ReseaRch—Best PRactices Putting Research into …€¦ · ReseaRch—Best PRactices Putting...

Research & Math BackgroundContents Planning

Dr. Karen C. Fuson, Math Expressions Author

ReseaRch—Best PRactices

Putting Research into Practice

From Our Curriculum Research Project: Analyzing the Structure and Language of Word Problems

In this unit, children analyze a variety of word problem structures: Add To, Take From, Put Together/Take Apart, and Compare. They also analyze problems with not enough information, problems with extra information, problems with hidden information, and problems that require two steps to solve.

• Add To and Take From problems provide a quantity which is modified by a change—something is added or subtracted—which results in a new quantity.

• Put Together/Take Apart problems have all of the quantities of objects present from the start and nothing is introduced or removed.

• Compare problems involve someone or something that has more or less of something than someone or something else.

Throughout, children model, draw, or act out the actions or relations presented in the word problems as a strategy for understanding and solving them.

From Current Research: Using Mathematical Drawings

Mathematical drawings focus on the mathematically important features and relationships, such as quantity and operations, and can use small circles or other simple shapes. These representations can can evolve into schematic numerical drawings that show relations or operations.

Fuson, Karen C., Clements, Douglas H., Beckman, Sybilla. Focus in Grade 2: Teaching with Curriculum Focal Points, National Council of Teachers of Mathematics, 2011. 5

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Carpenter, Thomas P., Fennema, E., Franke, M.L., Empson, S.B., & Levi, L.W. Children's Mathematics: Cognitively Guided Instruction. Portsmouth, NH: Heinemann, 1999.

O’Daffer, Phares, et al. 2nd edition. Mathematics for Elementary School Teachers. Boston: Addison-Wesley, 2002. 73–87.

From Current Research: The Use of Questioning to Focus Learning and Promote Connections

To teach for depth understanding, teachers need to understand what their students are thinking and be able to support and extend that thinking. A teacher's use of questioning plays a vital role in focusing learning of foundational mathematical ideas and promoting mathematical connections. Such reasoning questions as "Why?" and "How do you know that?" posed during a lesson are great starters, but teachers also need to incorporate questioning techniques into their planning by thinking about specific questions to ask related to the particular topic being studied. When planning instruction, teachers must also anticipate the kinds of answers they might get from students in response to the questions posed.

Mirra, Amy. Focus in Prekindergarten–Grade 2: Teaching with Curriculum Focal Points, National Council of Teachers of Mathematics, 2011. 18

Other Useful References: Addition and Subtraction

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ACTIVITY 2ACTIVITY 2


Getting Ready to Teach Unit 1Using the Common Core Standards for Mathematical PracticeThe Common Core State Standards for Mathematical Content indicate what concepts, skills, and problem solving children should learn. The Common Core State Standards for Mathematical Practice indicate how children should demonstrate understanding. These Mathematical Practices are embedded directly into the Student and Teacher Editions for each unit in Math Expressions. As you use the teaching suggestions, you will automatically implement a teaching style that encourages children to demonstrate a thorough understanding of concepts, skills, and problems. In this program, Math Talk suggestions are a vehicle used to encourage discussion that supports all eight Mathematical Practices. See examples in Mathematical Practice 6.

Mathematical Practice 1Make sense of problems and persevere in solving them.

Children analyze and make conjectures about how to solve a problem. They plan, monitor, and check their solutions. They determine if their answers are reasonable and can justify their reasoning.

TeaCher ediTion: examples from Unit 1

MP.1 Make Sense of Problems Analyze the Problem Children can also solve Put Together problems in which both addends are unknown. Direct children’s attention to Problem 5 on Student Activity Book page 38.

• How is this problem different from the problems we just solved on page 37? Possible answers: The problem has only one number and we need to find two numbers. We know the total, but we don’t know either addend. There is more than one correct answer to this problem.

Lesson 13

MP.1 Make Sense of Problems Analyze the Problem As a class, determine the hidden or first-step question for Problem 3. How many tomatoes are left after Mari makes the sauce? Write the first-step question on the board.

Repeat for Problem 4 and 5.

Leave the first-step questions on the board. Use the Solve and Discuss structure to complete the solutions for Problems 3–5. Children should show an equation and a drawing for each step.

Lesson 19

Mathematical Practice 1 is integrated into Unit 1 in the following ways:Make Sense of ProblemsAnalyze Relationships

Analyze the ProblemDescribe Relationships

Look for a Pattern

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Mathematical Practice 2Reason abstractly and quantitatively.

Children make sense of quantities and their relationships in problem situations. They can connect diagrams and equations for a given situation. Quantitative reasoning entails attending to the meaning of quantities. In this unit, this involves addition and subtraction within 20.

TeacheR edITIoN: examples from Unit 1

MP.2 Reason abstractly and Quantitatively Help children more fully understand the strategy by asking these questions.

• How do you know when you can use the Doubles Plus 2 or Doubles Minus 2 strategy to add two numbers? The partners or addends are just two numbers apart.

Circle each of the totals from the exercises you just worked through.

