Unit 7 Measurement and Data: Conversion for CC-Edition... · Unit 7 Measurement and Data: Conversion ... a ruler, and explain to students that a millimeter is a unit of measurement

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H-1Measurement and Data

Unit 7 Measurement and Data: Conversion

IntroductionIn this unit, students will learn about metric units of length and distance, and metric units of mass. Students will reinforce the notion of place value by noting similarities in the relationships between different units and the relationships between different place values. As they convert and compare measurements throughout the unit, students will add, subtract, multiply, and divide numbers, including decimals to thousandths. For details, see the prior knowledge required section for each lesson.

To help students visualize the relative size of the units they are learning, create a class measurement display and add to it every time you introduce a new unit. Make sure the items are clearly labeled, stating the unit and which dimension is equal to the unit. Example: “Thickness = 1 mm”

Examples:

1 mm – thickness of cardboard, width of some beads, thickness of some buttons

1 cm – thickness of some books, width of a centimeter cube, width of a tens block

1 m – length of some scarves, length of a piece of yarn

1 km – length of a spool of fishing line, length of several spools of yarn

Mass vs. weight. At this level, we use the terms “weight” and “mass” interchangeably. Weight technically refers to the force of gravity pulling an object down (that’s why people weigh less in lower gravity). Pan balances and playground seesaws compare weight. Mass technically refers to the amount of matter in an object, and it does not change in outer space where the force of gravity is lower. It is appropriate at this level to use the word “mass” to mean “how strongly an object pulls down when you try to lift it.”

Teacher’s Guide for AP Book 5.1

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MD5-1 Centimeters and MillimetersPages 166–167

Introduce centimeters. Point out the centimeter marks on a ruler, and explain to students that a centimeter is a unit of measurement. Write the word “centimeter” on the board, circle the letters “c” and “m,” and write the abbreviation “cm.” Explain that these are the two ways to write centimeter.

Show students some objects that they can use as benchmarks for centimeters, such as centimeter cubes, buttons, a book that is 1 cm thick. You can keep a display of such objects, with labels such as “thickness = 1 cm,” for students to use as a reference.

Draw a number line and explain that counting on a number line is just like using a ruler. Demonstrate this by asking a volunteer to use a number line to count (by “hopping”) to five. Then ask another volunteer to demonstrate how to measure 5 cm with the ruler.

ACTIVITY

Students use rulers to find items that have the given length, width, or thickness.

a) 6 cm b) 10 cm c) 1 cm d) greater than 15 cm e) between 12 cm and 15 cm f) about 35 cm long

Introduce lines and line segments. Explain that we use the word line for straight lines that can be drawn with a ruler. A line can be extended at both ends, as much as we want. However, when we are interested in a piece of line that has a specific length (such as 5 cm), we call that piece of line a line segment.

0 cm 1 2 3 4 5

1 2 3 4 5

STANDARDS 5.MD.A.1, 5.NBT.A.1

VoCABUlARY centimeter (cm) convert line line segment measurement millimeter (mm)

GoalsStudents will measure lengths in centimeters and millimeters, and convert between centimeters and millimeters.

PRIoR KNoWlEDGE REQUIRED

Understands the concept of linear measurement Understands that smaller units require more units Can multiply whole numbers by 10 Is familiar with decimals to tenths Can locate a number on a number line, including decimal tenths

MATERIAlS

centimeter and millimeter rulers benchmarks for centimeters and millimeters

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H-3Measurement and Data 5-1

Drawing a line segment. Follow the steps below to demonstrate how to draw a line segment of length 2 cm.

Step 1: Draw a vertical dash to mark where the line segment starts. Place the ruler so that the dash is at the zero mark.

Step 2: Count forward from zero by hops (two hops for 2 cm, three hops for 3 cm, and so on). Make sure not to move the ruler.

Step 3: Draw a second vertical dash to mark the other end of the line segment. (For 2 cm, draw the dash at the 2 cm mark.)

Step 4: Draw a line segment connecting the dashes.

Exercises: Use the steps to draw a line segment of the given length. Have a partner measure your line segment.

a) 3 cm b) 5 cm c) 8 cm

Check that students do not move the ruler while counting by hops, and that they place the ruler properly in Step 1. Students who have difficulty performing the first three steps together might also benefit from using a pre-drawn line, where they only need to mark the beginning and the end of the line segment.

Introduce millimeters. Identify the length and marking of a millimeter on a ruler, and explain to students that a millimeter is a unit of measurement that is smaller than a centimeter. Write the word “millimeter” on the board, circle the two “m”s, and write the abbreviation “mm.” Explain that these are the two ways to write millimeter. Add benchmarks representing 1 mm to the display.

Converting centimeters to millimeters. Draw a 5 cm ruler and explain that there are 10 millimeters in 1 centimeter. If available, show two rulers side by side—one that shows centimeters only and another that shows millimeters only—to show how each measurement in centimeters corresponds to a measurement in millimeters, as on p. 166 in the AP Book.

ASK: How many millimeters are there in 2 centimeters? Explain to students that one way to find the answer is by skip counting. Using the 5 cm ruler, skip count by tens from 0 cm to 2 cm. Say “20” at the 2 cm mark. Extend the ruler to 10 cm and skip count by tens as a class to determine how many millimeters are in 10 cm.

On the board, draw the T-table as shown in the margin. Ask students to fill in the measurements in millimeters. Extend the table a few more rows and have students continue the pattern.

Then ASK: How can you get the number in the right-hand column from the number in the left-hand column? What mathematical operation takes 1 to 10, 2 to 20, and so on? (multiply by 10)

Exercises: Convert to millimeters.

a) 3 cm b) 8 cm c) 15 cm d) 50 cm

Bonus: 2,345 cm

cm mm

1 10

2 20

3 30

4 40

(MP.7)


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Answers: a) 30 mm, b) 80 mm, c) 150 mm, d) 500 mm, Bonus: 23,450 mm

Comparing measurements. ASK: Which is longer, 2 hours or 5 minutes? Which number is greater, 5 or 2? Point out that even though 5 is more than 2, 2 hours is still longer than 5 minutes. The units are different, and we need to take the size of the units into account when comparing measurements.

Write on the board:

7 cm 30 mm

ASK: Which number is greater, 7 or 30? Does this mean that 30 mm is longer than 7 cm? (no) Explain that to compare measurements correctly, you need to convert them to the same unit. It is usually more convenient to use the smaller unit, so convert the measurement in the larger unit. Ask students to change 7 centimeters to millimeters. Which is longer, 70 mm or 30 mm? So which is longer, 7 cm or 30 mm? (7 cm)

Exercises: Which measurement is longer?

a) 35 mm or 2 cm b) 48 mm or 20 cm c) 765 mm or 90 cm d) 65 cm or 643 mm e) 46 cm or 3,456 mm

Bonus f ) 12,345 mm or 7,892 cm g) 654,234 mm or 8,893 cm

Answers: a) 35 mm, b) 20 cm, c) 90 cm, d) 65 cm, e) 3,456 mm, Bonus: f ) 7,892 cm, g) 654,234 mm

Measuring in millimeters. Explain to students that counting every millimeter in a measurement can take a long time, but there is a quick way to do it. Draw a ruler representing 30 mm and tell the class that you want to count 26 mm. Then demonstrate how to skip count by tens to the tens place value preceding the amount, and count on from there by ones. (10, 20, 21, 22, 23, 24, 25, 26)

Have volunteers demonstrate this shortcut method by counting to several different numbers. Then have students practice measuring different objects to the closest millimeter.

Review decimal tenths on number lines. Remind students that when one whole is divided into 10 equal parts, the parts are called tenths, and decimals are a convenient way to write fractions with denominator 10, 100, 1,000, and so on. We write 1/10 as 0.1, 3/10 as 0.3, and 5 4/10 as 5.4. Have students read and write a few decimal tenths.

Draw a number line divided into tenths, from 0 to 3. Remind students how to find a number on this number line. For each decimal below, point to several locations on the line and have students signal thumbs up if the location is correct, and thumbs down if it is not.

Exercises: Indicate the location of the decimal on a number line.

a) 0.4 b) 1.1 c) 2.8 d) 1.5

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Answers:

Converting millimeters to centimeters using a ruler. Point out that 1 mm is one tenth of a centimeter, so this means you can write a decimal measurement 1 mm = 0.1 cm. ASK: How can you write 2 mm in centimeters? (0.2 cm) PROMPT: 2 mm is how many tenths of a centimeter? (2/10) Repeat with 5 mm, 9 mm, then 16 mm. (0.5 cm, 0.9 cm, 1.6 cm)

Ask students to look at a millimeter ruler. How is the ruler the same as the number line you drew? How is it different? (each centimeter is divided into ten units, but there is a longer mark for every five millimeters, not every ten)

Have students locate 23 mm on a ruler. SAY: Think of 1 cm as one whole. What position on a number line does 23 mm show? (2.3 cm) Point out that as you count by tens of millimeters, you are counting by centimeters, so counting by millimeters is counting by tenths. Work through the next few Exercises as a class, then have students work individually.

