds ÷ 3 = 4 hundreds

3/25/2021 10:07:53 AM

EUREKA MATH2

4 ▸ M3 ▸ TA ▸ Lesson 1

Use Unit Form to Divide

Teacher Note

Students write multiples of 100 and 1000 in unit form to divide. Display the picture of the four expressions and use the Math Chat routine to engage students in mathematical discourse.

8 ones ÷ 4 8 tens ÷ 4

Give students 1 minute of silent think time to find all four quotients and write equations in standard form to match. Have students give a silent signal to indicate they are finished.

8 hundreds ÷ 4 8 thousands ÷ 4

Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Purposefully choose work that allows for rich discussion about connections between strategies.

The use of unit form is an intentional scaffold and conceptual skill in preparation for multi-digit division. It can help prevent common errors and misconceptions when students begin recording long division. For example, if students think about dividing 1,284 as dividing 12 hundreds 8 tens 4 ones, they may be more deliberate about aligning the value of the digits in the quotient with the units in the dividend and be less likely to misalign the digits in the quotient.

Then facilitate a class discussion. Invite the selected students to share their thinking with the whole group and record their reasoning. As students discuss, highlight thinking that relates division facts to dividing larger units. I pictured the disks. I pictured 8 hundreds disks divided into 4 groups. Each group has 2 hundreds. I know 8 ÷ 4 = 2, so I know 8 thousands ÷ 4 = 2 thousands. All the problems relate back to 8 ÷ 4. Each total is 8 of a different unit, so the quotient is 2 of that unit. Ask questions that invite students to make connections and encourage them to ask questions of their own. Invite students to turn and talk about how writing division expressions in unit form is similar to representing them with place value disks. Write 20 ÷ 4. Name two ways to express 20 ÷ 4 in unit form.

2 tens ÷ 4

Teacher Note Consider using place value disks to represent why thinking about 20 ÷ 4 in unit form as 2 tens ÷ 4 is not helpful to finding 20 ÷ 4. Invite students to show 2 tens disks and to divide by 4. Then invite them to show 20 ones disks and to divide by 4.

20 ones ÷ 4

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EUREKA MATH2

4 ▸ M3 ▸ TA ▸ Lesson 1

Write 2 tens ÷ 4 and 20 ones ÷ 4. Direct students to think–pair–share about which expression they would use to help them find 20 ÷ 4. I would use 20 ones ÷ 4. I don’t know how to divide 2 tens by 4. To divide 2 tens by 4, I would have to rename 2 tens as 20 ones, so I would use 20 ones ÷ 4.

Promoting the Standards for Mathematical Practice

What is 20 ones ÷ 4?

Students look for and express regularity in repeated reasoning (MP8) as they divide by using unit form and basic division facts.

5 ones

Ask the following questions to promote MP8:

Write 20 ones ÷ 4 = 5 ones. Provide 1 minute for students to turn and talk about how 2000 ÷ 4 can be written in unit form in a way that is helpful in finding the quotient. What expression did you think about to help you find 2000 ÷ 4? Why? We thought about 20 hundreds ÷ 4. First, we thought about 2 thousands ÷ 4. It was not helpful because we don’t know how to divide 2 by 4. When we thought of 2000 as 20 hundreds, we knew that 20 hundreds ÷ 4 = 5 hundreds. Invite students to write the equations

20 ones ÷ 4 = 5 ones and 20 hundreds ÷ 4 = 5 hundreds in standard and unit form.

20 ones ÷ 4 = 5 ones

20 ÷ 4 = 5

20 hundreds ÷ 4 = 5 hundreds

2000 ÷ 4 = 500

• What patterns do you notice when dividing in unit form? How can that help you find 2000 ÷ 4 more efficiently? • Does anything repeat when you use unit form to find 20 ÷ 4, 200 ÷ 4, and 2000 ÷ 4? How can that help you find 4000 ÷ 8 more efficiently?

How are the unit form and standard form equations similar and different? I see 20 in each total, but it has different place values. The divisor, 4, is the same in unit form and standard form. I see 5 in each quotient, but 5 has different place values, either ones or hundreds. Invite students to turn and talk about how writing problems in unit form can help them divide.

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EUREKA MATH2

4 ▸ M3 ▸ TA ▸ Lesson 1

Divide Multiples of 100 Students divide multiples of 100 by using a place value strategy. Present this statement: 7 times as much as ____ is 2100. Give partners 1 minute to complete the statement and show their work. Circulate as students work. Identify two students who solved the problem differently to share their work. Look for students who rename the total or use basic division facts to support their reasoning. Then facilitate a class discussion. Invite the selected students to share their reasoning with the whole group. Place Value Disks

UDL: Action & Expression Consider providing place value disks for students to use as they divide 2100 by 7 .

Unit Form

100

21 ÷ 7 = 3 21 hundreds ÷ 7 = 3 hundreds

7 × = 2100 2100 ÷ 7 = 300

Ask questions that invite students to make connections between the solutions. Where do you see 21 hundreds in the place value disks? I see 21 total disks. Each disk has a value of 1 hundred.

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EUREKA MATH2

4 ▸ M3 ▸ TA ▸ Lesson 1

Where do you see the quotient, 3 hundreds, in the place value disks? There are 3 hundreds disks in each group. What division fact helped you find the quotient? Where do you see the division fact in the work?

21 ÷ 7 = 3. In the place value disks, there are 21 hundreds disks divided into 7 groups with 3 hundreds disks in each group. 21 ÷ 7 = 3 helped me find the quotient. 21 hundreds ÷ 7 = 3 hundreds Display the picture of 2100 ÷ 3 represented in standard form and unit form.

2100 ÷ 3 = 21 hundreds ÷ 3 = 7 hundreds

Invite students to think–pair–share about how standard form and unit form are used in the equation.

= 700

I see 2100 ÷ 3 written as 21 hundreds ÷ 3. That helps me to see a division fact that is familiar, 21 ÷ 3. 21 hundreds ÷ 3 is 7 hundreds. 7 hundreds is written in standard form, 700.

Direct students to work with a partner to write a similar equation to find 4800 ÷ 6.

Invite students to turn and talk about how place value units are represented in the equation in both standard form and unit form.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

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EUREKA MATH2

4 ▸ M3 ▸ TA ▸ Lesson 1

Land Debrief

5 min

Objective: Divide multiples of 100 and 1000.

Math Past

Facilitate a discussion that emphasizes how thinking about place value units can support dividing multiples of 100 and 1000. How did we use division facts to divide multiples of 100 and 1000? We represented a problem with hundreds or thousands disks. The number of disks we divided was the same as the division facts we know. When we wrote a problem in unit form, we could see that the digits showed a division fact we know. Then we could think about the correct units.

The Math Past resource includes information about the history of calculating devices that led to the modern calculator. Students may be interested in examining the pictures of the various devices and relating the descriptions of their functionalities to the pictured device.

How can thinking about place value units help you to divide multiples of 100 and 1000? When I think of the number in unit form, I can use a division fact I know. When I see division problems with multiples of hundreds and thousands, I Iook for facts that I know. The division fact is the same, but the unit changes.

Exit Ticket

5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem. Consider incorporating the information about one or two devices at a time throughout the module as the purpose or functionality of the device corresponds to the lesson objective.

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EUREKA MATH2

4 ▸ M3 ▸ TA ▸ Lesson 1

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

Name

Date

EUREKA MATH2

4 ▸ M3 ▸ TA ▸ Lesson 1

Divide. Draw place value disks to help you. 5. 1200 ÷ 3 = 12 hundreds ÷ 3

Divide. Use the place value disks to help you. 1.

8 ÷ 2 = 8 ones ÷ 2 =

400

hundreds

100

80 ÷ 2 = 8 tens ÷ 2

ones

100

800 ÷ 2 =

hundreds ÷ 2

hundreds

400

EM2_0403TE_A_L01.indd 29

tens

1,000 1,000 1,000 1,000

Divide. Use unit form to help you. 6. 600 ÷ 2 = 6 hundreds ÷

1,000 1,000 1,000 1,000

8000 ÷ 2 =

thousands ÷

thousands

= 4000

300

hundreds

thousands ÷ 3

thousands

= 3000

8. 2400 ÷ 6 =

hundreds ÷

hundreds

400

7. 9000 ÷ 3 =

PROBLEM SET

9. 3000 ÷ 6 =

hundreds ÷

hundreds

500

3/25/2021 10:07:57 AM

EUREKA MATH2

4 ▸ M3 ▸ TA ▸ Lesson 1

EUREKA MATH2

4 ▸ M3 ▸ TA ▸ Lesson 1

Divide. 10. 400 ÷ 2 =

200

11. 6000 ÷ 3 = 2000

12. 200 ÷ 5 =

13. 4000 ÷ 8 =

500

14. A used car costs 9 times as much as new tires. The used car costs $6300. How much do the tires cost?

6300 ÷ 9 = 700 The tires cost $700.

PROBLEM SET

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LESSON 7

Angle Relationships and Unknown Angle Measures Identify and describe angle relationships given in diagrams. Write and solve equations that use angle relationships to find unknown angle measures.

EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

EXIT TICKET Name

Date

In the diagram, ∠BAD is a straight angle. Use this diagram for problems 1–4.

x°

132° A

1. Describe the relationship between ∠ BAC and ∠CAD. ∠BAC and ∠CAD are a linear pair, so their measures sum to 180°.

Lesson at a Glance This lesson uses familiar angle relationships as a springboard for writing and then solving equations by using if–then moves. Students use a variety of strategies to find an unknown angle measure. Then they are introduced to if–then moves through nonmathematical statements. After seeing the mathematical version of the if–then moves, students apply them to find other unknown angle measures. Through a mix of partner and independent work, as well as a sharing of strategies, students are encouraged to make the connection between an arithmetic strategy and an algebraic strategy. This lesson formalizes the term complementary angles.

