ReseaRch—Best PRactices Putting Research into Practice · From Current Research: Structuring an...

Research & Math BackgroundContents Planning

Dr. Karen C. Fuson, Math Expressions Author

ReseaRch—Best PRactices

Putting Research into Practice

From Current Research: Structuring an Array to Understand Area

Students need to structure an array to understand area as truly two-dimensional. Spatial structuring is the mental operation of constructing an organization or form for an object or set of objects in space.

National Council of Teachers of Mathematics (NCTM). Focus in Grade 2. Reston, Va.: NCTM, 2011. p. 113.

From Current Research: Adding and Subtracting Length Units

Common Core Standard 2.MD.6 relates length, the scale on rulers, and number-line diagrams that use the same scale. This standard does not require students to use number-line diagrams within 100 as a method to solve addition and subtraction within 100. We have discussed difficulties with such methods above. The standard does require students to understand the relationships of the lengths involved.

A major difficulty of using a number-line diagram for adding or subtracting within 100 is that the unit lengths must be small. The length model means that you cannot label each point.

Drawings that show the lengths of the numbers with elongated circles help children understand the number-line model as a length model.

National Council of Teachers of Mathematics (NCTM). Focus in Grade 2. Reston, Va.: NCTM, 2011. p. 80.

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Cross, C. T., Woods, T. A, Schweingruber, H. (Eds.) (2009). Mathematics learning in early childhood: Paths toward excellence and equity. Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

From Our Curriculum Research Project: Learning Paths

The activities in this unit help students structure an array, as recommended on the previous page. Squares forming an array are pushed together to form a rectangle, and the two colors of the squares allow children to see the rows or the columns forming the rectangle. Children fold and draw equal shares using rows or columns, but they also extend to seeing equal shares formed in other ways. These activities are foundational for relationships connecting multiplication, area, and unit fractions in Grade 3.

In this unit, children continue the vital focus on seeing and counting length units developed earlier in the year. They see larger numbers on number-line diagrams and encircle the lengths to see the length-units involved in adding and subtracting. The addends are embedded in the total in number-line diagrams. Children encircle all three numbers to be sure that these embedded relationships are understood. Because the use of number-line diagrams for adding and subtracting within 100 is difficult (see elaborations in Focus in Grade 2 and in the 2009 National Research Council report referenced below), children in this unit solve word problems involving lengths using numerical methods and problem representations they used for non-length problems throughout the year.

Another Useful Reference

UNIT 7 | Overview | 701M

ACTIVITY 1

ACTIVITY 1


Getting Ready to Teach Unit 7Using 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 7

MP.1 Make Sense of Problems Solve a Similar Problem Direct children to fold a second circle into 4 equal shares. If children are struggling with the task, have volunteers model how to partition the circle into fourths by folding. Suggest that they draw along the fold lines to make the parts more visible.

• How many equal shares did we create? four

• Each equal share is a fourth. There are 4 fourths in a whole.

• How do you know that the parts are equal shares? I can fold the parts over each other to see that they match exactly.

Lesson 2

MP.1 Make Sense of Problems Analyze the Problem Tell children they are going to solve word problems about these length units.

• When you read a problem, visualize how long the length units are so you will know whether your problem is about short length units or long length units. This will help you remember to write that length unit in your answer.

Lesson 3

Mathematical Practice 1 is integrated into Unit 7 in the following ways:

Make Sense of ProblemsLook for a Pattern

Solve a Similar ProblemUse a Different Method

Analyze the ProblemAnalyze Relationships

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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 100.

TeACHeR eDITION: examples from Unit 7

MP.2 Reason Abstractly and Quantitatively Connect Diagrams and Equations Ask children to use 6 of their tiles to build an array that is 2 rows of 3 tiles. Model the array on the board and write 3 + 3 = 6 and 2 + 2 + 2 = 6. Point to the rows as you say and write 3 + 3 = 6 and then to the columns as you say and write 2 + 2 + 2 = 6.

• These equations help you find the number of tiles in the array.

Do two or three such examples. Have some children make rows of alternating colors and others make columns of alternating colors. Children can now complete Exercises 3 and 4 by writing the number of objects in each row and in each column. They use these numbers to write the two addition equations.

Lesson 1

MP.2 Reason Abstractly and Quantitatively Discuss different strategies that could be used for solving problems with rectangles and squares. Remind children of the methods they have learned for adding three and four numbers.