• Are these totals even or odd? even

• Do you think any Doubles Plus 2 or Doubles Minus 2 total will be even? Why? Yes. Possible explanation: It is always 2 more or 2 less than another even number.

Lesson 7

MP.2 Reason abstractly Ask children if they think they could keep adding columns to the table and continue to fill them in. Ask children to describe how they would do this. Children should realize that they can continue counting by 1s to fill in the top row. For the bottom row, they always double the number directly above.

3 4 5 6 7 86 8 10 12 14 16

Lesson 21

Mathematical Practice 2 is integrated into Unit 1 in the following ways:

Reason Abstractly Reason Abstractly and Quantitatively

Connect Symbols and WordsConnect Diagrams and Equations

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Mathematical Practice 3Construct viable arguments and critique the reasoning of others.

Children use stated assumptions, definitions, and previously established results in constructing arguments. They are able to analyze situations and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others.

Children are also able to distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Children can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

MATH TALK is a conversation tool by which children formulate ideas and analyze responses and engage in discourse. See also MP.6 Attend to Precision.

TeaCher ediTion: examples from Unit 1

MP.3 Construct Viable arguments Compare Representations Explain that the longer rectangle is the total, the shorter rectangle is one of the addends, and the oval is the unknown addend. Have children discuss how this drawing is similar to, and different from, the matching drawings.

Lesson 14

What’s the Error? W H O L E C L A S S

MP.3, MP.6 Construct Viable arguments/Critique reasoning of others Puzzled Penguin Explain that Puzzled Penguin sometimes gets confused about math and that they will be helping Puzzled Penguin throughout the year. This Puzzled Penguin exercise addresses an error that children might make when counting by 2s. They note the pattern in the ones as 2, 4, 6, 8, 2, 4, 6, 8 rather than 2, 4, 6, 8, 0, 2, 4, 6, 8, 0. They fail to include the decade numbers. Children can discuss ways to help Puzzled Penguin. Some children may count by 2s by first saying 2, then saying 3 “in their head,” then saying 4, then saying 5 “in their head,” and so on. This would help them not miss any numbers.

Lesson 6


Critique Reasoning of OthersConstruct Viable Arguments

Compare MethodsPuzzled Penguin

Compare Representations

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Mathematical Practice 4Model with mathematics.

Children can apply the mathematics they know to solve problems that arise in everyday life. This might be as simple as writing an equation to solve a problem. Children might draw diagrams to lead them to a solution for a problem. Children apply what they know and are comfortable making assumptions and approximations to simplify a complicated situation. They are able to identify important quantities in a practical situation and represent their relationships using such tools as diagrams, tables, graphs, and formulas.

Teacher ediTion: examples from Unit 1

MP.1, MP4 Make Sense of Problems/Model with Mathematics Draw a Diagram For each problem, children draw and label a proof drawing, Math Mountain, or comparison bars. Then they write an equation for each problem as well. Choose children who drew different drawings or wrote different equations to share their solutions with the class.

Lesson 20

MP.4, MP.5 Use appropriate Tools/Model Mathematics MathBoard If none of the children draw comparison bars to solve, invite everyone to show the comparison bars on their MathBoards. Check that children draw the longer bar to represent Darryl. The 6 should be placed in an oval after the bar representing 8 for Sarah.

Lesson 21


Model with Mathematics Draw a Diagram

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Mathematical Practice 5Use appropriate tools strategically.

Children consider the available tools and models when solving mathematical problems. Children make sound decisions about when each of these tools might be helpful. These tools might include paper and pencil, a straightedge, a ruler, or the MathBoard. They recognize both the insight to be gained from using the tool and the tool’s limitations. When making mathematical models, they are able to identify quantities in a practical situation and represent relationships using modeling tools such as diagrams, grid paper, tables, graphs, and equations.

Modeling numbers in problems and in computations is a central focus in Math Expressions lessons. Children learn and develop models to solve numerical problems and to model problem situations. Children continually use both kinds of modeling throughout the program.


MP.5 Use appropriate Tools Dime Strips and Pennies Have children sit in Helping Pairs. Explain that they will use their Dime Strips, pennies, and coin covers to add numbers with teen totals. They will see how to separate the second partner into a chunk to make ten and a chunk over ten that makes the teen number. Have children who have used the Make-a-Ten strategy on previous days show how they do it. Move through the steps below with the pennies, eliciting as much as possible from children.

Lesson 3

MP.5 Use appropriate Tools Fingers Ask a child to demonstrate with fingers how to change 8 + 6 to be 10 + 4. There are several ways to do this. Ask other children if they have a different way. The easiest way is shown below, so demonstrate this method if a child doesn’t show it.

10 + 4 = 144 left over

8

8 + 6 = 14

Lesson 3


Use Appropriate ToolsMathBoardConnecting Cubes

Dime Strips and PenniesFingers

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Mathematical Practice 6Attend to precision.

Children try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose. They are careful about specifying units of measure to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. Children give carefully formulated explanations to each other.

