1 Place Value Concepts for Addition and Subtraction
2 Place Value Concepts for Multiplication and Division
3 Multiplication and Division of Multi-Digit Numbers
4 Foundations for Fraction Operations
5 Place Value Concepts for Decimal Fractions
6 Angle Measurements and Plane Figures
Before This Module
Grade 4 Module 2
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.
Overview
Multiplication and Division of Multi-Digit Numbers
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 to find products and quotients. Application of the associative property helps students to rewrite two-factor multiplication expressions as three-factor expressions so, again, they can 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.
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.
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.
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
Students use multiplicative relationships to convert units of time and customary units of weight and liquid volume to smaller units. They use conversion tables and number lines to express larger measurement units in terms of smaller units and recognize that the smaller units are all multiples 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.
Multiplication and Division of Multi-Digit Numbers
Why
Multiplication and Division of Multiples of Tens, Hundreds, and Thousands
Lesson 1
Divide multiples of 100 and 1000
Lesson 2
Multiply by multiples of 100 and 1000
Lesson 3
Multiply a two-digit multiple of 10 by a two-digit multiple of 10
Topic B
Division of Thousands, Hundreds, Tens, and Ones
Lesson 4
Apply place value strategies to divide hundreds, tens, and ones.
Lesson 5
Apply place value strategies to divide thousands, hundreds, tens, and ones.
Lesson 6
Connect pictorial representations of division to long division.
Lesson 7
Represent division by using partial quotients.
Lesson 8
Choose and apply a method to divide multi-digit numbers.
63
Topic C
Multiplication of up to Four-Digit Numbers by One-Digit Numbers
Lesson 9 .
Apply place value strategies to multiply three-digit numbers by one-digit numbers.
Lesson 10
Apply place value strategies to multiply four-digit numbers by one-digit numbers.
Lesson 11
Represent multiplication by using partial products.
Lesson 12
Multiply by using various recording methods in vertical form.
Topic D
66
Multiplication of Two-Digit Numbers by Two-Digit Numbers
Lesson 13
Multiply two-digit numbers by two-digit multiples of 10.
Lesson 14
Apply place value strategies to multiply two-digit numbers by two-digit numbers.
Lesson 15
Multiply with four partial products.
Lesson 16
Multiply with two partial products.
Lesson 17
Apply the distributive property to multiply.
Topic E
Problem Solving with Measurement
Lesson 18
Express units of time in terms of smaller units.
Lesson 19
Express customary measurements of weight in terms of smaller units.
Lesson 20
Express customary measurements of liquid volume in terms of smaller units.
Topic F
Remainders, Estimating, and Problem Solving
Lesson 21
Find whole-number quotients and remainders.
Lesson 22
Represent, estimate, and solve division word problems.
Lesson 23
Solve multi-step word problems and interpret remainders.
Lesson 24
Solve multi-step word problems and assess the reasonableness of solutions.
Acknowledgments
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 of module 2. In module 2, students multiply and divide with tens and ones. Division follows multiplication and students relate the methods used for division with those they use for multiplication (e.g., distributive property, area model). Module 2 ends with a focus on factors and multiples. To continue the progression of learning from 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.
What is the intent of using vertical form for multiplication and division?
In grade 4, students multiply and divide by using strategies based on place value and the properties of operations. This module uses the representations of place value charts, area models, and vertical form because they are based on place value understanding. The intent of vertical form is to provide 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 with addition and subtraction, 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.
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.
4.Mod3.AD1
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.
4.OA.A.3
4.Mod3.AD2
Multiply whole numbers of up to four digits by one-digit whole numbers, and multiply 2 two-digit whole numbers.
4.Mod3.AD3
Divide whole numbers of up to four digits by one-digit whole numbers.
4.Mod3.AD4
Express larger units of time, customary units of weight, and liquid volumes in terms of a smaller unit by using tables.
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.
4.NBT.B.5
4.NBT.B.6
4.MD.A.1
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
4.Mod3.AD2 Multiply whole numbers of up to four digits by one-digit whole numbers, and multiply 2 two-digit whole numbers. 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.
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. Which model shows 3 × 1,078?
