Assess Embedded Opportunities to Inform Instruction, 3–5
A Story of Units ®
Before This Module
Grade 2 Modules 1 and 2
In grade 2, students describe and apply place value concepts to two- and three-digit numbers. They count and bundle ones, tens, and hundreds up to 1,000. Students read and write numbers in standard, unit, and expanded forms and apply place value understanding to add and subtract two- and three-digit numbers by using a variety of strategies. Simplifying strategies consist of composing and decomposing tens and hundreds to make problems easier to compute mentally and developing various written methods to record student thinking.
Students also estimate and measure length by using a variety of tools and units in the customary and metric systems of measurement.
Grade 3 uses familiar place value concepts to expand student understanding of metric measurement of weight and liquid volume and to develop fluency in addition and subtraction within 1,000.
Overview
Place Value Concepts Through Metric Measurement
Topic A
Understanding Place Value Concepts Through Metric Measurement
Students estimate and measure weight and liquid volume by using grams, kilograms, liters, and milliliters. Familiar concepts, such as composing and decomposing place value units, are applied to the relationship between the smaller unit (gram or milliliter) and the unit that represents 1,000 (kilogram or liter). The number line is applied as a familiar model for understanding how to read circular and vertical measurement scales. Students solve one-step word problems that have measurement contexts.
Topic B
Rounding to the Nearest Ten and Hundred
Reading temperatures on a thermometer provides an initial context for students to round to the nearest ten or hundred. To round a number, students determine the two consecutive tens or hundreds that the number is between and then determine which ten or hundred is closer by thinking about the number in relation to the halfway mark. The number line, when presented vertically, provides a new perspective on a familiar tool and is used to help students to round numbers. Students then apply their rounding skills to estimate sums and differences.
Topic C
Simplifying Strategies to Find Sums and Differences
Students apply their knowledge of the vertical number line to the scale on a scaled bar graph. They represent data in a scaled bar graph and solve addition and subtraction problems related to scaled bar graphs. They explore a variety of addition and subtraction strategies based on place value, the properties of operations, and the relationship between addition and subtraction. The strategies include adding and subtracting like units, making the next ten or hundred, taking from a ten or hundred, and using compensation. Emphasis is placed on thinking flexibly and building toward the use of mental math. Throughout the topic, students share and critique solution strategies for addition and subtraction problems and explain how their choice in strategy helps them to add or subtract efficiently.
After This Module
Grade 4 Module 1
In grade 4, students apply their understanding of measurement units to convert weight, liquid volume, and length measurements from larger units to smaller units.
Students generalize place value and rounding concepts and relationships to larger, multi-digit numbers. They add and subtract multi-digit numbers by using the standard algorithms for addition and subtraction.
Topic D
Two- and Three-Digit Measurement Addition and Subtraction
Students use concrete and pictorial place value models alongside vertical form to represent and record the work of the standard algorithm for addition and the standard algorithm for subtraction. They compose and decompose units as needed and estimate to assess the reasonableness of their answers. Students apply their computation skills to select an appropriate strategy and solve one- and two-step word problems involving measurement contexts and units.
Topic C Simplifying Strategies to Find Sums and Differences
Topic C explores a variety of familiar addition and subtraction strategies based on place value, the properties of addition, and the relationship between operations. Each strategy is familiar from grade 2. Topic C also extends grade 2 work with data to include scaled bar graphs.
Experience from topic B with scales that have intervals other than 1 transfers to scaled bar graphs. Students relate the scales on scaled bar graphs to the scales used on measurement tools and the vertical number line they used for rounding. Answering how many more and how many less questions about the data helps students move from reading and creating graphs to finding sums and differences.
Students use place value understanding to demonstrate properties when they add and subtract like units, make or take from a benchmark number (i.e., a ten or hundred), and use compensation. Familiar models and recording methods, including number bonds, the arrow way, and tape diagrams, are used to represent student thinking. Emphasis is placed on developing flexibility and fluency in finding sums and differences and building toward the use of mental math. Students evaluate the types of problems that are well-suited to each strategy. Students are not expected to name and master every strategy; rather, they are expected to purposefully approach problems in ways that make each problem easier for them, even if it is a different approach than one selected by someone else. Strategy work from grade 2 is advanced by the use of parentheses to record the decompositions shown in number bonds.
In topic D, students use place value models and the standard algorithms for addition and subtraction. They continue to be encouraged to use strategies that make sense to them by evaluating problems to determine when a simplifying strategy is a more efficient approach than a place value chart and the standard algorithm. Students use addition and subtraction strategies to solve problems involving measurement contexts and measurement units.
Progression of Lessons
Lesson
13
Collect and represent data in a scaled bar graph and solve related problems.
Lesson
14
Use place value understanding to add and subtract like units.
Lesson
15
Use the associative property to make the next ten to add.
