research—best practices putting research into practice1 minute = ____ seconds ... activity 2...

Research & Math BackgroundContents Planning

Dr. Karen C. Fuson, Math Expressions Author

ReseaRch—Best PRactices

Putting Research into Practice

From Our Curriculum Research Project Perimeter and Area

Initial classroom research showed that students made two kinds of errors in solving perimeter and area of rectangle problems. The design of area and perimeter activities was refined to help students avoid these errors by deepening their understanding.

1. Visualizing and differentiating units

Many students had difficulty visualizing the length units for perimeter and the square units for area and remembering which was which, because area and perimeter problems are typically presented by showing a rectangle with a number on two sides. Students need to see the same rectangle with length units around all the sides to show the perimeter and the same rectangle filled with square units to show the area. It is important for students initially to see and to draw two versions of a rectangle: one with the perimeter length units and the other with the area square units.

2. Calculating only part of the perimeter

Students need to become aware that opposite sides of a rectangle are congruent, and to write the lengths of all sides on the figure initially to visualize and understand perimeter. Such experiences also help students avoid adding only the two numbers shown on the rectangle to find the perimeter.

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Lehrer, Richard. "Developing Understanding of Measurement" A Research Companion to Principles and Standards for School Mathematics. NCTM Reston, VA 2003 Chapter 12: 179–192.

National Council of Teachers of Mathematics. Learning and Teaching Measurement (NCTM 2003 Yearbook). Ed. Douglas H. Clements. Reston: NCTM, 2003.

Van de Walle, John A., Karp, Karen M., Bay-Williams, Jennifer M. Elementary and Middle School Mathematics: Teaching Developmentally (Seventh Edition). Allyn & Bacon, 2009.

Ferrer, Bellasanta B., Bobbie Hunter, Kathryn C. Irwin, Maureen J. Sheldon, Charles S. Thompson, Catherine P. Vistro-Yu. “By the Unit or Square Unit?” Mathematics Teaching in the Middle School 7.3 (Nov. 2001).

National Council of Teachers of Mathematics. Mathematics Teaching in the Middle School (Focus Issue: Measurement) 9.8 (Apr. 2004).

From Current Research

Measurement is a common daily activity performed throughout the world and in all sectors of society. As a well-conceived, logical system of units, the metric system provides models that reinforce concepts and skills involving numeration, decimal relationships, and estimation, connecting mathematics to the rest of the pre-K–12 curriculum. For example:

• Conversion between metric units is facilitated by the decimal nature and consistent prefixes of the system, which lead to a high degree of accuracy in measurements.

• Units for volume, capacity, and mass are interrelated (for example, one cubic decimeter of water, which is one liter, has a mass of one kilogram).

On an international level in the scientific and industrial worlds, the metric system is the standard system of measurement.

Given that the customary system of measurement is well established in the United States and that the metric system is used throughout the rest of the world, the pre-K–12 curriculum should include both systems.

Other Useful References

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Getting Ready to Teach Unit 5Using the Common Core Standards for Mathematical PracticeThe Common Core State Standards for Mathematical Content indicate what concepts, skills, and problem solving students should learn. The Common Core State Standards for Mathematical Practice indicate how students 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 students 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.

Students 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 5

MP.1 Make Sense of Problems Analyze Relationships Point out to students that converting units of time is similar to converting other units of measure. On the board, write:1 minute = ____ seconds

12 minutes = _____ seconds

• What operation would you use to change a larger unit to a smaller unit? multiplication

• What would you multiply 12 by to find the number of seconds in 12 minutes? multiply 12 by 60

Lesson 3

MP.1 Make Sense of Problems Some word problems involve more than one step. Ask the following questions to help the students solve Problem 1 on Student Book page 187.

• What helping question can help you solve this problem? How many meters long is each piece of string?

