ReseaRch—BesT PRacTIces Putting Research into · PDF fileReseaRch—BesT PRacTIces...

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Putting Research into Practice

Dr. Karen C. Fuson, Math Expressions Author

From Our Curriculum Research Project: Learning Path for Measurement

Grade 2 length measurement with rulers begins the formal use of measurement tools that use a length scale in which numbers indicate the total number of unit lengths from the zero of the scale. Such length scales are number-line diagrams that are also used as bar graph scales and as scales on a line plot (dot plot). Two National Research Council reports (see references at the end) recommend that work with such scales not begin before Grade 2 because of the conceptual difficulties described above in using such scales: Many students count the numerals on the scale rather than seeing and counting the length units.

To overcome this difficulty, Math Expressions emphasizes marking and counting the length units. Initially, children draw marks to show the length units on things they are measuring, and then they count these length units. Through such activities, they mathematize length as consisting of small units of the same size that compose the total length.

This drawing and then counting of the length units also helps children extend the work they did in earlier grades with seeing different kinds of measurement units for different kinds of measures. This is the basis for future work with other units: square units compose and are counted to make an area (initially of rectangles), and cubes compose and are counted to make volumes (initially of rectangular prisms).

UNIT 3 | Overview | 279O

Research & Math BackgroundContents Planning

From Current Research: Development of Length Measurement Concepts

Concepts of length measurement, even those that appear obvious to adults, are not well developed by many second graders. For example, some begin counting at the numeral 1 on a ruler. To these students, the numerals on a ruler may signify when to start counting, rather than an amount of space that has already been covered. They need to develop the notion that “one” is the space from the beginning of the ruler to the hash mark.

The tricky part about the scale on a ruler is that the numbers tell the number of units so far, but they distract students, leading them to count the hash marks above the numbers instead of counting the length-unit. Discussion must focus on what the numbers on the ruler are counting (the length-units so far) and on what students need to count when they are measuring length (the number of unit lengths). The relation between the size and number of length-units can then be explicitly developed.

National Council of Teachers of Mathematics (NCTM). Focus in Grade 2. Reston, Va.: NCTM, 2011. pp. 109 and 111.

Kilpatrick, J., Swafford, J., Findell, B. (Eds.) (2001). Adding it Up: Helping Children Learn Mathematics. Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

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.

Other Useful References

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Getting Ready to Teach Unit 3Using 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 3

Measure Heights S M A L L G R O U P S

MP.1 Make Sense of Problems Act It Out Invite a volunteer to the station to demonstrate how to measure height. Discuss with children how they would measure the height of someone who is taller than a meter stick. Children can act out how to do so. They will see that if a child is taller than 100 centimeters, they use two meter sticks and add the measure on the top stick to 100 centimeters to find the height.

Lesson 6

MP.1 Make Sense of Problems Justify Reasoning

MATH TALKin ACTION

Children discuss Exercise 6.

Did anyone else get 4 cm for XY and 5 cm for WX?

Ryan: I did! And I knew I did not need to measure the other two sides at all.

Emir: Why is that?

Ryan: In a rectangle, the sides across from each other have the same length.

Lesson 3

Mathematical Practice 1 is integrated into Unit 3 in the following ways:

Draw a DiagramMake Sense of Problems

Look for a PatternJustify Reasoning

Act It Out

UNIT 3 | Overview | 279Q

ACTIVITY 2

ACTIVITY


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 length measurements.

TeacheR ediTion: examples from Unit 3

MP.2 Reason abstractly and Quantitatively Connect Diagrams and Equations Children then complete the equation next to the square to find the distance around the square.

When children are finished, discuss Exercise 3 together, and have volunteers share ways to find the distance around the square. You can use these questions to elicit the idea that you need to measure only one side of the square.

• What is the length of the side between E and F? 4 cm

• Without measuring the other sides, do you know their lengths? yes

• How do you know the lengths of the other three sides? All four sides of a square are equal in length.