Exercises: Use a ruler to convert to centimeters.

a) 14 mm b) 32 mm c) 27 mm d) 45 mm

Answers: a) 1.4 cm, b) 3.2 cm, c) 2.7 cm, d) 4.5 cm

Extensions1. Draw each object to the given measure.

a) a shoe 6 cm long b) a tree 5 cm high c) a glass 3 cm deep

2. a) i) Draw a collection of alligators, each one being 1 cm longer than the previous one.

ii) A pencil shrinks when it is sharpened. Draw a collection of pencils, each one being 1 cm shorter than the previous one.

iii) Draw a sequence of toboggans in which each one is 2 cm longer than the last one.

iv) A carrot shrinks when it is eaten. Draw a collection of carrots, each one being 2 cm shorter than the previous one.

b) Write a story about one of the growing or shrinking items in part a) or invent your own! Tell a story about how the item grows or shrinks. How does the carrot get eaten? Who wants to ride the toboggans? Write the story to go with the pictures that you have drawn.

3. A nickel is about 2 mm thick. Josie has a stack of nickels 10 mm high. How much money does Josie have?

Answer: 5 nickels = 25¢

0 1 2 3

0.4 1.1 1.5 2.8


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MD5-2 Centimeters and Millimeters (Advanced)Pages 168–169

Review relevant prior knowledge. Remind students that there are 10 millimeters in each centimeter, so if an object is 4 cm long, it is also 40 mm long. Review how students can multiply by 10 to convert measurements in centimeters to measurements in millimeters. Have students convert several measurements in centimeters to measurements in millimeters.

Exercises: Convert to millimeters.

a) 6 cm b) 12 cm c) 89 cm d) 103 cm

Answers: a) 60 mm, b) 120 mm, c) 890 mm, d) 1,030 mm

Measuring to the closest centimeter and to the closest millimeter. On the board, draw a centimeter ruler and a bar that goes from 0 cm to a point between 5 cm and 6 cm (say about 52 mm). ASK: How long is the bar? Is it longer or shorter than 5 cm? (longer) Is it longer or shorter than 6 cm? (shorter) SAY: We can see that the bar is not exactly 5 cm long and not exactly 6 cm long—the bar is between 5 and 6 cm long. ASK: Is the bar closer to 5 cm long or closer to 6 cm long? Write the words “closer to” on the board. Highlight the distances between the end of the bar and the 5 cm and 6 cm marks on the ruler. Explain that the bar ends closer to the 5 cm mark, so we say the bar is “about 5 cm long” or “5 cm long when measured to the closest centimeter.” Write both expressions on the board.

Add millimeters to the ruler. ASK: What is the length of the bar in millimeters? (52 mm) How many millimeters are in 5 cm? (50) In 6 cm? (60) Is 52 closer to 50 or to 60? (50) What are we doing when we change 52 to 50 because it is closer to 50 than to 60? (rounding) When you are rounding 52, will you round up to 60 or down to 50? (down to 50) Finally, write on the board:

The bar is 52 mm long. The bar is about 5 cm long.


VoCABUlARY centimeter (cm) convert measurement millimeter (mm) round down round up rounding

GoalsStudents will convert between centimeters and millimeters.


Understands the concept of linear measurement Understands that smaller units require more units Can multiply and divide decimals and whole numbers by 10 Can locate a number on a number line, including decimal tenths Knows that 10 mm = 1 cm Can convert whole centimeters to millimeters Can use a ruler to convert millimeters to centimeters Knows that measurements can be in fractions of a unit

MATERIAlS

rulers variety of objects to measure

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H-7

Draw a bar of a different length (say, 37 mm) and have students raise the number of fingers that correspond to the length of the bar in centimeters. Then have students say the length in millimeters. Have students write two sentences about the length of the bar as shown in the previous example.

Draw a bar 45 mm long. Is 45 closer to 40 or closer to 50? (neither) When you round 45 to the nearest 10, will you round down to 40 or up to 50? (up to 50) Explain that you do the same when determining the length to the closest centimeter—you round up by saying that the bar is 5 cm long when measured to the closest centimeter.

ACTIVITY

Give students various objects that have lengths between exact numbers of centimeters. Students measure them to the closest centimeter and closest millimeter. Partners exchange objects and check each other’s answers.

Identifying measurements between other measurements. Write these pairs of measurements on the board:

A: 8 cm and 9 cm B: 9 cm and 10 cm C: 10 cm and 11 cm

Ask students to say which pair of measurements (A, B, or C) the measurement 87 mm is between. (A) Repeat the question for 93 mm. (B) Ask students to find another millimeter measurement between each pair of lengths.

Now write these pairs of measurements in millimeters on the board. Ask students to find a measurement in whole centimeters that is between the lengths in each pair:

A: 56 mm and 63 mm B: 78 mm and 86 mm C: 102 mm and 114 mm

Sample answers: A: 6 cm, B: 8 cm, C: 11 cm

Review multiplying and dividing decimals and whole numbers by 10. Remind students that they shift the decimal point 1 place right to multiply and 1 place left to divide. Remind them that they can write zeros after the decimal point and the number will not change. For example, 6.7 = 6.70. ASK: Why is that true? (the decimal part is 7/10 = 70/100) As well, remind students that even though we do not write the decimal point in whole numbers, we can still add it without changing the number. For example, 12 = 12.0. Use the questions below as a test to make sure that all students can perform the required multiplication and division.

Exercises: Multiply or divide.

a) 25 ÷ 10 = b) 3 ÷ 10 = c) 122 ÷ 10 = d) 32.4 ÷ 10 = e) 0.4 × 10 = f ) 5.3 × 10 = g) 1.89 × 10 = h) 0.08 × 10 =

Answers: a) 2.5, b) 0.3, c) 12.2, d) 3.24, e) 4 = 4.0, f ) 53, g) 18.9, h) 0.8

(MP.3)

Measurement and Data 5-2


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Using multiplication and division to convert between centimeters and millimeters. Remind students that they multiplied by 10 to convert a measurement in centimeters to millimeters. ASK: What operation would you use to convert a measurement in millimeters to centimeters? (divide by 10) PROMPT: Which operation is the opposite of multiplication? Convert the first few measurements in each Exercise as a class, then have students work individually.

Exercises

1. Convert to centimeters.

a) 6 mm b) 12 mm c) 89 mm d) 103 mm

Answers: a) 0.6 cm, b) 1.2 cm, c) 8.9 cm, d) 10.3 cm

2. Convert to millimeters.

a) 7.2 cm b) 0.8 cm c) 0.89 cm d) 10.3 cm

Answers: a) 72 mm, b) 8 mm, c) 8.9 mm, d) 103 mm

Solving problems requiring conversions. Work through the following problems as a class. Encourage multiple solutions. For example, for the first problem, students can convert to millimeters and perform operations with decimals.

Exercises

a) An eastern box turtle is 14.8 cm long. A New Mexico whiptail lizard is 192 mm long. Which reptile is longer? How much longer? Write the answer in centimeters and in millimeters.

b) A dime is about 18 mm wide. Amy laid out 7 dimes side by side. How wide is the row in centimeters?

Answers: a) The lizard is 44 mm = 4.4 cm longer, b) 12.6 cm

Extensions1. To change a measurement in millimeters to a mixed measurement

in centimeters and millimeters, remind students that the number of centimeters is the number of tens of millimeters. Therefore all they need to do is to separate the ones and the tens, or imagine the number made from ones and tens blocks. For example, there are 17 tens and 3 ones in 173, so there are 17 cm 3 mm in 173 mm. In other words: 173 mm = 170 mm + 3 mm = 17 cm + 3 mm = 17 cm 3 mm.

Change each measurement to a mixed measurement in centimeters and millimeters using the least number of millimeters.

a) 17 mm b) 47 mm c) 125 mm d) 2 cm 25 mm e) 32 cm 425 mm f ) 421 cm 1,232 mm

Answers: a) 1 cm 7 mm, b) 4 cm 7 mm, c) 12 cm 5 mm, d) 4 cm 5 mm, e) 74 cm 5 mm, f ) 544 cm 2 mm

(MP.4)

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2. To convert from a mixed measurement to millimeters, change the centimeters to millimeters, then add the leftover millimeters. Example: 33 cm 2 mm = 330 mm + 2 mm = 332 mm.

Convert to millimeters.

a) 5 cm 8 mm b) 78 cm 9 mm c) 44 cm 1 mm d) 67 cm 89 mm e) 79 cm 56 mm f) 60 cm 1,234 mm

Answers: a) 58 mm, b) 789 mm, c) 441 mm, d) 759 mm, e) 846 mm, f ) 1,834 mm

3. Find the difference in length between the lizard and the turtle.

lizard Turtle

a) 2 cm 5 mm 3 cm 1 mm

b) 7 cm 8 mm 12 cm 3 mm

c) 2 cm 17 mm 5 cm 26 mm

Answers: a) 6 mm, b) 45 mm = 4.5 cm, c) 39 mm = 3.9 cm


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H-10

MD5-3 Meters and CentimetersPages 170–171

Introduce meters. Identify the length and marking of a meter on a ruler or measuring tape, and explain to students that a meter is another unit of measurement. Write the word “meter” and the abbreviation “m,” and explain that these are the two ways to write meter.