• What operation will you use to find the number of meters in each piece of string? division

• What operation will you use to convert the 2 meters into centimeters? multiplication

Lesson 7

Mathematical Practice 1 is integrated into Unit 5 in the following ways:

Make Sense of ProblemsAnalyze Relationships

Analyze the ProblemUse a Different Method

Describe RelationshipsCheck Answers

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Mathematical Practice 2Reason abstractly and quantitatively.

Students 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 recognizing the relative sizes of units of measure and relating formulas for area and perimeter to diagrams.

TeACHeR eDITION: examples from Unit 5

MP.2 Reason Abstractly and Quantitatively Connect Diagrams and Equations Use the following example to help students understand how to use the area formula to find an unknown side. Draw a rectangle on the board and label the top side 15 ft. Write A = 105 square feet below the rectangle. Write the area formula on the board and have a volunteer come to the board to replace letters in the formula with the known numbers. Discuss with students how they are going to find the unknown side length. The students should point out that the area formula is a multiplication formula, so to find the unknown side length, they are going to have to use the inverse of multiplication, which is division.

Lesson 7

MP.2 Reason Quantitatively Students can reason that because there are 100 centimeters in every meter, they can find the number of meters given a certain number of centimeters. The students can reason that if an object is 500 centimeters long, that the object is also 5 meters long.

Lesson 8


Reason QuantitativelyConnect Diagrams and Equations

Reason Abstractly and QuantitativelyConnect Symbols and Words

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

Students 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.

Students 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. Students 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 students formulate ideas and analyze responses and engage in discourse. See also MP.6 Attend to Precision.

TeaCher ediTion: examples from Unit 5

MP.3 Construct a Viable argument Justify Conclusions As a class, discuss Exercise 6. Have students justify their choice of weight. Although the weight of the ship would most commonly be expressed in tons, students should be encouraged to share why they might choose another measure to express the weight.

Lesson 5

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 On the board, draw a rectangle with the sides labeled 3 in., 3 in., 5 in., and 5 in. Under the rectangle write A = 3 × 3 × 5 × 5 = 225 sq in.

• Puzzled Penguin used this equation to find the area of the rectangle. Is it correct? No

• How do you know? The area formula multiplies length times width. Puzzled Penguin multiplied four values. It looks like the Puzzled Penguin confused the formula for area with the formula for perimeter.

Ask for a volunteer to show how to find the area correctly.

Lesson 6


Critique the Reasoning of OthersConstruct a Viable Argument

Puzzled Penguin Justify Conclusions

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Mathematical Practice 4Model with mathematics.

Students 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. Students might draw diagrams to lead them to a solution for a problem. Students 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 Build a Model Hand each student a copy of Make a Yard (TRB M26). Ask them to cut out the four parts and glue or tape them together to make a strip that is 1 yard in length. There are small tabs at the ends of each part to help align the sections properly. Discuss with students the measurement divisions on their paper strip.

• Find 1 inch.

• Find 1 foot. How many inches are equal to 1 foot? 12 in.

• What is the length of the entire strip? 1 yd

• How many feet are equal to 1 yard? 3 ft

• How many inches are equal to 1 yard? 36 in.

Lesson 4

MP.1, MP.4 Make Sense of Problems/Model with Mathematics Draw a Diagram Encourage students having difficulty with perimeter and area problems to draw the rectangle on a sheet of paper. Then have them label the rectangle with the information from the problem. Students should label both known sides of the rectangle, so they can see all unknown side lengths. From there, they should be able to more easily solve the problem now that they have a visual representation of what they need to find.

Lesson 7


Model with MathematicsWrite an Equation

Draw a DiagramDraw a Model

Build a Model

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Mathematical Practice 5Use appropriate tools strategically.

Students consider the available tools and models when solving mathematical problems. Students 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. Students learn and develop models to solve numerical problems and to model problem situations. Students continually use both kinds of modeling throughout the program.