TeAcher ediTion: examples from Unit 1

MP.6 Attend to Precision Review the structure of Compare problems that children have solved in this lesson. They are situations in which one amount is more or less than another amount. Then ask children to write a word problem using the word more. They can work in pairs or in small groups. Have children exchange problems to solve.

Lesson 15

MP.6 Attend to Precision Explain a Solution Use the Solve and Discuss structure to solve Problem 2. Pairs of children work at their seats while three to six children work at the board. Children make a labeled drawing and equation for each part. Ask several children, whose solutions look different, to explain their drawings and solutions and answer the questions posed by the class.

Lesson 19

MATH TALKin ACTION

Below is a sample classroom discussion of the different ways to determine whether 9 is even or odd.

Allison: I know 9 is odd. When I count by 2s, I skip right over it: 2, 4, 6, 8, . . . 10. I went right from 8 to 10. When we count by 2s, we say the even numbers.

Marcus: It’s odd for sure. When I draw pairs of counters, I can’t make 9 without having one counter not in a pair.

Jorge: I know another way to show 9 is odd. Suppose I have 9 marbles. I can’t share 9 marbles equally with my friend. One of us will get one more marble than the other. You just can’t divide 9 marbles so it’s fair or even. So 9 must be odd.

Darius: If 9 is even, we could write an addition double for it. I know 4 + 4 = 8 and 5 + 5 = 10. There is no addition double for 9. So, 9 has to be odd.

Lesson 6


Attend to PrecisionDescribe Methods

Explain a SolutionPuzzled Penguin

Explain a Method

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Mathematical Practice 7Look for structure.

Children analyze problems to discern a pattern or structure. They draw conclusions about the structure of the relationships they have identified.


MP.7 Look for Structure Identify Relationships Ask children to give word problems for the Math Mountain and for the equation (Exercise 2 on Student Activity Book page 3). Any situation in which the addends are 8 and 6 and the total is unknown is fine. Ask children to tell in their own words a word problem another child gives; the problem should describe the same situation. This helps children learn to listen as other children talk and to vary the math language in the question (in all, in total, altogether, then at the end, and so on.)

Lesson 1

MP.7 Look for Structure Identify Relationships Direct children’s attention to Student Activity Book page 55. Ask a child to read Problem 1 aloud.

• What information does the problem ask for? the number of cans Matt brings in for the food drive

• Before we can answer that question, we need to answer a different, or hidden question. Read the problem again. What is that hidden question? How many cans does Olivia bring?

• The first question below the green box shows us the hidden question in the problem.

Lesson 19


Look for Structure Identify Relationships

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Class Activity

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► Math and Healthy FoodBeren and her friends are making funny face pizzas.

1. Darryl uses 2 green olives for eyes and 9 black olives to make a big smile. How many olives does he use?

label

2. Sarah uses 6 fewer mushroom slices than Darryl. Sarah uses 8 slices. How many slices does Darryl use?

label

3. When they start making the pizzas, there are a dozen small tomatoes. Darryl uses 2 tomatoes. Beren and Dawn each use 1 tomato. No one else uses any. How many tomatoes are left?

label

olives11

slices14

tomatoes8

UNIT 1 LESSON 21 Focus on Mathematical Practices 65

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Class Activity

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► Problems with Extra InformationSolve. Cross out the information you do not need.

4. Beren makes a fruit salad. She uses 2 strawberries, 8 blueberries, 7 raspberries, and 3 apples. How many berries does she use?

label

5. Darryl makes a snack mix. He uses 2 cups of cereal, 4 cups of raisins, 3 cups of dried cherries, and 2 cups of walnuts. How many more cups of dried fruit does he use than cups of nuts?

label

► Write and Solve a Problem“Ants on a Log” is a snack made with celery, peanut butter, and raisins.

Show your work.

6. Use the pictures. On a separate sheet of paper, write a problem.

Exchange with a classmate. Solve each other’s problem.

Beren’s Snack Darryl’s Snack Sarah’s Snack

berries17

more cups

Children’s problems will vary.

5

66 UNIT 1 LESSON 21 Focus on Mathematical Practices

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Mathematical Practice 8Look for and express regularity in repeated reasoning.

Children use repeated reasoning as they analyze patterns, relationships, and calculations to generalize methods, rules, and shortcuts. As they work to solve a problem, children maintain oversight of the process while attending to the details. They continually evaluate the reasonableness of their intermediate results.

TeACHeR eDITION: examples from Unit 1

MP.8 Use Repeated Reasoning Generalize Have children suggest addition and subtraction equations from the Math Mountains. This work with Math Mountains helps children use these patterns in solving teen additions and subtractions. Look vertically at the 9 + Math Mountains (in the column on the left) to see the pattern of the teen ones being one less than the non-9 addend.

Lesson 2

MP.8 Use Repeated Reasoning Draw Conclusions After children complete Exercises 25–28, help them relate equal groups to addition doubles.

• Can you write an addition double for an odd number? No. Possible explanation: You need to have two equal groups to write an addition double.