Multiply whole numbers of up to four digits by one-digit whole numbers, and multiply 2 two-digit whole numbers. Multiply. 3 × 1,078 =
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.
Casey found 3 × 1,078. Her work is shown.
3,000 + 2,10 0 + 24 = 5,12 4
Casey made a mistake. Explain the mistake and what she should do to correct it.
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.
Progression of Lessons
Lesson 1
multiples of 100 and 1000.
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.
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, 4 × 500 = 4 × (5 × 100) = (4 × 5) × 100 = 2000 .
Lesson 3
Multiply a two-digit multiple of 10 by a two-digit multiple of 10.
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.
Divide multiples of 100 and 1000 .
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
• How can thinking about place value units help us to divide multiples of 100 and 1000?
Achievement Descriptor
4.Mod3.AD3 Divide whole numbers of up to four digits by one-digit whole numbers. (4.NBT.B.6)
Agenda Materials
Fluency 10 min
Launch 5 min
Learn 35 min
• Represent Division with Place Value Disks
• Use Unit Form to Divide
• Divide Multiples of 100
• Problem Set
Land 10 min
Teacher
• Place value disks set
• Teacher computer or device*
• Projection device*
• Teach book*
Students
• Place value disks set
• 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.
Lesson Preparation
Gather at least 12 thousands disks, 12 hundreds disks, 12 tens disks, and 12 ones disks for each student and the teacher.
Fluency
Happy Counting by Hundreds
Students visualize a number line while counting aloud to develop familiarity with Happy Counting.
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.
Teacher Note
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.
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
Display the answer.
Write the equation in standard form.
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.
Display the equation.
Continue with 6 tens ÷ 2 = tens, displaying the answer after each question or prompt.
What is 6 tens ÷ 2?
Write the equation in standard form.
Repeat the process with the following sequence:
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.
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:
Launch
Students relate dividing one-dollar bills, ten-dollar bills, and hundred-dollar bills.
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.
Display the picture of the bills.
Invite students to think–pair–share about how Liz can divide the one-dollar bills.
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.
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. 5
Language Support
Consider using strategic, flexible grouping throughout the module.
• Pair students who have different levels of mathematical proficiency.
• Pair students who have different levels of English language proficiency.
• Join pairs to form small groups of four.
As applicable, complement any of these groupings by pairing students who speak the same native language.
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 hundreds ÷ 2
6 thousands ÷ 2
Display the picture of the completed representations.
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.
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.
Teacher Note
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.
Use Unit Form to Divide
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.
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 ones ÷ 4
8 tens ÷ 4
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.
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
20 ones ÷ 4
Teacher Note
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.
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.
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.
What is 20 ones ÷ 4?
5 ones
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.
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.
Promoting the Standards for Mathematical Practice
Students look for and express regularity in repeated reasoning (MP8) as they divide by using unit form and basic division facts.
Ask the following questions to promote MP8:
•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?
Copyright
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.
UDL:
Action & Expression
Consider providing place value disks for students to use as they divide 2100 by 7 .
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.
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.
Invite students to think–pair–share about how standard form and unit form are used in the equation.
2100 ÷ 3 = 21 hundreds ÷ 3 = 7 hundreds = 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.
Land
Debrief
5 min
Objective: Divide multiples of 100 and 1000.
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.
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.
Math Past
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.
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.
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
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
Tell time to the hour and half hour by using digital and analog clocks.
Lesson at a Glance
Students analyze an analog clock and a digital clock that show the same time. They learn how each clock represents the hour and minutes and practice reading the time to the hour and half hour on both kinds of clocks.
Key Question
• What time is it? How do you know?
Achievement Descriptor
1.Mod5.AD10 Tell time to the hour and half hour on analog and digital clocks. (1.MD.B.3)
Agenda Materials Lesson Preparation
Fluency 10 min
Launch 5 min
Learn 35 min
• Hours and Minutes
• Tell Time to the Hour and Half Hour
• Match: Time
• Problem Set
Land 10 min
Teacher
• Computer with internet access*
• Projection device*
• Teach book*
• 100-bead rekenrek
Students
• Dry-erase marker*
• Pencil*
• Personal whiteboard*
• Personal whiteboard eraser*
• Learn book*
• Match: Time cards removable (1 per student pair, in the student book)
* These materials are only listed in lesson 1. Ready these materials for every lesson in this module.