Show students a meter stick and explain that it is a tiny bit longer than one meter long. It has about 1 cm of extra length at both ends, just as many rulers do.

Introduce estimation. Explain that sometimes you want to know how long an object is, but you do not need to know the precise length. Giving a number that is about the right length is called estimating the length, and the number itself is called an estimate. Explain that one possible tool to produce a good estimate of a distance in meters is giant steps.

ACTIVITY

large steps are about a meter long. Using tape, make two lines one meter apart on the floor. Students stand with their heels to one of the lines and make a step so that the heel of the front foot touches the other line. Students perform the step several times to make sure they can all do it.

Divide students into pairs. One partner takes five steps they think will be about a meter each. The other partner measures the distance using


VoCABUlARY centimeter (cm) conversion convert estimate estimating measurement meter (m) meter stick millimeter (mm)

GoalsStudents will estimate and measure in meters, and convert measurements between meters and centimeters.


Can measure in centimeters and millimeters Understands that smaller units require more units Can multiply and divide decimals and whole numbers by 10 Is familiar with decimals to hundredths Can locate a number on a number line, including decimal hundredths Can convert between centimeters and millimeters Knows that measurements can be in fractions of a unit

MATERIAlS

rulers measuring tapes meter sticks pieces of string items 1 m long

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a meter stick. Emphasize that it is much more difficult to take five big steps all in a row than to take five steps walking normally. Suggest that students lead off with the same foot each time: take one giant step; bring your back foot forward and place it next to your other foot; then take another giant step. Students repeat the exercise to make the distance of 5 steps as close to 5 meters as possible.

SAY: I want to know about how many meters long the classroom is. Have students take giant steps across the room to help you find out. Remind them to take identical-sized steps and to lead off with the same foot each time. Then have a volunteer use a meter stick to measure the length of the classroom. ASK: Did we get the same answer both ways? Were our answers close? Which way was faster? Explain that when we are in a hurry and we just need to know about how long a room is in meters, we can use giant steps instead of meter sticks to find out. Discuss situations in which it is important to know exact measurements (examples: in a race, making a ruler to sell, choosing paper to put in a book, making legs for a table).

Encourage students to suggest items that are 1 m long (a scarf, a pool noodle, etc.) for the class measurement display (see Lesson MD5-1). Then measure the items with a meter stick to verify their length.

ACTIVITY

Students use a string and a ruler to measure the lengths of various body parts (such as arm length, arm span, height from floor to armpit, distance around the head), and compare the measurements to 1 meter. Encourage students to predict the answers before measuring.

Converting meters to centimeters. Explain that there are 100 centimeters in 1 meter. Draw a number line on the board and label the increments above the number line 0 m, 1 m, 2 m, and so on to 10 m. Label the first two increments below the number line 0 cm and 100 cm. ASK: How many centimeters would be in 2 meters? (200) What number will we skip count by to find the answer? (100) Have students skip count by hundreds and invite volunteers to write the corresponding numbers of centimeters on the number line below the increments.

On the board, draw the T-table in the margin and ask students to fill in the measurements in centimeters. Extend the table a few more rows and have students continue the pattern. Then ASK: How can you get the number in the right-hand column from the number in the left-hand column? What mathematical operation takes 1 to 100, 2 to 200, and so on? (multiply by 100) ASK: How is converting meters to centimeters the same as converting centimeters to millimeters? (we also multiply) How is it different? (we multiply by 10 to get millimeters from centimeters and by 100 to get centimeters from meters) Summarize on the board:

m cm

1

2

3

4


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H-12

× 100 × 10

m cm cm mm

Exercises: Convert from meters to centimeters.

a) 4 m b) 16 m c) 52 m Bonus: 2,775 m

Answers: a) 400 cm, b) 1,600 cm, c) 5,200 cm, Bonus: 277,500 cm

Comparing measurements. ASK: Which measurement is longer, 2 cm or 5 mm? Which number is greater, 5 or 2? Remind students that even though 5 is more than 2, 2 cm is longer than 5 mm. The units are different, and we need to take the size of the units into account when comparing measurements. It is necessary to convert measurements into the same unit before comparing them.

Exercises: Point your thumbs toward the larger measurement.

a) 300 cm or 7 m b) 305 cm or 2 m c) 480 cm or 2 m d) 765 cm or 90 m e) 6,500 cm or 643 m f) 89 m or 34,567 cm

Bonus g) 120,345 cm or 7,892 m h) 654,234 cm or 89,893 m

Answers: a) 7 m, b) 305 cm, c) 480 cm, d) 90 m, e) 643 m, f ) 34,567 cm, Bonus: g) 7,892 m, h) 89,893 m

Determining measurements between other measurements. On the board, draw a number line from 2 m to 5 m. ASK: Where would 453 cm be on this number line? (between 4 m and 5 m) To prompt students to see the answer, have volunteers write the measurements in centimeters below the measurements in meters. Repeat the Exercise with 231 cm (between 2 m and 3 m) and 379 cm (between 3 m and 4 m). Then ask students to give a different example of a measurement in each of the intervals. Finally, ask students what measurement in an exact number of meters is between 283 cm and 341 cm. (3 m)

Review writing decimal hundredths: 1 cm = 0.01 m. Remind students how to read numbers such as 0.67, 2.34, and 0.08. Then ASK: If 1 m = 100 cm, what fraction of 1 meter is 1 centimeter? (1 hundredth) How can we write this with decimals? (1 cm = 0.01 m) Write on the board:

1 m = 100 cm, so 1 cm = 0.01 m

ASK: How many hundredths of 1 m is 2 cm? (2 hundredths) On the board, draw the table in the margin and have students help you fill it in up to 10 cm.

Exercises: Convert to meters.

a) 9 cm b) 17 cm c) 324 cm d) 607 cm

Answers: a) 0.09 m, b) 0.17 m, c) 3.24 m, d) 6.07 m

ASK: How many hundredths of 1 meter is 40 centimeters? (40 hundredths)

cm m

1

2

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H-13

Write on the board:

40 cm = 0.40 m

ASK: What other way can we write 0.40? (0.4) Remind students that they can remove zeros or write zeros after the last non-zero digit to the right of the decimal point. So 40 cm = 0.40 m = 0.4 m.


a) 90 cm b) 50 cm c) 320 cm d) 2,670 cm

Answers: a) 0.90 m = 0.9 m, b) 0.5 m, c) 3.2 m, d) 26.7 m

Reverse the task. ASK: How many centimeters are in 0.8 m? To prompt students to see the answer, ask them to write 0.8 as hundredths. (0.80) Write on the board (and read as you are writing):

0.8 m = 0.80 m = 80 cm

Repeat with 0.7 m, and 1.3 m. (70 cm, 130 cm)

Exercises: Convert to centimeters.

a) 0.5 m b) 5.4 m c) 3.21 m d) 2.67 m

Answers: 50 cm, b) 540 cm, c) 321 cm, d) 267 cm

Extensiona) A CD case is 142 mm long. How many CDs will fit on a display shelf

that is 3.5 m long?

b) Ron stacks CDs on a display shelf. He stacks a book 21 cm wide between every 5 CDs and leaves open ends so there are 5 CDs at each end. How many books and CDs can he stack on the 3.5-m long shelf?

Answers: a) 24 CDs, b) 20 CDs and 3 books

(MP.4)



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H-14

MD5-4 Meters and Centimeters (Advanced)Pages 172–173

Review multiplying and dividing decimals and whole numbers by 100. Remind students that they shift the decimal point 2 places right to multiply and 2 places left to divide. Remind them that they can write zeros after the decimal point, and the number will not change. For example, 6.7 = 6.70. ASK: Why is that true? (the decimal part is 7/10 = 70/100) As well, remind students that even though we do not write decimal point in whole numbers, we can still add it without changing the number. For example, 12 = 12.00. Use the questions below as a test to make sure all students can perform the required multiplication and division.


a) 25 ÷ 100 = b) 3 ÷ 100 = c) 12.2 ÷ 100 = d) 387 ÷ 100 = e) 0.4 × 100 = f ) 5.3 × 100 = g) 1.89 × 100 = h) 0.08 × 100 =

Answers: a) 0.25, b) 0.03, c) 0.122, d) 3.87, e) 40, f ) 530, g) 189, h) 8

Using multiplication and division to convert between centimeters and meters. Remind students that they multiplied by 100 to convert a measurement in meters to centimeters. ASK: What operation would you use to convert a measurement in centimeters to meters? (divide by 100) PROMPT: Which operation is the opposite of multiplication? Convert the first few measurements in each Exercise as a class, then have students work individually.

Exercises

1. Convert to meters.

a) 26 cm b) 52 cm c) 158 cm d) 532 cm e) 6 cm f) 0.7 cm g) 80 cm h) 60 cm

STANDARDS 5.MD.A.1, 5.NBT.A.1, 5.NBT.A.2

VoCABUlARY centimeter (cm) conversions convert line measurement meter (m) millimeter (mm) mixed measurement

GoalsStudents will convert between centimeters and meters.