Teacher ediTion: examples from Unit 5

MP.5 Use appropriate Tools Give students the opportunity to choose the best measurement unit and measurement tool to use in various situations. Write the following on the board:

What customary unit of measure would you use to express the measure of

• the amount of milk in a recipe. fluid ounces; cup

• the width of a desk. inches; ruler, yard stick

• the weight of a puppy. pounds, ounces; scale

• the floor of a closet for carpeting. square feet; ruler, yard stick, tape measure

Lesson 5

MP.5 Use appropriate Tools Use a Ruler Direct students’ attention to the inch ruler shown in Exercise 11 on Student Book page 178. Point out that there are 8 spaces between each whole inch that are separated by tick marks. Each section is 1 _ 8 of an inch. Since we typically start at 0 and measure from left to right, we can count the number of whole inches and tick marks to represent the whole inch and fractional part of an inch that is measured. Discuss the different lengths of the tick marks and point out to students that 1 _ 2 is half of 1 whole, 1 _ 4 is half of 1 _ 2 , and 1 _ 8 is half of 1 _ 4 .

Lesson 4


Use Appropriate ToolsUse a Ruler

Use a CalculatorDraw a Diagram

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Mathematical Practice 6Attend to precision.

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


MP.6 Attend to Precision Explain a Solution Students work in Small Groups to complete Exercises 3–5. Have students discuss how they completed the table in Exercise 3. Then apply their reasoning to describe the pattern in Exercise 4.

Lesson 5

MP.6 Attend to Precision Explain a Representation Direct students to read Problem 19 on Student Book page 174 and complete the line plot based on the data provided in the table. Then, ask the following questions:

• How do you know which labels to put on the line plot? I put the labels 1 _ 4 , 1 _ 2 , 3 _ 4 , and 1 on the line plot because these are the labels from the table. I’ll show how many students read for each interval of time.

• How do you represent the number of students for each time on the line plot? For each student response of each time, I mark a dot. Five students chose 1 _ 2 , so I put five dots above the 1 _ 2 on the line plot.

Lesson 3

MATH TALK Have students complete Exercises 11–12 in Small Groups. To ensure that students are multiplying by the correct multiple of ten, have them explain their method for completing the table and double number line.

Lesson 2

MATH TALKin ACTION

Students discuss their methods for solving Problem 4.

What is the first step in solving the problem?

Mike: I wrote down the information I knew and what I needed to find to solve the problem.

Shelly: Since we need to give the answer in ounces and the measures are in pounds, I know that I’m going to have to convert from pounds to ounces. I did this conversion first before I did anything else.

Brendon: I did not convert first. Instead, I divided the number of pounds, 4, by 2, to get how many pounds were in each bag. Then I converted.

Lesson 7


Attend to PrecisionDescribe a MethodExplain a Representation

Puzzled PenguinExplain a Method

Explain a SolutionVerify Solutions

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Mathematical Practice 7Look for structure.

Students 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 5

MP.7 Look for Structure Identify Relationships Ask students to look at their meter strips, especially at the beginning and end. Explain what the vertical lines at the top represent and how they relate to each other. Students should be able to describe what each set of markings represents, starting with the meter marking at the bottom of the strip.

960 970 980 990 1000

96

10

97 98 99 100

1

10 20 30 40 50 60 70

1

mm

cm 2 3 4 5

dm

m

6 7

80

8

90 100

9

1

10

Lesson 1

MP.7 Look for Structure Have students examine the chart on Student Book page 169. Together, discuss the names of the units of liquid volume and compare them to the names of the units of length. Be sure students recognize the following:

• The base unit is different from that of length, but the prefixes remain the same.

• The relationship between neighboring units remains the same.

Lesson 2


Look for Structure Identify Relationships

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Class Activity5-8

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► Math and GardensGardens come in all shapes and sizes and can include flowers, vegetables, and many other plants. A Dutch garden is a type of garden that is often a rectangle made up of smaller rectangles and squares, called flowerbeds. A hedge or wall is often placed around the perimeter of the garden. Dutch gardens are known for having very colorful, tightly packed flowers. The Sunken Garden is a famous Dutch garden at Kensington Palace in London, England.