On the board, draw the rectangle and square from Problems 4 and 6 on Student Activity Book page 314.

• What shape is each of these figures? How do you know? They are both rectangles, because a rectangle has opposite sides that are the same length. A rectangle has all right angles. The second figure is also a square because all sides have the same length.

• Why might it be easier to find the distance around a rectangle than a four-sided figure that has no equal sides? In a rectangle, you can add the two sides that are different, and then just add the total to itself.

• How about finding the distance around a square? Is there a quicker way to add? First add two sides. You know the other two sides will have the same total. Now all you have to do is add the total to itself to get the distance around the square.

Lesson 4


Reason Abstractly Reason Abstractly and QuantitativelyConnect Diagrams and Equations

Connect Symbols and ModelsCompare Strategies

UNIT 7 | Overview | 701O

ACTIVITY 1

ACTIVITY 2


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 7

What’s the Error? W H O L E C L A S S

MP.3, MP.6 Construct Viable arguments/Critique the reasoning of others Puzzled Penguin Draw the following on the board.

• Puzzled Penguin wants to show thirds of a rectangle. This is what Puzzled Penguin draws. Puzzled Penguin has made a mistake.

• What was Puzzled Penguin’s mistake? Puzzled Penguin made three shares, but they are not equal.

Lesson 2

MP.3 Construct a Viable argument Compare Methods If any of the methods described above do not come up in your classroom discussion, bring them up. Encourage your more-advanced children to try adding all sides together so that you can discuss how sometimes you get more than one new ten. (35 + 38 + 24 + 26 gives you 2 new tens in the ones total of 23.)

Lesson 4


Critique the Reasoning of OthersConstruct Viable ArgumentsPuzzled Penguin

Compare StrategiesCompare Methods

Compare RepresentationsJustify Conclusions

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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.


MP.4 Model with Mathematics Number Line Diagram Ask children to tell what addition or subtraction the number line diagram shows. 28 - 17 = 11 Then invite children to make up word problems that could be solved using the number line diagram.

Lesson 3

Is the Statement True?Write the following on the board.

Amelia and Ben each have sandwiches that are exactly the same size and shape. Amelia cuts her sandwich into 2 halves. Ben cuts his sandwich into 2 halves. All four pieces are the same size and the same shape. False; they are the same size but may or may not be the same shape.

MP.3 Construct a Viable Argument Children should be able to explain whether or not they agree that the statement is true.

MP.4 Model with Mathematics Make a Model or a Drawing Children should model the situation by cutting paper or drawing representations.

Lesson 6


Model with MathematicsNumber Line DiagramMake a Model or Drawing

UNIT 7 | Overview | 701Q

ACTIVITY 1

ACTIVITY 2


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.

Teacher ediTion: examples from Unit 7

MP.5 Use appropriate Tools Square-Inch Tiles Demonstrate at the board or with an overhead projector the step-by-step process of building an array with the tiles. Placing one tile at a time (purple, orange, purple, orange, and so on as shown below), children will eventually make 2 rows of 10 squares. The top row will be purple, and the bottom row will be orange. If tiles are not available, use the paper squares provided on Student Activity Book page 299.

even

10 + 10 = 20

Lesson 1

MP.5 Use appropriate Tools Number Line Diagrams Look at Student Activity Book page 312.

• What do the number lines remind you of? Possible responses: centimeter rulers, meter sticks, inch rulers Why? because they have little marks with numbers under them

• (Display a centimeter ruler.) What do these numbers tell us? how many centimeters long something is

• (Point to the number 5 on the ruler.) I see 5 on the ruler. Is that point 5 centimeters? No, 5 centimeters is the length from the beginning of the ruler to the 5.

Lesson 3


Use Appropriate ToolsSquare-Inch TilesMathBoard Modeling

RulerPaper ModelNumber Line Diagrams

StraightedgeDrawing

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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, and express numerical answers with a degree of precision appropriate for the problem context. Children give carefully formulated explanations to each other.


What’s the Error? W H O L E C L A S S

MP.3, MP.6 Construct Viable Arguments/Critique the Reasoning of Others Puzzled Penguin Exercise 13 on Student Activity Book page 320 addresses the common error of looping the point representing the number rather than the length representing the number. Discuss with children what Puzzled Penguin did wrong in this exercise and how they can help. Have children cross out Puzzled Penguin’s number line diagram and show the correct way to use the number line diagram to find the total.