Understands the concept of linear measurement Understands that smaller units require more units Can multiply and divide decimals and whole numbers by 100 Knows that 1 m = 100 cm Can convert whole meters to centimeters Can compare decimals to hundredths Can convert between dollar and cent notations

MATERIAlS

meter sticks grid paper

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Answers: a) 0.26 m, b) 0.52 m, c) 1.58 m, d) 5.32 m, e) 0.06 m, f ) 0.007 m, g) 0.80 m = 0.8 m, h) 0.60 m = 0.6 m

2. Convert to centimeters.

a) 7.25 m b) 30.82 m c) 0.09 m d) 0.03 m e) 0.065 m f) 0.2 m g) 20.9 m h) 0.3 m

Answers: a) 725 cm, b) 3,082 cm, c) 9 cm, d) 3 cm, e) 6.5 cm, f ) 20 cm, g) 2,090 cm, h) 30 cm

Comparing money to meters and centimeters. SAY: 1 m = 100 cm. What other unit do we know that has 100 smaller units in it? (a dollar has 100 cents in it) Point out that the relationship between meters and centimeters is very similar to the relationship between dollars and cents. Conversions work very similarly: 625¢ = $6.25, and 625 cm = 6.25 m. Point out that when students convert money amounts in cents to dollars, they divide by 100. ASK: How are dollars and cents different from meters and centimeters? (there is a unit—millimeters—that is smaller than centimeters, but there is no money unit smaller than cents)

Using mixed measurements. Explain that we often think of prices such as $5.75 as 5 dollars and 75 cents. This measurement combines dollars and cents. Similarly, we can express a measurement using a combination of meters and centimeters (e.g., 5 m 75 cm). This is called a mixed measurement. To write the length of a desk in that form, we could measure the desk in meters, write how many whole meters fit along the desk (1 meter, replace the meter stick along the desk as you did when making the 1 m mark), then measure just the remainder in centimeters (show how to do this). Model measuring another object in meters and centimeters.

ACTIVITY

Students measure several objects in the classroom (e.g., their own height) in meters and centimeters. For example, to measure the height of a student, draw a vertical line on the wall and have the student stand against the wall, with their back to the line. Make a mark on the line showing the height of the student. Then use a meter stick to make a mark 1 m from the ground, and measure the rest of the height in centimeters. Have students record their measurements two ways, for example, as 1 m 14 cm and as 114 cm.

Writing measurements in different ways: meters and mixed measurements. ASK: How many dollars and cents are in $5.34? (5 dollars 34 cents) Remind students that meters and centimeters work like dollars and cents. How many meters and centimeters are in 5.34 m? (5 m 34 cm) Emphasize that centimeters are hundredths of meters, so the whole number is the number of meters and the hundredths are the number of whole centimeters.

(MP.7)


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H-16

Exercises: Convert the measurement in meters to a mixed measurement.

a) 3.14 m b) 5.08 m c) 12.89 m d) 24.01 m

Answers: 3 m 14 cm, b) 5 m 8 cm, c) 12 m 89 cm, d) 24 m 1 cm

Remind students that we can write zeros to the right of the decimal point, without changing the number. For example, 2.3 = 2.30. ASK: How many meters and centimeters are in 2.3 m? (2 m 30 cm) Why not 2 m 3 cm? (2.3 m = 2.30 m, not 2.03 m) Repeat with 4.6 m and 7.5 m. (4 m 60 cm, 7 m 50 cm)

Reverse the task. Do the first three examples as a class, then have students work individually. You can use the money analogy again: converting 5 m 78 cm to meters is the same as writing 5 dollars 78 cents in dollar notation.

Exercises: Convert the mixed measurement to a measurement in meters.

a) 5 m 78 cm b) 8 m 20 cm c) 2 m 9 cm d) 24 m 16 cm e) 47 m 72 cm f) 6 m 8 cm g) 78 m 40 cm Bonus: 1 m 23.4 cm

Answers: a) 5.78 m, b) 8.20 m = 8.2 m, c) 2.09 m, d) 24.16 m, e) 47.72 m, f ) 6.08 m, g) 78.40 m = 78.4 m, Bonus: 1.234 m

Writing measurements in different ways: centimeters and mixed measurements. Now have students think of how they will convert a mixed measurement into centimeters. Point out that part of the measurement is already in centimeters, so all they need to do is convert the part in meters into centimeters. Demonstrate how to use grid paper to convert the measurements as shown in the margin.

Exercises: Write the measurement in centimeters only.

a) 5 m 9 cm b) 2 m 8 cm c) 3 m 12 cm d) 123 m 78 cm

Answers: a) 509 cm, b) 208 cm, c) 312 cm, d) 12,378 cm

Ask students to compare these measurements for the same object: 1 m 24 cm and 124 cm. ASK: Where do the centimeters in the mixed measurement appear in the measurement in centimeters? (as the tens and ones digits) Where do the meters in the mixed measurement appear in the measurement in centimeters? (as the hundreds digit) What is the connection between the hundreds in the measurement in centimeters and the meters in the mixed measurement? (100 cm = 1 m) Explain that this means we have a way to convert between mixed measurements and measurements in centimeters. To convert from a measurement in centimeters to a mixed measurement, take the tens and the ones and leave them as centimeters, and write the hundreds digit as the numbers of meters. Example: 346 cm = 300 cm + 46 cm = 3 m 46 cm. Have students convert each centimeter measurement to a mixed measurement. Have students raise their fingers to show how many whole meters are in each mixed measurement.

(MP.3)

4 m = 400 cm, so 4 m 6 cm

= 4 0 0 cm

+ 6 cm

4 0 6 cm

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Exercises: Convert to a mixed measurement.

a) 362 cm b) 540 cm c) 723 cm d) 984 cm

Answers: a) 3 m 62 cm, b) 5 m 40 cm, c) 7 m 23 cm, d) 9 m 84 cm

Ask students to convert 1,352 cm to meters and centimeters. SAY: I know a student who thinks 1,352 cm = 3 m 52 cm because the hundreds digit of 1,352 is 3. Is the answer correct? (no) Have students explain the error. Point out that this question is similar to telling how many dollars and cents are in 1,352¢. We know that 1,352¢ = $13.52, or 13 dollars and 52 cents. Since centimeters and meters are related in the same way as cents and dollars, then 1,352 cm = 13 m 52 cm. Also, the number of meters in a measurement in centimeters is exactly the number of hundreds, because 1 m is 100 cm. Summarize: to convert a measurement in centimeters to a mixed measurement, separate the ones and tens from the rest and convert the remaining hundreds to meters: 1,352 cm = 1,300 cm + 52 cm = 13 m + 52 cm.

ASK: How many meters are in 305 cm? (3) How many centimeters are left over? (5 cm)

Exercises: Write the mixed measurements.

a) 567 cm b) 408 cm c) 304 cm d) 801 cm e) 6,789 cm f) 1,230 cm Bonus: 123,456 cm.

Answers: a) 5 m 67 cm, b) 4 m 8 cm, c) 3 m 4 cm, d) 8 m 1 cm, e) 67 m 89 cm, f ) 12 m 30 cm, Bonus: 1,234 m 56 cm

Extensions1. Express each measurement in meters and centimeters using the least

number of centimeters.

a) 1 m 327 cm b) 4 m 927 cm c) 2 m 1,022 cm d) 12 m 1,272 cm

Answers: a) 4 m 27 cm, b) 13 m 27 cm, c) 12 m 22 cm, d) 24 m 72 cm

2. a) 0.01 m refers to a unit that is one hundredth of a meter, and that fits in 1 meter a hundred times. Which unit is that?

b) $0.01 refers to a unit that fits into $1 a hundred times. Which unit is that?

c) How many times does 0.001 m fit into a meter? Which unit is that? Hint: Use a meter stick.

Answers: a) centimeter, b) cent, c) 1,000 times, millimeter


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H-18

MD5-5 Meters and KilometersPages 174–175

Introduce kilometers. Explain to students that a kilometer is a unit of measurement that is much larger than a meter or centimeter. Write the word “kilometer” on the board, circle the letters “k” and “m,” and write the abbreviation “km.” Explain that these are the two ways to write kilometer.

Explain that there are one thousand meters in one kilometer. Add a spool of fishing line or a pack of yarn representing a kilometer to the measurement display. (The packaging always labels the length of fishing line or yarn on each spool in meters; one large spool of line or two large spools of yarn should suffice.)

Explain that 1 kilometer is about the distance a person can walk in 15 minutes, the length of about 10 football fields, or the length of about 10 small city blocks. If there is a daily school event that takes about 15 minutes, such as a recess break, refer students to this event to help them to understand how long a 15-minute walk would take. If there is a familiar place close to school that is about 1 km away, mention that place to help students understand how far they would have to walk.

Exploring how many meters are in a kilometer. SAY: A soccer field is about 100 m long. About how many meters long are two soccer fields? Three soccer fields? How can you find out? (e.g., skip count by 100s, use a T-table) Use both methods to see how long 6 soccer fields are in meters. (600) Then ask students to look for a pattern in the T-table. How many soccer fields will be in a kilometer? (10)

Converting from whole kilometers to meters. Write on the board:

10 mm = 1 cm 100 cm = 1 m 1,000 m = 1 km

Review with students how to use multiplication to convert centimeters to millimeters and meters to centimeters. Summarize on the board:

(MP.7)

× 10 × 100 ?

mm cm m km

STANDARDS 5.MD.A.1, 5.NBT.A.1, 5.NBT.A.2

VoCABUlARY centimeter (cm) conversion convert kilometer (km) measurement meter (m) millimeter (mm)

GoalsStudents will convert between meters and kilometers.