Jared looked up some information about taking care of a garden. He discovered that you need to water and fertilize a garden regularly to help the plants grow. The information he found is shown in the table at the right.

Use the diagram below to answer the questions. It shows a planned flowerbed for a Dutch garden. The perimeter of the flowerbed is 200 feet.

Gardening Information

Fertilizer 4 ounces per 100 sq feet

Water 30 gallons every 3 days

10 ft

?

1. What is the length of the unknown side?

2. What is the area of the garden?

3. How many ounces of fertilizer should be used on the flowerbed?

4. How many cups of water are used every 9 days?

Name Date

Show your work.90 feet

900 square feet

36 ounces

1,440 cups

UNIT 5 LESSON 8 Focus on Mathematical Practices 189

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► Rectangles in GardensPadma wants to create a rectangular shaped garden in her backyard. She wants to have a total of threeflowerbeds, two of which will be the same size. She drew a diagram of how she wants the garden to look. Use the diagram to answer the questions below.

5. What is the perimeter of the entire blue section of flowerbeds?

6. Which section of the garden has a greater perimeter, the green section or the entire blue section? How much greater?

7. What is the area of the whole garden?

8. Padma decides to plant tulips in one of the blue flowerbeds and roses in the green flowerbed. Compare the area of the tulip flowerbed and the rose flowerbed using >, <, or =.

Name Date

Show your work.

P = 44 meters

blue section; 14 meters greater

168 square meters

5,600 square meters = 5,600 square meters

190 UNIT 5 LESSON 8 Focus on Mathematical Practices

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STUDeNT eDITION: LeSSON 8, pAGeS 189–190

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

Students use repeated reasoning as they analyze patterns, relationships, and calculations to generalize methods, rules, and shortcuts. As they work to solve a problem, students 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 Encourage students to summarize their conclusions. They should be able to verbalize that they multiply by a multiple of 10 to convert a larger unit to a smaller unit.

Lesson 1

Mp.8 Use Repeated Reasoning Generalize Challenge students to work in Small Groups to adapt the equations for perimeter and area of rectangles to squares.

• How can you apply the formula for perimeter to a figure with four congruent sides? 4 × s = P

Lesson 6


Use Repeated Reasoning Generalize

Focus on Mathematical practices Unit 5 includes a special lesson that involves solving real world problems and incorporates all 8 Mathematical Practices. In this lesson students use what they know about measurement, area, and perimeter to plan and analyze Dutch gardens.


Math Expressions VOCABULARY

As you teach the unit, emphasize

understanding of this term.

• prefixesSee the Teacher Glossary.

Getting Ready to Teach Unit 5Learning Path in the Common Core StandardsIn this unit, students explore customary and metric measurement, time concepts, and area and perimeter. An understanding of the base ten structure of the place value system is used to help students understand the metric system of measurement. This system of measurement is an important focus, as it is the system of measurement used in the sciences. It is also the everyday system of measurement used in almost all other countries in the world. Frequent exposure to metric units will help students think about measurement in the metric system.

Students also build on their understanding of measuring distance, weight, capacity, and time using the U.S. customary system, including measuring to the 1 _ 8 inch. They apply the skills they develop to real-world measurement situations. They also develop their ability to make estimates and to use measurements to solve problems.

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

In this unit, we use Puzzled Penguin to show typical errors that students make. Students enjoy explaining Puzzled Penguin’s error and teaching Puzzled Penguin the correct way to convert measurements and find perimeter and area. The following common errors are presented to the students as letters from Puzzled Penguin and as problems in the Teacher Edition that were solved incorrectly by Puzzled Penguin:

→ Lesson 2: Choosing the incorrect multiplier when converting liters to milliliters

→ Lesson 6: Using the incorrect formula to find the area of a rectangle

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 MEASUREMENT AND DATA

Measurement Data As the name

suggests, measurement data comes

from taking measurements. For

example, if every child in a class

measures the length of his or her

hand to the nearest centimeter,

then a set of measurement data

is obtained.

