Supplemental Materials - Teach: Effective Instruction with Eureka Math Squared, K-5
1 Place Value Concepts Through Metric Measurement and Data · Place Value, Counting, and Comparing Within 1,000 Module
2 Addition and Subtraction Within 200
3 Shapes and Time with Fraction Concepts
4 Addition and Subtraction Within 1,000
5 Money, Data, and Customary Measurement
6 Multiplication and Division Foundations
Before This Module
Grade 1 Module 6 Part 2
Grade 1 students deepen their problemsolving skills as they use tape diagrams and drawings to represent and solve more complex problems within 20, which include start unknown problem types.
Students build on their work in module 5 by extending addition strategies to larger numbers within 100. The focus is on making easier problems by decomposing one or both addends. Students may add like units, add tens then ones, or vice versa, or make the next ten. They use various tools and recording methods, such as number bonds, the number path, and the arrow way to support their strategy work.
Overview
Addition and Subtraction Within 200
Topic A
Simplifying Strategies for Addition
Students use place value understanding, properties of operations, and the relationships between numbers to make simpler problems. A variety of addition simplifying strategies promote flexibility, accuracy, and efficiency. Students use number bonds, the open number line, and the arrow way to record their thinking. As they build a toolbox of strategies, students begin to consider the effectiveness and efficiency of each strategy in different situations. Students synthesize their learning by applying addition strategies to solve put together and add to word problem types.
Topic B
Strategies for Composing a Ten and a Hundred to Add
Students build upon their understanding of place value strategies and of making a unit of ten or a hundred. As they move fluidly between concrete, pictorial, and abstract levels of representation, students develop a conceptual understanding of addition. They use place value disks and drawings to systematically add like units and compose a new unit of ten or a hundred when they have more than 9 of a unit. Students relate place value drawings to expanded form written recordings and to totals below recordings, both of which clearly show the units of hundreds, tens, and ones that are added.
Topic C
Simplifying Strategies for Subtraction
Students add a variety of subtraction simplifying strategies to their toolboxes. They use familiar models and recording methods from topic A as they progress toward mental computation.
Students build on work from grade 1 by representing and solving take from with result unknown and change unknown problem types by using tape diagrams and equations to make sense of part–total relationships. Count Back with
Topic D
Strategies for Decomposing a Ten and a Hundred to Subtract
Students build upon their understanding of place value strategies and of taking from a unit of ten or a hundred. As they move fluidly between concrete and pictorial levels of representations, students develop a conceptual understanding of subtraction. They use place value disks and drawings to systematically decompose a larger value unit when they need more in the ones place or in the tens place to subtract. Students relate place value drawings to unit form recordings in which the total, or minuend, does not change, but is simply renamed.
After This Module
Grade 2 Module 4
Students use place value understanding, the properties of operations, and the relationship between addition and subtraction to add and subtract within 1,000. They expand their toolbox of Level 3 simplifying strategies to include adding and subtracting tens and hundreds, making or taking from a ten or a hundred, and using various forms of compensation.
Finally, students advance their problem-solving skills by using the relationship between addition and subtraction to represent and solve add to and take from with start unknown problems. Students also work with two-step word problems, some of which involve single-digit addends.
Students continue to relate place value disks and drawings to the standard vertical form, systematically recording up to two compositions or decompositions within 1,000 as needed. When subtracting, students also decompose multiples of 100 or numbers with 0 in the tens place, renaming the total in one or two steps. Students use addition to explain why their subtraction strategies work.
Throughout the module, students use the Read–Draw–Write process and apply various addition and subtraction strategies to solve one- and two-step word problems.
Contents
Addition and Subtraction Within 200
Why
Achievement Descriptors: Overview
Topic A
Simplifying Strategies for Addition
Lesson 1
Reason about addition with four addends.
Lesson 2
Break apart and add like units.
Lesson 3
Use compensation to add within 100.
Lesson 4
Use compensation to add within 200.
Lesson 5
Make a ten to add within 100.
Lesson 6
Make a ten to add within 200.
Lesson 7 .
Solve word problems by using simplifying strategies for addition.
Topic B
Strategies for Composing a Ten and a Hundred to Add
Lesson 8 . . . .
Use concrete models to compose a ten.
114
Lesson 9
Use place value drawings to compose a ten and relate to written recordings.
Lesson 10
Use concrete models to compose a hundred.
Lesson 11
Use math drawings to compose a hundred and relate to written recordings.
Lesson 12
Use place value drawings to compose a ten and a hundred with two- and three-digit addends. Relate to written recordings.
Topic C
Simplifying Strategies for Subtraction
Lesson 13
Represent and solve take from word problems.
Lesson 14
Use addition and subtraction strategies to find an unknown part.
Lesson 15
Use compensation to subtract within 100.
Lesson 16
Use compensation to subtract within 200.
Lesson 17
Take from a ten to subtract within 200.
118
Lesson 18
Take from a hundred to subtract within 200.
Lesson 19 . . .
Solve word problems with simplifying strategies for subtraction.