• The Match: Time cards removables must be torn out of student books. Cut out the cards on the removable to make a set. Each student pair needs one set of cards.
• Prepare the digital interactive clock for the lesson.
Fluency
Whiteboard Exchange: 4 as an Addend
Students find a total and use the commutative property to write a related addition sentence to build addition fluency within 20.
After each prompt for a written response, give students time to work. When most students are ready, signal for them to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.
Display 1 + 4 = .
Write the equation, and then find the total.
Display the completed addition sentence: 1 + 4 = 5.
Change the order of the addends to write a related addition sentence. (Point to the addends.)
Display the related addition sentence: 4 + 1 = 5.
Repeat the process with the following sequence:
5-Groups: 10 and Some More
Students recognize a group of dots and say the number two ways to prepare for identifying a given set with all the tens composed in lesson 3.
Display the 5-group cards that show 11.
How many dots? Raise your hand when you know.
Wait until most students raise their hands, and then signal for students to respond.
We can say 11 is 1 ten 1 one. On my signal, say it with me. Ready? 11 is 1 ten 1 one.
Repeat the process with the following sequence:
As students are ready, challenge them to recognize the groups of dots more quickly by showing each set of 5-group cards for a shorter time.
Counting on the Rekenrek by Tens and Ones
Materials—T: 100-bead rekenrek
Students count to a specified number the Say Ten way, then in standard form to prepare for recording the units of tens and ones in a given set in lesson 2.
Show students the rekenrek. Start with all the beads to the right side.
Let’s count to 41 the Say Ten way.
Say how many beads there are as I slide them over.
Slide over 10 beads in each row all at once as students count to 4 ten.
1 ten, 2 ten, 3 ten, 4 ten
Slide over 1 more bead as students count to 4 ten 1.
4 ten 1
Student View
Slide all the beads back to the right side.
Let’s count to 41 the regular way.
Say how many beads there are as I slide them over.
Repeat the process as students count by tens and ones to 41. 10, 20, 30, 40, 41
Repeat the process as students count by tens and ones the Say Ten way and in standard form to the following numbers:
Language Support
Consider using strategic, flexible grouping throughout the module.
• Pair students who have different levels of mathematical proficiency.
• Pair students who have different levels of English language proficiency.
• Join pairs of students to form small groups of four.
As applicable, complement any of these groupings by pairing students who speak the same native language.
Launch
Students listen to sounds of different durations and learn that 60 minutes make an hour.
Tell students to prepare to listen carefully. Play the first sound, which lasts for 1 second.
What do you notice?
It is very short.
Play the next sound, which lasts for 10 seconds.
What do you notice this time?
It was longer this time, so I can tell it is music.
The first sound is a short part. (Show your palms facing each other, close together.) The sound we just heard is longer. (Increase the distance between your palms.)
Play the third sound, which is 60 seconds long.
Is this sound shorter (show palms close together) or longer (expand the distance between palms) than the first one?
Longer
That last sound was a minute long!
Help students recall their learning about time by asking the following questions.
Which is longer, a minute or an hour? How do you know?
An hour is longer. A minute goes by fast, but an hour takes more time.
You are right. An hour is longer because 60 minutes go together to make an hour.
Transition to the next segment by framing the work.
Today, we will look at a clock and see how minutes make an hour.
Promoting the Standards for Mathematical Practice
Students attend to precision (MP6) when they use minutes and hours appropriately to describe lengths of time. Appropriate use of units begins with understanding that a minute is a short length of time and an hour is a longer length of time. It extends to discussing how larger units are made out of smaller ones.
Reasoning with units in different contexts (e.g., time, measurement, or shapes) helps students to work with numerical units (tens, hundreds, etc.) later on.
Learn
Hours and Minutes
Students count minutes to see how analog and digital clocks represent the time.
Show students 1 o’clock on the analog clock only.
(Point to the red hand.) The short hand is the hour hand. It tells the hour.