Measurement Data

Lessons

1 2 3 4 5 7

Metric and Customary Systems of Measure In this unit, students describe the attributes of objects, convert from one unit to another, and solve problems within the metric and customary systems of measure. Using units from both systems allows students to develop a deeper understanding of each attribute as well as the relationship between the two systems.

Metric Units Metric units are derived from the base units: meters, liters, and grams. Prefixes are used to name larger and smaller units. Emphasis is placed on the metric units that are most commonly used in everyday life around the world and units with which students may already be familiar. For example, students may know about 2-L bottles or 10-km Olympic races.

Estimation The ability to estimate measurements is an important skill. Often it can replace finding actual measurements or be used to check a measurement or the solution to a measurement problem. Benchmarks are used frequently throughout the unit to help students conceptualize the relative size of different measurements. Students who develop strong personal benchmarks, for example: my uncle is about 6 feet tall, my dog weighs about 9 kilograms, or our juice pitcher holds about 12 cups, are more likely to be able to estimate accurately.

Tables and Double Number Lines Throughout the unit, tables and double number lines are tools used to model unit equivalencies of metric and customary units of measure. Both models help students to convert measurements by facilitating the recognition of patterns in the conversions. The examples below show how students use a table and a double number line to convert customary units of capacity.

Quarts Fluid Ounces

1 32

2 64

3 96

4 128

5 160

6 192

quarts

cups

0 1 2 3 4

0 4 8 12 16

The use of the double number lines is beneficial because it enables students to use their understanding of the properties of a ruler to help conceptualize measurement conversions. Note that the distance between each quart is divided into 4 equal sections to show that there are 4 cups in a quart. Students can see that since 3 aligns with 12, 3 quarts = 12 cups.

UNIT 5 | Overview | 451Y


Linear Measurement

Lessons

1 4

Metric Units of Linear Measurement In Lesson 1, students begin their exploration of measurement by investigating the metric units of linear measure. A ruler is used to help students visualize both the size of metric units and how they are related to each other. A ruler like the one below shows that: 1 decimeter = 10 centimeters = 100 millimeters.

10 20 30 40 50 60 70

1

mm

cm 2 3 4 5

dm

m

6 7

80

8

90 100

9

1

10

As students explore the properties of a ruler, they are given the opportunity to also investigate the following properties of measurement. Iteration: measuring tools have units that repeat. Partitioning: measuring tools have large units divided into smaller units that are the same size. Compensatory Principle: more smaller units than larger units are needed to measure any distance. Transitivity: the relationship among three elements, for example, if Object A is longer than Object B and Object B is longer than Object C, then Object A is longer than Object C.

Students also continue to develop a sense of the relative size of metric units as they identify the appropriate unit to measure common lengths. Common benchmarks can help students develop a sense of the sizes of metric units. For example, things that are typically measured in feet or yards are measured in meters. Greater lengths, such as distances between cities, are measured in kilometers. Shorter lengths, such as the length of a pencil, are measured in centimeters, and very small lengths are measured in millimeters.

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Fractional Lengths [Students]

use their developing knowledge

of fractions and number lines

to extend their work from the

previous grade by working with

measurement data involving

fractional measurement values.

To help students understand the relationship among the linear units of metric measure, a table like the one below is presented.

Units of Length

kilometer hectometer decameter meter decimeter centimeter millimeter

km hm dam m dm cm mm

10 × 10 × 10 × 10 × 10 × 10 × 1 m 10 × 10 × 10 × 10 × 10 × 10 ×larger larger larger smaller smaller smaller

1 km 1 hm 1 dam 10 dm 100 cm 1,000 mm= 1,000 m = 100 m = 10 m = 1 m = 1 m = 1 m

Students are able to apply their understanding of place value and powers of ten to convert from one measurement to another. The meter is the base unit. The larger units of measure are to the left of the meter and the smaller units are to the right. As students move from left to right on the table, they multiply. As they move from right to left, they divide. Students learn that the prefixes of each measurement indicate its magnitude.