Lesson 6


Use Repeated Reasoning Generalize

Draw Conclusions

Focus on Mathematical PracticesUnit 1 includes a special lesson that involves solving real world problems and incorporates all 8 Mathematical Practices. In this lesson children use what they know about addition and subtraction to solve problems about healthy foods.

STUDeNT eDITION: LeSSON 21, PAGeS 65–66


Math Expressions VOCABULARY

As you teach this unit, emphasize

understanding of these terms.

• partner• Math Mountain• teen number• equation chain

See the Teacher Glossary.

Getting Ready to Teach Unit 1Learning Path in the Common Core StandardsIn this unit, children work toward building fluency with addition and subtraction within 20 and mastering all addition and subtraction word problem subtypes.

Visual models and real world situations are used throughout the unit to help children understand the meaning of addition and subtraction.

Help Children Avoid Common ErrorsMath Expressions gives children opportunities to analyze and correct errors, explaining why the reasoning was flawed.

In this unit we use Puzzled Penguin to show typical errors that children make. Children enjoy teaching Puzzled Penguin the correct way, why this way is correct, and why Puzzled Penguin made the error. Common errors are presented in the Puzzled Penguin feature in the following lessons:

→ Lesson 6: Eliminating decade numbers when counting by 2s

→ Lesson 8: Misunderstanding the meaning of the equal sign when the expression after the equal sign includes more than one number and an operation sign

→ Lesson 15: Incorrectly labeling comparison bars

→ Lesson 20: Not recognizing extra information in a word problem

In addition to Puzzled Penguin, there are other suggestions listed in the Teacher Edition to help you watch for situations that may lead to common errors. As a part of the Unit Test Teacher Edition pages, you will find a common error and prescription listed for each test item.

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from THE PROGRESSIONS FOR THE COMMON CORE STATE STANDARDS ON OPERATIONS AND ALGEBRAIC THINKING

Relate Addition and Subtraction

Diagrams used in Grade 1 to show

how quantities in the situation

are related continue to be useful

in Grade 2, and students continue

to relate the diagrams to situation

equations. Such relating helps

students rewrite a situation

equation like - 38 = 49 as

49 + 38 = because they see that

the first number in the subtraction

equation is the total. Each addition

and subtraction equation has

seven related equations. Students

can write all of these equations,

continuing to connect addition and

subtraction, and their experience

with equations of various forms.

Relate Addition and Subtraction

Lessons

1 2

Math Mountains and Equations Math Mountains are used in Math Expressions to show how addition and subtraction are related. A Math Mountain shows a total on top and two partners (addends) at the bottom. In Lesson 1, children relate Math Mountains to addition and subtraction equations and to real world problems.

Math Mountain Equation

8 6

14

8 + 6 = 14

Real World Problem

There are 8 flowers in a vase. There are 6 flowers in a glass. How many flowers are there altogether?

Math Mountain Equations

15

69

9 + 6 = 15

15 - 9 = 6

Real World Problems

There were 9 children playing in the park. Some more children came. Now there are 15 children playing. How many children came to the park?

There were 15 children playing in the park. Nine went home. How many children are still playing?

UNIT 1 | Overview | 1EE


Related Equations Children discuss why the eight equations below come from the Math Mountain. In their discussion, they informally discuss properties of addition and of equality, and the relationship between addition and subtraction.

9 3

12 �9�+�3�=�12� 12�=�9�+�3

�3�+�9�=�12� 12�=�3�+�9

�12�-�9�=�3� 3�=�12�-�9

�12�-�3�=�9� 9�=�12�-�3

Math Mountain Cards Children use Math Mountain Cards to practice addition and subtraction. The cards make clear how addition, subtraction, and finding an unknown addend are related. The cards reinforce that the same process is used for subtraction and for finding an unknown addend.

4 5

I think: “4 + 5 = 9.”

+

– –

4 5

4 5

9

9

I think: “9 - 5 = ?”or "5 + ? = 9."

I think: “9 - 4 = ?”or "4 + ? = 9."

+

– –

+

– –

4 5

4 5

9

9

I think: “9 - 5 = ?”or "5 + ? = 9."

I think: “9 - 4 = ?”or "4 + ? = 9."

+

– –

+

– –

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Count On for Addition or Subtraction Children at Level 1 for addition count all; children at Level 1 for subtraction take away. Children at Level 2 count on. At Grade 2, children should be using either Level 2 or Level 3 strategies for addition and subtraction. Level 2 strategies are reviewed in Lesson 1. Level 3 strategies are reviewed in Lessons 3–5. (At Level 3, children decompose an addend and compose a part with another addend.)

As the examples below illustrate, the process for counting on to find a total and counting on to find a partner look the same. Only the solver knows which problem is being solved.

9 + 3 = 12

Already 9

3 more tomake 12

101112Stop when

I hear 12.

9 + 3 = 12

Already9 10 11 12

12 - 9 = 3

I took 9away.

3 more tomake 12

101112

3 more tomake 12

I took 9away.

12 – 9 = 3

101112

In counting on to find the total 12, you keep track of the second addend 3 and stop when your fingers show 3. The words tell the total 12.