Scaled bar graphs use a scale other than 1 to show data containing large quantities. Scaled bar graphs help represent the data so that we can see similarities and differences and answer questions about the data.
Breaking apart numbers into tens and ones helps me add and subtract in my head. I add tens to tens and ones to ones. There are many ways I can represent my thinking.
When one addend is close to a benchmark number, I can compute in my head by breaking apart the other addend and making a ten or hundred.
Lesson 16
Use compensation to add.
98 + 56 = 154
56 + 100- 2 156154
When one addend is close to a benchmark number, I can use compensation to compute in my head by adding the benchmark number and then subtracting the extra that I added.
Lesson 17
Use place value understanding to subtract efficiently using take from a ten.
50 - 17 = 33 20 - 17 = 3 30 + 3 = 33 30 20 3
When the number I subtract is close to a ten, I can break apart the total into two parts to make one of the parts a ten so I can subtract in my head. After I subtract from the ten, I add the answer to the other part to find the difference in the original problem.
Lesson 18
Use place value understanding to subtract efficiently using take from a hundred. 230 - 96 = 134 100 - 96 = 4 130 + 4 = 134 130100 4
- 96 = 130 + (100 - 96) = 130 + 4 = 134
When the number I subtract is close to a hundred, I can break apart the total into two parts to make one of the parts a hundred so I can subtract in my head like I did when the number was close to a ten. Then I add the other part.
Use compensation to subtract.
Lesson at a Glance
Students use compensation as a strategy to subtract more efficiently. Students intentionally select a subtraction strategy and explain their reasoning.
Key Questions
• How does compensation help make it easier to subtract mentally?
• When is compensation a useful subtraction strategy?
Achievement Descriptor
3.Mod2.AD2 Add and subtract within 1,000 fluently using strategies based on place value, properties of operations, or the relationship between addition and subtraction. (3.NBT.A.2)
3.Mod2.AD2 Add and subtract within 1,000 fluently using strategies based on place value, properties of operations, or the relationship between addition and subtraction.
RELATEDCCSSM
3.NBT.A.2 Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
Partially Proficient
Add and subtract within 1,000 fluently with no more than one instance of regrouping using a strategy based on place value, properties of operations, or the relationship between addition and subtraction.
Add or subtract.
328 + 239 =
570 155 =
Proficient
Add and subtract within 1,000 fluently with two instances of regrouping using a strategy based on place value, properties of operations, or the relationship between addition and subtraction.
Add or subtract.
458 179 =
278 + 364 =
Highly Proficient
Add and subtract within 1,000 fluently using multiple strategies.
Which of these show a correct way to find the unknown number in 500 = 197 + ?
Circle the two correct answers
Use place value understanding to add and subtract like units.
Lesson at a Glance
Students use place value understanding to add and subtract mentally. Unit form is used to support composing and decomposing like units. Students record and explain their thinking with models, including number bonds and the arrow way.
Key Questions
• How does place value help us add and subtract?
• How are models such as the number bond and the arrow way helpful in recording and explaining our thinking?
Achievement Descriptor
3.Mod2.AD2 Add and subtract within 1,000 fluently using strategies based on place value, properties of operations, or the relationship between addition and subtraction. (3.NBT.A.2)
Use place value understanding to subtract efficiently using take from a ten.
Lesson at a Glance
Students use the take from a ten strategy to subtract efficiently. They apply the strategy to two- and three-digit subtraction problems.
Key Questions
• How might the take from a ten strategy simplify a problem?
• When is the take from a ten strategy useful?
Achievement Descriptor
3.Mod2.AD2 Add and subtract within 1,000 fluently using strategies based on place value, properties of operations, or the relationship between addition and subtraction. (3.NBT.A.2)
Use place value understanding to subtract efficiently using take from a hundred.
Lesson at a Glance
Students build on their understanding of the take from a ten strategy to take from a hundred. They apply the strategy when subtracting from three-digit numbers when the subtrahend is close to a hundred. Students describe when the strategy is useful.
Key Questions
• How can the take from a hundred strategy be used when subtracting from three-digit numbers?
• How is take from a hundred a useful strategy for some subtraction problems?
Achievement Descriptor
3.Mod2.AD2 Add and subtract within 1,000 fluently using strategies based on place value, properties of operations, or the relationship between addition and subtraction. (3.NBT.A.2)
Use compensation to subtract.
Lesson at a Glance
Students use compensation as a strategy to subtract more efficiently. Students intentionally select a subtraction strategy and explain their reasoning.
Key Questions
• How does compensation help make it easier to subtract mentally?
• When is compensation a useful subtraction strategy?
Achievement Descriptor
3.Mod2.AD2 Add and subtract within 1,000 fluently using strategies based on place value, properties of operations, or the relationship between addition and subtraction. (3.NBT.A.2)
Module 2 Topic C Quiz
This assessment has five items. If you administer this assessment on paper, use this scoring guide and the answer key from the assessment book to score and grade each student’s assessment. No partial credit is given unless otherwise indicated in the notes.