• Can you find the distance around a square by measuring just one side? yes

• Describe how you can find the distance around this square using the measure of one side. I can add four 4-cm lengths together.

Lesson 3

MP.2 Reason abstractly and Quantitatively Ask children to complete Exercises 16 and 17 by measuring the lengths of different objects in feet and yards.

Have children complete Exercise 18 and invite them to share their answers. Elicit the idea that since feet are smaller than yards, there are more of them.

• Are the measures more exact when you measure in feet or in yards? feet

Lesson 7


Reason Abstractly Reason Abstractly and Quantitatively

Connect Diagrams and EquationsConnect Symbols and Models

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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 to 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.


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

If I measure the length of the classroom in inches and then in feet, the number of inches will be greater than the number of feet.

Children decide if the statement is true or false and use words and models to support their position. The statement is true.

MP.3 Construct a Viable Argument Children should be able to explain whether they agree that the statement is true or whether they think it is false.

MP.3 Construct a Viable Argument Justify Conclusions Children should be able to explain how they know they are correct.

Lesson 9

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 Tell children that Puzzled Penguin looked at the 6-cm line segment on Student Activity Book page 117 and said that the segment was 7 cm long.

• What mistake do you think Puzzled Penguin has made? Puzzled Penguin counted the centimeter marks instead of the 1-cm lengths. Puzzled Penguin should count the spaces between the marks rather than the marks themselves.

Lesson 1


Critique the Reasoning of OthersPuzzled Penguin Construct Viable Arguments

Compare RepresentationsJustify Conclusions

UNIT 3 | Overview | 279S

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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 3

MP.4, MP.5 Model with Mathematics/Use appropriate Tools Children measure the length of each pencil by placing it directly on the inch scale on Student Activity Book page 149. Have them measure the length to the nearest inch and make a dot on the line plot to show the length of the pencil.

Lesson 8

Make a Class Line Plot in Inches

MP.4 Model with Mathematics Make a Graph If time permits, make a line plot that shows the heights of all children in the class in inches. Place a measuring tape horizontally across the board. Each child places a dot above his or her height in inches.

Lesson 7


Model with MathematicsMake a Graph

Draw a ModelRuler

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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 MathBoard Have children draw two squares and two rectangles on their MathBoards. Ask them to label each shape using letters.

Lesson 3

MP.4, MP.5 Model with Mathematics/Use Appropriate Tools Ruler Children estimate and then measure each book. Children can measure the width of the book by placing it directly on the inch scale on Student Activity Book page 150. Children can measure to the nearest inch and add a dot to the line plot to represent each book. Encourage children to predict whether the number of centimeters will be greater than or less than the number of inches. Then repeat the process using centimeters.

Lesson 8


Use Appropriate ToolsMathBoard

RulerMeter StickUse a Concrete Model

UNIT 3 | Overview | 279U

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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 3

MATH TALK MP.6 Attend to Precision Explain a Representation Allow children time to discuss what they think this line plot means. At the end of the discussion, children should understand that the line plot shows the heights of five children and that each dot stands for the height in centimeters of one of those children. Discuss the line plot using questions such as the following.

• How many children’s heights are shown on the line plot? 5

• Where would a dot for 109 centimeters be? Between 108 cm and 110 cm

• What is the height of the tallest child in the group? 115 cm

• What is the height of the shortest child in the group? 108 cm

• Do any children have the same height? Yes How do you know? There are two dots over 110.

Lesson 6

Identify Shapes W H O L E C L A S S MATH TALK

MP.6 Attend to Precision Explain a Representation Children identify orally the shapes shown on Student Activity Book page 132. Then discuss why many of the shapes can be named in several ways. Encourage children to find all the ways to identify the shape before moving on to the next shape.

Lesson 5

MATH TALKin ACTION

Would anyone like to share how they added 4 cm + 4 cm + 5 cm + 5 cm?

Tia: I added 4 cm and 5 cm and got 9 cm, and then I added 4 cm and 5 cm again to get another 9 cm. Then 9 cm + 9 cm = 18 cm.