(Point to the blue hand.) The long hand is the minute hand. It tells the minutes.
Turn on the digital clock as well.
This is a different type of clock. It shows the time using only numbers. The numbers to the left of the dots tell the hour. (Point to the 1 on the digital clock.) The numbers to the right of the dots tell the number of minutes. (Point to the 00 on the digital clock.)
Tell students that both clocks show 1 hour and 0 minutes. Help them read the time on each clock as 1 o’clock. Tell students that as time passes, the hands on the first clock (the analog clock) move, but the numbers on the second clock (the digital clock) just change.
Watch and see what happens on each clock as time passes. Let’s count minutes.
Slowly move the minute hand from 1:01 to 1:30. Have the class chorally count the minutes. 1 minute, 2 minutes, … , 29 minutes, 30 minutes
Point to the analog clock that shows 1:30.
What did the hands do on this clock as we counted?
The minute hand moved a little bit at a time. It went from the top of the clock to the bottom of it.
The hour hand only moved a little. Now it’s past the 1, but not to the 2.
Reset the clock to 1 o’clock.
As time passes, the minute hand moves forward one tick mark at a time. Each tick mark represents 1 minute. Watch the blue minute hand.
Slowly move the minute hand from 1:00 to 1:05.
The hour hand moves too. As minutes go by, the hour hand moves slowly from one number on the clock to the next. Watch the red hour hand.
Slowly move the minute hand to 1:30.
Now the hour hand is between two numbers, or hours. The minute hand points straight down to the 6. When the hands are in this position, we say the first number for the hour and read the time as one thirty.
Point to the digital clock that also shows 1:30.
What did the numbers on this clock do as we counted?
They changed. The minutes went from 00 up to 30. They went up by 1 each time.
What time does the clock show?
1:30
Both clocks started at 1 o’clock. We counted 30 minutes from 1 o’clock to 1:30. Now the clocks show 1:30.
Let’s keep counting until the minute hand moves all the way around the clock. (Point to the picture of the analog clock.)
Slowly move the minute hand from 1:30 to 2 o’clock. Have the class chorally count the minutes.
What time do the clocks show now? How do you know?
2 o’clock
The hour hand is pointing at the 2 and the minute hand is on the 12.
There is a 2 and two zeros on the clock with only numbers.
Confirm that both clocks show 2 o’clock.
(Point to the digital clock.) On this clock, after the minutes show 59, they start over at 0. It shows 1:59 and then 2:00. This happens because there are 60 minutes in 1 hour. How many minutes are in an hour?
60 minutes
(Point to the analog clock.) On the other clock, when the minute hand goes all the way around the clock, the hour hand arrives at the next number, or hour. Now the hour hand is pointing to 2.
Tell Time to the Hour and Half Hour
Students practice telling time to the hour and half hour by using an analog clock.
Have students practice telling time to the hour and half hour. Consider having students stand.
Use only the analog clock set to 2 o’clock.
What time is it?
2 o’clock
Move the minute hand to show 2:30. Provide a moment of wait time.
What time is it?
2:30
Continue to show various times to the hour and half hour (3:00 and 3:30, 11:00 and 11:30, etc.) until students say the time to the half hour with confidence.
Match: Time
Materials—S: Match: Time cards
Students match cards that use different formats of time to show time to the hour and half hour.
Demonstrate and explain how to play Match: Time by using the following directions:
• Place all of the Match: Time cards faceup.
• One partner finds two cards that match because they show the same time.
• They tell their partner how they know the cards match.
• Partners take turns finding matches until all the matches have been found.
Differentiation: Support
If needed, provide additional support for students with telling time to the half hour.
• The hour hand is between 2 and 3. When the hour hand is between two hours, or numbers, we say the first hour. The hour hand is getting closer to the next hour, but it’s not there yet.
• When the minute hand points straight down at the 6, 30 minutes of the hour have passed.
• This clock shows 2:30. The hour hand is still in the two o’clock hour and 30 minutes have passed.
Circulate as students play and ask the following questions:
• (Point to a card.) Where do you see the hour? Where do you see the minutes?
• Show me a match. What is the time on both cards?
Problem Set
Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.