Lesson 5

MATH TALKin ACTION

Children share and discuss the different ways they made 4 equal shares in Exercise 3.

Kendra: I made a line up and down in the middle to make halves, and then I made a line up and down in each half to make 4 equal shares.

Alexa: I did the same thing, but I went across. I think this will work for all equal shares. Look, it works to make two halves and three thirds. I think we could make more equal shares with lines across.

Carlton: I think so, too, and the lines up and down can be used for any equal shares, I think.

Dennis: I put these together and made half up and down and half across.

Lexi: I did the same thing, but I connected the corners of the rectangle.

Francisca: I made two different equal shares by making two halves and then drawing different slanted lines (diagonals) inside those halves.

Lesson 2


Attend to PrecisionExplain a Representation

Explain a SolutionPuzzled Penguin

UNIT 7 | Overview | 701S

ACTIVITY 2ACTIVITY 2


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.

Teacher ediTion: examples from Unit 7

MP.7 Look for Structure Identify Relationships After the discussion of fourths, ask children to think of another word for a fourth. If no one volunteers the information, write quarter on the board and ask a child to read the word. Discuss the different contexts in which children have used the word quarter. Children may be familiar with quarters from their work with money and time. Help children see that four quarters make a dollar or that a quarter hour is a fourth of an hour. Have volunteers explain in their own words how a quarter is the same as a fourth.

Lesson 2

Identify a Pattern PA IRS MATH TALK

Draw this pattern on the board.

1 square 4 squares 9 squares

MP.7 Look for Structure Identify Relationships Have volunteers write the addition equations that can be used to determine the total number of squares for the 2-by-2 square and the 3-by-3 square. 2 + 2 = 4, 3 + 3 + 3 = 9

Lesson 6


Look for Structure Identify Relationships

701T | UNIT 7 | Overview

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ACTIVITY 2

Class Activity7-6 Name

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► Math and FlagsShips can use flags to send messages. A flag can be used alone to send a message.A group of flags can be used to spell out a message.

This flag means “I have a pilot on board.” It can also be used for the letter H.

This flag means “Return to ship.” It can also be used for the letter P.

1. How many parts does the flag have?

parts

2. Does the flag show equal parts?

yes no

3. How many parts does the flag have?

parts

4. Does the flag show equal parts?

yes no

2

2

UNIT 7 LESSON 6 Focus on Mathematical Practices 321

2_MNLESE824529_U07L06.indd 321 12/04/12 9:49 AM

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► Rectangular Flags

7. Show 4 equal shares that are rectangles.

8. Show 4 equal shares that are triangles.

9. On a separate sheet of paper, design your own flag. Use equal parts. Color your flag.

5. Draw a square flag. Show halves. Color the flag. Color a half of the flag blue.

6. Draw a square flag. Show thirds. Color the flag. Color a third of the flag red.

► Square Flags

Children’s drawings may vary. Children’s drawings may vary.

322 UNIT 7 LESSON 6 Focus on Mathematical Practices

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STUDeNT eDITION: LeSSON 6, pAGeS 321–322

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.


Mp.8 Use Repeated Reasoning Generalize Now have children compare the halves formed by each of the three different fold lines (using three different rectangles). Children can discuss their findings to conclude that there are a number of ways to partition a rectangle into 2 equal shares. The equal shares from different methods do not always have the same shape.

Lesson 2

Mp.8 Use Repeated Reasoning Give children 10 minutes to work in groups to continue the pattern. Have them use centimeter grid paper to draw the squares. Have them write an addition equation for each.

Lesson 6


Use Repeated Reasoning GeneralizeDraw Conclusions

Focus on Mathematical practices Unit 7 includes a special lesson that involves solving real world problems and incorporates all 8 Mathematical Practices. In this lesson, children look at nautical flags to find equal shares.


Getting Ready to Teach Unit 7Learning Path in the Common Core StandardsIn this unit, children will learn about arrays and equal shares and solve problems involving the addition and subtraction of lengths.

Visual models and real world situations are used throughout the unit to help children understand important fraction and measurement concepts.

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 2: Not showing equal shares

→ Lesson 5: Looping the numbers on a number line diagram instead of the lengths

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.

Math Expressions VOCABULARY

As you teach this unit, emphasize

understanding of this term:

• number line diagramSee the Teacher Glossary.