Understands the concept of linear measurement Understands that smaller units require more units Can multiply and divide decimals and whole numbers by 1,000 Can convert between meters and centimeters Can compare decimals to thousandths

MATERIAlS

spool of fishing line or pack of yarn to represent 1 km

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SAY: You multiply by 10 to get from centimeters to millimeters. Why? (because there are 10 mm in 1 cm) You multiply by 100 to get from meters to centimeters. Why? (because there are 100 cm in 1 m) What operation would you use to get from kilometers to meters? (multiplication) What number will you multiply by? (1,000) Why? (there are 1,000 m in 1 km) Replace the question mark above with “× 1,000.”

Exercises: Convert the measurements in kilometers to meters.

a) 3 km b) 8 km c) 15 km d) 39 km

Bonus e 100 km f ) 25 km = m = cm = mm

Answers: a) 3,000 m, b) 8,000 m, c) 15,000 m, d) 39,000 m, Bonus: e) 100,000 m, f ) 25,000 m = 2,500,000 cm = 25,000,000 mm

Comparing measurements. Write on the board:

5 km 3,000 m

ASK: Which measurement is greater? Remind students that to compare measurements correctly you need to convert them to the same unit, and that it is usually more convenient to use the smaller unit. Ask students to convert 5 km to meters, then decide which measurement is longer. (5 km)

Exercises: Point your thumb toward the larger measurement.

a) 304 m or 2 km b) 4,090 m or 2 km c) 4,605 m or 90 km d) 653 km or 6,500 m e) 89 km or 330,567 m Bonus: f ) 120,345 m or 765 km g) 123,765 cm, 3,432 m, or 2 km

Answers: a) 2 km, b) 4,090 m, c) 90 km, e) 653 km, e) 330,567 m, Bonus: f ) 765 km, g) 3,432 m

Converting between meters to kilometers using place value. Remind students how to read numbers such as 0.467, 2.304, and 0.008. Also, remind them that numbers such as 0.05 can be written as 0.050. Then ASK: If 1 km = 1,000 m, how many times does 1 meter fit into 1 kilometer? (1,000) What fraction of 1 kilometer is 1 meter? (1 thousandth) How do we write this with decimals? (1 m = 0.001 km) Write on the board:

1 km = 1,000 m, so 1 m = 0.001 km

ASK: How many thousandths of 1 km is 2 m? (2 thousandths) Start a table as in the margin and have students help you fill it in up to 10 m.

Exercises: Convert to kilometers.

a) 9 m b) 17 m c) 324 m d) 607 m e) 500 m f) 250 m

Answers: a) 0.009 km, b) 0.017 km, c) 0.324 km, d) 0.607 km, e) 0.500 km = 0.5 km, f ) 0.250 km = 0.25 km

m km

1

2

...


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H-20

Remind students that they can remove the zeros to the right of the decimal point, so 0.500 km can be written as 0.5 km. If students still have zeros in their answers, have them rewrite the answers without the zeros at the end.

Reverse the task. ASK: How many meters are in 0.748 km? (748 m) How many thousandths are in 0.06? (60) How do you know? (0.06 = 0.060) How many meters are in 0.06 km? (60 m)


a) 0.512 km b) 5.474 km c) 3.281 km d) 24.617 km e) 8.002 km f) 0.003 km g) 0.7 km h) 26.28 km

Answers: a) 512 m, b) 5,474 m, c) 3,281 m, d) 24,617 m, e) 8,002 m, f ) 3 m, g) 700 m, h) 26,280 m

Review multiplying and dividing decimals and whole numbers by 1,000. Use the same process you used to review multiplying decimals by 10 and 100. Use the questions below as a test to make sure all students can perform the required multiplication and division.


a) 258 ÷ 1,000 = b) 35 ÷ 1,000 = c) 2 ÷ 1,000 = d) 4,387 ÷ 1,000 = e) 0.004 × 1,000 = f ) 5.356 × 1,000 = g) 1.89 × 1,000 = h) 0.08 × 1,000 = i ) 0.3 × 1,000 =

Answers: a) 0.258, b) 0.035, c) 0.002, d) 4.387, e) 4, f ) 5,356, g) 1,890, h) 80, i ) 300

Using multiplication and division to convert between meters and kilometers. SAY: You multiplied a measurement in kilometers by 1,000 to get the measurement in meters. What operation would you use to go in the opposite direction, to convert a measurement in meters to kilometers? (divide by 1,000) PROMPT: Which operation is the opposite of multiplication? Convert the first few measurements in each Exercise as a class, then have students work individually.

Exercises

1. Convert to kilometers.

a) 263 m b) 502 m c) 1568 m d) 1500 m e) 6 m f) 79 m g) 80 m h) 600 m

2. Convert to meters.

a) 7.259 km b) 30.825 km c) 0.009 km d) 0.03 km e) 0.065 km f) 0.2 km g) 20.89 km h) 1.3 km

Answers: 1. a) 0.263 km, b) 0.502 km, c) 1.568 km, d) 1.5 km, e) 0.006 km, f ) 0.079 km, g) 0.08 km, h) 0.6 km 2. a) 7,259 m, b) 30,825 m, c) 9 m, d) 30 m, e) 65 m, f ) 200 m, g) 20,890 m, h) 1,300 m

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Extensions1. a) An adult female Komodo dragon is about 2 m long. How many

female Komodo dragons lined up end to end will make 1 km?

b) An adult male Komodo dragon is about 3 m long. About how many adult male Komodo dragons lined up end to end will make 1 km?

c) A newly hatched Komodo dragon is 49 cm long. About how many Komodo dragons hatchlings lined up end to end will make 1 km?

d) There are 18 eggs in an average Komodo dragon nest. What is the total length of the hatchlings from a single nest?

e) Scientists found 75 Komodo dragon nests. Is the total length of the hatchlings in these nests more than 1 km?

Answers: a) 500, b) 333, c) 2,000, d) 882 cm, e) The total length of the hatchlings from one nest is slightly less than 9 m. The total length of the hatchlings from 75 nests is slightly less than 675 m or 0.675 km, so the answer is no.

2. a) The table lists the highest mountain on each continent. Write the height of each mountain in meters. Rank the mountains from highest (1) to lowest (7).

Mountain location Height (km and m)

Height (m) Ranking

AconcaguaSouth America

(Argentina)6 km 962 m

Denali (McKinley)

North America (United States)

6 km 195 m

ElbrusEurope (Russia)

5 km 633 m

EverestAsia

(Nepal/China)8 km 848 m

KilimanjaroAfrica

(Tanzania)5 km 963 m

Kosciuszko Australia 2 km 228 m

Vinson Massif Antarctica 4 km 897 m

b) Calculate the difference in height between the highest and the lowest mountains.

Answers: a) 1. Everest 8,848 m, 2. Aconcagua 6,962 m, 3. Denali 6,195 m, 4. Kilimanjaro 5,963 m, 5. Elbrus 5,633 m, 6. Vinson Massif 4,897 m, 7. Kosciuszko 2,228 m; b) 8,848 – 2,228 = 6,620 m


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H-22

MD5-6 PerimeterPages 176–177

Introduce perimeter. Write the word perimeter and explain to students that perimeter is the measurement around the outside of a shape. Illustrate the perimeters of some classroom items by running your hand along the outside of a desk, the blackboard, or a blackboard eraser. Write the phrase “the measurement around the outside of a shape.”

Draw on the board the figure in the margin. Explain that each side of each square represents 1 cm, and that perimeter is calculated by totaling the outside edges. Demonstrate a method for calculating the perimeter by marking or crossing out each edge as you count it. Demonstrate this several times.

Exercises: Each side of a square is 1 cm long. Find the perimeter of the figures.

a) b) c)

Answers: a) 10 cm, b) 10 cm, c) 16 cm

Finding perimeter by adding side lengths. Use one of the shapes from the previous Exercise to demonstrate the method for calculating perimeter by counting the entire length of each side and creating an addition statement. Write the length of each side on the picture.

Exercises: Each square is 1 cm long. Write the side length on each side. Then write the addition statement and find the perimeter.

a) b) c)

Answers: a) 1 cm + 3 cm + 3 cm + 1 cm + 2 cm + 2 cm = 12 cm, b) 12 cm, c) 14 cm

1 cm

STANDARDS 5.MD.A.1, 5.NBT.B.7

VoCABUlARY centimeter (cm) conversion convert measurement meter (m) perimeter

GoalsStudents will solve problems related to perimeter of shapes.


Can multiply and divide decimals and whole numbers by 100 Can convert between meters and centimeters Can compare decimals to hundredths Can add decimals

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NoTE: Students will often miss one of the sides that is 1 unit long when two such sides meet making a reflex angle, as the two highlighted sides show in part b). Have students count the sides of the shape and make sure that the number of sides they add to find the perimeter matches the number of sides in the shape. You can also spot some incorrect calculations by looking for answers that are odd numbers. The perimeter of a shape made of squares is always an even number.