Topic D
Strategies for Decomposing a Ten and a Hundred to Subtract
Lesson 20 .
Reason about when to unbundle a ten to subtract.
Lesson 21 .
Use concrete models to decompose a ten with two-digit totals.
Lesson 22
Use place value drawings to decompose a ten and relate them to written recordings.
Lesson 23
Use concrete models and drawings to decompose a hundred.
Lesson 24
Use place value drawings to decompose a hundred and relate them to written recordings.
Lesson 25
Use place value drawings to subtract with two decompositions.
Lesson 26
Solve add to and take from with start unknown word problems.
Lesson 27
278
Solve two-step word problems within 100. Module Assessment
Achievement Descriptors: Proficiency Indicators
Observational Assessment Recording Sheet
302
318
330
346
Works Cited
Acknowledgments
Why
Addition
and Subtraction Within 200
Why are two topics devoted to simplifying strategies for addition and subtraction?
By the end of grade 2, students are expected to add and subtract fluently within 100 by using strategies based on place value, properties of operations, and the relationship between addition and subtraction. Fluency means being able to operate with numbers flexibly, efficiently, and accurately.
Because students are not expected to work fluently with the standard addition and subtraction algorithms until grade 4, topics A and C are intentionally devoted to Level 3 addition and subtraction methods, in which students use simplifying strategies to make simpler problems. This gives students time to work through and to make connections between various strategies. As students apply place value understanding from module 1 and leverage familiar tools, they develop confidence and flexibility. While students are not expected to master all of the Level 3 strategies, they are expected to reason about the numbers in a problem and to consider efficient solution paths by using tools and written recordings. This builds their capacity toward mental math.
Addition and subtraction problems are presented horizontally throughout grade 2. A vertical presentation implies use of the standard vertical form notation. In contrast, a horizontal presentation is more conducive to students thinking flexibly about number relationships to choose the most efficient strategy.
Why is the standard vertical form for addition and subtraction not introduced in this module?
After much consideration of our students’ learning, teachers’ input, and a review of the research around how students learn and how mathematical concepts progress, we decided it makes the most sense to hold off introducing the standard vertical form until module 4. Why?
1. This module focuses on the conceptual understanding of addition and subtraction through the use of concrete models, drawings, and strategies. By intentionally delaying
Methods for Addition and Subtraction
Level 1: Count all
Level 2: Count on by ones
Level 3: Make a simpler addition or subtraction problem. These methods often use the associative property:
• Decompose addends to add or subtract like units
• Use benchmark numbers to count on or count back
• Use compensation to adjust numbers
• Decompose addends to make or take from a ten or a hundred
• Think of subtraction as an unknown addend problem.
the introduction of the vertical form, we create more time and space for students to explore a variety of strategies, which encourages them to reason about number relationships and efficiency, rather than jump to one specific strategy.
2. Grade 2 math standards call for students to relate their strategies to a written method. When students compose or decompose units by using models or drawings, they connect actions and language to corresponding steps in a written recording (e.g., expanded form, totals below, unit form). In contrast to the vertical form, these written methods have one thing in common—they explicitly highlight place value units.
Why do some word problems in lesson 27 involve single-digit addends?
Lesson 27 is the students’ first formal experience with two-step word problems. Due to the added cognitive lift of solving a multi-step problem, many two-step problems involve single-digit addends. By using single-digit quantities and easier problem types, students can focus on representing number relationships with a drawing and with an equation. This builds their confidence as problem solvers.
Ms. Bell wants to give each student a pencil in their favorite color. How many pencils does Ms. Bell need for the class?
She already has 5 yellow pencils. How many more pencils does Ms. Bell need?
Colors in Ms. Bell’s class
1
Which word problem types, or addition and subtraction situations, are used in this module?
The table shows examples of addition and subtraction situations.1 Darker shading in the table indicates the four kindergarten problem types. Students in grades 1 and 2 work with all problem types. Grade 2 students reach proficiency with the unshaded problem types.
Grade 2 students are expected to master all addition and subtraction problem types by the end of the year. This module focuses on the following problem types.
• Add to with result unknown: Both parts are given. An action joins the parts to form the total.
27 cars are in the parking lot. 39 more cars pull into the lot. How many cars are in the lot now? (Lesson 7)
• Take from with result unknown: The total and one part are given. An action takes away one part from the total.
63 people are on a bus. 48 people get off the bus at the park. How many people are still on the bus? (Lesson 19)
• Put together with total unknown: Both parts are given. No action joins or separates the parts. Instead, the parts may be distinguished by an attribute like type, color, size, or location.
125 students are sitting in the cafeteria. 69 students are standing in the lunch line. How many students are there in all? (Lesson 7)
• Compare with difference unknown: Two quantities are given and compared to find how many more or how many fewer.
The vet checks the pets. She checks 74 dogs and 28 cats. How many more dogs than cats does she check? (Lesson 16)
Take from with change unknown: The total and the resulting part are given. An action takes away an unknown part from the total. The situation equation (e.g., 57 – = 28) can be rewritten as a related solution equation (e.g., 28 + = 57 or 57 – 28 = ).