Key Questions • What does it mean for two angles to be complementary?

2. Write an equation for the angle relationship and solve for x .

x + 132 = 180 x + 132 − 132 = 180 − 132 x = 48

• How does knowing about angle relationships help us find unknown angle measures?

The solution is 48.

• How can we use if–then moves to find unknown angle measures?

3. Check your solution to the equation. Because 48 + 132 = 180 is a true number sentence, we know 48 is a solution to the equation.

Achievement Descriptors 7.Mod3.AD8 Compare algebraic solutions to arithmetic solutions

4. What is the measure of ∠ BAC ?

of word problems. (7.EE.B.4.a)

m∠ BAC = 48°

7.Mod3.AD12 Write and solve equations to find unknown angle

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

Agenda

Materials

Fluency

Teacher

Launch Learn

5 min

30 min

• None

Students

• Angle Measure

• None

• If–Then Moves

Lesson Preparation

• Strategies

• None

Land

10 min

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

Fluency Solve One-Step Equations Students solve one-step equations to prepare for finding unknown angle measures. Directions: Solve for the variable. 1.

x-4 = 7

8 + x + 7 = 25

5.2 + z = 13

7.8

4 a = 20

b 3

=21

5 71

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EUREKA MATH2

Launch

7 ▸ M3 ▸ TB ▸ Lesson 7

Students identify angles and angle relationships. Display problem 1. Tell students they have 2 minutes to work on problem 1 with a partner. Do not expect students to list all the relationships as there are many angle relationships in the diagram. Some examples are given as student responses.

1. In the diagram, DB and EC intersect at point A. List all the angles you see in the diagram. Note any angle relationships that you notice.

F Teacher Note

Before grade 7, students refer to complementary angles as “two angle measures that sum to 90°.”

E Sample: Angles:

∠BAC , ∠BAF , ∠BAD , ∠BAE , ∠CAF , ∠CAD , ∠CAE , ∠FAD , ∠FAE , ∠DAE Angle relationships: Angles on a line

∠BAE and ∠DAE ∠EAD , ∠DAF , and ∠FAC ∠EAD and ∠DAC

Linear pair

∠DAF and ∠FAB ∠DAC and ∠CAB ∠EAD and ∠DAC ∠CAF and ∠FAE

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

∠DAB

Straight angle

∠EAC Angle measures that sum to 90°

∠BAC and ∠CAF

Adjacent angles

∠EAD and ∠DAF ∠BAC and ∠EAB

Angles at a point

∠BAF , ∠FAE , and ∠EAB

After about 2 minutes, ask groups to share with the class the angles and angle relationships they identified. As students share, point to the angles mentioned. Create a list of the different types of angle relationships in the diagram. Ensure the following points are addressed during the share: • Some pairs of angles have measures that sum to 90°. Students are introduced to the term complementary angles later in the lesson.

UDL: Representation Trace the paths of the angles as the students name the points and vertices. This helps clarify the naming convention and the relationships between angles.

• There is more than one way to name an angle. For example, ∠EAD is the same angle as ∠DAE. • When an angle is named, the middle letter in the angle name is the vertex of the angle. Display the following two angles, which highlight the difference between an angle that needs to be identified with three letters and an angle that only needs one letter. Use the prompts that follow those angles to provide reasoning for the naming conventions.

B A

I J K

Language Support Consider creating an anchor chart of familiar angle relationships such as angles on a line, adjacent angles, linear pair, straight angles, angles at a point, and angles that sum to 90°. Throughout this topic, other angle relationships are introduced, so leave space for them to be added.

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

Can ∠HKI be named ∠K? No. Point K is the vertex of three angles: ∠HKJ, ∠HKI, and ∠IKJ. It would be unclear which angle was being referred to. Three points need to be in the name. Do we always need three points to name these angles? Why? It depends on the diagram. If naming an angle with just one point is unclear as to which angle is being named, then the name needs to include three points. Can ∠BAC be named with just one letter? Explain.

∠BAC can be named ∠A because only one angle has a vertex at point A. Display the image from problem 1.

F C

E If ∠BAC has a measure of 50° , can we determine the measure of any other angles? Turn and talk to your partner. Today, we will find unknown angle measures by writing and solving equations.

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

Learn

Promoting the Standards for Mathematical Practice

Angle Measure Students find an unknown angle measure. Display problem 2. Have students complete problem 2 independently. Circulate and identify students who use different methods: an arithmetic strategy, an algebraic strategy, or a tape diagram. Do not prompt or specify which strategy students need to use. All strategies are discussed later in the lesson. As students finish, have them compare their strategy with a partner. 2. What is the measure of

∠BGA?

B 30°

Ask the following questions to promote MP5: • Which tools would be the most helpful in finding the unknown angle measure? • How can you estimate the angle measure? Does your estimate sound reasonable?

Differentiation: Support

x°

Students use appropriate tools strategically (MP5) when they choose among arithmetic strategies, algebraic strategies, and tape diagrams to determine unknown angle measures.

If students need support with identifying ∠BGA, move them forward by helping them trace the angle. Have them place their pencil on point B, trace the angle until point G, and continue to point A. Students can also highlight the interior of ∠BGA.

Arithmetic: 90 - 30 = 60

m∠BGA = 60°

Teacher Note

After they discuss strategies with a partner, have students return to the whole group. Did you and your partner find the same measurement for ∠BGA? Did you use the same strategy? My partner and I both found that the measure of ∠BGA is 60°. I used an equation, but my partner drew a tape diagram.

In grade 6, students are introduced to the symbol m∠ABC, which refers to the measure of ∠ABC.

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

Ask a few students to display their work and explain their thinking. If the following strategies are not displayed by students, model them for students. Tape Diagram

Algebraic

Let x represent the number of degrees of the measure of ∠BGA.

x + 30 = 90

x + 30 - 30 = 90 - 30 x = 60

30 90 - 30 = 60

m∠BGA = 60°

m∠BGA = 60° Facilitate a class discussion by using the following prompts. What are some similarities and differences between the strategies? All strategies lead to the same measurement of 60°. All strategies use the measure of ∠BGC. All strategies use the angle relationship of ∠BGA and ∠BGC having measures that sum to 90°. The tape diagram is another visual representation of the situation. The algebraic strategy requires us to solve an equation. The arithmetic strategy is straightforward and simple because it doesn’t have a variable. How do you know that the measures of ∠BGA and ∠BGC sum to 90°?

The little square by the vertex shows us that their measurements sum to 90°.

By definition, two angles with measures that sum to 90° are called complementary angles.

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

Invite a few students to go to the board to sketch other examples of complementary angles. Ask students to explain how they know they have drawn complementary angles. If students only draw adjacent angles, present the following pair of angles to the class.

70° Language Support

20°

Consider adding complementary angles to the angle relationship anchor chart.

Ask students if these angles are complementary. If students answer no because the angles are not adjacent, refer them to the definition. Does our definition say they must be adjacent? No. It says only that the sum of the angle measures needs to be 90°. Are these complementary angles? Why? Yes. Because 70 plus 20 is 90, they are complementary angles. Display the following diagram.

20° 40° 30°

Are these complementary angles? No. The definition states that complementary angles are two angles with measures that sum to 90°. The diagram shows three angles with measures that sum to 90°.

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

If–Then Moves Students use if–then moves to find unknown angle measures. Display the following statements: • If you live in Kansas, then you live in the United States. • If it is Wednesday, then it is a weekday. • If you are in the United States, then you are in North America. • If it is a weekday, then it is Wednesday. What do you notice? All the statements are in the form of If …, then …. Are all the if–then statements true? Explain your thinking. The first statement is true because Kansas is in the United States. The second statement is true because Wednesday is a weekday. The third statement is true because the United States is in North America. The fourth statement is false because there are other weekdays besides Wednesday. Now let’s look at mathematical if–then statements. Read and display the following statement to the class:

Teacher Note

If a = b, then a + c = b + c. Assume a, b, and c are numbers. Give a thumbs-up if you agree with the statement and a thumbs-down if you disagree with the statement. Have students discuss what the statement means with a partner. Invite a few pairs to share their reasoning. If two quantities are equal, I can add the same amount to both quantities and they will still be equal. For example:

If students question the limitations of c being a nonzero number in the last if–then statement, ask them the following questions. • What is the unknown factor equation that represents 4 ÷ 0? • What number times 0 gives us a product of 4? Because dividing by 0 is undefined, c cannot be 0 in that statement.

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

Repeat the thumbs-up or thumbs-down process, partner discussions, and sharing for the remaining three if–then moves. If a = b, then a - c = b - c. If a = b, then a ⋅ c = b ⋅ c. If a = b and c ≠ 0, then a ÷ c = b ÷ c. These statements are known as if–then moves. Let’s take another look at the algebraic method used to solve problem 2. Display the algebraic method from problem 2. Let x represent the number of degrees of the measure of ∠BGA.

x + 30 = 90

Teacher Note

x + 30 - 30 = 90 - 30 x = 60

In grade 6, students solve equations by using the following properties of operations:

Did we use any if–then moves there? Explain your thinking.

• a+b−b=a

Yes. We subtracted 30 from both sides of the equation.

• a−b+b=a

Now, let’s look at another problem that uses if–then moves. Direct students to problem 3. Have students work in pairs to complete parts (a) and (b). 3. In the diagram, ∠DAB is a straight angle.

23°

• When b ≠ 0, a ÷ b ⋅ b = a. The if–then moves help prepare students for the properties of equality in grade 8.

C x°

• When b ≠ 0, a ⋅ b ÷ b = a.

A D a. Describe the relationship between ∠DAC and ∠CAB.