Mohammed: I added 5 cm and 5 cm to make 10 cm. Then I added 4 cm and 4 cm to get 8 cm. 10 cm + 8 cm is 18 cm. Same answer!

Lesson 3


Attend to PrecisionPuzzled Penguin

Explain a Representation

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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.


Sort Shapes S M A L L G R O U P S

MP.7 Look for Structure Identify Relationships Distribute copies of Identify Shapes (TRB M90). This page shows the same shapes as on Student Activity Book page 132. Invite children to cut out the 12 shape cards. Children can then sort the shapes into two categories. Discuss with children the different categories they may have chosen. Point out that they could use 2-dimensional shapes and 3-dimensional shapes as their categories.

Encourage some children to sort the 2-dimensional shapes based on the number of sides. Other children should sort the 2-dimensional shapes based on whether all the sides are equal. Children should use the most specific names when discussing each shape. Invite children to share how they sorted the shapes.

• Which 2-dimensional shape makes up the faces of the cubes shown? the square

Lesson 5

The Ruler as a Group of Lengths IND IV IDUALS

MP.7 Look for Structure Work with children as they go through Exercises 8–12 on Student Activity Book page 119. These exercises help children see that the numbers on rulers represent the line segments of different lengths; they also provide an opportunity for children to practice drawing line segments using their rulers.

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The major goal in measuring lengths throughout this year is to help children see that a length measure is the number of unit lengths making a total length. Sometimes children focus on the number and think of that point, instead of thinking of the number as telling the number of unit lengths so far. Numbers on number lines, ruler scales, and bar graph scales all tell the total length so far.

Lesson 1


Look for Structure Identify Relationships

UNIT 3 | Overview | 279W

ACTIVITY 2ACTIVITY 2

Class Activity

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► Make Quilts 2. Continue the pattern to complete this quilt square.

3. Use centimeter grid paper. Draw a square that is 16 centimeters long on each side. Use your square to make a quilt pattern. Color your quilt square.

When you finish, count the shapes.

triangles

pentagons

quadrilaterals

hexagons

Children’s patterns may vary.

Answers will vary.

154 UNIT 3 LESSON 9 Focus on Mathematical Practices

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Class Activity

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► Math and QuiltsA patchwork quilt is made by sewing pieces of cloth together. Look for shapes in these patchwork quilts.

1. Color each shape a different color.

Shape triangle quadrilateral pentagon hexagon

Color red orange blue yellow

UNIT 3 LESSON 9 Focus on Mathematical Practices 153

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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 3

MP.8 Use repeated reasoning Generalize Fill in the table with children’s answers for the 4-cm line segment and have them try to find a pattern. Continue with examples until the pattern becomes clear.

Centimeters Marks 1-cm Lengths 4 5 4 7 8 7 11 12 11 16 17 16

Lesson 1

Identify a Pattern MATH TALK

1 2 3 4 512 24 36 48 60

MP.8 Use repeated reasoning

• (Point to the remaining empty cell in the bottom row.) What number should we place in this cell? 48 Why? Following the pattern rule, add another 12 to 36 to show the sum of four 12s, or 48.

Lesson 9


Use Repeated Reasoning Generalize

Focus on Mathematical Practices Unit 3 includes a special lesson that involves solving real world problems and incorporates all 8 Mathematical Practices. Children use what they know about patterns and shapes in a lesson about quilts.

STUdenT ediTion: LeSSon 9, PageS 153–154

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Math Expressions VOCABULARY

As you teach this unit, emphasize

understanding of this term.

• partner lengthsSee the Teacher Glossary.

Getting Ready to Teach Unit 3Learning Path in the Common Core StandardsIn this unit, children learn to measure and estimate lengths, relate addition and subtraction to length, represent lengths on line plots, and reason with shapes and their attributes.

Visual models and real world situations are used throughout the unit to help children understand shapes and length measurements.

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 1: Counting the marks rather than the spaces between the marks on a centimeter ruler

→ Lesson 8: Incorrectly relating the size of the unit and the number of units

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.