In counting on to find the unknown addend 3, you keep track of the words you say and stop when you hear 12. The number of fingers tell the unknown addend 3.

Using the Commutative Property of Addition The Commutative Property of Addition states that two addends can be added in either order and the sum remains the same. So, for addition, children could begin counting on from either addend. However, children should be encouraged to count on from the greater addend. In this way, there will be fewer numbers to count on and less chance of errors.

UNIT 1 | Overview | 1GG



Addition and Subtraction Within 20

The deep extended experiences

students have with addition and

subtraction in Kindergarten and

Grade 1 culminate in Grade 2 with

students becoming fluent in single-

digit additions and the related

subtractions using the mental

Level 2 and 3 strategies as

needed. So fluency in adding and

subtracting single-digit numbers

has progressed from numbers

within 5 in Kindergarten to within

10 in Grade 1 to within 20 in

Grade 2. The methods have also

become more advanced.

Make-a-Ten Strategies

Lessons

3 4 5

At Level 3, children decompose an addend and compose a part with another addend. Make a Ten is a Level 3 strategy. The goal at Grade 2 is for children to be using mental Level 2 and Level 3 strategies.

Make-a-Ten Strategy for Addition The difficult part of this strategy is separating the smaller addend into two parts: the amount that when added to the greater addend makes 10 and "the rest." Children model this strategy using coin strips and fingers.

8 + 6 =

10 + 4 = 14

6 gives 2 to 8 to make 10.We know there are 4 left.We add the 4 to the 10 and get 14.

6

partners of 62 4

8 + 6

8 + 6 =

10 + 4 = 144 left over

8

8 + 6 = 14

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Make-a-Ten Strategy for Subtraction The Make-a-Ten strategy for subtraction is the same as for finding an unknown addend. While the problem situations and the equations are different, in both problems, the total is known and one addend (or partner) is unknown.

8 + = 14

Step 1

Think: I already have 8.

Step 2

Put up 2 to make 10.

9

10

Step 3

Put up 4 more to make 14.

9

10 4 make 14

Step 4

Find the unknown partner.

2

4

2 + 4 = 6

Using Drawings for the Make-a-Ten Strategy Children use drawings to show the Make-a-Ten strategy. Children may generate any of a variety of drawings to illustrate this strategy. Two samples are shown below.

8 + 6 = 8 + = 148 + 6 8 + 68 + 6 8 + 6

UNIT 1 | Overview | 1II


Even and Odd Numbers

Lesson

6

Children explore several methods for deciding whether a number is even or odd. In this Math Talk in Action, children share those methods.

MATH TALKin ACTION

Below is a sample classroom discussion of the different ways to determine whether 9 is even or odd.

Let’s talk about how we can decide whether 9 is an even number or an odd number.

Allison: I know 9 is odd. When I count by 2s, I skip right over it: 2, 4, 6, 8, . . . 10. I went right from 8 to 10. When we count by 2s, we say the even numbers.

Marcus: It’s odd for sure. When I draw pairs of counters, I can’t make 9 without having one counter not in a pair.

Jorge: I know another way to show 9 is odd. Suppose I have 9 marbles. I can’t share 9 marbles equally with my friend. One of us will get one more marble than the other. I can draw a picture to show what I mean.

1 3 75

2 4 6 8

Me

My friend

9

You just can’t share 9 marbles so it’s fair or even. So 9 must be odd.

Darius: If 9 is even, we could write an addition double for it. I know 4 + 4 = 8 and 5 + 5 = 10. There is no addition double for 9. So, 9 has to be odd.


Odd and Even Numbers Standard

2.OA.3 relates doubles additions

up to 20 to the concept of odd and

even numbers and to counting by

2s by pairing and counting by 2s

the things in each addend.

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Strategies Using Doubles

Lesson

7

Doubles Plus/Minus 1 Children can use either the Doubles Plus 1 or the Doubles Minus 1 strategy to add two numbers that are 1 apart.

7 + 6 =

Using a Double Plus 1 or Using a Double Minus 1

6 + 6 = 12, so 7 + 7 = 14, so

7 + 6 = 13, 1 more than 12 7 + 6 = 13, 1 less than 14

Doubles Plus/Minus 2 Children can use either the Doubles Plus 2 or the Doubles Minus 2 strategy to add two numbers that are 2 apart.

8 + 6 =

Using a Double Plus 2 or Using a Double Minus 2

6 + 6 = 12, so 8 + 8 = 16, so

8 + 6 = 14, 2 more than 12 8 + 6 = 14, 2 less than 16


Doubles Strategies Another Level

3 method that works for certain

numbers is a doubles ±1 or ± 2

method:

6 + 7 = 6 + (6 + 1) = (6 + 6) + 1 = 12 + 1 = 13.

These methods do not connect with

place value the way make-a-ten

methods do.

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Equations, Equation Chains, and Vertical Form

Lesson

8

Equations Children work with equations that have operations on both sides of the equal sign, for example, 6 - 1 = 4 + 1. This reinforces that = means "is the same as," not "makes" or "results in."