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

b. Write an equation for the angle relationship and solve for x.

x + 23 = 180 x + 23 - 23 = 180 - 23 x = 157 The solution is 157. Have students return to the whole group. Model how to check the solution to an equation by completing part (c) together. Emphasize that substituting the correct solution for the variable creates a true number sentence. If students end up with a false number sentence, then they need to determine whether they made an error when substituting the value or when solving the equation. Tell students that they are expected to check their solutions to equations from now on. c. Check your solution to the equation.

157 + 23 = 180

Teacher Note Problem 3(c) shows each step associated with checking the solution. Moving forward, only the sentence describing how the answer is a solution is provided. But it is expected that students will always complete such steps to check their solutions.

180 = 180 Because 157 + 23 = 180 is a true number sentence, we know 157 is a solution to the equation. Have students continue to work with their partner to complete problems 3(d)–6. Circulate as students work and listen for instances where students identify if–then moves. d. What is the measure of ∠DAC?

m∠DAC = 157° 4. ∠A and ∠B are complementary angles. The measure of ∠A is 34.5°. Write and solve an equation to determine the measure of ∠B. Check your solution. Let x represent the number of degrees of the measure of ∠B.

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5. Consider the diagram.

D 25° x °

25°

A a. Describe the relationship among ∠CAE, ∠EAD, and ∠DAB.

∠CAE, ∠EAD, and ∠DAB have measures that sum to 90° b. Write an equation for the angle relationship and solve for x.

25 + x + 25 = 90 x + 50 = 90 x + 50 - 50 = 90 - 50 x = 40 The solution is 40. c. Check your solution to the equation. Because 25 + 40 + 25 = 90 is a true number sentence, we know 40 is a solution to the equation. d. What is the measure of ∠EAD?

m∠EAD = 40°

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7 ▸ M3 ▸ TB ▸ Lesson 7

6. In the diagram, ∠CAB is a straight angle.

E C

x°

x° x° A

(3x)° B

a. Describe the relationship among ∠CAE, ∠EAD, ∠DAF, and ∠FAB.

∠CAE, ∠EAD, ∠DAF, and ∠FAB are angles on a line, so the sum of their measures is 180° b. Write an equation for the angle relationship and solve for x.

x + x + x + 3x = 180 6x = 180 6x ÷ 6 = 180 ÷ 6 x = 30 The solution is 30. c. Check your solution to the equation. Because 30 + 30 + 30 + 3(30) = 180 is a true number sentence, we know 30 is a solution to the equation. d. What are the measures of ∠CAE, ∠EAD, ∠CAD, and ∠FAB?

m∠CAE

m∠EAD

m∠CAD

m∠FAB

x = 30

x + x = 30 + 30

3x = 3(30)

= 60

= 90

m∠CAD = 60°

m∠FAB = 90°

m∠CAE = 30°

m∠EAD = 30°

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7 ▸ M3 ▸ TB ▸ Lesson 7

When students are finished, confirm their answers. Then facilitate a class discussion by using the following prompts. What angle relationships are in problems 3–6? Problem 3 has a linear pair. Problem 4 has complementary angles. Problem 5 has adjacent angles with measures that sum to 90°. Problem 6 has angles on a line. When did you use if–then moves? As we solved each equation, we used an if–then move. Each time, we used it directly after we wrote the equation. How did you check your solution to each equation? I took the solution I got and substituted it for the variable in the original equation. When it made a true number sentence, I knew the solution to the equation was correct.

Strategies Students find the value of x. Direct students to problem 7. Consider having them work independently for about 2 minutes. If students solve arithmetically, challenge them to write and solve an algebraic equation or use a tape diagram. Allow for productive struggle. Students practice solving equations of this form throughout the topic, so it is not imperative that all students exhibit mastery in using an algebraic method now. When the struggle is no longer productive, or when most students have finished, tell them to compare their work with a partner. Identify specific students to share their strategy with the class.

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7. In the diagram, ∠AFE is a straight angle. What is the measure of ∠BFA? Check your answer.

C 36° 36° x°

Students should not assume that angles are right angles. However, students may recognize that ∠AFC and ∠EFC both have a measure of 90°. That is because they are made up of identical pairs of angles and each pair forms half of a straight angle. As long as students can justify their thinking, encourage them to use this information to help them solve the problem.

x° F

Teacher Note

Sample: Arithmetic strategy:

180 - 36 - 36 = 108 108 ÷ 2 = 54 The measure of ∠BFA is 54°. Because 54 + 36 + 36 + 54 = 180 is a true number sentence, we know that 54° is the measure of ∠BFA. Invite students who used varied solution strategies to share their work with the class. Ask students to look for connections among the strategies. For example, a student who used an arithmetic strategy had to divide 108 by 2 to get 54. We see this same move in the algebraic strategy as 2x = 108.

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7 ▸ M3 ▸ TB ▸ Lesson 7

Tape Diagram

Algebraic

2x + 36 + 36 = 180

180 x

2x + 72 = 180 2x + 72 - 72 = 180 - 72 2x = 108

180 x

2x ÷ 2 = 108 ÷ 2

x = 54 m∠BFA = 60°

72 180 x

180 – 72

72 180 - 72 = 108

180 x

x 108

72 x

108 m∠BFA = 60°

2 units = 108 1 unit = 108 ÷ 2 1 unit = 54 x = 54

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7 ▸ M3 ▸ TB ▸ Lesson 7

Land Debrief

5 min

Objectives: Identify and describe angle relationships given in diagrams. Write and solve equations that use angle relationships to find unknown angle measures. Facilitate a class discussion by using the following prompts. Encourage students to add to their classmates’ responses. What does it mean for two angles to be complementary? Two angles are complementary if their measures sum to 90°. How does knowing about angle relationships help us find unknown angle measures? The angle relationships help us write an equation to represent the situation. For example, when we have complementary angles, we need to write an equation that shows the two angle measures sum to 90°. When we have a linear pair, we need to write an equation that shows that the two angle measures sum to 180°. How can we use if–then moves to find unknown angle measures? If–then moves help us solve the equation that represents the angle relationship. If we add, subtract, or multiply by any number on both sides of the equation, the sides remain equal. We can also divide both sides of the equation by any nonzero number and have the sides remain equal.

Exit Ticket

5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

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Topic B Unknown Angle Measurements In topic A, students use the distributive property to write expressions in equivalent ways. Students build on this work in topic B when they write and solve equations to find unknown angle measurements. Students begin the topic by recalling how to name angles and identify angle relationships from a diagram. Students use prior knowledge of angle relationships and their understanding of a new term, complementary angles, to write equations. As students look for efficient methods to solve these equations, they are introduced to the use of if–then moves. Students practice checking solutions to equations by using the original equation and the problem context. In a digital lesson, students continue to write equations to represent angle relationships. Students apply their understanding of a new term, supplementary angles. Students find unknown angle measures by using tape diagrams and equations. Students determine that when solving an equation, they can divide both sides of the equation by a nonzero number or multiply both sides of the equation by its multiplicative inverse and get the same result. Students continue to write and solve equations that represent angle relationships. They are introduced to two new angle relationships: vertical angles and angles at a point. In the last lesson, students apply the new learning from the topic to solve more challenging, multi-step angle relationship problems. In topic C, students continue working with equations, gaining fluency in solving equations containing all types of rational numbers. In grade 8, the if–then moves are formalized as the properties of equality. Those properties are used in solving more rigorous equations throughout the high school grades.

If–Then Moves Assume a, b, and c are numbers. If a = b, then a + c = b + c. If a = b, then a - c = b - c. If a = b, then a ⋅ c = b ⋅ c. If a = b and c ≠ 0, then a ÷ c = b ÷ c.

Vertical Angles and Angles at a Point

D (5x)° E

(7x + 96)° A

35°

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Progression of Lessons Lesson 7

Angle Relationships and Unknown Angle Measures

Lesson 8

Strategies to Determine Unknown Angle Measures

Lesson 9

Solving Equations to Determine Unknown Angle Measures

Lesson 10 Problem Solving with Unknown Angle Measures

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Before This Module

Overview

Grade 6 Module 4

Expressions, Equations, and Inequalities

In grade 6, students apply properties of operations to solve one-step equations of the forms x + p = q and px = q for cases in which p, q, and x are nonnegative rational numbers. In this module, students extend this work to negative rational numbers and to the forms px + q = r and p(x + q) = r. In grade 6, students write inequalities of the form x > c or x < c to represent a constraint in a problem. Such inequalities have infinitely many solutions that can be represented on a number line. In this module, students extend that learning to include solving inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Students solve inequalities that are less than or equal to and greater than or equal to and interpret the solution set in the context of the problem.

Topic A Equivalent Expressions Topic A centers on writing expressions in equivalent forms. Students move from doing familiar work with numerical expressions to determining when algebraic expressions are equivalent. Through the application of properties of operations—namely the distributive property—as well as the use of the tabular model, students multiply and factor expressions with rational and negative numbers. Modeling the Distributive Property with the Tabular Model

–2

–16

8(l - 2) = 8l - 16

Showing Factoring with the Tabular Model

–4

15x

–12

15x - 12 = 3(5x - 4)

Topic B Unknown Angle Measurements In topic B, students use familiar and new angle relationships to write and solve equations that help determine unknown angle measures. Students continue to use properties of operations and visual models to solve equations. They are introduced to a new strategy for solving equations: if–then moves.

If–Then Moves Assume a, b, and c are numbers. If a = b, then a + c = b + c. If a = b, then a - c = b - c. If a = b, then a ⋅ c = b ⋅ c. If a = b and c ≠ 0, then a ÷ c = b ÷ c.

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Topic C

After This Module

Solving Equations Students continue to use if–then moves in topic C to fluently solve equations of the forms px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Students engage in historical mathematics to determine the advantages and disadvantages of presenting problems rhetorically and symbolically. Later, students use the structure of an equation to make a simpler problem. Throughout this topic, students participate in activities and puzzles to simulate play when fluently solving equations. They end the q x topic by exploring a new type of equation, p = r , to foreshadow work with proportional reasoning in module 5.