Directions may be read aloud.
Land
Debrief
5 min
Objective: Tell time to the hour and half hour by using digital and analog clocks.
Show the analog clock set to 4:30.
What time is it? How do you know?
4:30
The hour hand is a little past 4 and the minute hand points to 6.
Show 4:30 with both clocks. Have students consider, discuss, and share the similarities and differences between the two clocks.
How are the clocks the same?
Both have numbers and tell the time.
Both tell you the time is 4:30.
Both clocks show 4 hours and 30 minutes.
UDL: Engagement
Foster collaboration and help students to engage successfully in the sorting game by assigning clear roles for each partner. Review the activity goal, directions, and group norms before groups begin.
Consider embedding any socio-emotional skills students are learning in other areas of their day, such as sharing, taking turns, and disagreeing respectfully, into the activity.
How are the clocks different?
One only has numbers. It does not have hands.
One has hour and minute hands.
For 30 minutes, one has the number 30. The other one shows 30 minutes at the 6.
These clocks show the time in different ways. They both make a new hour every 60 minutes.
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
Distributed practice with telling time helps students master the skill. Each day, consider pausing periodically at the hour and half hour to ask: What time is it?
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
3.Write what you do and the time you do it.
Topic A Grouping Units in Tens and Ones
Topic A builds on work with tens and ones that students completed in modules 3 and 4. At first students reason about units in the context of time. They learn through experience that smaller units can compose larger units. For example, they discover that 1 hour is made of 60 minutes and 1 half hour is made of 30 minutes.
Lessons build on the idea that smaller units can be used to compose larger units by considering the place value units of tens and ones. Students work with sets of objects to compose groups of 10 and represent two-digit totals in different ways. For example, 26 ones can also be represented as 1 ten 16 ones or 2 tens 6 ones. Working with numbers that have more than 9 ones prepares students for adding and subtracting with larger numbers in later grades. However, students come to understand that the digits we use to write a number, such as the 2 and the 6 in 26, show how many tens and ones there are when the number is expressed in its “most composed” form. This leads to recognition that the value of digits can be determined based on their place in the number. The value of the digit 2 can be expressed as 2 tens or 20. The value of the digit 6 can be expressed as 6 ones or simply 6. Students compose (or decompose) a total such as 26 by place value units: 20 and 6 or 2 tens 6 ones. Using different representations of the same total invites students to consider equivalence and deepens their number sense.
Several place value models that increase in complexity help students internalize the basic understanding that 10 ones are equivalent to 1 ten.
50 pennies
50 ones 5 dimes 5 tens
Groupable
Students group 10 ones to compose a new unit of ten. They may put 10 bears into 1 cup, link 10 cubes into a stick, or circle 10 donuts to represent a box of 10. Students can see and manipulate the individual units within the new larger unit. The size of the new unit is proportionally larger than the base unit.
Pregrouped
Students group 10 ones and trade them for a new item that represents 1 ten. For example, given 23 centimeter cubes, students trade 10 cubes for 1 ten-centimeter stick. The new item is proportionally larger than the base unit.
In this topic, students also add 10 and take 10 all at once from numbers. They add to and subtract from numbers in sequence: 54 + 10 = 64, 64 + 10 = 74, and so on. From this a pattern becomes visible: the digit in the tens place changes by 1, but the digit in the ones place remains the same.
Nonproportional
Students trade 10 ones for a new item that represents 1 ten, but the ten looks different from the 10 ones. An example is trading 10 pennies for 1 dime. These models are nonproportional because the new unit, in this case a dime, is not visually 10 times larger than the base unit, or penny. Nonproportional models prepare students to work with place value disks in grades 2–5.
Progression of Lessons
Lesson 1
Tell time to the hour and half hour by using digital and analog clocks.
4:30
The minute hand is pointing at the 6, and the hour hand is not at 5 yet, so it is 4:30.
Lesson 2
Count a collection and record the total in units of tens and ones.
We composed tens by making groups of 10. We had 5 groups of ten and 2 extras. That is 52 bears.
Lesson 3
Recognize the place value of digits in a two-digit number.