Equation Chains Children write equation chains. Equations chains have more than one equal sign. To help children see the equivalent expressions, they loop them within an equation chain.

5 = 2 + 3 = 3 + 2 = 1 + 4 = 4 + 1 = 6 - 1

Equation chains are used in Quick Practices to help children generate several addition and subtraction expressions for a number.

Vertical Form Children rewrite equations in vertical form. Children label the partners (P) and the total (T) to connect the two forms.

T9+4= 9+

4

_ P

PT

P P T

9 4 P P

T

9+ =13 9+

_

13

PP

T

13P P T

9

P P

T

13- = 9 13

-

_

9

TP

P

13T P P

9

P P


Equations Equations with totals on

the left help children understand

that = does not always mean

“makes” or “results in” but always

means “is the same number as.”

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Add Three or Four Addends

Lesson

9

Using the Associative Property of Addition When adding three or four addends, children develop their own strategies for grouping the addends to make the addition easier. They make use of place value (making one or more tens) and the Associative Property of Addition, which states that addends can be grouped in any way and the sum stays the same. By sharing and discussing their strategies, children develop effective and efficient strategies for adding three or more numbers.

MATH TALKin ACTION

Again invite children who used different methods to share them.

Who would like to share their method for finding 8 + 8 + 5 + 2?

Carl: I added the 8 and 2 and got 10. Then I added the 8 and 5 and got 13. To add 10 and 13, I just thought about 10 + 10 (which is 20) and 3 more, which is 23.

Isabella: I added 8 + 8 and got 16. Then I broke the 5 into 4 + 1 so I could use the 4 to make 20. Then I just added in the 1 and the 2. So, my answer is the same, 23.

Aiden: I added 8 + 8 and got 16. Then I added 5 + 2 and got 7. I didn't know what 16 + 7 was, so I counted on from 16. (Uses fingers to keep track of 7 numbers.) 17, 18, 19, 20, 21, 22, 23. The total is 23.

Adding three or four 1-digit numbers prepares children for adding up to four 2-digit numbers. [CC.2.NBT.6]

from THE PROGRESSIONS FOR THE COMMON CORE STATE STANDARDS ON NUMBERS AND OPERATIONS IN BASE TEN

Fluency with Place Value and

Properties Both general

methods and special strategies

are opportunities to develop

competencies relevant to the NBT

standards. Use and discussion

of both types of strategies offer

opportunities for developing

fluency with place value and

properties of operations, and

to use these in justifying the

correctness of computations (MP.3).

UNIT 1 | Overview | 1MM


Add To and Take From Word Problems

Lessons

10 11

The Math Expressions program teaches children to solve word problems using a learning progression that was developed through classroom research during the Children’s Math Worlds NSF-funded research project headed by the author.

See page T8 of this Teacher's Edition for a summary of all the problem types children need to master in Grade 2.

Add To and Take From Problems In Lesson 10, children represent and discuss solution methods for solving these problems. In Lesson 11, they extend their work to include writing their own word problems.

Add To and Take From problems start with a quantity that is then modified by a change—something added to or taken from the original quantity—that results in a new quantity. In such problems, the start, the change, or the result may be unknown, so there are six subtypes: unknown start, unknown change, and unknown result for both Add To and Take From situations. Children apply reasoning or diagrams they have learned for adding and subtracting to solve these problems. The work done with these problem types in Grade 1 provides a solid foundation for continuing and extending problem solving in Grade 2.

Using the Solve and Discuss participant structure to work through problems is an excellent way for children to see various ways to solve these problems and for you to see how well children can represent and solve the problems.


Grade 2 Grade 2 students build

upon their work in Grade 1 in two

major ways. They represent and

solve situational problems of all

three types, which involve addition

and subtraction within 100 rather

than within 20.

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Grade 2 Diagrams used in Grade

1 to show how quantities in the

situation are related continue to

be useful in Grade 2, and students

continue to relate the diagrams to

situation equations.

Put Together/Take Apart Word Problems

Lessons

12 13

In Math Expressions, children solve word problems by understanding the problem situation so that they can decide the best way to solve the problem. Rather than using word clues or arbitrary rules, they first find a way to understand and represent the problem situation. This gives them a basis for deciding how to solve the problem.

Put Together/Take Apart Problems In Lesson 12, children represent and discuss solution methods for solving these problems. In Lesson 13, they extend their work to include problems with group names and problems that have two unknown addends.

In Put Together/Take Apart problems, all objects are present from the start, and nothing is introduced or taken away. The situations involve describing groupings within the total or conceptually putting objects together or taking them apart. In such problems, the total, one addend, or two addends may be unknown, so there are three subtypes: unknown total, unknown addend, and two unknown addends.

In Lesson 13, children work with problems where the two addends may have different names but belong to the same group. Part of solving these problems involves deciding what the group is. Children have been encouraged to use labels in the problem solving lessons in this unit; as they encounter more complex problems, this practice becomes even more important.