Review conversion between meters and centimeters using multiplication. Remind students that they can multiply by 100 to convert measurements in meters to centimeters. Use the Exercises below to make sure all students can perform the conversions.

Exercises: Convert to centimeters.

a) 2.35 m b) 4.5 m c) 0.08 m

Answers: a) 235 cm, b) 450 cm, c) 8 cm

Finding perimeter of shapes with sides given in meters and centimeters. On the board, draw a rectangle and mark two of the sides as 0.75 m and 44 cm. Ask students to determine the length of the opposite sides. Then explain that you want to find the perimeter of the rectangle. ASK: Does it make sense to add 0.75 + 44 + 0.75 + 44 to find the perimeter? (no) Why not? (the lengths of the sides are given in different units) Remind students that when comparing lengths in different units, they need to convert the lengths to the same unit before comparing. When performing mathematical operations on measurements, the situation is similar: they all need to be converted to the same unit. It is usually easier to convert all measurements to the smaller unit.

Have students convert 0.75 m to centimeters, then have them find the perimeter. (75 cm + 44 cm + 75 cm + 44 cm = 238 cm)

Exercises: Convert the measurements to the smaller unit, then find the perimeter.

a) b)

2.5 m

85 cm

1.2 m

72 cm

c) d)

2 m

2.5 m150 cm

1.2 m

1.3 m

50 cm

Answers: a) 670 cm, b) 384 cm, c) 600 cm, d) 300 cm


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H-24

Review converting centimeters to meters by dividing by 100. Remind students that division by 100 looks like shifting the decimal point 2 places to the left. Review adding decimals. Then have students convert the measurements in the Exercises above to meters and find the perimeter. Ask students to convert the answers to centimeters to make sure they got the same answers for the same shapes.

Answers: a) 6.7 m, b) 3.84 m, c) 6 m, d) 3 m

Extensions1. Distribute one piece of string, about 30 cm long, and a geoboard to

each student. Have them tie the ends of the string together to form a loop, then create a variety of shapes on the geoboard with the string. Explain that the shapes will all have the same perimeter because the length of the string, which forms the outside edges, is fixed. How many different shapes have the same perimeter?

2. Distribute Pentamino pieces (a set of twelve figures each made of five squares; see BlM Pentomino, p. H-37) to students and have them calculate the perimeter of each figure and fill in the table below.

Figure Perimeter Number of Inside Edges

a) Which shape is different from all other shapes?

b) Double the number of inside edges and add to the perimeter for all shapes. What do you notice? Try to explain your findings.

Answers: a) All shapes except shape F (shown in the margin) have perimeter 12 and 4 inside edges. Shape F has perimeter 10 and 5 inside edges; b) The answer is always 20. Explanation: Each inside edge belongs to 2 squares. Each edge in the perimeter belongs to 1 square. The answer is the total number of edges on all squares. There are 5 squares, so the total number of edges is 5 × 4 = 20.

3. Have students investigate perimeters of shapes made from squares using BlM Patterns in Perimeters (p. H-38).

(MP.3, MP.7)

(MP.3)

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MD5-7 Changing Units of lengthPages 178–179

Review conversion from centimeters to millimeters and from meters to centimeters. Remind students that there are 10 mm in 1 cm, so if an object is 4 cm long, it is also 40 mm long. Remind students that they can multiply by 10 to convert from centimeters to millimeters. Multiplying by 10 looks like shifting the decimal point one place to the right.

Exercises: Convert the measurements in centimeters to millimeters.

a) 6 cm b) 12 cm c) 8.9 cm d) 0.6 cm Bonus: e) 2,345 cm f) 123,456.789 cm

Answers: a) 60 mm, b) 120 mm, c) 89 mm, d) 6 mm, Bonus: e) 23,450 mm, f ) 1,234,567.89 mm

Remind students that there are 100 cm in 1 m. So if an object is 3 m long, it is also 300 cm long. Remind them that they can multiply by 100 to convert measurements in meters to centimeters. This multiplication looks like shifting the decimal point two places to the right.

Introduce decimeters. Point out that 100 = 10 × 10, so it would be nice to have a relationship like this:

× 10 × 10 × 10

mm cm m

10 mm = 1 cm 10 cm = 1 10 = 1 m

Explain that there is a unit of measurement that goes in the blanks and it is called a decimeter. Write the word “decimeter” on the board (circle the letters "d" and "m") and the abbreviation “dm,” and explain that these are two ways to write decimeter. Have students fill in the blanks above with the new term.

STANDARDS 5.MD.A.1, 5.NBT.A.1, 5.NBT.A.2, 5.NBT.B.7

VoCABUlARY centimeter (cm) conversion convert decimeter (dm) kilometer (km) measurement meter (m) millimeter (mm)

GoalsStudents will convert between metric units of length.


Can multiply and divide decimals and whole numbers by 10, 100, and 1,000 Understands that smaller units require more units Can convert between metric units of length measurement Can add and subtract decimals

MATERIAlS

BlM Measurement Cards (pp. H-39–40)


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H-26

Explain that decimeters are similar to dimes in money. There are 100 cm in a meter, and 100 cents in a dollar. A decimeter is between centimeters and meters, the same as a dime is between cents and dollars. There are 10 cm in a decimeter, and 10¢ in a dime. Similarly, there are 10 dm in a meter, and 10 dimes in a dollar. You might point out that just as people usually do not say that something costs 7 dimes, even though it would be a clear indication of cost, people also do not usually say that something is 7 dm long, even though it is a clear indication of length.

On the board, work through Questions 1 and 2 on p. 178 in the AP Book and have students work simultaneously in their books.

Exploring the relationship between the number of units in a measurement and the size of the unit. Remind students that when they measure an object in centimeters and measure the same object in millimeters, the number of centimeters in the measurement is always smaller than the number of millimeters. Why does this happen? (millimeters are smaller, so you need more of them) SAY: John and Kyla each have $2. Kyla has $2 in dimes, and John has $2 in pennies. Who has more coins? (John) Why? (because pennies have a smaller value) Point out that to get the same amount of money, you need more coins of the smaller value (or unit), and fewer coins of the larger value (or unit). The larger the unit, the fewer number of units you need to get the same amount or to cover the same distance.

Converting between centimeters and millimeters. Remind students that they multiply by 10 to convert a measurement in centimeters to a measurement in millimeters. ASK: Why do you multiply and not divide to convert centimeters to millimeters? How is that connected to the size of the new unit? (the new unit is smaller so you need more units, so you multiply by 10, not divide by 10) Write on the board:

Change 4.2 cm to mm:

The new units are times ,

so I need units.

I by .

ASK: What are the new units? (mm) What is larger—centimeters or millimeters? (centimeters) Are the new units smaller or larger than the old units? (smaller) How many times smaller? (10 times) Fill in the first two blanks (10, smaller) Do I need more units or fewer units? (more) Fill in the next blank (more). If I need more units, do I need to multiply or to divide? (multiply) By how much? (10) Fill in the last two blanks. (multiply, 10)

Then perform the calculation: 4.2 cm × 10 = 42 mm. PROMPTS: Which way should we move the decimal point? (right) If I move the decimal point left, how can I tell this is wrong? (0.42 is smaller than 4.2, but I am looking for a larger number) Is 42 mm correct? (yes) Finally, write the answer: 4.2 cm = 42 mm.

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Repeat with 350 mm to cm, 26 cm to mm, 8.42 cm to mm, 86.9 mm to cm. (35 cm, 260 mm, 84.2 mm, 8.69 cm) Provide blanks as above and have volunteers fill them in. Provide help as needed. Then have students individually practice solving more problems the same way.

Exercises: Convert the measurement.

a) 1,340 mm to cm b) 2.6 cm to mm c) 0.48 cm to mm

Answers: a) 134 cm, b) 26 mm, c) 4.8 mm

Converting between different metric units. On the board, draw the first diagram in the margin. ASK: As you go down the steps, do the units become larger or smaller? (smaller) How many times smaller with each step? (10 times) If you go 2 steps down, how many times smaller is the new unit? (10 × 10 = 100 times) What if you go 3 steps down? (10 × 10 × 10 = 1,000 times) So when you go down the stairs, do you need more units or fewer units? (more units) When you go down the stairs, will you multiply or divide? (multiply) Add arrows to the diagram going down the stairs and label them.

Repeat the questions for going up the stairs, and finish by adding the arrows. The finished diagram will look like the second picture in the margin. Solve several conversion questions together as a class. Do the first question yourself, but have volunteers help you to solve others (ASK: What do I do now?) Then have students practice individually. For more practice, use the Activities below.