There are 57 tacos in the lunchroom. Then some tacos are eaten. Now there are 28 tacos left for the next class. How many tacos were eaten? (Lesson 13)
• Put together/take apart with addend unknown: The total and one part are given. No action joins or separates the parts.
Hope picks 63 apples. 47 of the apples are green. How many apples are not green? (Lesson 21)
• Compare with smaller (quantity) unknown: The bigger quantity and the difference between the quantities are given.
There are 28 fewer plums than lemons in a bin. There are 73 lemons. How many plums are in the bin? (Lesson 19)
The following problem types tend to be among the most challenging subtypes for grade 2 students.
• Add to with start unknown: The total and one part are given. An action joins one part with the unknown start to form the total. Since one part is unknown, the problem can be thought of as a subtraction problem. The situation equation (e.g., + 35 = 90) can be rewritten as a related solution equation (e.g., 35 + = 90 or 90 – 35 = ).
Alex has some money in his bank. He finds 35 cents. Now he has 90 cents in his bank. How much money did have Alex have in his bank to start? (Lesson 26)
• Take from with start unknown: Both parts are given. The starting quantity, or total, is unknown. There is an action that takes away one part from the unknown total. Since the total is unknown, the problem can be thought of as an addition problem. Note that the situation equation (e.g., – 25 = 20) can be rewritten as a related solution equation (e.g., 25 + 20 = ). Add to and take from with start unknown problems are two of the most challenging subtypes for students.
Ming has some money. She spends a quarter on an eraser and has 20 cents left. How much money did she have before she bought the eraser? (Lesson 26)
Achievement Descriptors: Overview
Addition and Subtraction Within 200
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 (recording sheet provided in the module resources),
• data from other lesson-embedded formative assessments,
• Exit Tickets,
• Topic Tickets, and
• Module Assessments.
This module contains the six ADs listed.
2.Mod2.AD1
Represent and solve one-step addition and subtraction word problem types within 100 by using drawings and equations with a symbol for the unknown.
2.OA.A.1
2.Mod2.AD2
Add up to 4 two-digit numbers by using strategies based on place value or properties of operations.
2.Mod2.AD3
Add within 200 by using concrete models or drawings, strategies based on place value, or properties of operations.
2.NBT.B.6
2.NBT.B.7
2.Mod2.AD4
Subtract within 200 by using concrete models or drawings, strategies based on place value, properties of operations, or the relationship between addition and subtraction.
2.Mod2.AD5
Rename 10 of a smaller unit as 1 of a larger unit.
2.Mod2.AD6
Rename 1 of a larger unit as 10 of a smaller unit.
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.
2.NBT.B.7
2.NBT.B.7
2.NBT.B.7
ADs have the following parts:
2.Mod2.AD2 Add up to 4 two-digit numbers by using strategies based on place value or properties of operations.
• 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 2 module 2 is coded as 2.Mod2.AD1.
RELATED CCSSM
2.NBT.B.6 Add up to four two-digit numbers using strategies based on place value and properties of operations.
• AD Language: The language is crafted from standards and concisely describes what will be assessed.
Partially Proficient Proficient Highly Proficient
• AD Indicators: The indicators describe the precise expectations of the AD for the given proficiency category.
Add up to 3 two-digit numbers by using strategies based on place value or properties of operations.
Add. Show how you know.
15 + 22 + 35 =
Add 4 two-digit numbers by using strategies based on place value or properties of operations.
• Related Standard: This identifies the standard or parts of standards from the Common Core State Standards that the AD addresses.
Add. Show how you know.
15 + 22 + 17 + 35 =
2.Mod2.AD3 Add within 200 by using concrete models or drawings, strategies based on place value, or properties of operations.
RELATED CCSSM
2.NBT.B.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
115 + 48 = AD Code Grade.Module.AD#
Partially Proficient Proficient Highly Proficient
Add within 100 by using concrete models or drawings, strategies based on place value, or properties of operations.
Add. Show how you know.
52 + 29 =
Add within 200 by using concrete models or drawings, strategies based on place value, or properties of operations.
Add. Show how you know.
Achievement Descriptors: Proficiency Indicators
2.Mod2.AD1 Represent and solve one-step addition and subtraction word problem types within 100 by using drawings and equations with a symbol for the unknown.
RELATED CCSSM
2.OA.A.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1
1See [CCSSM] Glossary, Table 1.
Represent and solve one-step kindergarten and grade 1 addition and subtraction word problem types* within 100 by using drawings and equations with a symbol for the unknown.
Read
Alex rides her bike for 45 minutes on Sunday.
On Monday, she rides for 17 minutes.
How many total minutes does Alex ride?
Draw Write
Represent and solve one-step grade 2 addition and subtraction word problem types† within 100 by using drawings and equations with a symbol for the unknown.
Read
Alex rides her bike on Sunday.
On Monday, she rides for 17 minutes.
She rides 62 total minues.