UNIT 3 | Overview | 279Y


Concepts of Length Measurement

Lesson

1

Iterative Process As children begin their work with measurement, one goal is to help children see that measurement is an iterative process. When a child finds that a pencil is 8 centimeters long, it is important that he or she understands that the length is 8 centimeters not because the end of the pencil lines up with the 8 mark on a centimeter ruler, but because the length is 8 iterations (or repetitions) of a 1-centimeter length.

The exploration of the meaning of the marks on a ruler is intended to support this concept. Children can think of a ruler as a succession of very thin consecutive lengths layered on top of each other.

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This important measurement concept is not limited to measurement of length but applies to all units of measure.

Measuring Tools It is important for children to realize that the units on measuring tools must always be the same; that is, they must be standard units. Without standardization, there would be no way to compare measurements, as one person’s ”meter” might be longer than another person’s. Even when using nonstandard units in earlier grades, everyone used the same unit: for example, Inch Cubes or paper clips.

In the first part of this unit, children use 25-centimeter rulers and measure in centimeters. Later in the unit, children make and use meter sticks, yardsticks, and inch rulers. When appropriate, point out that because all the measuring tools are the same, the measurements can be compared to each other.

from THE PROGRESSIONS FOR THE COMMON CORE STATE STANDARDS ON MEASUREMENT AND DATA

Grade 2 Students in Grade 2

measure lengths to generate a set

of measurement data. For example,

each student might measure

the length of his or her arm in

centimeters, or every student might

measure the height of a statue

in inches. (Students might also

generate their own ideas about

what to measure.)

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Attributes of 2-Dimensional Shapes

Lessons

2 3 4

Ever since Kindergarten, children have been exploring shapes. At this grade level, they will use attributes to identify and draw shapes.

Angles Children learn that the “corners” or “vertices” of shapes are called angles. In geometry, angle has three meanings: the shape formed by the intersection of two line segments, the measure of that shape, and the rotation required to form that shape. At this grade level, children learn only about angles as shapes that together with sides make up two-dimensional closed figures.

Attributes Children extend the work they did with shapes in Grade 1 to now use attributes to identify and draw shapes with 3, 4, 5, and 6 sides and angles. They learn that the number of sides and the number of angles a shape has tells what that shape is.

triangle 3sides 3angles

quadrilateral 4sides 4angles

pentagon 5sides 5angles

hexagon 6sides 6angles

Children also measure the difference in side lengths of squares and rectangles to explore how squares and rectangles are different.

As children work with these shapes, they will draw them. Not all shapes need to be drawn with rulers. Model sketching shapes quickly so that the attributes are apparent and can be discussed. Point out that it is more important to have time to discuss the shapes than to spend a great deal of time to draw the shapes perfectly.

UNIT 3 | Overview | 279AA


Estimate and Measure Side Lengths in 2-Dimensional Shapes

Lessons

3 4

Measuring side lengths of geometric figures brings together concepts from measurement and from geometry. Children continue to use centimeter rulers in these two lessons.

Estimation Since children have measured lengths in centimeters in the previous lessons, they now extend their measuring skills to include estimating the side lengths of figures before measuring. This is a practical skill that is useful in many real world situations. Practice is the best way to help children improve their ability to estimate lengths. Encourage discussion of ways that children use to estimate lengths. Some children may know that the width of one of their fingers is 1 centimeter or that the width of their palm is about 5 centimeters. Some may have a visual sense of 1 centimeter and can imagine moving along a line and counting off centimeters.

Measure Lengths Children measure the side lengths of a figure to find the distance around the figure. This gives children practice in measuring and prepares them for next year’s work with perimeter. For some of the exercises, children measure by marking off each centimeter length. This helps reinforce the idea that measurement is iterative. Children also measure lengths when they are asked to draw figures with given side lengths.

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Attributes of 3-Dimensional Shapes

Lesson

5

Rectangular Prisms Each child constructs a rectangular prism and two cubes using nets for these shapes. Children discuss and determine the attributes of these shapes. They learn that the plane figures that make up the outside of these shapes are called faces.