Exercises: Say how many times larger or smaller the new unit is. Then convert.

a) 1.6 mm to cm b) 72.3 cm to mm c) 0.409 m to mm d) 74 mm to m e) 3.8 dm to cm f ) 22 cm to dm g) 0.01 dm to mm h) 1,290 mm to dm i) 0.24 m to mm j) 345 mm to cm k) 0.35 mm to cm l ) 5.4 cm to m

Answers: a) 10 times larger, 0.16 cm; b) 10 times smaller, 723 mm; c) 1,000 times smaller, 409 mm; d) 1,000 times larger, 0.074 m; e) 10 times smaller, 38 cm; f ) 10 times larger, 2.2 dm; g) 100 times smaller, 1 mm; h) 100 times larger, 12.9 dm; i ) 1,000 times smaller, 240 mm; j ) 10 times larger, 34.5 cm; k) 10 times larger, 3.5 cm; l ) 100 times larger, 0.054 m

ACTIVITIES 1–2

Give students BlM Measurement Cards. Have them cut out the cards and shuffle the deck well before each of the activities below. The activities increase in difficulty; weaker students might spend their time doing the first activity, whereas stronger students should spend more time doing the third activity.

1. Sort the cards into groups of 4, so all cards in each group have the same measurement expressed in different units.

m

dm

cm

mm

m × 10

dm × 10

÷ 10 cm × 10

÷10 mm

÷10



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2. Place the cards face down in a rectangle. Player 1 turns over two cards. If the cards show the same measurement, they place the cards in the common pile. Player 2 checks that the decision about the cards is correct. If the cards do not show the same measurement, Player 1 turns the cards back over and Player 2 takes a turn.

3. Advanced: Students work in partners. Player 1 picks a random pair of cards and adds the lengths on the cards. Player 2 subtracts the lengths. Students will need to convert the measurements to the same unit most of the time. Partners check each other’s answers, then switch roles.

Extensions1. Write a measurement in whole decimetres that is between the given two

measurements. Explain how you found the answer.

a) 320 mm and 437 mm b) 507 mm and 622 mm c) 1 1/2 m and 1 3/4 m

Answers: a) 4 dm, b) 6 dm, c) 16 dm

2. John has a strip of paper 1 dm long and 2 cm wide. He folds the strip so that it has a crease down its center (see sample in margin). How can John use this strip as a benchmark to make each of these measurements?

a) 5 cm b) 3 cm c) 1 cm

Answers: a) half the strip from the crease to the end is a benchmark for 5 cm; b) draw a 5 cm line segment, then mark off 2 cm from one end, using the width of the strip as a benchmark; the 5 cm line segment is divided into 2 cm and 3 cm line segments; c) draw a 3 cm line segment as in part b) and mark off 2 cm from its end; the line segment is divided into 2 cm and 1 cm

(MP.3, MP.7)

(MP.1)

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MD5-8 Problems with lengthPages 180–181

Finding missing sides in a rectangle given the perimeter and one side. Draw a rectangle on the board. Write a length on one of the longer sides. Ask students what the length of the opposite side should be. Add the length of one of the shorter sides and ask students what the length of the last side should be. ASK: What is the perimeter of the rectangle? Ask a volunteer to write the addition equation for the perimeter.

Draw another rectangle and say that its perimeter is 26 m. Say that the length of one side is 4 m. Ask students to find the lengths of the other sides. PROMPTS: What is the length of the opposite side? (4 m) What is the total length of the two remaining sides? (26 m – 8 m = 18 m) How long is one of these sides? (9 m)

To prompt students to see a different solution, draw a big dot on one corner of the rectangle and ask students to imagine walking around the rectangle, starting from the dot. Trace your finger along the entire perimeter to illustrate walking, and stop at the opposite corner. ASK: How far did you go? (13 m) How do you know? (this is halfway around the rectangle, so the distance is half the perimeter) If these two sides make 13 m, and one side is 4 m, how long is the other side? (9 m)

Exercises: Find the missing side.

a) perimeter 24 cm b) perimeter 22 m

5 cm

5 m

Answers: a) 7 cm, b) 6 m

(MP.2, MP.3)


VoCABUlARY centimeter (cm) conversion convert kilometer (km) measurement meter (m) millimeter (mm) perimeter

GoalsStudents will solve problems involving conversions between metric units of length.


Can multiply and divide decimals and whole numbers by 10, 100, and 1,000 Can convert between metric units of length measurement Can add and subtract decimals

MATERIAlS

grid paper BlM Measurement Cards (pp. H-39–40)


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Review adding and subtracting decimals. Remind students to align the decimal points when performing operations with decimals.

Exercises: Add or subtract.

a) 2.6 + 3.5 b) 1.3 + 8.7 c) 0.85 + 2.567 d) 3.5 – 2.1 e) 3.5 – 1.9 f ) 2.45 – 1.2 g) 8.02 – 7.88 h) 2.2 – 1.87

Answers: a) 6.1, b) 10, c) 3.417, d) 1.4, e) 1.6, f ) 1.25, g) 0.14, h) 0.33

Then have students find the missing sides of rectangles with sides given in decimals.

Exercises: Find the missing side.

a) perimeter 21 cm b) perimeter 12.4 m

3.5 cm

2.2 m

c) perimeter 6.7 cm d) perimeter 2.3 km

1.35 cm

0.15 km

Answers: a) 7 cm, b) 4 m, c) 2 cm, d) 1 km

Now tell students that you would like to check the answers to the previous Exercises by solving the problems using a different method, so that the calculation does not include decimals. Ask them to convert the measurements into smaller units. Then have them find the perimeter of the rectangles in the new units and check that their answers are the same using both methods.

Using organized search to find the dimensions of a rectangle with the given perimeter. Ask students to find all the rectangles with a perimeter of 14 units and sides with lengths in whole units (e.g., 3 units, not 3.5). If students find only one (or zero) rectangles, show them a systematic method of finding the answer. On grid paper, draw a line segment 1 unit long, and ask students to finish the rectangle so that the perimeter is 14 units. Repeat with a side 2 units long. ASK: What side length would you try next? (3 units) When students draw a rectangle starting with 4 units, ask them if this rectangle is different from the previous one. It is not, so explain that we can stop, because we have stopped producing new rectangles. Students can try to produce a rectangle starting with sides 5, 6, and 7, to see that they are not producing anything new. (Students will see that a rectangle with perimeter 14 and sides 7 is impossible. Prompt them to compare rectangles with sides 5 or 6 with rectangles drawn earlier.

(MP.1)

(MP.1)

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They should see that they have already drawn these rectangles, but in a different orientation.)

Use the problems on p. 181 of the AP Book to review the material learned in this unit.

ACTIVITY

Students work in pairs. They will need cards from BlM Measurement Cards. Player 1 picks two cards at random and decides which of the measurements is longer, and if it is a lot longer than the other. The player draws a rectangle that is roughly to scale and marks the lengths of the sides as given by the cards. (For example, if one side is 9 m and the other is 1.23 cm, the player draws a long thin rectangle.) Player 2 finds the perimeter of the rectangle. Players then switch roles.

ExtensionA square has perimeter 0.8 m. What is the length of each side? How do you know?

Solution: 0.8 m = 80 cm; the sides of a square are equal, so each side is 80 cm ÷ 4 = 20 cm


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MD5-9 MassPages 82–83

Introduce mass. Ask students how people can check how heavy an object is. (weigh it, put it on a scale) Explain that in science we usually use the word mass to describe how heavy objects are, and we say that the heavier object has the greater mass.

Name some pairs of objects that have similar mass: a pen and a pencil, a full water bottle and a book, a truck and a bus. Then name several everyday objects and have students name an object with a similar mass: What has a mass similar to an eraser? (e.g., a glue stick)

Introduce grams. ASK: In what units is weight measured? Students are likely to be familiar with pounds, but may remember other units (ounces, kilograms, grams) from previous grades. Explain that in the metric system, the mass of small objects is measured in grams. Write “gram” on the board, circle the letter “g,” and write the abbreviation “g.” Explain that these are two ways to write grams. Explain that a large paperclip and a large chocolate chip weigh 1 gram each. A nickel weighs 5 grams.

Point out that since 1 gram is such a small weight, there are very few everyday items that weigh less than that. For example, grains of rice, pills, and most insects, such as ants, weigh less than 1 gram. Only very large insects weigh more than 1 gram.

ACTIVITY

Give students a collection of everyday items that have their weight listed in metric units (canned food, plastic spice jars, and so on).


VoCABUlARY conversion convert gram (g) kilogram (kg) mass meter (m) mixed measurement

GoalsStudents will convert metric units of mass.


Can multiply and divide decimals and whole numbers by 1,000 Can compare and sort objects by weight (heavier/lighter) Understands the concept of measuring weight Can add and subtract decimals Can multiply 2-digit numbers by 2-digit numbers Can multiply and divide numbers up to 4 digits by 1-digit numbers Can solve word problems using the four operations Can convert between metric units of length measurement

MATERIAlS

a collection of items such as dry food with weights marked in grams and kilograms grid paper

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Have them sort the objects into three groups by mass: less than 100 g, between 100 g and 500 g, more than 500 g. Then show more objects, without telling students the mass, and have them signal which group these objects should be sorted into. Students can show hands at desk level for the group with mass less than 100 g, hands at shoulder level for the medium-sized group, and hands up for the objects heavier than 500 g. This will help students to develop a feel for the size of grams.

Introduce kilograms. Point out a few larger objects in the “heavy” group. ASK: Would 1 gram be a convenient unit to measure the weight of a laptop, a pile of books, or a human? (no) Why not? (the number of grams will be very large) Explain that the unit used to measure the mass of objects of that size is the kilogram. Write “kilogram” on the board, circle the letters “k” and “g,” and write the abbreviation “kg.” Explain that these are two ways to write kilogram.