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Equal Shares

Lessons

1 2

Some Common Core State Standards for Operations and Algebraic Thinking (2.OA.3, 4), Measurement and Data (2.MD.1), and Geometry (2.G.2, 3) connect ideas about making arrays and dividing rectangles with the understandings needed for work with area, multiplication, and fractions in Grade 3. These two lessons are rich in mathematical concepts, so encourage children to share, question, and explain freely.

Arrays Children use Square-Inch Tiles to make arrays. They learn to break an array apart into rows or into columns and can find the total number of squares in the array by counting in various ways or by using addition equations.

1. Looptherows. 2. Loopthecolumns.

Children then measure rectangles to partition them into rows and columns of same-size squares and count to find the total number of squares. They use divided rectangles to show halves, thirds, and fourths and see that they can divide the rectangles into these fractions in different ways. As much as possible, encourage children to discuss the different ways and to discover that all ways represent the same part of the rectangle—a half, a third, or a fourth.

Measureincentimeters.Drawrowsandcolumns.

13. Shadetoshowhalvestwodifferentways.

14. Shadetoshowfourthstwodifferentways.

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

The Progression in Operations

and Algebraic Thinking deals with

the basic operations—the kinds

of quantitative relationships they

model and consequently the kinds

of problems they can be used to

solve as well as their mathematical

properties and relationships.

UNIT 7 | Overview | 701W


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

Although most of the standards

organized under the OA heading

involve whole numbers, the

importance of the Progression

is much more general because it

describes concepts, properties, and

representations that extend to

other number systems, to measures,

and to algebra.

Paper Folding By using this tactile and kinesthetic approach to working with equal shares, children develop a intuitive feeling for working with fractions in later grades.

Using circles and rectangles, children fold to make equal shares. They see that making halves and fourths is relatively easy, but making thirds presents them with a mathematical problem. Let children try their own ideas for doing this before guiding them to find a way as described in Lesson 2.

Making Drawings After the paper folding explorations, children are asked to draw lines in shapes to make equal shares. Again they should discover through their discussion of their work that a whole can be divided into equal shares in many ways and that all the ways do not consist of the same shapes.

4. Make2equalshares.Showdifferentways.Shadehalfofeachsquare.

5. Make3equalshares.Showdifferentways.Shadeathirdofeachsquare.

6. Make4equalshares.Showdifferentways.Shadeafourthofeachsquare.

Childrenmayshadeanyequalshare.

Childrenmayshadeanyfourth.

Working with arrays and equal shares in these ways prepares children for the area, multiplication, and fraction concepts they will learn in Grade 3.

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

Use place value understanding

and properties of operations

to add and subtract Students

become fluent in two-digit

addition. Representations such

as manipulative materials and

drawings may be used to support

reasoning and explanations about

addition with three-digit numbers.

Length Word Problems

Lessons

3 4 5

Word Problems As children solve the word problems involving lengths in these lessons, they learn that using or making a drawing can help them understand what to do to solve the problem. they see that their answers must be labeled with the appropriate unit. they should also recognize that to solve these problems involving measures they can use the same problem-solving approaches they have used throughout the year.

26 feet

24 feet

38 feet

35 feet

Meat Counter

Farm Stand

Door

Bakery

3. Hereistherouteacustomertakeswhileshoppingatthestore.Howfardoesthecustomerwalkaltogether?

unit

123 feet

this informal introduction to perimeter prepares children for the more formal approach in Grade 3. Use the term distance around rather than perimeter when discussing these problems.

Number Line Diagrams

Lessons

3 5

Number Line Diagrams Children learn that addition and subtraction can be represented on a number line diagram.

0 5 10 15 20 25 30 35 40 45 55 6550 60 70 75 80 85 90 95 100

T

16. Loop67.Add26toit.LoopthetotalT.

Howlongisit? 93units

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

D

15. Loop38and84.LoopthedifferenceD.

Howlongisit? 46units

UNIt 7 | Overview | 701Y


Children relate number line diagrams to rulers but must understand that the distance between two numbers on a number line is not an actual unit of length. It is important that children learn how to represent these computations on a number line diagram, but they will likely find it easier to use one of the computation methods they have learned this year rather than using a number line diagram to find a total or a difference.

Focus on Mathematical Practices

Lesson

6

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 apply what they have learned about equal shares to solve problems about nautical flags and to design their own flags.

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ReseaRch—Best PRactices Putting Research into Practice · From Current Research: Structuring an...

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