1 3.Mod2.AD2 PP 1 × 3 = 3 Polytomous Each correct response is worth 1 1 2 points.
2 3.Mod2.AD6 P 1 × 2 = 2 Polytomous Each correct response is worth 1 2 point.
3 Part A 3.Mod2.AD7 PP 1 × 3 = 3 Dichotomous The correct response is worth 3 points.
3 Part B 3.Mod2.AD7 P 1 × 2 = 2 Dichotomous The correct response is worth 2 points.
4 Part A 3.Mod2.AD2 HP 1 × 1 = 1 Dichotomous Both correct responses must be identified to earn 1 point.
4 Part B 3.Mod2.AD2 P 1 × 2 = 2 Dichotomous The correct response is worth 2 points.
5 3.Mod2.AD2 HP 1 × 1 = 1 Polytomous Each correct response is worth 1 3 point.
To compute the adjusted score for each student,
• divide a student’s total earned points by the total possible points (i.e., 14 total possible points for this Topic Quiz),
• enter that result into the adjusted score converter on the digital platform, and
• use the value from the adjusted score converter to determine a grade.
Topic D
Two- and Three-Digit Measurement Addition and Subtraction
Topic D is devoted to using place value to model the standard algorithm for addition and the standard algorithm for subtraction with numbers within 1,000. Students use vertical form in conjunction with a place value model to allow them to better recognize like units and the value of numbers when they are not on the place value chart. Representations move from concrete to pictorial to abstract, and they serve to solidify the understanding of the composition and decomposition of units. Students learn that although standard algorithms work for all problems, there might be a more efficient strategy to use depending on the numbers being added or subtracted. Students apply learning from topic C to analyze addition and subtraction problems, many with a measurement context, and determine whether to use a simplifying strategy or the place value chart and standard algorithm for each.
Students begin the topic by using the place value chart and the standard algorithm for addition. They manipulate place value disks on a chart to represent the algorithm and record their work in vertical form. The concrete place value disks are then replaced with drawings on place value charts, where dots are used to represent the addends and the sum. Students bundle once and then twice and record the composition of new units vertically in different ways.
For subtraction, students manipulate place value disks on a chart before transitioning to drawing on the place value chart. Students get ready to subtract by checking each place value and unbundling as necessary before performing any subtractions. They record their work in vertical form with one and then two decompositions. Before subtracting, students estimate the answer by rounding the minuend and subtrahend and finding their difference. After subtracting, students use their estimate to assess the reasonableness of their answer. Students see that they can use addition to check their subtraction.
Within the topic, students apply their computation skills to solve one-step addition and subtraction word problems involving measurement contexts. To conclude the topic, students solve two-step word problems involving all four operations.
In future lessons, fluency activities provide ongoing practice toward developing fluency with addition and subtraction strategies and algorithms. Use of the standard algorithms for addition and subtraction extends to numbers within 1,000,000 in grade 4 module 1 and to decimals in grade 5.
Progression of Lessons
Lesson 20
Add measurements using the standard algorithm to compose larger units once.
Lesson 21
Add measurements using the standard algorithm to compose larger units twice.
Lesson 22
Subtract measurements using the standard algorithm to decompose larger units once.
Place value disks help to show addition and to transition to the vertical form. The standard algorithm is a process I can use to add like units. When I need to make a new unit, I record the new group on the line.
Instead of using actual place value disks, I can draw dots on a place value chart to represent the disks. When I need to make a new unit, there are different ways I can record it. New groups below shows the renaming on the line. Totals below shows the total for each place value below the addends.
Place value disks help me understand how to get ready to subtract, and they help me to see how one of a larger unit can be unbundled to ten of a smaller unit. When I subtract by using the standard algorithm, I show my work in vertical form. Estimating the answer helps me decide if my answer is reasonable.
Lesson 23
Subtract measurements using the standard algorithm to decompose larger units twice.
Lesson 24
Subtract measurements using the standard algorithm to decompose larger units across two place values.
Lesson 25
Solve two-step word problems.
Representing a subtraction problem by using dots on the place value chart helps me to see how to get ready to subtract. Sometimes I need to unbundle twice to be ready to subtract. I can see this on the place value chart and in vertical form. For some problems, another subtraction strategy is a better choice. After I subtract, I can check my answer by using addition.
When I need more ones and there are 0 tens, I need to unbundle twice to get enough ones. I can show that on the place value chart and in vertical form.
There are many ways to show my thinking and solve two-step word problems. I choose the way that makes sense to me. I can draw tape diagrams to represent the problems. They help me see what operations I should use to solve the problems. It is important to check each step to see if my answer is reasonable as I work.
Assess: Embedded Opportunities to Inform Instruction, 3–5
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.