Customary Units of Linear Measurement In Lesson 4, students continue their work with linear measurement as they use a ruler to measure and convert among customary units of measure. They extend their measurement skills as they identify measurements through the nearest 1 _ 8 inch. For example, students identify the length of the line segment below as being at 3 7 _ 8 inches.

1 2 3 4 5 60inches

They realize that measuring to the nearest 1 _ 8 inch is more precise than measuring to the nearest 1 _ 2 or 1 _ 4 inch.

Students use standard equivalencies to convert among inches, feet, yards, and miles. They use tables and double number lines as tools to help them.

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Capacity, Mass, and Weight

Lessons

2 5

Metric Units of Liquid Capacity Capacity is a measure of volume usually associated with liquid measurement. Students use benchmarks to help them determine the size of metric units of capacity. For example, one liter is a little more than one quart. Bottled water, juices, and carbonated drinks often come in one-liter or two-liter containers. Smaller quantities of liquids are typically labeled in milliliters. Milliliters and grams are used in this country for prescriptions. As with metric units of length, a table is presented to show the relationship among metric units of liquid capacity. The liter is the base unit.

Units of Liquid Volume

kiloliter hectoliter decaliter liter deciliter centiliter milliliter

kL hL daL L dL cL mL

10 × 10 × 10 × 10 × 10 × 10 × 1 L 10 × 10 × 10 × 10 × 10 × 10 ×larger larger larger smaller smaller smaller

1 kL 1 hL 1 daL 10 dL 100 cL 1,000 mL= 1,000 L = 100 L = 10 L = 1 L = 1 L = 1 L

Customary Units of Capacity In the customary system, as students see in Lesson 5, the volume of liquids is measured in fluid ounces, cups, quarts, and gallons. One cup of milk, for example, will fit into a container with a capacity of one cup. Teaspoon and tablespoon conversions are also presented. Students use standard conversions among the various measurement units, as well as use their estimation skills to determine the appropriate unit to measure common containers.

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Weight and Mass Weight is dependent on the effects of gravity, but mass is a measurement independent of gravity. A person who weighs 150 pounds on Earth weighs less on the moon, but still has the same mass. Because we all live on Earth, we sometimes talk in everyday terms about something “weighing 100 grams.”

Metric Units of Mass The gram is the basic unit of mass. One gram is a very small unit. A paper clip or peanut has a mass of about one gram. Students use a table like the ones for meters and liters to convert among the metric units of mass.

Units of Mass

kilogram hectogram decagram gram decigram centigram milligram

kg hg dag g dg cg mg

10 × 10 × 10 × 10 × 10 × 10 × 1 g 10 × 10 × 10 × 10 × 10 × 10 ×larger larger larger smaller smaller smaller

1 kg 1 hg 1 dag 10 dg 100 cg 1,000 mg= 1,000 g = 100 g = 10 g = 1 g = 1 g = 1 g

Customary Units of Weight In Lesson 5, students explore the customary units of weight, ounces, pounds, and tons. The ton is a new unit for students. To help them understand the magnitude of the customary units of weight, students find benchmarks for each measurement unit. They use standard equivalencies to convert among the units.

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Time

Lesson

3

Time Conversions In Lesson 3, students use time unit equivalencies to convert among time units. The units students explore include seconds, minutes, hours, days, weeks, months, and years.

Time, Fractions, and Line Plots Working with time provides opportunities to review what students have learned about fractions. It is common to relate fractions of an hour to minutes in order to determine that 1 _ 4 hour is 15 minutes and 3 _ 4 hour is 45 minutes. Students apply this understanding as they interpret line plots that include fractions of an hour.