Grade 7 Modules 4 and 5 In module 4, students extend solving equations to applying formulas to determine area, circumference, and surface area. Students solve percent equations in module 5.

Grade 8 If–then moves are formalized as the properties of equality in grade 8 and are used to solve more rigorous equations throughout high school.

Topic D Inequalities In topic D, students apply what they know about solving equations to solve inequalities. They begin by graphing boundary numbers and testing numbers to determine the correct region on the number line to shade. Students determine whether solving by using if–then moves is more efficient. Through experimentation, students notice that when both sides of an inequality are multiplied or divided by a negative number, the inequality sign must be reversed to maintain a true number sentence. This discovery necessitates additional if–then moves for inequalities.

The work done in this module supports work with linear equations in one and two variables in grade 8. Students also extend their understanding of equivalent expressions when they solve systems of linear equations in grade 8.

If–Then Statements for Inequalities Let a, b, and c represent numbers. If a < b, then a + c < b + c. If a < b, then a - c < b - c. If a < b and c is a positive number, then a ⋅ c < b ⋅ c. If a < b and c is a negative number, then a ⋅ c > b ⋅ c. If a < b and c is a positive number, then a ÷ c < b ÷ c. If a < b and c is a negative number, then a ÷ c > b ÷ c. Versions of these statements can be written to begin with >, ≤, or ≥ instead of <. Copyright © Great Minds PBC

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Why Expressions, Equations, and Inequalities I notice that this module does not encourage the use of manipulatives. Why not? The use of manipulatives can support student engagement and provide differentiation and equity. Manipulatives can promote student thinking and aid in communicating about the mathematics being learned. Manipulatives often bridge learning from the conceptual stage to the pictorial or abstract stages of learning. However, students may lose the chance to deepen their understanding of concepts if manipulatives are used in isolation from mathematical connection. Algebra tiles are a widely used manipulative to engage students in understanding the “rules” for solving equations. Although the tiles can be helpful in representing variables and constants, these and other difficulties may occur when students employ these manipulatives: 1. Students must assign numbers and variables to a set of colored tiles. Students are then tasked with remembering which color and shape of tile represents a positive number, which color and shape represents a negative number, and which color and shape represents a variable. 2. Use of colored tiles connects easily to addition, but some learners cannot conceptualize subtracting a negative number or multiplying when the first factor is negative. Some interpretations of division cannot be modeled by using the tiles. 3. Tiles do not represent non-integer rational numbers and cannot be used to model arithmetic with rational numbers.

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Further, use of algebra tiles often imposes procedural directions that mask the mathematics happening when an equation is being solved. The properties of operations, and, in this module, the if–then moves, are not clear when focus is placed on the movement of the manipulative rather than on the solving of the equation.

I notice that this module includes standards for geometry. Why are these standards addressed in this module? Students understand and apply angle relationships to determine unknown angle measures. These relationships necessitate equivalence. To determine whether angles are complementary, students understand that the two angle measures must sum to 90°. To determine whether angles are supplementary, students understand that the two angle measures must sum to 180°. A natural approach to determine unknown angle measures in these and other cases is to solve for the unknown by using an equation. Determining unknown angle measures drives the need to solve equations. Students use equations to show why angles are equal in measure.

Why is the word simplified not used in this module? The word simplified has multiple meanings depending on the different situations in which it is used. For example, when asked to simplify a fraction, students perform a different task than they would when they simplify an expression. When a student is directed to simplify in any situation, a specific description of the term should be provided that is appropriate for that situation. If you choose to have students simplify expressions in this module, we recommend that you define simplify for each case in which it is used, and we encourage you to accept all equivalent forms of the expression as correct.

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Supplemental Materials

Lead: Facilitating Successful Implementation of Eureka Math2, K–9

Credits Great Minds® has made every effort to obtain permission for the reprinting of all copyrighted material. If any owner of copyrighted material is not acknowledged herein, please contact Great Minds for proper acknowledgement in all future editions and reprints of this handout.

Works Cited Great Minds. Eureka Math2TM. Washington, DC: Great Minds, 2021. https://greatminds.org/math.

Supplemental Materials

Lead: Facilitating Successful Implementation of Eureka Math2, K–5

Appendix A Implementation Support Tool (IST)

Implementation Support Tool K–5 A Story of Units®

Implementation Support Tool K–5

Overview Using Eureka Math2 to its fullest potential takes time, reflection, and continuous intentional preparation. This process is accelerated with the partnership between a teacher and an instructional coach (e.g., administrators, district level coaches, facilitators). The Implementation Support Tool (IST) supports both teachers and coaches in creating the optimal experience for every student in a Eureka Math2 classroom. It describes teacher practices that are essential to each component of a Eureka Math2 lesson.

Understanding the Implementation Support Tool The IST has the look of an evaluation rubric and observation checklist; however, it is not intended to be either. Understanding the IST and its structure helps to properly inform coaching, reflection, and preparation. The bullets below outline these important understandings. •

There is not a linear progression across the IST. Teacher actions described in the rightmost column do not necessarily have greater instructional impact than teacher actions described in the leftmost column. Always consider the impact that the teacher actions described on the IST will have on student learning independently or in direct comparison with other teacher actions.

•

The teacher actions described on the IST are not all-or-nothing actions. Each can be performed with varying degrees of effectiveness.

•

The teacher actions described on the IST are like the keys on a piano. Just as playing a song requires playing the right notes at the right times, exemplary instruction requires using the correct teacher actions at the appropriate times for appropriate reasons. Not every teacher action is appropriate at all times.

•

Most teacher actions on the IST are strengthened by the presence of other teacher actions. Before focusing on the development of a teacher action, check if there is a prerequisite teacher action that should be developed first.

Using the Implementation Support Tool Like most tools, the IST is most effective when used as intended. This tool is not meant to be used in an evaluative fashion or as a checklist during teacher observation. Instead, it supports implementation the following ways. •

Coaches can use the IST to develop a deeper understanding of Eureka Math2 during early implementation observations.

•

Coaches can use the IST to support their analysis of observation data following its collection. It is not recommended for use during classroom observations because it can distract coaches from collecting as much specific, objective observation data as possible.

•

Coaches can use the IST to name instructional priorities for targeted professional development, PLCs, and 1:1 coaching cycles.

•

Coaches can refer to the IST for shared language during feedback conversations with teachers following observation.

•

Coaches can use the IST with teachers to prompt goal setting and self-reflection.

•

Individual teachers or teams can use the IST during lesson preparation to look for opportunities to leverage prioritized teacher actions.

•

Individual teachers can develop their practice by using the IST to reflect on their current instruction. This reflection can be supported by using the IST to analyze recorded classroom footage.

Note: For an explanation of the structure of the IST, refer to Appendix I: Structure.

April 2022 58

Implementation Support Tool K–5 Fluency Fluency uses activities that solidify and build students’ ability to use mathematical procedures flexibly, accurately, efficiently, and appropriately. Students become familiar with fluency routines because of their consistent use across modules and grade levels, allowing for efficient teaching and learning. All fluency routines benefit from the general fluency indicators below. The most common routines—Whiteboard Exchange, Choral Response, Counting, and Sprints—have additional sets of indicators to support implementation.

Choral Response

Whiteboard Exchange

General Fluency Indicators

Core Implementation

Instructional Habits

Adaptive Implementation

Meeting Component Purpose a. Completes fluency activity or activities within or near the total indicated time b. States directions clearly and includes all necessary details for engagement (e.g., topic/task, timeframe, cue) c. Prepares the environment to optimize engagement and learning (e.g., location of the activity, materials, concrete manipulatives, visuals)

Engaging and Monitoring indicators vary by routine and are included for specific fluency routines only.

Using Routines a. Uses the suggested sequence to build fluency through repeated practice with a targeted concept or skill b. Gives immediate, concise, and specific feedback to each student based on the accuracy of their response (e.g., asks questions, prompts to include the unit) c. Returns to students with incorrect initial response to validate their corrections

Engaging d. Provides appropriate work time before signaling students to show their whiteboards e. Establishes expectations for participation to maintain efficiency and ensure student accountability Monitoring f. Scans every student’s whiteboard to notice and note trends, varied solutions, exemplars, and misconceptions

Responding g. Adjusts the sequence of problems (e.g., complexity, level of abstraction) based on the existence of pervasive trends

Using Routines a. Uses the suggested sequence to build fluency through repeated practice with a targeted concept or skill b. Uses established cues (hand signals or verbal indicators) to prompt every student to respond in unison c. Displays or states correct responses

Engaging d. Provides appropriate think time before signaling students to chorally respond Monitoring e. Listens to students’ responses to confirm accuracy and recognize outliers (e.g., mistakes, lack of participation)

Responding f. Prompts students to use precise mathematical language g. Adjusts the sequence of prompts (i.e., decreased or increased complexity) to improve access or provide challenge h. Responds to errors with scaffolds such as questioning or concrete or pictorial supports to improve access

Promoting Discourse indicators are excluded from the Fluency section of the IST due to the nature of the lesson component.