How many minutes does Alex ride on Sunday?
Draw Write
Alex rides for total minutes.
Alex rides for minutes on Sunday.
* Common Core Standards Writing Team, Progressions for the Common Core, 2011–2015.
Common Core Standards Writing Team, Progressions for the Common Core, 2011–2015.
Topic D Strategies for Decomposing a Ten and a Hundred to Subtract
In topic D, students build on their understanding of place value strategies and taking from a unit of ten or a hundred. Consistent use of place value disks and place value drawings helps students systematically model the steps they take when they decompose a larger value unit. When students relate models to the unit form, it primes them to use the vertical form in module 4. As with addition, students are not expected to master the standard subtraction algorithm until grade 4.
Throughout the topic, students move through concrete and pictorial representations to develop a conceptual understanding of subtraction. First, students use place value disks to represent the total and subtract numbers concretely. Then they use place value drawings to represent subtraction problems. Students see they can decompose a unit of ten, and later a hundred, when they need more ones in the ones place or tens in the tens place to subtract. Problems gradually increase in complexity as students decompose once, and then twice, to subtract. As it was in topic C, the language is intentionally consistent and repetitive. This secures familiarity with the representations and anchors students’ understanding as they move toward work with more abstract numbers.
Students relate place value drawings to more abstract recordings that show the minuend and subtrahend in unit form. While students are not expected to write unit form recordings independently, they make connections that deepen their place value understanding. For example, when they unbundle 1 ten into 10 ones in a drawing, students see how 1 hundred 2 tens 6 ones can be renamed as 1 hundred 1 ten 16 ones. The fact that the value of the minuend does not change when it is renamed is a key understanding in grade 2.
When students rename a three-digit total to subtract, they recognize that they can rename the total as tens and ones, write it in unit form, and subtract like units. For example, 167 – 82 can be thought of as 16 tens 7 ones – 8 tens 2 ones. Students are prompted to rename the total before they subtract within each place value. This encourages them to see that the total value remains the same, but it can be described by using different units. Renaming the total gives students flexibility with the order in which they subtract units.
Students use part–whole thinking and the relationship between addition and subtraction to solve take from with start unknown and add to with start unknown word problems. Start unknown problems are new for grade 2 students and present a greater cognitive lift because it is more challenging to start with an unknown, such as – 25 = 20.
These problems use only numbers within 100 so students can focus on the relationships presented. As they read carefully and draw what they know, students recognize the value in using a tape diagram to represent problems.
Topic D culminates with one- and two-step word problems. These include familiar problem types to give students the chance to apply their now well-stocked toolboxes of solution strategies. Students use number relationships to make decisions about whether to add or subtract, which bolsters their confidence as they prepare to work within 1,000 in module 4.
Progression of Lessons
Lesson 20
Reason about when to unbundle a ten to subtract.
Lesson 21
Use concrete models to decompose a ten with two-digit totals.
Lesson 22
Use place value drawings to decompose a ten and relate them to written recordings.
I can find 61 – 6 by unbundling a ten and drawing 10 ones on the chart. I can rename 61 as 5 tens 11 ones so I know I have enough ones in the ones place to subtract.
I can find 35 – 17 by using place value disks. Since I don’t have enough ones in the ones place to subtract 7 ones from 5 ones, I can decompose a unit of ten into 10 ones. Now I have 2 tens 15 ones. When I take away 1 ten 7 ones, there are 1 ten 8 ones left.
I can find 126 – 19 by making a place value drawing. Since I don’t have enough ones in the ones place to subtract 9 ones, I can rename 2 tens 6 ones as 1 ten 16 ones. Now I have enough to subtract in the ones and tens places.
Lesson 23
Use concrete models and drawings to decompose a hundred.
Lesson 24
Use place value drawings to decompose a hundred and relate them to written recordings.
Lesson 25
Use place value drawings to subtract with two decompositions.
To find 148 – 65, I can exchange a hundreds disk for 10 tens disks. Now I have 0 hundreds 14 tens 8 ones, and I can more easily subtract 6 tens 5 ones.
To find 108 – 32, I can rename 1 hundred as 10 tens. I am ready to subtract in the ones and tens places. I know 8 ones – 2 ones = 6 ones and 10 tens – 3 tens = 7 tens. The answer is 76.
To find 154 – 87, I can decompose 1 ten so I have enough to subtract in the ones place. Then I can decompose 1 hundred so I have enough to subtract in the tens place. I can rename 154 as 14 tens 14 ones. Now I am ready to subtract.
Lesson 26
Solve add to and take from with start unknown word problems.
Lesson 27
Solve two-step word problems within 100.
I can draw a tape diagram to represent the problem. The diagram shows 90 cents as the total and 35 cents as a part. The other part is the unknown. I can write ___ + 35 = 90 to match the problem. Then I count up from 35 to 90 to find the answer, 55 cents.
I can draw a tape diagram that has parts for each pencil color, so I know the total for the first step is 19. Then I can make a new tape with a total of 19 and show one part is 5 and the unknown part is 14.