Write on the board:

1 kilometer = meters 1 km = m

ASK: What number goes in the blanks? (1,000) Invite a volunteer to circle the common parts in the words “kilometer” and “meter.” Ask students to guess what the part “kilo” means. Explain that kilo means 1,000 in Greek. When you see kilo in a measurement unit, you know right away that there are 1,000 smaller units in the large unit. Write on the board:

1 kilogram = grams 1 kg = g

ASK: What number goes in the blanks? (1,000)

Have students look through the objects in the group “more than 500 g” to find any objects that weigh more than 1,000 g. If there are no such objects, ask why that could be. Show an object such as a large package of flour. Point out that it is more convenient to write the weight of such a heavy object in kilograms, using decimals, than to write it in grams.

Converting kilograms to grams. Remind students that they multiplied by 1,000 to convert the measurement in kilometers to meters. ASK: What number would we multiply the number of kilograms by to get the number of grams? (1,000) Why? (there are 1,000 g in 1 kg) As well, remind students that they can shift the decimal point 3 places to the right to multiply decimals by 1,000.

Exercises: Convert to grams.

a) 7 kg b) 12 kg c) 3.545 kg d) 2.27 kg e) 34.5 kg f ) 0.8 kg

Answers: a) 7,000 g, b) 12,000 g, c) 3,545 g, d) 2,270 g, e) 34,500 g, f ) 800 g


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Comparing measurements in different units. Remind students that when they compare two measurements in different units, they need to convert one of the measurements so the units become the same. For example, 7 km is larger than 300 m, even though 7 is smaller than 300, because 7 km = 7,000 m, and 7,000 m > 300 m.

Exercises: Convert the measurement in kilograms to grams, then compare the measurements. To signal the answers, students can show a sideways “V” with their fingers open in the direction of the larger measurement to resemble a “<” or a “>” sign.

a) 354 g or 3 kg b) 78 kg or 7,800 g c) 24,897 g or 20 kg d) 37,890 g or 3.789 kg e) 2.78 kg or 278 gBonus: f ) 456,789 kg or 30,000 g g) 1,000,000 g or 10,000 kg

Answers: a) 354 g < 3 kg, b) 78 kg > 7,800 g, c) 24,897 g > 20 kg, d) 37,890 g > 3.789 kg, e) 2.78 kg > 278 g, Bonus: f) 456,789 kg > 30,000 g, g) 1,000,000 g < 10,000 kg

Converting grams to kilograms. ASK: If 1 kg = 1,000 g, what fraction of a kilogram is 1 gram? (one thousandth) Write on the board:

1 g = 0.001 kg

Make a table as shown and ask students to fill it in up to 10 g.

g 1 2 3 4 …

kg 0.001

Remind students that they can remove zeros to the right of the non-zero digits after the decimal point, so 0.010 kg can be written as 0.01 kg.

ASK: If we multiply by 1,000 to convert from kilograms to grams, what operation would you use to convert from grams to kilograms? (divide by 1,000)

Exercises: Convert to kilograms.

a) 7,846 g b) 893 g c) 45 g d) 2 g e) 34,905 g f ) 800 g g) 70 g h) 34,300 g

Answers: a) 7.846 kg, b) 0.893 kg, c) 0.045 kg, d) 0.002 kg, e) 34.905 kg, f ) 0.8 kg, g) 0.07 kg, h) 34.3 kg

Mixed measurements. Remind students that they can write length as a mixed measurement, for example, a shelf is 5 m 23 cm long. Explain that mass can also be written as a mixed measurement, such as 3 kg 456 g. Point out that since grams are thousandths of kilograms, the conversion from kilograms to mixed measurements is easy: all you need to do is to write the measurement in kilograms up to thousandths. The whole part then becomes the number of kilograms, and the number of thousandths after the decimal point is the number of grams. Show an example:

2.35 kg = 2.350 kg = 2 kg 350 g

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Exercises: Write as a mixed measurement.

a) 3.789 kg b) 78.089 kg c) 5.78 kg d) 2.4 kg

Answers: a) 3 kg 789 g, b) 78 kg 89 g, c) 5 kg 780 g, d) 2 kg 400 g

Remind students that to convert a mixed measurement such as 5 m 23 cm to centimeters, they need to convert meters to centimeters and add the remaining centimeters:

5 m = 500 cm, so 5 m 23 cm = 500 cm + 23 cm = 523 cm

ASK: How can you convert 3 kg 456 g to grams? Take up different ideas, then lead students to the idea of converting the part in kilograms and adding the remaining grams. Students should perform the addition on a grid, as shown in the margin. Do the first two Exercises below as a class, then have students individually convert the measurements to grams.

Exercises: Convert to grams.

a) 2 kg 4 g b) 13 kg 67 g c) 5 kg 159 g d) 2 kg 391 g e) 52 kg 604 g f ) 643 kg 600 g g) 8 kg 15 g h) 30 kg 1 g

Bonus: i ) 100 kg 100 g j ) 1,000 kg 10 g

Answers: a) 2,004 g, b) 13,067 g, c) 5,159 g, d) 2,391 g, e) 52,604 g, f ) 643,600 g, g) 8,015 g, h) 30,001 g, Bonus: i ) 100,100 g, j ) 1,000,010 g

Now reverse the task. Write on the board:

23,765 g = kg g

ASK: How many whole kilograms are in 23,765 g? (23) Have a volunteer fill in the first blank. SAY: This is the number of thousands in 23,765, because there are 1,000 g in 1 kg. Have a volunteer fill in the second blank. (765)

Exercises: Convert to mixed measurements.

a) 7,893 g b) 82,081 g c) 35,950 g d) 100,009 g

Bonus: 7,123,456 g

Answers: a) 7 kg 893 g, b) 82 kg 81 g, c) 35 kg 950 g, d) 100 kg 9 g, Bonus: 7,123 kg 456 g

Solving word problems with masses. Work through the following problems as a class. Invite and encourage as many students as possible to contribute answers, explanations, and suggestions.

Exercises

a) A newborn Siberian tiger cub weighs about 1 kg. It gains 700 grams per week. How much weight does it gain in 4 weeks? How much does the tiger weigh after 4 weeks? Express your answer in grams, in kilograms, and as a mixed measurement.

b) A raccoon weighs 7.45 kg. A beaver is 3,800 g heavier than the raccoon. How much does the beaver weigh?

3 kg = 3,000 g, so 3 kg 456 g

= 3, 0 0 0 g

+ 4 5 6 g

3, 4 5 6 g

(MP.4)


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c) A paper coffee cup with a lid weighs about 8 grams. Express your answer in grams, in kilograms, and as a mixed measurement.

i) Ken buys one cup of coffee every day. What mass of garbage does he generate in a week from paper cups? In a year?

ii) Blanca buys coffee 3 times a day. What mass of garbage from paper cups does she generate in a year?

d) A small bag of rice weighs 1,816 g, and a big bag of rice weighs 3 kg. Three big bags cost as much as 5 small bags. Which combination is a better buy? Hint: Find the weight of 3 big bags and of 5 small bags first. Which gives more rice for the same price?

Answers: a) The tiger cub gains 2,800 g in 4 weeks, so its weight is 3,800 g = 3.8 kg = 3 kg 800 g; b) 11,250 g = 11.25 kg = 11 kg 250 g; c) i) Ken generates 56 g of garbage from paper cups in a week, 2,912 g = 2.912 kg = 2 kg 912 g in a year; ii) Blanca generates 8,736 g = 8.736 kg = 8 kg 736 g of garbage from paper cups in a year; d) 3 big bags weigh 9 kg, and 5 small bags weigh 9,080 g = 9.08 kg, so 5 small bags are a better buy

Extensions1. A small box of rice weighs 1,362 g, and costs $5.60. A large bag of rice

weighs 2.27 kg and costs $9. How much do 5 small boxes weigh and cost? How much do 3 large bags weigh and cost? Which combination is a better buy?

Answer: 5 small boxes weigh 6.81 kg and cost $28. 3 large bags weigh 6.81 kg and cost $27. Three large bags are a better buy.

2. Milligrams. We measure very small weights in milligrams (mg). 1 g = 1,000 mg

a) Convert to milligrams.

3 g = 7 g = 12 g =

b) Which is heavier: 3 g or 456 mg? 5 g or 5,890 mg?

c) How many milligrams are in 1 kilogram?

Answers: a) 3,000 g, 7,000 g, 12,000 g; b) 3 g > 456 mg, 5 g < 5,890 mg; c) 1,000,000 mg

3. Metric Tonnes. We measure very large weights in metric tonnes (t). 1 t = 1,000 kg

a) Convert to kilograms.

4 t = 15 t = 100 t =

b) A male elephant weighs 5 metric tonnes. A female elephant weighs 3,897 kg. How much heavier is the male than the female?

Answers: a) 4,000 kg, 15,000 kg, 100,000 kg; b) 1,103 kg

(MP.4)

Unit 7 Measurement and Data: Conversion for CC-Edition... · Unit 7 Measurement and Data: Conversion ... a ruler, and explain to students that a millimeter is a unit of measurement

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