Customizing d. Administers alternate fluency activity to increase engagement or in response to a demonstrated student need e. Scaffolds the sequence with additional problems to provide access (e.g., lower complexity initial problems, intermediate problems to ease complexity, extension problems) f. Allots time appropriately across Fluency activities to meet student needs (e.g., prepare for current or upcoming lesson, practice grade level content, or maintain content from previous grades) Responding g. Models or thinks aloud about a parallel problem or sequence to address a widespread, small-gap misconception and then re-engages students in a similar sequence (excluding Sprints)

Implementation Support Tool K–5 Fluency, Continued

Sprints

Counting

Core Implementation

Instructional Habits

Using Routines a. Uses concise, clear signals to guide students to count in unison (upward, downward, and stopping) b. Repeats practice of counting up and down to help students commit sequences to memory and recognize patterns c. States unit, starting number, and ending number prior to activity

Engaging d. Uses appropriate pace for the complexity and familiarity of the count

Using Routines a. Directs students to complete the sample problems to ensure their understanding of the purpose of the sprint b. Frames the sprints as an opportunity for growth (e.g., encouraging and celebrating effort, success, and improvement) c. Provides 1 minute for students to complete as many problems as possible, in order without skipping d. Reviews the correct answers to each sprint at a brisk pace and with an established verbal cue that allows for all students to follow along e. Provides an opportunity for students to complete more problems on sprint A or to analyze and discuss the patterns in sprint A to increase success on sprint B f. Directs students to record their performance and to calculate and celebrate growth after sprint B

Engaging g. Provides appropriate think time for students to internalize patterns before transitioning to sprint B

Monitoring e. Listens for accurate counting by all students f. Listens to key junctures or anticipated points of struggle within the sequence (e.g., crossing over ten or hundred) to assess for fluency

Monitoring h. Circulates during the sprint to provide encouragement, ensure students are completing the problems in order, and look for areas where fluency breaks down i. Listens to student responses during correction to identify where a discussion about patterns is warranted j. Listens to peer-to-peer discussions about patterns in sprint A to ensure students have named high-leverage takeaways to apply in sprint B

Adaptive Implementation

Responding g. Alters the pace of the count based on student responses h. Focuses on counting up and down at key junctures or known points of struggle within the sequence i. Responds to errors by pausing the count to ask targeted questions or to provide concrete/pictorial supports j. Gradually removes scaffolds to increase academic ownership

Responding k. Prompts students to analyze specific sequences of problems in preparation for current and future lessons l. Names or has students name the high-leverage patterns identified within peer-to-peer discussions prior to sprint B

Students Expectations

Math2

During fluency activities, students in a Eureka classroom should do the following: • Participate fully (verbally, on whiteboards, with hands when appropriate, etc.) • Respond to established signals and prompts • Include units in responses, when appropriate • Make corrections, when appropriate

In addition to the indicators for all fluency activities, during sprints, students in a Eureka Math2 classroom should also do the following: • Complete problems in order, with urgency, for 60 seconds • Look for and communicate about the patterns in the sprint • Work to improve fluency by looking for patterns and applying them

Implementation Support Tool K–5 Launch Launch creates an accessible entry point to the day’s learning through activities that build context, create a need for new learning, or activate prior knowledge. Every Launch ends with a transition statement that sets the goal for the day’s learning. Core Implementation

Meeting Component Purpose a.

Aligns facilitation to the purpose statement of Launch

Provides opportunities indicated by lesson for students to notice; wonder; apply; and articulate strategies, choice of models, and understandings

Honors the amount of teacher-to-student discourse, studentto-student discourse, and whole-class discourse within Launch

Concludes Launch by using the given transition statement to set the goal for and make an explicit connection to the day’s learning

Completes the Launch within or near the time indicated

Using Structures, Routines, and Activities f. States directions clearly and includes all necessary details for engagement (e.g., topic/task, thinking job, timeframe, cues, peer interaction) g. Prepares the environment to optimize engagement, collaboration, and learning (e.g., desk arrangement, Thinking Tool, Talking Tool, materials, visuals)

Instructional Habits

Adaptive Implementation

h. Provides appropriate think time for students to process questions or displayed content and form responses

Customizing p. Uses the margin notes to improve access (e.g., UDL, Differentiation: Support, Language Support, strategic pairings/groupings)

Engaging

Encourages flexible thinking (e.g., varied models and strategies) and connections to prior knowledge to access content

Monitoring j. Circulates to recognize and encourage application of mathematical understanding k. Listens to student discourse for understanding, connections to prior knowledge, and evolving reasoning l. Monitors student work to notice and note trends, varied models and strategies, exemplars, and misconceptions to leverage during class discussion Promoting Discourse m. Prompts students to elaborate (e.g., by saying “Why?” “How do you know?” or “Tell me more”) to make student thinking more visible, clear, and complete n. Asks questions that invite students to make connections between solutions, models, strategies, and previous lessons, modules, or grade level o. Prompts students to restate, build on, or evaluate other students’ responses to enhance discussion and strengthen habits of discussion Student Expectations

During Launch, students in a Eureka Math2 classroom should do the following: • • • • •

Generate, test, share, critique, and refine their ideas Engage in student-to-student and whole-class discourse Process presented images and information by noticing and wondering Formulate and articulate their thinking regarding solution pathways, strategies, models/representations, and understandings Ask and answer questions when engaging in routines and activities

Responding q.

Calls on students or selects student work strategically to highlight trends, varied solutions, connections to prior content, and misconceptions

Models the use of or directs students to use specific sections of the Talking Tool and Thinking Tool to support discourse and metacognition

Implementation Support Tool K–5 Learn Learn presents new learning related to the lesson objective, usually through a series of instructional segments. This lesson component takes most of the instructional time. Suggested facilitation styles vary and may include explicit instruction, guided instruction, group work, partner activities, and digital elements. Core Implementation

Meeting Component Purpose a.

Focuses the progression of questioning, thinking, and discourse in alignment with the purpose

Instructional Habits

Engaging k.

statement of each Learn segment b.

Honors the ratio of questioning to direct instruction, the amount of discourse, and the format of the discourse so that students generate, test, share, critique, and refine their ideas

Releases responsibility to students while working on lesson pages, as indicated by the lesson and student performance

Uses tools and models accurately to develop student understanding of the lesson’s

Monitoring l.

Completes Learn within or near the total time indicated, allowing time for instruction, lesson

thinking job, timeframe, cues, peer interaction) h.

understanding of the day’s objective and to articulate their strategies, models, and

Recording Sheet (OARS) for K–2) o.

Prompts students to elaborate or clarify responses (e.g., by saying “Why?” “How do you know?” or “Tell me more”), and use precise language to advance student thinking

Asks questions that invite students to make connections between solutions; models; strategies; and previous questions, lessons, modules, or grade levels

Student Expectations

During Learn, students in a Eureka Math2 classroom should do the following: •

Apply concepts, skills, models, and strategies connected to the lesson objective to solve problems

•

Make thinking visible by using numbers, words, models, and tools

•

Attempt to use accurate mathematical language in discourse and writing

•

Ask questions to clarify their thinking or understand the reasoning of others

•

Articulate understanding in whole-class, peer-to-peer, and teacher-to-student discourse Make connections between concepts and skills and between current and previous content

•

Use the Read–Draw–Write process to solve word problems

Calls on students or selects student work strategically to highlight trends, varied solutions, misconceptions, and broader mathematical understanding

Models or directs students to use specific sections of the Talking Tool and Thinking Tool to support metacognition and discourse

Verifies possible solutions throughout Learn by inviting students to share work or revealing objective-aligned

Promoting Discourse

Incorporates Read–Draw–Write process (grades 1–5) for solving word problems

•

understanding to existing problems in the sequence

misconceptions (uses the Observational Assessment

understandings

Monitors student work to notice and note trends,

Creates and inserts problems that bridge or reinforce

Responding

varied models and strategies, exemplars, and

Prepares the environment to optimize engagement, learning, and collaboration (e.g., desk Uses structures, routines, and activities as opportunities for students to develop their

pairings/groupings) r.

Assessment Recording Sheet (OARS) for K–2) n.

arrangement, Thinking Tool, Talking Tool, materials, visuals) i.

Listens to student discourse to gauge students’ ability

Uses the margin notes to improve access (e.g., UDL, Differentiation: Support, Language Support, strategic

mathematical terminology (uses the Observational

Using Structures, Routines, and Activities States directions clearly and includes all necessary details for engagement (e.g., topic/task,

Circulates to recognize and encourage application of

to articulate their understanding and use precise

pages, and Problem Set g.

mathematical understanding m.

Uses the mathematical language of the lesson or related language (e.g., decompose and unbundle) to demonstrate precise yet accessible terminology

Customizing

Provides appropriate think time for students to process questions or displayed content and form responses

strategies and objective e.

Adaptive Implementation

exemplars v.

Addresses learner variance by challenging or supporting students (e.g., modeling, targeted questioning, connecting to prior learning, adjusting complexity)

w. Adjusts pacing within Learn segments to meet the needs of students while honoring the objective

Implementation Support Tool K–5 Problem Set The Problem Set—a part of Learn—is an opportunity for independent practice. Core Implementation

Meeting Component Purpose a. Allots time for every student to work on the Problem Set independent of teacher guidance b. States directions clearly and includes all necessary details for engagement (e.g., topic/task, timeframe, peer interaction) c.

Prioritizes any Problem Set problems that are referenced in Land

Instructional Habits

Engaging d. Limits time with individual students to prioritize monitoring every student’s work e. Asks students questions to prompt the continuation of work while maintaining the cognitive lift Monitoring f. Monitors every student’s work to notice and note trends, varied models and strategies, exemplars, and misconceptions Promoting Discourse indicators are excluded from the Problem Set section of the IST due to the nature of the lesson component.

Adaptive Implementation

Customizing g.

Designates the order of problems to complete to ensure practice with the most important components of the lesson objective

Creates and assigns additional problem sequences, as needed, to provide access or advance learning

Responding i. Directs students to leverage classroom resources (e.g., anchor charts, other examples in their own work, Talking Tool, Thinking Tool, Practice Partner) j.

Student Expectations

While working on a given Problem Set, students in a Eureka Math2 classroom should do the following: • • • • •

Engage in practice, independent of the teacher, for the allotted time Think about, make sense of, and represent problems before solving them Show work and explain their thinking as indicated Refer to resources (e.g., anchor charts, other examples from the lesson, Thinking Tool, Practice Partner) for support before asking for help Adjust existing work based on teacher feedback

Responds to pervasive misconceptions by questioning or modeling with a parallel example to provide a scaffold in real time

Implementation Support Tool K–5 Land Land helps facilitate a brief discussion to close the lesson and provides students with an opportunity to complete the Exit Ticket. Suggested questions, including key questions related to the objective, help students synthesize the day’s learning. The Exit Ticket provides a window into what student understand that can be used to inform instructional decisions. Core Implementation

Instructional Habits

Meeting Component Purpose a.

