ReseaRch—BesT PRacTIces Putting Research into Practice · UNIT 7 Resea R ch ReseaRch—BesT...

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Putting Research into Practice

Dr. Karen C. Fuson, Math Expressions Author

From Current Research: Compose Shapes

The abilities involved in putting together and taking apart shapes are important for many reasons. These geometric competencies are at the foundation of geometry, but also arithmetic (e.g., composing and decomposing numbers and arrays in multiplication), measurement, and higher-order geometric work, as well as such fields as architecture and the visual arts. Creating and then iterating units and higher-order units in the context of constructing patterns, measuring, and computing are established bases for mathematical understanding and analysis.

Students can develop the ability to intentionally and systematically combine shapes to make new shapes and complete puzzles. They do so with increasing anticipation on the basis of the shapes’ attributes, and thus they develop mental imagery of the component shapes.

Concepts of Length

Length is a characteristic of an object or path found by quantifying how far it is between endpoints of the object. “Distance” is often used similarly to quantify how far it is between two points in space. Measuring length or distance consists of two aspects, identifying a unit of measure and subdividing the object by that unit: placing that unit repeatedly from one end to the other (iterating the unit) alongside the object (or placing multiple copies of that unit alongside).

Fuson, Karen C., Clements, Douglas H., Beckmann, Sybilla. Focus in Grade 1: Teaching with Curriculum Focal Points. Reston, VA: NCTM, 2011. 66, 67, 71.

Cross, C. T., Woods, T. A, Schweingruber, H. (Eds.) Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity. Center for Education, Division of Behavioral and Social Sciences and Education. 2009. National Academy Press. Washington, DC

Clements, Douglas H. “Teaching and Learning Geometry.” A Research Companion to Principles and Standards for School Mathematics. Reston, VA: NCTM, 2003. 151–178.

Other Useful References

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Research & Math BackgroundContents Planning

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

Share with Halves and Fourths IND IV IDUALS

MP.1, MP.4 Make Sense of Problems/Model with Mathematics Make a Drawing Work together to solve the problems on Student Activity Book page 222. After you read each problem, discuss with children how they will partition and color the shape.

• What shape is the sandwich? square How will you cut the sandwich? I will draw to cut it into four pieces. Why? There are four people eating the sandwich. What is an easy way to cut something that is shaped like a square into fourths? Cut it into halves. Then cut both halves in half.

Lesson 8

Discuss Problems with Equal Shares MATH TALK

MP.1 Make Sense of Problems Analyze the Problem Discuss the pictures on Student Activity Book page 235 with children. Ask them to describe each food item and name its shape. burger: circle; cornbread: square; cheese: rectangle Explain to children that they are going to draw lines on the pictures to show equal shares and solve the problems.

Lesson 14

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

Make Sense of ProblemsMake a Drawing

Analyze the Problem

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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 includes partitioning rectangles and circles into two and four equal shares and recognizing that decomposing into more equal shares results in smaller shares.

TeACHeR eDITION: examples from Unit 7

MP.2 Reason Abstractly and Quantitatively Discuss the ways children can cut the sandwich into fourths. Be sure children understand that the square sandwich was cut into four equal shares called fourths (or quarters). Each friend will get one equal share, which is one fourth of (or one quarter of) the whole sandwich. Each share is the same size, so each person will get an equal share.

Lesson 8

MP.2 Reason Abstractly and Quantitatively For Exercise 7, be sure children understand that they will compare the lengths of the picnic food in Exercises 4–6 and order them from longest to shortest. Explain that they will write the names of the food in this order. Invite children to share their strategies.

Lesson 14


Reason Abstractly and QuantitativelyReason Abstractly

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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.

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 reasoning of others Puzzled Penguin Discuss the Puzzled Penguin problem.

• Puzzled Penguin drew the hour hand between the 12 and 1. Is Puzzled Penguin correct? No. What did Puzzled Penguin do wrong? Puzzled Penguin showed the time 12:30, not 1:30.

Children cross out the incorrect hour hand and discuss how they can help Puzzled Penguin. Responses should include that 1:30 means that it is 30 minutes after 1:00. Children draw the correct hour hand half way between the 1 and 2 to match the time on the digital clock.

Lesson 4

MP.3 Construct a Viable argument Compare Representations Draw each figure on the board as you ask questions to the class.

• (Draw a large open figure resembling a triangle.) What shape is this? How do you know? It looks like a triangle, but it’s not; it’s open, not closed.

• (Draw the corner on the same figure to make it closed.) What shape is this? How do you know? It is a triangle; it is closed with three sides and three corners.