Reason about when to unbundle a ten to subtract.
Lesson at a Glance
Students use place value understanding to reason about when they need to unbundle a ten to subtract. They draw to represent subtraction problems and unbundle a ten to subtract.
Key Question
• When do we need to unbundle a ten to subtract?
Achievement Descriptors
2.Mod2.AD4 Subtract within 200 by using concrete models or drawings, strategies based on place value, properties of operations, or the relationship between addition and subtraction. (2.NBT.B.7)
2.Mod2.AD6 Rename 1 of a larger unit as 10 of a smaller unit. (2.NBT.B.7)
Name
Miss Wells has
Use concrete models to decompose a ten with two-digit totals.
Lesson at a Glance
Students use place value disks to model a subtraction problem. They determine when to rename a ten as smaller units to subtract. Students make connections between concrete models and place value drawings.
Key Question
• Does the total change when we exchange or rename units?
Achievement Descriptor
2.Mod2.AD6 Rename 1 of a larger unit as 10 of a smaller unit. (2.NBT.B.7)
Use place value drawings to decompose a
ten and relate them to
written recordings.
Lesson at a Glance
Students use a place value drawing to represent subtraction problems. They decompose a ten and rename it as 10 ones. Students relate a place value drawing to a written recording in which the total and known part are written in unit form.
Key Questions
• How do place value drawings help us subtract?
• How does a place value drawing relate to a written recording?
Achievement Descriptors
2.Mod2.AD4 Subtract within 200 by using concrete models or drawings, strategies based on place value, properties of operations, or the relationship between addition and subtraction. (2.NBT.B.7)
2.Mod2.AD6 Rename 1 of a larger unit as 10 of a smaller unit. (2.NBT.B.7)
Use concrete models and drawings to decompose a hundred.
Lesson at a Glance
Students use place value disks to decompose a hundred and rename a three-digit total to subtract. Students learn that they can rename the total in tens and ones, write it in unit form, and subtract like units. Students make connections between concrete models and place value drawings.
Key Questions
• How can we unbundle a hundred to help us subtract?
• How can unit form help us subtract like units?
Achievement Descriptors
2.Mod2.AD4 Subtract within 200 by using concrete models or drawings, strategies based on place value, properties of operations, or the relationship between addition and subtraction. (2.NBT.B.7)
2.Mod2.AD6 Rename 1 of a larger unit as 10 of a smaller unit. (2.NBT.B.7)
Use place value drawings to decompose a hundred and relate them to written recordings.
Lesson at a Glance
Students use place value drawings to represent subtraction problems. They decompose a hundred and rename it as 10 tens. Students relate place value drawings to written recordings.
Read Tam has 148 gumdrops. Nick has 74 gumdrops.
How many more does Tam have than Nick? Draw
Key Questions
• How do place value drawings help us to subtract?
• How does a place value drawing relate to a written recording?
Achievement Descriptors
2.Mod2.AD4 Subtract within 200 by using concrete models or drawings, strategies based on place value, properties of operations, or the relationship between addition and subtraction. (2.NBT.B.7)
2.Mod2.AD6 Rename 1 of a larger unit as 10 of a smaller unit. (2.NBT.B.7)
Use place value drawings to subtract with two decompositions.
Lesson at a Glance
Show an efficient strategy to subtract. Sample:
Students subtract from three-digit totals with two decompositions by using both place value drawings and written recordings. They rename the totals in both standard and unit form to subtract when there are not enough ones and tens.
Key Questions
• How do place value drawings help us subtract when there are not enough tens and ones?
• How does renaming help us subtract?
Achievement Descriptors
2.Mod2.AD4 Subtract within 200 by using concrete models or drawings, strategies based on place value, properties of operations, or the relationship between addition and subtraction. (2.NBT.B.7)
2.Mod2.AD6 Rename 1 of a larger unit as 10 of a smaller unit. (2.NBT.B.7)
Solve add to and take from with start unknown word problems.
Lesson at a Glance
Students use part–total thinking and the relationship between addition and subtraction to solve add to with start unknown and take from with start unknown word problems. They use tape diagrams to represent the word problems.
Key Questions
• How can a tape diagram help make sense of a word problem?
• How can part–total thinking and the relationship between addition and subtraction help find the unknown?
Achievement Descriptor
2.Mod2.AD1 Represent and solve one-step addition and subtraction word problem types within 100 by using drawings and equations with a symbol for the unknown. (2.OA.A.1)
Solve two-step word problems within 100.
Lesson at a Glance
Students represent and solve two-step word problems. They reason about the problem to determine an efficient strategy to find the unknowns. They solve a two-step problem that requires interpreting a graph.
There are 67 books in the red bin.
There are 48 fewer books in the green bin. How many books are in the green bin?
Sample:
How many books are in the green bin?
67 - 48 = 19
There are 19 books in the green bin.
Key Questions
• How do we solve a word problem with more than one step?
• How does a tape diagram help us make sense of a word problem?