Engaging

Asks questions that elicit student thinking and

cause them to synthesize the day’s learning (e.g., suggested questions, key questions)

b. Focuses student discussion to align with the

Debrief

lesson objective, highlighting important concepts and vocabulary c.

Honors the amount of student-to-student discourse and whole-class discourse within Land

Adaptive Implementation

Completes the Debrief within or near the indicated time

Customizing

Provides adequate think time for students to process questions or displayed content and to

collected data, that promote synthesis of the lesson or topic Responding

Monitoring Listens to student discourse to gauge understanding and identify opportunities to push for

k. Affirms accuracy and efficiency in student responses while clarifying any inaccuracies and

clearer thinking

deepening understanding

Promoting Discourse

h. Prompts students to elaborate or clarify responses (e.g., by saying “Why?” “How do you

Revisits questions from earlier in the lesson to check for growth in understanding

know?” or “Tell me more”), or to use precise language to advance student thinking

Creates questions, in response to previously

Provides students with supports (e.g., Talking Tool, mathematical terminology) to improve access to discussion

form responses

Prompts students to restate, build on, or evaluate other students’ responses to enhance discussion and strengthen habits of discussion

Meeting Component Purpose

Engaging

Exit Ticket (Grades 1–5)

a. Allows an amount of time within or near the

indicated time allotted for students to

Responding

Makes tools (e.g., anchor charts, concrete manipulatives, Thinking Tool) available to support independence

independently complete as much of their Exit Ticket

b. Collects Exit Tickets to analyze qualitative data to identify strengths to leverage in future lessons, varied solution paths, and misconceptions to

Designates the order of problems for students to complete to ensure students answer the problems

Monitoring

that highlight the most important components of

d. Circulates room while students work on their Exit Tickets to ensure student accountability

as possible

the lesson objective (within the given time)

and gather formative data around unproductive struggle

e. Monitors every student’s work to note trends and confirm the effectiveness of the lesson to inform customization of the next day’s lesson Promoting Discourse indicators are excluded from the Exit Ticket section of the IST due to the

address

nature of the lesson component. Student Expectations

During the Debrief component of Land, students in a Eureka

Math2

classroom should refer to the day’s activities to do the

following:

During the Exit Ticket component of Land, students in a Eureka Math2 classroom should do the following: •

Solve a problem representative of the lesson objective

•

Attempt to use accurate mathematical language when synthesizing the lesson’s key understandings

•

Make thinking visible by using numbers, words, and models

•

Engage in discourse to articulate understanding, gain the insights of others, and evaluate errors

•

Complete as much of the Exit Ticket as possible

•

Reflect on their own thinking following analysis of peer work and discourse

Implementation Support Tool K–5

Appendix I: Structure The Implementation Support Tool (IST) includes a section for each lesson component of Eureka Math2 lessons (Fluency, Launch, Learn, and Land), as well as a section dedicated to the Problem Set—a part of Learn.

Each section of the IST contains a series of indicators, which are brief statements that together describe exemplary implementation of the corresponding section. Indicators are lettered for quick reference (e.g., Launch indicator (a) or Exit Ticket indicator (d)).

The indicators for each lesson component are organized into four primary categories—Core Implementation, Instructional Habits, Adaptive Implementation, and Student Expectations— which are described below. The Core Implementation indicators describe facilitation of the lesson component as designed. The curriculum materials (e.g., Teach book, digital platform) contain all the resources necessary for teachers to model the Core Implementation indicators. The Instructional Habits indicators describe practices that, when used routinely, provide teachers with opportunities to optimize engagement, gather formative data, and develop students’ ability to articulate themselves mathematically. The Adaptive Implementation indicators describe ways to effectively adapt Eureka Math2 in response to formative data, both during lesson preparation and in the moment. The Student Expectations indicators describe behaviors students exhibit when the learning in each lesson component is optimized.

Implementation Support Tool K–5

Appendix I: Structure, Continued In most cases, the columns for Core Implementation, Instructional Habits, and Adaptive Implementation contain subcategories that group indicators by their purpose within instruction. Indicators have been categorized based on their primary purpose; however, the implementation of a category’s indicators often positively impacts instruction across other categories. The following table describes the seven subcategories and provides corresponding examples of each from the Launch section of the IST. Core Implementation

Component Purpose indicators describe individual actions that are essential to the component or a culmination of actions that embody the intent of the component. Ex. Provides opportunities indicated by lesson for students to notice; wonder; apply; and articulate strategies, choice of models, and understandings

Structures, Routines, and Activities indicators describe essential actions for optimizing student participation during structures, routines, and activities (e.g., turn and talks, think–pair–shares, partner work, group work, class discussions, choral response, instructional routines, Read–Draw–Write, notice and wonders, context videos, and digital interactives). Ex. Prepares the environment to optimize engagement, collaboration, and learning (e.g., desk arrangement, Thinking Tool, Talking Tool, materials, visuals)

Instructional Habits

Adaptive Implementation

Engaging indicators describe teacher habits that maximize the number of students actively engaging at a given moment.

Customizing indicators describe ways to adapt Eureka Math2 lessons prior to facilitation in service of students’ needs based on sources including, but not limited to, previous Problem Sets, Exit Tickets, and Topic Tickets/Quizzes.

Ex. Provides appropriate think time for students to process questions or displayed content and form responses

Monitoring indicators describe teacher habits that maximize a teacher’s awareness of student engagement and understanding. Ex. Listens to student discourse for understanding, connections to prior knowledge, and evolving reasoning

Promoting Discourse indicators describe teacher habits that maximize clarity and precision as students articulate themselves mathematically. Ex. Prompts students to elaborate (e.g., by saying “Why?” “How do you know?” or “Tell me more”) to make student thinking more visible, clear, and complete

Ex. Uses the margin notes to improve access (e.g., UDL, Differentiation: Support, Language Support, strategic pairings/groupings)

Responding indicators describe ways to adapt Eureka Math2 lessons in the moment in response to formative data collected during lesson facilitation. Ex. Calls on students or selects student work strategically to highlight trends, varied solutions, connections to prior content, and misconceptions

Implementation Support Tool 6–9 A Story of Ratios® A Story of Functions®

Implementation Support Tool 6–9

•

The teacher actions described on the IST are not all-or-nothing actions. Each can be performed with varying degrees of effectiveness.

•

Coaches can use the IST to develop a deeper understanding of Eureka Math2 during early implementation observations.

•

Coaches can use the IST to name instructional priorities for targeted professional development, PLCs, and 1:1 coaching cycles.

•

Coaches can refer to the IST for shared language during feedback conversations with teachers following observation.

•

Coaches can use the IST with teachers to prompt goal setting and self-reflection.

•

Individual teachers or teams can use the IST during lesson preparation to look for opportunities to leverage prioritized teacher actions.

•

Individual teachers can develop their practice by using the IST to reflect on their current instruction. This reflection can be supported by using the IST to analyze recorded classroom footage.

Implementation Support Tool 6–9

Fluency (Optional) In A Story of Ratios and A Story of Functions, Fluency is not part of the 45-minute lesson structure. If lesson time permits, a fluency activity can be administered as a bell ringer or adapted as one of the teacher-led routines below. Fluency uses activities that solidify and build students’ ability to use mathematical procedures flexibly, accurately, efficiently, and appropriately. Students become familiar with fluency routines because of their consistent use across modules and grade levels, allowing for efficient teaching and learning. All fluency routines benefit from the general fluency indicators below. The most common routines—Whiteboard Exchange, Choral Response, Count By, and Sprints—have additional sets of indicators to support implementation.

Choral Response

Whiteboard Exchange

General Fluency Indicators

Core Implementation

Instructional Habits

Adaptive Implementation

Meeting Component Purpose a. Completes fluency activity within or near suggested time (10 minutes for Sprints, 3–5 minutes for all other routines) b. States directions clearly and includes all necessary details for engagement (e.g., topic/task, timeframe, cue) c. Prepares the environment to optimize engagement and learning (e.g., location of the activity, materials, concrete manipulatives, visuals)

Engaging and Monitoring indicators vary by routine and are included for specific fluency routines only.

Responding g. Adjusts the sequence of problems (e.g., complexity, level of abstraction) based on the existence of pervasive trends

Promoting Discourse indicators are excluded from the Fluency section of the IST due to the nature of the lesson component.

Customizing d. Administers alternate fluency activity to increase engagement or in response to a demonstrated student need e. Scaffolds the sequence with additional problems to provide access (e.g., lower complexity initial problems, intermediate problems to ease complexity, extension problems) Responding f. Models or thinks aloud about a parallel problem or sequence to address a widespread, small-gap misconception and then re-engages students in a similar sequence (excluding Sprints)

Implementation Support Tool 6–9

Fluency, Continued

Sprints

Count By

Core Implementation

Instructional Habits

Engaging d. Uses appropriate pace for the complexity and familiarity of the count

Using Routines a. Directs students to complete the sample problems to ensure their understanding of the purpose of the sprint b. Frames the sprints as an opportunity for growth (e.g., encouraging and celebrating effort, success, and improvement) c. Provides 1 minute for students to complete as many problems as possible, in order without skipping d. Reviews the correct answers to each sprint at a brisk pace and with an established verbal cue that allows for all students to follow along e. Provides an opportunity for students to complete more problems on sprint A or to analyze and discuss the patterns in Sprint A to increase success on sprint B f. Directs students to record their performance and to calculate and celebrate growth after sprint B

Engaging g. Provides appropriate think time for students to internalize patterns before transitioning to sprint B

Adaptive Implementation

Students Expectations

During fluency activities, students in a Eureka Math2 classroom should do the following: • Participate fully (verbally, on whiteboards, with hands when appropriate, etc.) • Respond to established signals and prompts • Include units in responses, when appropriate • Make corrections, when appropriate

Implementation Support Tool 6–9

Launch Launch creates an accessible entry point to the day’s learning through activities that build context, create a need for new learning, or activate prior knowledge. Every Launch ends with a transition statement that sets the goal for the day’s learning. Core Implementation

Meeting Component Purpose a.