Shapes with Cubes After children build their cubes, they work together to explore 3-dimensional shapes constructed with cubes. They construct shapes from front, top, and side view drawings of a shape, and they make front, top, and side view drawings for a given shape. Both aspects of the activity—visualizing the shape and drawing the views—support children in developing spatial sense.

top view side viewfront view

Attributes Children use attributes to distinguish between 2-dimensional shapes and 3-dimensional shapes. Identifying these differences help children to construct hierarchical ideas based on properties. As children use differences in shapes to recognize, draw, and investigate shapes, they build a foundation for future work with perimeter, area, and similarity.

UNIT 3 | Overview | 279CC


Tools for Measuring Length

Lessons

6 7 8

Either the metric system or the customary system is used for measuring length in almost all countries. Although most countries use the metric system, the United States is the only major country still using the customary system.

Meter Stick As children construct meter sticks, they explore the relationship between centimeters and meters and compare this relationship to that between cents and dollars in our monetary system. Although decimeters are in common use in many countries, the children will not work with them at this time. That the relationship of decimeters to meters is like that of dimes to dollars may be noted as a way to help children relate smaller metric units to a meter. After making meter sticks, children will measure heights in centimeters to collect data for creating line plots. (See the next page for more information on line plots.)

Inch Ruler and Yardstick Children build and use inch rulers and yardsticks. As they measure with both, they may notice that the length of an object contains more inches than feet and more feet than yards. They use these tools to gather height data in inches.

Compensatory Principle When children compare the data they collect for their heights in centimeters and in inches, they see that more centimeters than inches are needed to record the same height. This informal exposure to the Compensatory Principle, which states that when a smaller unit is used to measure a length, more units will be needed to measure the length, helps children build an intuitive understanding of the principle.

Children do not need to know the name of the principle but rather to understand the concept that more smaller units than larger units are needed to measure the same length.


Grade 2 In Grade 2 the only

requirement of the Standards

dealing with measurement

data is that students generate

measurement data and build line

plots to display the resulting data

sets. (Students do not have to

generate the data every time they

work on making line plots. That

would be too time-consuming.

After some experiences in

generating the data, most work

in producing line plots can be

done by providing students with

data sets.)

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Grade 2 A display of measurement

data must present the measured

values with their appropriate

magnitudes and spacing on the

measurement scale in question

(length, temperature, liquid

capacity, etc.). One method for

doing this is to make a line plot.

Line Plots

Lessons

6 7 8

The definition of a line plot given in the Glossary of the Common Core State Standards for Mathematics is “a method of visually displaying a distribution of data values where each data value is shown as a dot or mark above a number line. Also known as a dot plot.”

Generate Data Measurement is a way to generate data—most measurements are made for a reason. That reason may be to explore relationships, to test ideas, or to see whether an experiment can be duplicated. In this unit, children learn to use measurement to generate data and then to display and analyze those data.

Make Line Plots Making a line plot is a simple process. A number line of appropriate length is drawn and labeled to match the scale of the measuring tool used to gather the data. Then a dot (or an X) is placed above the tick mark for a given measurement.

For example, suppose that 6 children recorded the lengths of their shoelaces in inches to generate this data set: 24, 0, 24, 0, 36, 24. The line plot representing these data looks like this:

Length of Shoelaces (inches)

4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 360 2

Analyze Line Plots The data points in a line plot show the shape of the data. The spread of the data, clusters of points, outliers—all are easy to see and interpret, though children will not use these terms. For example, the line plot above shows that three children had 24-inch shoelaces, that two children had no shoelaces, and that one child had much longer shoelaces. Discussion of what these results mean might lead to comments such as “some shoes do not have shoelaces” or “it seems as if most shoes have 24-inch shoelaces.”

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Focus on Mathematical Practices

Lesson

9

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, students apply what they have learned about patterns, modeling, and using appropriate tools and what they have learned about identifying and making geometric shapes as they work through activities involving quilts.

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