3 05 05 3
After we composed all of the tens, there were 5 tens and 3 ones. The digits 5 and 3 make 53. The value of the digit 5 is 50 and the value of the digit 3 is 3.
Lesson 4
Represent a number in multiple ways by trading 10 ones for a ten.
Lesson 5
Reason about equivalent representations of a number.
I can trade 10 pennies for a dime and use dimes and pennies to show different ways to make 30 cents.
I would rather have 30 loose crayons so I can see all the colors, but both pictures have the same number of crayons. 30 ones is 30 and 3 tens is 30.
Lesson 6
Add 10 or take 10 from a two-digit number.
When I add ten, the digit in the tens place is 1 more. When I subtract ten, the digit in the tens place is 1 less.
30 crayons 3 boxes of crayons
Before This Module
Kindergarten Module 6
Students begin to develop place value understanding when they come to see that teen numbers are composed of 10 ones and some more ones. They do not formalize the notion of “a ten” as a unit. Students also count to 100 by tens and by ones.
Grade 1 Module 3
Students rename groups of ten ones as units of ten. They come to see that all two-digit numbers are composed of tens and ones.
Grade 1 Module 4
Students use 10-centimeter sticks (tens) and centimeter cubes (ones) to measure lengths. They state total lengths in terms of tens and ones.
Overview
Place Value Concepts to Compare, Add, and Subtract
Topic A
Grouping Units in Tens and Ones
Topic A builds on work with tens and ones from modules 3 and 4. Lessons develop the idea that smaller units, such as ones, compose larger units, such as tens. The following place value models, which increase in complexity, help students to internalize the equivalence of 10 ones and 1 ten and to understand that two-digit numbers represent amounts of tens and ones.
Groupable
Physically group 10 ones to compose a new unit of ten. The size of the new unit is proportionally larger than the base unit.
Pre-grouped Nonproportional
Tens are composed by trading 10 ones for a new item that represents 1 ten. The new item is proportionally larger than the base unit.
10 ones are traded for a new item that represents 1 ten. However, the new item is non-proportional.
Visually, it is not 10 times larger than the base unit.
Why
Place Value Concepts to Compare, Add, and Subtract
Why are there so many strategies and tools for adding large numbers?
One concept unifies the many ways to add two-digit numbers: make an equivalent yet easier problem. Break the addends into parts and combine the parts in ways that make sense. Exposing students to more than one way to make an easier problem increases number sense, encourages flexibility, and provides choice.
Presenting different strategies acknowledges that learners think about numbers in different ways. Also, the addends may lend themselves to certain strategies. For example, to solve 29 + 21 efficiently, simply add the 1 from 21 to 29 to make 30. Then add 30 and 20. However, to solve 32 + 26, it may be easier to think of 26 in terms of tens and ones, solve 32 + 20, and add 6 more to make 58. Students who develop a toolbox of strategies will begin to select them intentionally.
Students also select different tools or models, such as manipulatives, drawings, number paths, number bonds, and number sentences. How they solve a problem is less important than an accurate solution, the written record of their thinking, and their explanation of their reasoning. Sharing such representations and engaging in discussion to analyze various ways to solve are critical features of instruction that deepen understanding of place value, operations, and quantity.
What skills do students need to add two-digit numbers by the end of the year?
Students need several skills to add 2 two-digit numbers. As they work through module 5 topic E and module 6 part 2, they continue to develop these embedded skills through practice. However, if students are still working toward proficiency with some of the first skills in the following chart, then they may benefit from directly modeling 2-digit addition problems using concrete or pictorial tools.
How do students deepen and advance their understanding of the equal sign?
Students evaluate number sentences such as 90 – 40 = 20 + 30. They determine whether the number sentence is true or false by calculating the value of the expressions on either side of the equal sign. This provides practice with adding and subtracting tens while giving students opportunities to engage with uncommon and complex equation types.
Students also evaluate number sentences with equivalent expressions that represent different ways to make an easier problem. For example, the expression on either side of the equals sign in 10 + 10 + 2 + 6 = 12 + 10 + 6 shows one way to break up the addends in 12 + 16. Students find the expressions have the same value and conclude that the number sentence is true. This helps confirm that there are several valid ways to solve a problem.