Aligns facilitation to the purpose statement of Launch

Provides opportunities indicated by lesson for students to notice; wonder; apply; and articulate strategies, choice of models, and understandings

Honors the amount of teacher-to-student discourse, student-tostudent discourse, and whole-class discourse within Launch

Concludes Launch by using the given transition statement to set the goal for and make an explicit connection to the day’s learning

Completes the Launch within or near the time indicated

Instructional Habits

Engaging h. Provides appropriate think time for students to process questions or displayed content and form responses i.

Encourages flexible thinking (e.g., varied models and strategies) and connections to prior knowledge to access content

Monitoring j. Circulates to recognize and encourage application of mathematical understanding k. Listens to student discourse for understanding, connections to prior knowledge, and evolving reasoning l. Monitors student work to notice and note trends, varied models and strategies, exemplars, and misconceptions to leverage during class discussion Promoting Discourse m. Prompts students to elaborate (e.g., by saying “Why?” “How do you know?” or “Tell me more”) to make student thinking more visible, clear, and complete n. Asks questions that invite students to make connections between solutions, models, strategies, and previous lessons, modules, or grade levels o. Prompts students to restate, build on, or evaluate other students’ responses to enhance discussion and strengthen habits of discussion Student Expectations

During Launch, students in a Eureka Math2 classroom should do the following: • • • • •

Adaptive Implementation

Customizing p. Uses the margin notes to improve access (e.g., UDL, Differentiation: Support, Language Support, strategic pairings/groupings) Responding q.

Calls on students or selects student work strategically to highlight trends, varied solutions, connections to prior content, and misconceptions

Models the use of or directs students to use specific sections of the Talking Tool and Thinking Tool to support discourse and metacognition

Implementation Support Tool 6–9

Learn Learn presents new learning related to the lesson objective, usually through a series of instructional segments. This lesson component takes most of the instructional time. Suggested facilitation styles vary and may include explicit instruction, guided instruction, group work, partner activities, and digital elements. Core Implementation

Instructional Habits

Engaging

Meeting Component Purpose a. b.

Focuses the progression of questioning, thinking, and discourse in alignment with the purpose statement of each Learn segment Honors the ratio of questioning to direct instruction, the amount of discourse, and the format of the Releases responsibility to students while working on lesson pages, as indicated by the lesson and

Uses tools and models accurately to develop student understanding of the lesson’s strategies and

Completes Learn within or near the total time indicated, allowing time for instruction, lesson pages,

Monitors student work to notice and note

States directions clearly and includes all necessary details for engagement (e.g., topic/task, thinking job, timeframe, cues, peer interaction)

Promoting Discourse o.

Prepares the environment to optimize engagement, learning, and collaboration (e.g., desk

you know?” or “Tell me more”), and use precise language to advance student thinking

Uses structures, routines, and activities as opportunities for students to develop their

Asks questions that invite students to make connections between solutions; models;

Incorporates Read–Represent–Solve–Summarize (RRSS) process (grades 6–8) or Modeling Cycle

strategies; and previous questions, lessons,

(Algebra I) for solving word problems

modules, or grade levels Student Expectations

During Learn, students in a Eureka Math2 classroom should do the following: •

Apply concepts, skills, models, and strategies connected to the lesson objective to solve problems

•

Make thinking visible by using numbers, words, models, and tools

•

Attempt to use accurate mathematical language in discourse and writing

•

Ask questions to clarify their thinking or understand the reasoning of others

•

Articulate understanding in whole-class, peer-to-peer, and teacher-to-student discourse

•

Make connections between concepts and skills and between current and previous content

•

Use the Read–Represent–Solve–Summarize or Modeling Cycle process to solve word problems

Calls on students or selects student work strategically to highlight trends, varied solutions, misconceptions, and broader mathematical understanding

Models or directs students to use specific sections of the Talking Tool and Thinking Tool to support metacognition and discourse

Verifies possible solutions throughout Learn by inviting students to share work or revealing objective-aligned

Prompts students to elaborate or clarify responses (e.g., by saying “Why?” “How do

understanding of the day’s objectives and to articulate their strategies, models, and understandings

trends, varied models and strategies,

arrangement, Thinking Tool, Talking Tool, materials, visuals) i.

understanding to existing problems in the sequence

exemplars, and misconceptions

Using Structures, Routines, and Activities

Creates and inserts problems that bridge or reinforce

Responding

use precise mathematical terminology n.

and Practice Problems g.

pairings/groupings) r.

ability to articulate their understanding and

Uses the mathematical language of the lesson or related language (e.g., ratio relationship and value of the ratio) to demonstrate precise yet accessible terminology

Listens to student discourse to gauge students’

Uses the margin notes to improve access (e.g., UDL, Differentiation: Support, Language Support, strategic

Circulates to recognize and encourage application of mathematical understanding

objectives e.

form responses

Monitoring

student performance d.

Customizing

to process questions or displayed content and

discourse so that students generate, test, share, critique, and refine their ideas c.

Provides appropriate think time for students

Adaptive Implementation

exemplars v.

Addresses learner variance by challenging or supporting students (e.g., modeling, targeted questioning, connecting to prior learning, adjusting complexity)

w. Adjusts pacing within Learn segments to meet the needs of students while honoring the objective

Implementation Support Tool 6–9

Practice Problems Practice problems may be assigned for completion outside of class or used during additional instructional time as an opportunity for in-class independent practice. The indicators below should be used when a teacher has assigned Practice problems to be used during class time. Core Implementation

Meeting Component Purpose a. Allots time for every student to work on the Practice problems independent of teacher guidance b. States directions clearly and includes all necessary details for engagement (e.g., topic/task, timeframe, peer interaction)

Instructional Habits

Engaging c. Limits time with individual students to prioritize monitoring every student’s work d. Asks students questions to prompt the continuation of work while maintaining the cognitive lift Monitoring e. Monitors every student’s work to notice and note trends, varied models and strategies, exemplars, and misconceptions Promoting Discourse indicators are excluded from the Practice problem section of the IST due to the nature of the lesson component.

Adaptive Implementation

Customizing f. Designates the order of problems to complete to ensure practice with the most important components of the lesson objective g.

Responding h. Directs students to leverage classroom resources (e.g., anchor charts, other examples in their own work, Talking Tool, Thinking Tool, Lesson Recap) i.

Student Expectations

While working on given Practice problems, students in a Eureka Math2 classroom should do the following: • • • • •

Engage in practice, independent of the teacher, for the allotted time Think about, make sense of, and represent problems before solving them Show work and explain their thinking as indicated Refer to resources (e.g., anchor charts, other examples from the lesson, Thinking Tool, Lesson Recap) for support before asking for help Adjust existing work based on teacher feedback

Creates and assigns additional problem sequences, as needed, to provide access or advance learning

Responds to pervasive misconceptions by questioning or modeling with a parallel example to provide a scaffold in real time

Implementation Support Tool 6–9

Land Land helps facilitate a brief discussion to close the lesson and provides students with an opportunity to complete the Exit Ticket. Suggested questions, including key questions related to the objective, help students synthesize the day’s learning. The Exit Ticket provides a window into what student understand that can be used to inform instructional decisions. Core Implementation

Instructional Habits

Meeting Component Purpose a.

Engaging

Asks questions that elicit student thinking and

cause them to synthesize the day’s learning (e.g., suggested questions, key questions)

Debrief

lesson objectives, highlighting important concepts and vocabulary Honors the amount of student-to-student discourse Completes the Debrief within or near the indicated time

improve access to discussion g.

Creates questions, in response to previously collected data, that promote synthesis of the lesson or topic

Provides students with supports (e.g., Talking Tool, mathematical terminology) to

Responding

k. Affirms accuracy and efficiency in student responses

Monitoring

and whole-class discourse within Land d.

Customizing

Provides adequate think time for students to process questions or displayed content and to form responses

b. Focuses student discussion to align with the

Adaptive Implementation

while clarifying any inaccuracies and deepening

Listens to student discourse to gauge understanding and identify opportunities to push for clearer thinking

understanding

Promoting Discourse

Revisits questions from earlier in the lesson to check for growth in understanding

h. Prompts students to elaborate or clarify responses (e.g., by saying “Why?” “How do you know?” or “Tell me more”), or to use precise language to advance student thinking

Prompts students to restate, build on, or evaluate other students’ responses to enhance discussion and strengthen habits of discussion

Meeting Component Purpose

Engaging

a. Allows an amount of time within or near the

indicated time allotted for students to

support independence

independently complete as much of their Exit Ticket

Exit Ticket

Responding

Makes tools (e.g., anchor charts, concrete manipulatives, Thinking Tool) available to

b. Collects Exit Tickets to analyze qualitative data to identify strengths to leverage in future lessons, varied solution paths, and misconceptions to

Designates the order of problems for students to complete to ensure students answer the problems

Monitoring

that highlight the most important components of the

d. Circulates room while students work on their Exit Tickets to ensure student

as possible

lesson objective (within the given time)

accountability and gather formative data around unproductive struggle

address

nature of the lesson component. Student Expectations

During the Debrief component of Land, students in a Eureka

Math2

classroom should refer to the day’s activities to do the

following:

During the Exit Ticket component of Land, students in a Eureka Math2 classroom should do the following: •

Solve a problem representative of the lesson objective

•

Attempt to use accurate mathematical language when synthesizing the lesson’s key understandings