• Now I will draw a smaller shape. (Draw a small triangle.) What shape is this? How do you know? This is a triangle; it has three sides and three corners.

• If we color this shape red, will it still be a triangle? (Color it red.) Yes, color does not change the shape.

Lesson 7


Construct Viable ArgumentsCompare Representations

Critique Reasoning of OthersJustify Conclusions

Puzzled Penguin

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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 Draw a Picture Hand out Student Activity Book page 203. Direct children to cut out the small analog clock that matches their time and activity in the chart. They should glue the clock to the top of a sheet of paper, and then write the time using hour:minutes (such as 7:00).

• Below your clock, write the time that is shown on your clock.

• Draw a picture showing what you do at that time.

Encourage children to fill the page with color and details, showing things that would be around them during the activity.

Lesson 3

Solve Problems with Equal Shares

MP.4, MP.5 Model with Mathematics/Use Appropriate Tools Paper Model Point to the burger that Jay wants to share. Invite children to suggest methods to solve the problem. Encourage them to fold a circle (TRB M70) to show two equal shares.

Discuss the pictures for Problems 2 and 3. Children should be able to explain different ways to fold paper models of a square and rectangle to show equal shares in order to solve the problems.

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Model with MathematicsPaper Model

Draw a Picture

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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.

Teacher ediTion: examples from Unit 7

MP.5 Use appropriate Tools Use a Clock Model As children follow along with their own analog clocks, move the hour hand on the demonstration clock to different numbers and have children say the time with you. Write several “o’clock” times on the board, and have children show the times on their clocks.

Lesson 2

Measure with Same-Size Units PA IRS MATH TALK

MP.5 Use appropriate Tools Use a Concrete Model Give each pair a handful of small paper clips. Model how to lay the paper clips end-to-end to measure the length of a classroom object, such as an unsharpened pencil.

• How many paper clips long is the (pencil)? Explain. Possible response: We lay 6 paper clips down along the bottom of the pencil. We count each paper clip: 1, 2, 3, 4, 5, 6. So, the pencil is 6 paper clips long.

Repeat by having each pair measure the length of another object using paper clips. Ask children to share their results and methods.

Lesson 13


Use Appropriate ToolsUse ManipulativesUse Grid Paper

Model the MathUse a Concrete ModelDraw a Diagram

Use a Clock ModelPaper FoldingPaper Model

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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.


MP.6 Attend to Precision Move the hour hand to 8 on the demonstration clock and have children tell the time as a group. Ask a volunteer to draw the minute and hour hands for 8 o’clock on the Time Poster. Repeat for all the hours on the clock. When children are drawing the minute and hour hands on the Time Poster, invite them to share how they know the minute and hour hands are pointing to the correct numbers on the clock.

Lesson 1

MP.6 Attend to Precision Explain an Example Ask volunteers to explain their reasoning for crossing out each figure. For example, children might identify the hexagon and explain that it is not a rectangle or square because it has six sides, six corners, and the corners are not square.

MATH TALK Discuss the shapes children might draw for Exercises 2 and 3. Some children may draw a square for both exercises. Other children may draw a rectangle that is not a square for Exercise 2 and a rectangle that is a square for Exercise 3. Discuss why both answers are correct.

Lesson 6


Attend to PrecisionExplain an Example

Explain a RepresentationDescribe a Method

Puzzled PenguinExplain Solutions

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Mathematical Practice 7Look for and make use of 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

Read the Clock and Write the Time W H O L E C L A S S

MP.7 Look for Structure Identify Relationships Hand out Student Activity Book page 201, and tell children to look at the clocks in Exercise 1. Guide them in discussing the two types of clocks and the time they show. Point out to children that the minute hand on the analog clock points to 12 and the minute numbers in the digital clock are 00. Then lead children through the other exercises on the page.

Lesson 2

Introduce the Rectangular Prism S M A L L G R O U P S

MP.7 Look for Structure Identify Relationships Give each group the rest of the rectangular prisms from the 3-Dimensional Shape Set to share. Invite them to explore the shapes and discuss what they see. Lead children to see that these additional shapes are made from cubes.

Lesson 10


Look for StructureIdentify Relationships

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► Math and a Picnic

Jay and his family are going on a picnic. Draw lines to show equal shares.

1. Jay wants to share his burger with his mom. How can he cut his burger into two equal shares?

2. Jay and his three sisters want to share a pan of corn bread. How can he cut the bread into four equal shares?

3. Jay’s mom and his three sisters want to share a block of cheese. How can they cut the block of cheese into four equal shares?