Why is telling time included in a module on place value?
Module 5 centers on the idea that smaller units such as ones compose larger unit such as tens. Time also involves composing units. For example, minutes make up hours and half hours, and hours make up days. Work with units in various contexts (e.g., time, measurement, and shapes) develops reasoning that helps students understand numerical units such as ones, tens, and hundreds. Students revisit telling time to the half hour in module 6.
Lead: Facilitating Successful Implementation of Eureka Math2, K–5
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.
Implementation Support Tool
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.
Implementation Support Tool
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
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)
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
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 and Monitoring indicators vary by routine and are included for specific fluency routines only.
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)
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
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
g. Adjusts the sequence of problems (e.g., complexity, level of abstraction) based on the existence of pervasive trends
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
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
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
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
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
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
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
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
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.
a. Aligns facilitation to the purpose statement of Launch
b. Provides opportunities indicated by lesson for students to notice; wonder; apply; and articulate strategies, choice of models, and understandings
c. Honors the amount of teacher-to-student discourse, studentto-student discourse, and whole-class discourse within Launch
d. Concludes Launch by using the given transition statement to set the goal for and make an explicit connection to the day’s learning
e.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 engag ement, collaboration, and learning (e.g., desk arrangement, Thinking Tool, T alking Tool, materials, visuals)
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 level
o. Prompts students to restate, build on, or evaluate other students’ responses to enhance discussion and strengthen habits of discussion
Student Expectations
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
r. Models the use of or directs students to use specific sections of the Talking Tool and Thinking Tool to support discourse and metacognition
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
Implementation Support Tool
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 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
c. Releases responsibility to students while working on lesson pages, as indicated by the lesson and student performance
d. Uses tools and models accurately to develop student understanding of the lesson’s strategies and objective
e. Uses the mathematical language of the lesson or related language (e.g., decompose and unbundle) to demonstrate precise yet accessible terminology
f. Completes Learn within or near the total time indicated, allowing time for instruction, lesson pages, and Problem Set
Using Structures, Routines, and Activities
g. States directions clearly and includes all necessary details for engagement (e.g., topic/task, thinking job, timeframe, cues, peer interaction)
h. Prepares the environment to optimize engagement, learning, and collaboration (e.g., desk arrangement, Thinking Tool, Talking Tool, materials, visuals)
i. Uses structures, routines, and activities as opportunities for students to develop their understanding of the day’s objective and to articulate their strategies, models, and understandings
j. Incorporates Read–Draw–Write process (grades 1–5) for solving word problems
Engaging
Instructional Habits
k. Provides appropriate think time for students to process questions or displayed content and form responses
Monitoring
l. Circulates to recognize and encourage application of mathematical understanding
m. Listens to student discourse to gauge students’ ability to articulate their understanding and use precise mathematical terminology (uses the Observational Assessment Recording Sheet (OARS) for K–2)
n. Monitors student work to notice and note trends, varied models and strategies, exemplars, and misconceptions (uses the Observational Assessment Recording Sheet (OARS) for K–2)
Promoting Discourse
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
p. 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
Adaptive Implementation
Customizing
q. Uses the margin notes to improve access (e.g., UDL, Differentiation: Support, Language Support, strategic pairings/groupings)
r. Creates and inserts problems that bridge or reinforce understanding to existing problems in the sequence
Responding
s. Calls on students or selects student work strategically to highlight trends, varied solutions, misconceptions, and broader mathematical understanding
t. Models or directs students to use specific sections of the Talking Tool and Thinking Tool to support metacognition and discourse
u. Verifies possible solutions throughout Learn by inviting students to share work or revealing objective-aligned 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
Problem Set
The Problem Set a part of Learn is an opportunity for independent practice.
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
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.
Customizing
g. Designates the order of problems to complete to ensure practice with the most important components of the lesson objective
h. 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. Responds to pervasive misconceptions by questioning or modeling with a parallel example to provide a scaffold in real time
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
Implementation Support Tool
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.