Time Spent Reading Each Night (in hours)

0 114

12

34

Students use the line plot to determine, for example, that the greatest number of students spent 1 _ 2 hour a night reading.

Elapsed Time In the previous grade, students found elapsed time in hours and minutes and used these skills to solve real world problems. Students review these skills in Lesson 3. The principle that clocks are comprised of iterated units, like all measuring tools, is reinforced as students count the sectors through which the clock hands have traveled to find elapsed time.

Using subtraction to find elapsed time is also presented. For example, to find the time that a student practiced his instrument if he started at 8:21 a.m. and ended at 9:35 a.m. can be found this way:

9:35 a.m.  – 8:21 a.m.

 

1 hr 14 min

Finding elapsed time across noon and midnight is also addressed. Students realize that they have to find the elapsed time between the starting time and either noon or midnight, then find the elapsed time from noon or midnight to the end time. These times, added together, give the total elapsed time.


Line Plots The Standards call for

students to represent measurement

data with a line plot. This is a

type of display that positions the

data along the appropriate scale,

drawn as a number line diagram.

These plots have two names in

common use, “dot plot” (because

each observation is represented

as a dot) and “line plot” (because

each observation is represented

above a number line diagram).

The number line diagram in a line

plot corresponds to the scale on

the measurement tool used to

generate the data. In a context

involving measurement of liquid

volumes, the scale on a line plot

could correspond to the scale

etched on a graduated cylinder.

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Perimeter and Area

Lesson

Perimeter The presentation of perimeter in Lesson 6 begins with emphasizing a conceptual understanding of perimeter. Students learn that perimeter is the distance around a figure and it is measured in linear units. The following models help students conceptualize the linear units in the perimeter of figures.

Through a variety of experiences in which students find the perimeter of rectangles, they are able to generalize the formula: P = l + w + l + w. Students’ basic knowledge about rectangles lays the conceptual foundation for this simple formula development. Students are encouraged to discuss how multiplication can be used to write this formula in a shorter way; for example: P = (l ⋅ 2) + (w ⋅ 2) or P = (l + w) ⋅ 2.

Area As students begin to explore area, they again concentrate on conceptualizing what area is and what units are used to measure area. Students learn that area is the number of same-sized units that cover a shape with no gaps or overlaps. Area is measured in square units. The following models help students visualize area.

The models above facilitate students understanding that they can find the area of the rectangle by calculating the number of squares in an array. Students call upon their prior understanding of the area model as pushed-together squares and that the number of squares in an array can be found by multiplying the number of squares in each row by the number of rows. They use this knowledge to help them develop and understand the area formula: A = l ⋅ w.

X Y

Z

X Y

Z

Key:

= 1 cm

Length = l

Width = w

Perimeter = P

Key:

= 1 sq cm

Length = l

Width = w

Area = A

6

UNIT 5 | Overview | 451EE


Problem Solving

Lessons

3 5 7

Problem Solving Plan In Math Expressions, a research-based problem-solving approach that focuses on problem types is used.

• Interpret the problem.• Represent the situation.• Solve the problem.• Check that the answer makes sense.

Time Problems Students apply their understanding of time concepts to solve problems involving elapsed time. A variety of problems are presented in which students need to find the start time, the end time, or the time that has elapsed.

Measurement, Perimeter, and Area Problems Students apply their understanding of measurement and operations as they solve problems in which they have to add, subtract, multiply, or divide measurements. They also solve problems involving perimeter and area. To successfully solve the problems presented in this unit, students need to understand the structure of the problem to identify the operation needed, as well as be able to use conversion equivalences to flexibly move from one unit to another.

Focus on Mathematical Practices

Lesson

8

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, students use what they know about measurement, area, and perimeter to solve problems involving gardening.

451FF | UNIT 5 | Overview

research—best practices putting research into practice1 minute = ____ seconds ... activity 2...

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