Class Activity

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Possible answers are shown.

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There will be lots of food at the picnic.Measure the food in small paper clips.

4. Orange slice How long? paper clips

5. Celery How long? paper clips

6. Cracker How long? paper clips

7. Order the picnic food from longest to shortest. Write the names.

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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.


MP.8 Use Repeated Reasoning Using the set of rectangles and squares, have pairs explore how to sort the shapes. Circulate as children work. As needed, give instructions leading them to sort the shapes into two groups: squares and rectangles that are not squares.

Lesson 6

MP.8 Repeated Reasoning Generalize Children should verbalize the following generalization:

→ You can order objects by length from longest to shortest, or from shortest to longest.

Lesson 12


Use Repeated ReasoningDraw Conclusions

GeneralizeConclude

Focus on Mathematical Practices Unit 7 includes a special lesson that involves solving real world problems and incorporates all eight Mathematical Practices. In this lesson children explore equal shares and measurement at a picnic.

STUDeNT eDITION: LeSSON 14, PAGeS 235–236

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Getting Ready to Teach Unit 7Learning Path in the Common Core StandardsThis unit builds on the geometry and measurement concepts and knowledge that children bring from Kindergarten. The focus of geometry is composing shapes. This helps children build mental images as they compose and describe the new shapes they are building. Children explore 2- and 3-dimensional shapes using unique manipulatives developed for Math Expressions. Children also learn some of the important basic concepts about measurement. Concepts are emphasized this year so that they will easily transition to using rulers next year. Measuring time units is also included in this unit. Children work with both analog and digital clocks as they read and tell time using clocks.

Children's work with composing and decomposing shapes, properties of shapes, and equal shares involves the basic reasoning children will apply to work with fractions, congruence, and symmetry in later grades.

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 an error. Common errors are presented in the Puzzled Penguin feature in the following lessons:

→ Lesson 2: Flips the hour hand and shows 9:00 rather than 3:00

→ Lesson 4: Draws the hour hand between the wrong two numbers and shows 12:30 rather than 1:30

→ Lesson 6: Has difficulty identifying squares and rectangles

→ Lesson 13: Does not line up paper clips properly when measuring an object

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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Measuring Time

Lessons

Time in the Real World Children come to school with understanding of time from their home environments. Some children may have read the numbers on a digital clock. However, the language used at home to express the time may vary greatly. Half past comes from reading an analog clock while nine thirty comes from reading a digital clock. Both types of clocks exist in our society but digital clocks are slowly replacing analog clocks. Many homes will only have digital clocks but children are still required to read an analog clock as a part of the mathematics curriculum. You may find that the first time children see an analog clock is when they arrive at school.

Time Poster In Lesson 1, children describe the parts of an analog clock. Math Expressions includes a Time Poster with the Grade 1 materials. It is most important that children can distinguish between the hour and minute hands. When you draw the hands be sure to draw the minute hand longer than the hour hand.

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Children are introduced to the expression o’clock, which means “of the clock.” This language is used for both analog and digital clocks.

Digital Notation The poster includes both an analog clock and a digital clock. This helps children connect the digital notation to the clockface of an analog clock.

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Make a Clock In Lesson 2, children cut out and make their own personal clock for which they can move the hands to show a given time. The hour hand is red and the minute hand is blue. The reason is that the most common error with telling time is to confuse the minute and hour hands. The color differences help children to see the difference in lengths of the two clock hands.

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After children make their clock, they begin reading the time in hours on an analog clock and writing the time on a digital clock. The clock hands continue to be blue and red.

Make a Time Book: Our Busy Day Children apply what they have learned about time in hours by making a time book and recording their daily activities on the hour by drawing and coloring pictures. This Lesson 3 activity gives children the opportunity to talk about time during the day and how activities change during the day.

The Hour Hand In Lesson 3, children also draw the hour hand on a clock given the time on a digital clock.

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Time in Half-Hours Lesson 4 introduces time to the half-hour. This is a difficult concept because this looks so different on an analog clock as compared to a digital clock. The minute hand on an analog clock points to 6 while the minutes on a digital clock read as 30.

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Children need practice with this skill to be successful. In Lesson 5, children practice time in half-hours as well as review time in hours.

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Analyzing 2-Dimensional Shapes

Lessons

6 7

Hands-On Manipulatives Math Expressions provides unique geometry manipulatives that enable children to analyze, compose, decompose, and reason about geometric properties of shapes.

Reasoning about Rectangles (including squares) After comparing different rectangles in Lesson 6, children observe the following attributes:

→ Closed: All rectangles are closed.