•

Make thinking visible by using numbers, words, and models

•

Engage in discourse to articulate understanding, gain the insights of others, and evaluate errors

•

Complete as much of the Exit Ticket as possible

•

Reflect on their own thinking following analysis of peer work and discourse

Supplemental Materials

Lead: Facilitating Successful Implementation of Eureka Math2, K–9

Appendix B Scenarios Scenario 1: Imagine that a 6th grade teacher comes to your office and says, “I’m used to teaching rational numbers at the beginning of the year, so I was going to adjust Module 2 and 3 to come before Module 1.” Sample Response: “The writers of the curriculum carefully crafted the scope and sequence, considering the placement of content down to the lesson level. There is a logical reason for the organization of the modules. Let’s go into the front matter of the modules to find out why Modules 2 and 3 come after Module 1.” Scenario 2: “My students didn't get Eureka Math2 last year, so they don't understand the way double number lines are introduced in this lesson. And tomorrow's lesson moves too quickly. I need to add a review day before I can teach this lesson and then after this lesson to prepare my students for tomorrow.” Sample Response: ““Let’s go back and revisit the module and topic overview to see how learning progresses throughout to create a plan to support students in accessing the day’s lesson. We only get a few days to use for responsive teaching, so we want to make sure those days are strategically placed, just in time.” In a scenario such as this, it may be helpful to revisit the module and topic to see where the lesson fits in to the trajectory of learning and return to the lesson to identify the purpose and role of that lesson in that trajectory of learning. Then, review the lesson to see how the learning strategically builds and creates multiple entry points for students. Identify support for learning that will allow students to access the day’s lesson. You may also want to investigate the multiple instances of double number line usage to ensure the teacher sees that students will have continued opportunities to use and become more proficient with using them. Scenario 3: You are completing a walkthrough and it is 15 minutes in to the 60-minute math block and the teacher is checking students’ homework while they are completing a math worksheet. Sample Response: Pacing will be impacted by the decision because there are only 30 minutes left in the block and 45 minutes of content in the lesson. The response to this will depend on the specific situation and purpose of the observation. You might want to determine the reason the teacher is using class time to review the previous day’s learning. You could say something like, “The way the Eureka Math2 lessons are designed, you can be confident that there are multiple entry points embedded into the lesson to help activate prior knowledge.” Scenario 4: The seventh-grade team has a 80-minute math block, on a block schedule. A teacher approaches you and says that they are unclear on how to fit two Eureka Math2 lessons in the allotted time. What do you do? Sample Response: It is recommended to reevaluate the schedule to a lot 60 minutes for second through fifth grade Eureka Math2 lessons. In order to complete two Eureka Math2 lessons in the same block, it is recommended to reevaluate the schedule to allot 90 minutes. We can use our

Supplemental Materials

Lead: Facilitating Successful Implementation of Eureka Math2, K–9 study of the trajectory of learning through module and topic level resources to make informed adjustments to the lessons. Understanding the purpose of each of the related lessons, and how they build, will allow us to make decisions about what to prioritize, shorten or even consolidate, to ensure that they fit within our time frame without sacrificing the big ideas, rigor and coherence of the lessons. We can also use our data from previous Exit Tickets to determine students’ entry points and help us make strategic decisions about the lessons.

Eureka Math2® Implementation Benchmarks Overview The Eureka Math2 curriculum aims to help all students think deeply about mathematics and become critical thinkers and problem solvers. A successful implementation of Eureka Math2 takes dedication from all stakeholders and progresses through the following four phases.

Engage

Experience

Enhance

Exemplify

Prior to Year 1

Year 1

~Years 2-3

~Years 4 and Beyond

Engage: Preparing to launch Eureka Math2

Experience: Learning and exploring Eureka Math2

Enhance: Extending knowledge of Eureka Math2

Exemplify: Skillfully implementing Eureka Math2

Teachers and leaders ensure that plans and materials are in place for the implementation of Eureka Math2. All members of the learning community understand the rationale behind the adoption and are invested in its success.

Teachers and leaders gain knowledge of Eureka Math2. They identify and explore key structures and aspects of the curriculum. As the learning community gains experience, implementation results may vary across classrooms and schools.

Teachers and leaders increase their understanding of and familiarity with Eureka Math2. They become more consistent in their pacing and lesson facilitation. Educators customize lessons to meet students’ needs while maintaining the curriculum’s rigor and intentionality.

Teachers and leaders facilitate a highly effective implementation of Eureka Math2 across classrooms and schools. Educators effectively maintain pacing and respond to student data when planning instruction.

Notes:  This resource is not intended as an evaluative tool. Instead, it should guide the progression of an implementation as each phase builds on previous phases. 

The timeline provided is a guideline. Previous experience with Eureka Math may lead to progressing through the phases at a faster pace.



In this resource, leaders may include, but are not limited to, district administrators, curriculum directors, principals, assistant principals, and instructional coaches. Teachers include, but are not limited to, general education teachers, special education teachers, intervention specialists, and paraprofessionals.

Eureka Math2® Implementation Benchmarks—Engage Phase (Prior to Year 1)1

Engage Preparing to launch Eureka Math2.

Experience

Leaders  

 



Identify individuals to lead and support implementation and define their roles. Ensure access to all print and digital curriculum materials for leaders, teachers, and students. Plan professional development for leaders and teachers. Participate in Lead: Facilitating Successful Implementation professional development. Introduce the learning community to Eureka Math2 and ensure buy-in for the implementation. Understand how Eureka Math2 assessments guide instruction.

Enhance

Teachers 

 

Participate in Launch: Bringing the Curriculum to Life or Power Up: Transitioning to Eureka Math Squared professional development.2 Preview print and digital curriculum resources. Organize materials such as teacher editions, student materials, and manipulatives.

Exemplify

Students 

There are no student actions during the Engage phase.

The timeline mentioned is a guideline. A specific implementation may move through these phases more or less quickly. The Power Up session is designed for teachers with prior Eureka Math experience.

Eureka Math2® Implementation Benchmarks – Experience Phase (Year 1)1

Engage

Experience

Leaders 



Set expectations for teachers regarding the implementation of Eureka Math2 (e.g., use Eureka Math2 daily). Use observations and feedback conversations to support teachers toward optimal implementation of Eureka Math2 . Develop a culture of curriculum study and provide supporting structures for teacher collaboration and planning. Celebrate teachers’ progress in attempts to incorporate new practices from the curriculum.

Exemplify

Teachers 

  



Enhance

Learning and exploring Eureka Math2.

Participate in Teach: Effective Instruction with Eureka Math2 and Assess: Embedded Opportunities to Support Instruction professional development sessions. Study curriculum materials to prepare for instruction. Follow the Fluency, Launch, Learn, and Land lesson structure. Learn the models and strategies emphasized in Eureka Math2 and use them as indicated. Maintain pacing, with the understanding that students develop proficiency over time. Introduce instructional routines (ie: What Doesn’t Belong, Math Chat) found in the curriculum.

Students 

   

Learn the models and strategies emphasized in Eureka Math2 and use them as indicated. Provide written and verbal explanations of mathematical problem solving. Ask and answer mathematical questions. Engage in mathematical discourse. Actively engage in instructional routines (ie: What Doesn’t Belong, Math Chat) found in the curriculum.

The timeline mentioned is a guideline. A specific implementation may move through these phases more or less quickly. 79

Eureka Math2® Implementation Benchmarks – Enhance Phase (~Years 2–3)1

Engage

Experience

Leaders 



Enhance

Plan professional development and additional support for new teachers or teachers new to a grade level. Use observations and feedback conversations to ensure lesson customizations maintain fidelity to the lesson objectives and support students’ needs. Establish structures for using assessments to effectively inform instructional decisions. Provide feedback that praises progress and pushes practice on content and instruction.

Exemplify

Extending knowledge of Eureka Math2.

Teachers 



Participate in the Adapt: Optimizing Instruction and Inspire: Discourse, Engagement, and Identity professional development. Effectively use the curriculum materials and student data to adapt daily instruction. Facilitate instructional routines, highlighting the Standards for Mathematical Practice, to promote student discourse. Encourage students to use models and strategies flexibly to solve problems and build their understanding of mathematics. Use data from assessment reports (Topic Quizzes, Module Assessments, etc.) to inform instructional choices.

Students 



Use models and strategies flexibly to solve problems and build understanding of mathematics. Use mathematical language in verbal and written communication about mathematics and problem solving. Demonstrate persistence in learning math and solving problems.

The timeline mentioned is a guideline. A specific implementation may move through these phases more or less quickly.

Eureka Math2® Implementation Benchmarks – Exemplify Phase (~Years 4 and Beyond)1

Engage



Enhance

Experience

Leaders Maintain systems for ongoing professional development, collaboration, and planning, accounting for varied experiences of teachers. Use observations and feedback conversations to ensure lesson customizations differentiate based on students’ needs. Analyze classroom and school data to identify inequity and implementation concerns, and make a plan to address them. Maintain systems for teacher development to ensure all teachers receive frequent support, observation, and feedback.

Exemplify Skillfully implementing Eureka Math2.

Teachers  

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Develop understanding of alignment with prior and successive grade levels. Make connections to prior or upcoming content, provide scaffolds and extend learning. Use data from assessments to inform instruction to meet whole class, small group, and individual student needs. Promote student-initiated and studentto-student discourse. Generously share experiences to mentor teachers new to the curriculum.

Students     

See themselves as mathematicians, expressing confidence in their ideas. Independently notice, wonder, and make connections to prior learning. Consistently provide clear explanations of problem solving processes. Offer critiques of others’ mathematical work. Demonstrate curiosity through mathematical wonderings.

The timeline mentioned is a guideline. A specific implementation may move through these phases more or less quickly.

Lead K – 9 IST and Supplemental Lesson Materials

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Topic: Grade 7 Module 3 Topic B