In some problems, children are guided to make up sets of objects for a problem situation, as shown in the example below. Note that apples and pears are subsets of the set pieces of fruit.

► You DecideComplete this problem.

4. Jenna has 4 and Bill

has 6 . How many

do they have altogether?

label

4 + =6 10

+A P

Answers will vary.

pieces of fruit

apples

pears

pieces of fruit

10

UNIT 1 | Overview | 1OO

14 15 16 18


Compare Word Problems

Lessons

In Math Expressions, children use what they know from reading and talking about a problem to draw and label representational diagrams, Math Mountains, number boxes, equations, and comparison bars to solve various types of word problems.

Compare Problems In Lessons 14, 15, and 18, children work with Compare problems, at first finding ways to represent these problems and then paraphrasing problem situations and finally working with more complex Compare situations involving comparisons in which the language used is opposite to the operation required. In Lesson 16, children work with the various problem types presented in the previous six lessons.

Compare problems involve finding how many more or less one quantity is than another quantity. Children draw number boxes or comparison bars and match objects between groups to help them compare. Children need practice with the more difficult comparison language by saying both forms of a comparison question and using equalizing language:

• How many fewer balloons does Sue have than Mike?• How many more balloons does Mike have than Sue?• How many balloons does Mike need to give away to have as many

as Sue?

One of the difficulties children may experience with comparison bars is thinking that they need to draw the bars to reflect the problem situation exactly. Help children see that they can draw the comparison bars first and then use the information from the problem to label the bars and fill in the numbers they know. So, for a problem in which Sue has 5 balloons and Mike has 7 balloons, children would start by labeling the longer bar Mike and the shorter bar Sue.

Mike

Sue

Then they add the information they know to the diagram and place a question mark to show the information they need to find out.


Some textbooks represent

all Compare problems with a

subtraction equation, but that is

not how many students think of

the subtypes. Students represent

Compare situations in different

ways, often as an unknown

addend problem. If textbooks and

teachers model representations of

or solution methods for Compare

problems, these should reflect the

variability students show. In all

mathematical problem solving,

what matters is the explanation

a student gives to relate a

representation to a context, and

not the representation separated

from its context.

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Find Appropriate Information in Word Problems

Lesson

17

In real world situations, problems do not have all the required information neatly organized in two or three sentences. Sometimes more information is given than is needed; sometimes not enough information is available, and sometimes information is hidden within the problem.

Problems with Too Much Information Because Math Expressions emphasizes understanding the problem situation before trying to solve a problem, children learn to recognize what information is important and what is not needed. For example, a problem may tell how many markers and crayons Shanna has and ask how many markers she has left after giving some to a friend. Children who understand the problem situation know that they do not need to know how many crayons Shanna has to solve the problem and so will cross out that information.

Problems with Not Enough Information Again the emphasis on understanding a problem situation helps children recognize when some information needed to solve a problem is not given in the problem. For example, knowing that Sam and his dog walked 15 blocks in two trips is not enough information to decide how many blocks they walked on one of the trips. Children may need to supply information or say that the problem cannot be solved.

Problems with Hidden Information Some problems have information embedded in quantity words, such as dozen, week, or pair. Children need to translate the quantity into a number and write it in the problem near the word for the quantity. For example, to solve the following problem, children must understand that there are 3 children in a set of triplets.

13. Thereare9childrenandasetoftripletsinthelibrary.Howmanychildrenareinthelibrary?

label

Tripletmeans3.

children12

library

UNIT 1 | Overview | 1QQ



Grade 2 Most two-step problems

made from two easy subtypes

are easy to represent with an

equation, …. But problems

involving a comparison or two

middle difficulty subtypes may

be difficult to represent with a

single equation and may be better

represented by successive drawings

or some combination of a diagram

for one step and an equation for

the other …. Students can make

up any kinds of two-step problems

and share them for solving.

Two-Step Word Problems

Lessons

19 20

Read Two-Step Problems Solving a variety of types of two-step problems is a strong focus of the Grade 2 Operations and Algebraic Thinking domain. Because two-step word problems require two steps to solve, it is useful for children to read and rephrase the word problem. As they do, they ask themselves, “What question does this problem want me to answer?” and “What do I need to know first to answer the question?”

Represent Two-Step Problems Children may use a combination of drawings, Math Mountains, and equations to represent the steps of two-step problems. The first step of a problem may be more easily represented with a drawing or Math Mountain that helps children find the information needed to proceed with the second step. Once the information is determined, the second step may often easily be represented with an equation. However, do not push children to use a representation they are not comfortable with.

Solve Two-Step Problems Using the Solve and Discuss structure for these problems provides exposure for all children to different ways of thinking about, representing, and solving two-step problems. As with all problem solving, it may be helpful to ask children to explain why an answer is reasonable or to rephrase the problem situation using the answer to see if it makes sense.

Distinguish Among Problem Types Lesson 20 provides practice with a variety of problem types to help children recognize the problem types and decide how best to represent and solve each type. Note that it is not necessary that children use the words from the Common Core State Standards document to identify problem types but rather they recognize the different types of situations and find ways to solve the problems.