Achievement Descriptor
This lesson supports 2.OA.A.1 and introduces two-step word problems. Its content is intended to serve as a formative assessment and is therefore not included on summative assessments in this module.
Observational Assessment Recording Sheet
Grade 2 Module 2
Addition
and Subtraction Within 200
Achievement Descriptors
2.Mod2.AD1
2.Mod2.AD2
2.Mod2.AD3
2.Mod2.AD4
2.Mod2.AD5
2.Mod2.AD6
Notes
Represent and solve one-step addition and subtraction word problem types within 100 by using drawings and equations with a symbol for the unknown.
Add up to 4 two-digit numbers by using strategies based on place value or properties of operations.
Add within 200 by using concrete models or drawings, strategies based on place value, or properties of operations.
Subtract within 200 by using concrete models or drawings, strategies based on place value, properties of operations, or the relationship between addition and subtraction.
Rename 10 of a smaller unit as 1 of a larger unit.
Rename 1 of a larger unit as 10 of a smaller unit.
Student Name
Dates and Details of Observations
Module Achievement Descriptors and Content Standards by Lesson
● Focus content ○ Supplemental content
Achievement Descriptor Aligned
2.Mod2.AD12.OA.A.1 ○○
2.Mod2.AD22.NBT.B.6 ●
2.Mod2.AD32.NBT.B.7 ●●●●●●●●●●● ○
2.Mod2.AD42.NBT.B.7 ○●●●●●●●●●●●○
2.Mod2.AD52.NBT.B.7 ●●●●●
2.Mod2.AD62.NBT.B.7 ●●●●●●
Use place value drawings to decompose a
ten and relate them to
written recordings.
Lesson at a Glance
Students use a place value drawing to represent subtraction problems. They decompose a ten and rename it as 10 ones. Students relate a place value drawing to a written recording in which the total and known part are written in unit form.
Key Questions
• How do place value drawings help us subtract?
• How does a place value drawing relate to a written recording?
Achievement Descriptors
2.Mod2.AD4 Subtract within 200 by using concrete models or drawings, strategies based on place value, properties of operations, or the relationship between addition and subtraction. (2.NBT.B.7)
2.Mod2.AD6 Rename 1 of a larger unit as 10 of a smaller unit. (2.NBT.B.7)
Agenda Materials
Fluency 10 min
Launch 5 min
Learn 35 min
• Rename a Ten to Subtract
• Connect Pictorial Models to a Written Recording
• Problem Set
Land 10 min
Teacher
• None Students
• None
Lesson Preparation
None
Fluency
Choral Response: Subtract in Unit and Standard Form
Students subtract tens in unit form and say an equation in standard form to prepare for decomposing a ten with two-digit numbers that represent totals.
Display 5 tens – 4 tens = .
What is 5 tens – 4 tens in unit form? Raise your hand when you know.
Wait until most students raise their hands, and then signal for students to respond.
1 ten
Display the answer.
On my signal, say the equation with the numbers in standard form.
50 – 40 = 10
Display the equation with the numbers in standard form.
Repeat the process with the following sequence:
Whiteboard Exchange: Model Numbers with Place Value Drawings
Students use place value drawings to model a two- or three-digit number, say the number in unit form, and write the number in expanded form to prepare for relating place value drawings to a written recording for subtraction.
After each prompt for a written response, 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 place value chart.
Draw a place value chart and label the hundreds place, the tens place, and the ones place.
Draw dots on the place value chart to show 24.
Display 24 modeled on the place value chart.
On my signal, say the number in unit form. Ready? 2 tens 4 ones
Display the number in unit form.
Write the number in expanded form.
Display the number in expanded form.
Repeat the process with the following sequence:
Choral Response: Rename Place Value Units
Students rename tens to build fluency with strategies that require unbundling larger units.
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 43 = 4 tens _____ ones and the dots on the place value chart.
43 is equal to 4 tens and how many ones?
3
Display the answer.
Display 43 = 3 tens _____ ones.
43 is equal to 3 tens and how many ones?
13
Display a ten unbundling into 10 ones on the place value chart, and then display the answer.
Repeat the process with the following sequence:
Launch
Students reason about how to rename a number by analyzing place value drawings.
Display the place value drawings alongside the number 206 to engage students in mathematical discourse.
Give them 2 minutes of silent think time to analyze the picture and look for connections. Invite students to write each representation in unit form on their personal whiteboards. Have students give a silent signal to indicate they are finished.
Invite students to discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Purposely choose thinking that allows for rich discussion about the connections between the number and the drawings.
Then facilitate a class discussion. Invite students to share their thinking with the whole group and record their reasoning.
As students discuss, highlight thinking that emphasizes renaming.
The first drawing shows 1 hundred 10 tens 6 ones. That’s the same as 2 hundreds 6 ones.
The second drawing shows 1 hundred 9 tens 16 ones. If you exchange 10 ones for a ten, you’ll have 1 hundred 10 tens 6 ones again. That means it’s also the same as 206.
I can take 1 of the hundreds from 206 and decompose it into 10 tens. Then I have 1 hundred 10 tens 6 ones.