Meeting Component Purpose
a. 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 lesson objective, highlighting important concepts and vocabulary
c. Honors the amount of student-to-student discourse and whole-class discourse within Land
d. Completes the Debrief within or near the indicated time
Meeting Component Purpose
a. Allows an amount of time within or near the indicated time allotted for students to independently complete as much of their Exit Ticket as possible
b. Collects Exit Tickets to analyze qualitative data to identify strengths to leverage in future lessons, varied solution paths, and misconceptions to address
Engaging
e. Provides adequate think time for students to process questions or displayed content and to form responses
f. Provides students with supports (e.g., Talking Tool, mathematical terminology) to improve access to discussion
Monitoring
g. Listens to student discourse to gauge understanding and identify opportunities to push for clearer thinking
Promoting Discourse
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
i. Prompts students to restate, build on, or evaluate other students’ responses to enhance discussion and strengthen habits of discussion
Engaging
c. Makes tools (e.g., anchor charts, concrete manipulatives, Thinking Tool) available to support independence
Monitoring
d. Circulates room while students work on their Exit Tickets to ensure student accountability 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 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:
• Attempt to use accurate mathematical language when synthesizing the lesson’s key understandings
• Engage in discourse to articulate understanding, gain the insights of others, and evaluate errors
• Reflect on their own thinking following analysis of peer work and discourse
Customizing
j. Creates questions, in response to previously collected data, that promote synthesis of the lesson or topic
Responding
k. Affirms accuracy and efficiency in student responses while clarifying any inaccuracies and deepening understanding
l. Revisits questions from earlier in the lesson to check for growth in understanding
Responding
f. Designates the order of problems for students to complete to ensure students answer the problems that highlight the most important components of the lesson objective (within the given time)
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
• Make thinking visible by using numbers, words, and models
• Complete as much of the Exit Ticket as possible
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.
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.
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)
Engaging indicators describe teacher habits that maximize the number of students actively engaging at a given moment.
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
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. 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
Appendix B
Scenarios
Scenario 1: Imagine that a 3rd grade teacher comes to your office and says, “I’m used to teaching multiplication and division at the beginning of the year, so I was going to adjust Module 2 to come after Module 4.”
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 Module 2 comes before Module 4 and the justification for that sequencing.”
Scenario 2: “My students didn't have Eureka Math2 last year, so aren’t familiar with the way fractions 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 the next lesson.”
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 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 supports for learning that will allow students to access the day’s lesson.
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 45 minutes left in the block and there are 60 minutes of content within the lesson. The response to this scenario will depend on the specific situation and purpose of the observation. The leader might want to determine the reason the teacher is using class time to review the previous day's learning. The leader could say something like, "The way the Eureka Math lessons are designed, you can be confident that there are multiple entry points embedded into the lesson to help activate prior learning."
Scenario 4: The grade 2 team has a 50-minute math block. A teacher tells you and says that they are struggling to fit Eureka Math2 lessons into 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.
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
Prior to Year 1
Engage:
Preparing to launch 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.
Notes:
Experience
Enhance
Experience:
Learning and exploring Eureka Math2
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.
Enhance:
Extending knowledge of Eureka Math2
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.
Exemplify
Exemplify:
Skillfully implementing Eureka Math2
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.
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.
Enhance Experience
Exemplify
Leaders Teachers Students
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.
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.
There are no student actions during the Engage phase.
1The timeline mentioned is a guideline. A specific implementation may move through these phases more or less quickly.
2 The Power Up session is designed for teachers with prior Eureka Math experience.
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.
Teachers
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.
1 The timeline mentioned is a guideline. A specific implementation may move through these phases more or less quickly.
Eureka Math2® Implementation Benchmarks
– Enhance Phase (~Years 2–3)1
Experience Engage
Exemplify Enhance
Leaders Teachers Students
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.
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.
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.
1The timeline mentioned is a guideline. A specific implementation may move through these phases more or less quickly.
Extending knowledge of Eureka Math2.
Eureka Math2® Implementation Benchmarks
– Exemplify Phase (~Years 4 and Beyond)1
Enhance Experience Engage
Exemplify
Skillfully implementing Eureka Math2.
Leaders Teachers Students
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.
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.
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.
1The timeline mentioned is a guideline. A specific implementation may move through these phases more or less quickly.