→ Four sides: All rectangles have four sides. All rectangles have opposite sides the same length. Squares are rectangles with all sides the same length.

→ Four corners: All rectangles have four corners called square corners.

Children also learn that sides and corners (angles) are properties of rectangles. However, orientation, size, and color are not defining properties.

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Sorting with Rectangles Children identify and sort rectangles into categories.

Squares Rectangles ThatAre Not Squares

Not Squaresor Rectangles

Reasoning about Triangles and Circles For Lesson 7, Math Expressions provides hands-on manipulatives for reasoning about triangles and circles.

After comparing different triangles children observe the following attributes:

→ Closed: All triangles are closed.

→ Three sides: All triangles have three sides.

→ Three corners: All triangles have three corners. Some triangles have a square corner. Others do not.

After comparing different circles, children observe the following attributes:

→ Closed: All circles are closed.

→ Circles are round, and have no corners.

Children revisit the idea that orientation, size, and color are not defining properties of triangles and circles.

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Composing and Decomposing Shapes

Lessons

8 9

Cutout Manipulatives Math Expressions includes cutouts for children to fold into halves and fourths. These simple shapes help children see how they could decompose a figure in different ways to show two or four equal shares in Lesson 8.

Explore Halves Children discover three possible ways to show two equal shares for rectangles. They find that there are many ways to show two equal shares for circles.

Many WaysThree Ways

Three Ways

Explore Fourths Children explore folding into fourths using the same figures.

Show One Half and One Fourth Children then use what they know about halves and fourths to draw and color one half and one fourth of a shape.

Real World Problems Children apply their reasoning to solve real world problems.

Four friends want to share a sandwich. How can they cut the sandwich into four equal shares? Draw lines. Color each share a different color.

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Compose 2-Dimensional Shapes In Lesson 9 children compose rectangles using hands-on manipulatives. They discover that they can make rectangles that are not squares in many different ways using different and same-sized pieces.

Using Rectangles That Are Not Squares

Using Squares

Using Squares and Other Rectangles

Compose Squares They also make squares using many different rectangles.

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Compose with Triangles and Circles Children continue to compose new shapes with triangles and circles.

3-Dimensional Shapes

Lessons

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Attributes of Geometric Solids Math Expressions provides a unique set of 3-dimensional solids for children to use as they explore solid geometry. In Lesson 10, children discuss the names of the shapes and their appearance, deciding whether a figure can roll, slide, or stack, and whether the surface is curved or flat. They learn the defining properties of a rectangular prism and compose rectangular prisms.

Compose Rectangular Prisms That Are Not Cubes Compose Rectangular Prisms That Are Cubes

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Compose Different Shapes In Lesson 11, children use different types of solids and describe the 3-dimensional shapes they compose.

Use Composite Shapes Children also learn that they can put together composite shapes to make a new shape.

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Concepts of Length

Lessons

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Compare and Order by Length In Lesson 12, children discuss the concept of length by comparing which of two items are longer.

Then children discuss how to place three paper strips in order from shortest to longest…

and also from longest to shortest.

Children next write numbers 1, 2, and 3 to order pictures of three objects from shortest to longest. Finally they draw three lines and label them 1, 2, and 3 to show the order from longest to shortest.

Transitivity Principle for Indirect Measurement Children use what they know about comparing lengths to reason about indirect measurement. This principle states that if the length of object A is greater than the length of object B, and the length of object B is greater than the length of object C, then the length of object A is greater than the length of object C.

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Measuring with Length Units The Grade 1 Common Core State Standards focus on iteration: expressing the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end. This is the focus of Lesson 13.

Measure with Paper Strips After the teacher draws a straight line on the board, a volunteer uses same-sized paper strips to measure the length of the line. The teacher demonstrates how to line up the first strip with the beginning of the line and helps the child tape the strips to the board below the line.

Be sure children understand that the strips should not have any gaps between them, and they should not overlap. This activity presents the following important concepts for measuring length:

→ Using the same-sized unit for measuring

→ Laying multiple copies of the measuring unit end-to-end without gaps or overlaps

Measure Objects with Paper Clips The next activity gives children an opportunity to measure real world objects with paper clips.

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Measure Pictures with Paper Clips Children now use paper clips to measure pictures on the Student Activity Book pages.

Measure in paper clips.

1. Red ribbon How long? ■ paper clips4

Focus on Mathematical Practices

Lesson

The standards for Mathematical Practice are included in every lesson of this unit. However, there is an additional lesson that focuses on all eight Mathematical Practices. In this lesson, children use what they know about geometry and measurement to solve real world problems at a picnic.

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