They all equal 206.
We can rename and represent numbers in different ways.
Transition to the next segment by framing the work.
Today, we will use place value drawings to rename a ten to subtract.
Learn
Rename a Ten to Subtract
Students rename a ten as ones to subtract by using place value drawings.
Write 40 – 25 = .
Let’s find 40 – 25 by using a place value drawing.
Guide students in drawing and labeling a two-column place value chart, as you do the same.
Invite students to think–pair–share about how they can represent the problem on a place value chart.
We can draw 4 dots in the tens place to show 40 because it’s the total.
You only draw 40 because it’s the number we need to subtract from.
25 is part of 40, so you just draw 40 and then take away the 25.
Draw 4 tens, as students do the same. Then guide students through the subtraction with the following sequence.
How many tens and ones do you see?
4 tens 0 ones
What is the part we are subtracting? Say it in unit form.
2 tens 5 ones
Look at the ones place. Can we subtract 5 ones from 0 ones? No.
Invite students to think–pair–share about where they can get more ones.
We can unbundle a ten.
We can rename 1 ten as 10 ones.
Promoting the Standards for Mathematical Practice
When students decompose a ten to rename a number so they have more ones, they look for and make use of the structure of place value units (MP7).
Ask the following questions to promote MP7:
• How does it help to decompose a ten when you do not have enough units to subtract from?
• Do we always need to decompose a ten to subtract?
Let’s rename 1 ten as 10 ones. Cross off a ten and draw an arrow from the tens place to the ones place. Then draw to represent the 10 ones. That shows how you rename 1 ten as 10 ones.
Draw to show the renaming as students do the same. Tell students to draw in 5-group formation.
How many tens and ones do we have now?
3 tens 10 ones
Invite students to think–pair–share about whether 4 tens 0 ones is the same as 3 tens 10 ones.
Yes. We just renamed 1 ten as 10 ones.
Yes. The ten is still there; it’s just decomposed into 10 ones.
You can see 10 in the ones place.
Are we ready to subtract in the ones place?
Yes. 10 ones is more than 5 ones.
Are we ready to subtract in the tens place?
Yes. 3 tens is more than 2 tens.
We are ready to subtract.
Ask the following questions and cross off dots as students do the same.
What is 10 ones – 5 ones?
5 ones
What is 3 tens – 2 tens?
1 ten
Direct students to chorally count how many tens and ones are left.
Read the completed equation.
40 – 25 = 15
Invite students to turn and talk about how the place value drawings help them subtract.
Connect Pictorial Models to Written Recordings
Students relate place value drawings to a written method and rename a ten to subtract.
Write 126 – 19 = .
Guide students to draw and label a three-column place value chart, as you do the same.
Let’s find 126 – 19 by using place value drawings.
Do we have to show 19 on the place value chart?
No. We need to find an unknown part, not the total.
No. We need to take 19 out of 126. Then we’ll see the part that is left and that’s the unknown.
Direct students to model 126 on their place value charts as you do the same.
How many hundreds, tens, and ones are in the total? Say it in unit form.
1 hundred 2 tens 6 ones
Record the unit form alongside the place value drawing.
You can record your thinking in different ways when you subtract. How does the recording match the drawing?
I see 126 drawn as 1 hundred 2 tens 6 ones and that’s the same as the unit form recording.
They both show 1 hundred 2 tens 6 ones.
What is the part we need to subtract? Say it in unit form.
1 ten 9 ones
Write 1 ten 9 ones and complete the unit form recording of the subtraction problem.
Teacher Note
Relating the place value drawing to the unit form recording helps students build toward conceptual understanding of the standard subtraction algorithm. Students are not expected to write the unit form. These representations will be revisited and applied to the vertical form in module 4. Mastery of the standard subtraction algorithm is not expected until grade 4.
Differentiation: Support
For students who need more concrete support, have materials such as bundles and place value disks readily available. Encourage students to use tools until they are comfortable with the subtraction process.
Look at the ones place. Can we subtract 9 ones from 6 ones? No.
Invite students to think–pair–share about where they can get more ones. We can decompose a ten into 10 ones. We can rename 1 ten as 10 ones.
Direct students to show the renaming as you do the same. Invite them to turn and talk to say the renamed units. Then direct their attention to the unit form recording.
The hundred stays the same. We renamed 2 tens 6 ones as 1 ten 16 ones.
Watch how I can show that in unit form.
Cross off 2 tens 6 ones and write 1 ten 16 ones.
Language Support
Students may need support to describe the renaming in detail. Consider modeling a think-aloud or posting sentence frames such as the following for students to refer to.
• We renamed hundreds tens ones as hundreds tens ones.
The unit form is a different way to represent the problem. It helps us describe and record what we did on the place value chart.
Are we ready to subtract in the ones place? Yes.
Are we ready to subtract in the tens place? Yes.
Are we ready to subtract in the hundreds place? Yes.
We are ready to subtract.