Focus on Mathematical Practices

Lesson

21

The Standards for Mathematical Practice are included in every lesson of this unit. However, the last lesson in every unit focuses on all eight Mathematical Practices. In this lesson, children use what they have learned about adding and subtracting to solve problems about healthy foods.

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Math Talk Learning CommunityResearch In the NSF research project that led to the development of Math Expressions, much work was done with helping teachers and children build learning communities within their classrooms. An important aspect of doing this is Math Talk. The researchers found three levels of Math Talk that go beyond the usual classroom routine of children simply solving problems and giving answers and the teacher asking questions and offering explanations.

Math Talk Level 1 A child briefly explains his or her thinking to others. The teacher helps children listen to and help others, models fuller explaining and questioning by others, and briefly probes and extends children’s ideas.

Example Word Problem: 9 birds are in the yard. 3 more birds join them. How many birds are in the yard now?

Who can tell us how many birds are in the yard now?

Billy: There are 12 birds.

How do you know?

Billy: I know that 9 and 3 more is 12.

Who found a different way to answer the question?

Lucy: I made a drawing. I wrote 9 and drew 3 circles. I counted on 3 more from 9: 10, 11, 12.

Math Talk Level 2 A child gives a fuller explanation and answers questions from other children. The teacher helps children listen to and ask good questions, models full explaining and questioning (especially for new topics), and probes more deeply to help children compare and contrast methods.

Example Word Problem: Snow has 10 marbles and 2 boxes. How many marbles can she put in each box?

How can we find the answer to this problem?

Jake: Is this a problem that has more than one answer?

Why do you ask that, Jake?

Jake: Because I know more than 1 way to find partners of 10.

Ruth: If we break apart 10, we can make a list of the ways. Do you think that is what we are supposed to do?

Nancy: I think so, because we know that there are many ways to make 10 with two addends.

Ruth: And it does not say that both boxes must have the same number of marbles.

What is one way to start the list?

UNIT 1 | Overview | 1SS


Math Talk Learning Community (continued)Caleb: Let’s start by writing the 10-partners 1 and 9, and then add 1 more to the first partner each time.

Nancy: I agree with Caleb.

Math Talk Level 3 The explaining child manages the questioning and justifying. Children assist each other in understanding and correcting errors and in explaining more fully. The teacher monitors and assists and extends only as needed.

Example Word Problem: Joe eats 4 green grapes and 8 purple grapes. How many grapes does he eat?

Who will show us how to find the answer?

Julia: I know that green grapes and purple grapes are both grapes, so I have to add 4 + 8. I know that is 12, so the answer is 12 grapes. I also made a drawing to be sure I was right.

4Bob: I think your answer is right, but your drawing only shows 11. You need to fix your drawing.

Nancy: Yes, when I count your circles, you are showing 4 + 7. So draw another circle.

How can we be sure that we make the right drawing?

Julia: I should have checked my drawing to be sure I showed 4 + 8. So we should always check what we do.

Nancy: I solved the problem in a different way. I started with the larger number, 8. I counted on 4 more. I counted 2 more to make 10, and then 2 more to get 12.

Julia: Did you make a drawing?

Nancy: I did. Here's my drawing.

10

8

Julia: I think starting with the larger number was a good idea. Drawing fewer circles makes it easier to get the drawing right.

Summary: Math Talk is important not only for discussing solutions to word problems but also for any kind of mathematical thinking children do, such as explaining why each number in a count sequence is 1 or 10 more than the number before it, or how to use a drawing to subtract, or how to put two triangles together to make a rectangle.

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NOTES:

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Building a Math Talk CommunityMATH TALK Frequent opportunities for students to explain

their mathematical thinking strengthen the learning community of your classroom. As students actively question, listen, and express ideas, they increase their mathematical knowledge and take on more responsibility for learning. Use the following types of questions as you build a Math Talk community in your classroom.

Elicit student thinking

• So, what is this problem about?

• Tell us what you see.

• Tell us your thinking.

Support student thinking

• What did you mean when you said _______?

• What were you thinking when you decided to _______?

• Show us on your drawing what you mean.

• Use wait time: Take your time…. We’ll wait….

Extend student thinking

• Restate: So you’re saying that

• Now that you have solved the problem in that way, can you think of another way to work on this problem?

• How is your way of solving like _______’s way?

• How is your way of solving different from _________’s way?

Increase participation of other students in the conversation

• Prompt students for further participation: Would someone like to add on?

• Ask students to restate someone else’s reasoning: Can you repeat what _______ just said in your own words?

• Ask students to apply their own reasoning to someone else’s reasoning:

• Do you agree or disagree, and why?

• Did anyone think of this problem in a different way?

• Does anyone have the same answer, but got it in a different way?

• Does anyone have a different answer? Will you explain your solution to us?

Probe specific math topics:

• What would happen if _______?

• How can we check to be sure that this is a correct answer?

• Is that true for all cases?

• What pattern do you see here?

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UNIT 1 | Overview | 1UU

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