Language Support
Consider facilitating a class discussion to clarify the relationship between a place value drawing and a written recording. Have students use their drawings to make connections to the written recording. Support the use of precise language with terms such as rename and decompose. Highlight similarities between the two recordings.
What is 16 ones – 9 ones?
7 ones
What is 1 ten – 1 ten?
0 tens
What is 1 hundred – 0 hundreds?
1 hundred
Write the remaining units in unit form on the recording.
Point to the place value drawing as students chorally count how many hundreds, tens, and ones are left.
Read the completed equation.
126 – 19 = 107
Invite students to turn and talk about connections between the place value drawing and the unit form recording.
Let’s use another strategy to check our work. 19 is close to 20. What other strategies can we use to check our work?
Because 19 is close to 20, we can use compensation to check our work.
Because 19 is close to 20 we can use take from a ten to check.
UDL: Action & Expression
When students listen to a classmate think aloud, they benefit from hearing the reasoning and decision-making that strategic learners engage in when problem solving.
Invite a student to think aloud about how to find the difference for 126 – 19 by using compensation, as you model the recording.
Invite students to think–pair–share about another way to check their work.
We can add 107 and 19. If we get 126, then we know we subtracted correctly. We can count on from 19.
Repeat the process to find 137 – 28, 153 – 22, and 186 – 47.
Problem Set
Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.
Differentiation: Challenge
Encourage students who demonstrate proficiency with the Problem Set to check their answers by using addition. This will strengthen their understanding of the relationship between addition and subtraction.
Land
Debrief 5 min
Objective: Use place value drawings to decompose a ten and relate them to written recordings.
Gather the class and facilitate a discussion that emphasizes the usefulness of place value drawings.
How do place value drawings help us subtract?
They help us see the total, so we know if we have enough of a unit to subtract.
They help us show how to rename units.
They help me keep track of what I’m doing.
How are a place value drawing and a written recording related?
They both show the total as units.
On the written recording, you show the total and the part you’re subtracting. On the place value drawing, you start with just the total but then you show the part you’re subtracting by crossing off units.
They both show how you rename tens and ones.
Direct students to look at their Problem Sets. Invite students to turn and talk about a problem in their Problem Sets that can be solved more efficiently by using another strategy.
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.
Sample Solutions
Read Alex has 84 one-dollar bills. He spends 46 one-dollar bills. How many one-dollar bills does Alex have now? Draw Write
8. Read
Ann has 142 dimes.
Jack has 17 fewer dimes than Ann. How many dimes does Jack have?
Draw
Write 142 – 17 = 125 Jack has 125 dimes.
Common Addition and Subtraction Problem Types 1
RESULT UNKNOWN CHANGE UNKNOWN START UNKNOWN
ADD TO
TAKE FROM
Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now?
���� + ���� = ?
Five apples were on the table. I ate two apples. How many apples are on the table now? ����−���� = ?
Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two?
���� + ?= ����
Five apples were on the table. I ate some. Then there were three apples. How many apples did I eat?
���� ?= ����
PUT TOGETHER / TAKE APART 3
COMPARE 4
Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before?
? + ���� = ����
Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before?
? ���� = ����
TOTAL UNKNOWN ADDEND UNKNOWN BOTH ADDENDS UNKNOWN 2
Three red apples and two green apples are on the table. How many apples are on the table?
���� + ���� = ?
Five apples are on the table. Three are red and the rest are green. How many apples are green?
+ ?= ����, ����−���� = ?
Grandma has five flowers. How many can she put in the red vase and how many in her blue vase?
DIFFERENCE UNKNOWN BIGGER UNKNOWN SMALLER UNKNOWN
(“How many more?” version): Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy?
(“How many fewer?” version): Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie?
+ ?= ����, ����−���� = ?
(Version with “more”): Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have?
(Version with “fewer”): Lucy has three fewer apples than Julie. Lucy has two apples. How many apples does Julie have? ���� + ���� = ?, ���� + ���� = ?
(Version with “more”): Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have?
(Version with “fewer”): Lucy has three fewer apples than Julie. Julie has five apples. How many apples does Lucy have?
= ?, ? + ���� = ����
Pink shading indicates the four Kindergarten problem types. Grade 1 and 2 students work with all types and variants. Blue shading indicates the four difficult problem types that students should work with in grade 1 but need not master until grade 2.
1 Adapted from Box 2–4 of Mathematics Learning in Early Childhood, National Research Council (2009, pp. 32, 33).
2 These take apart situations can be used to show all the decompositions of a given number. The Associated equations, which have the total on the left of the equal sign, help children understand that the = sign does not always mean makes or results in but always does mean is the same number as.
3 Either addend can be unknown, so there are three variations of these problem situations. Both Addends Unknown is a productive extension of the basic situation, especially for small numbers less than or equal to 10
4 For the Bigger Unknown or Smaller Unknown situations, one version directs the correct operation (the version using more for the bigger unknown and using less for the smaller unknown. The other versions are more difficult.
Credits
Great Minds. 2021. Eureka Math2®. Washington, DC: Great Minds. https://greatminds.org/math
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.
