Unit 7 Measurement and Data: Measuring Length in Metric Units

Transcription

1 Unit 7 Measurement and Data: Measuring Length in Metric Units Introduction In this unit, students will learn about measuring length, width, height, and distance. Materials Students will need a large number of 1 cm and 2 cm connecting cubes (also called snap cubes or linking cubes) and paper clips of two sizes, 3 cm and 5 cm. If you need to buy connecting cubes, be sure to get cubes that can link in different directions. In many lessons, you will need to draw images of specific length on the board or demonstrate working with a centimeter ruler. If you work in a setting that does not allow all students to clearly see regular-sized manipulatives (1 cm connecting cubes, centimeter rulers, or concrete rulers), use enlarged versions produced by photocopying. Alternatively, use an overhead or virtual manipulatives on an interactive whiteboard. Estimating Students will often be asked to estimate and then measure length. If you use the questions in the AP Book for assessment, make sure students estimate before they measure. When introducing concepts and solving problems as a class, encourage students to estimate the lengths of different objects and check by measuring. Comparing an estimate with the actual measurement of an object helps students develop knowledge of the length of a quantity, such as 1 m, and helps them improve their estimates. Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data H-1

2 MD2-1 Length, Width, and Height Pages Standards: preparation for 2.MD.A.1 Goals: Students will compare the lengths of straight objects directly by lining up the ends. Students will identify and compare widths and heights of objects that have an obvious front. Vocabulary: height, length, longer, longest, narrower, opposite, shorter, shortest, taller, wider, width Materials: colored pencils of different lengths (5 for each student, 10 for the teacher) a colored marker or a crayon 3 objects (e.g., index card, playing card, story book) for each student a toy car BLM Rectangles (p. H-31; see Extension 2) Introduce length. Explain that the distance from one end to the other end of an object is called the length. Demonstrate by showing the length of a pencil. Then, hold two colored pencils in one hand so that the bottoms are concealed in your fist. Stagger the pencils so that the longer pencil appears shorter. ASK: Which color of pencil looks longer? Reveal the bottoms of the two pencils and then align them. ASK: Now which color looks longer? Why did we get different answers? Which pencil is really longer? Have several colored pencils available. Align pencils two at a time and ask students to identify which pencil is longer. Then align the writing end of one pencil with the non-writing end of the other one and ask which one is longer. Introduce shorter. Align the end of a long red pencil with the end of a short blue pencil and SAY: The red pencil is longer than the blue pencil. We say the blue pencil is shorter than the red pencil. Shorter means not as long as. Longer and shorter are opposites. Write longer and shorter on the board. Show different pairs of properly aligned colored pencils. ASK: Which pencil is shorter? To see all students answers at the same time, ask them to hold up crayons or colored pencils of the correct color. Introduce longest and shortest. Using groups of three or more colored pencils, demonstrate the meanings of the words longest and shortest. Write longest and shortest on the board. Show different triples of properly aligned colored pencils and have students identify the color of the longest or the shortest pencil. H-2 Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data

3 Compare objects when the ends are not lined up. Show or draw the arrangement shown below: ASK: Which pencil is longer? (red) How can you tell even though the ends are not lined up? (The red pencil is longer because there is extra red at both ends.) Repeat with more arrangements. (MP.3) Show or draw the arrangement shown below: ASK: Which pencil will be longer when I line them up correctly? Explain how you know. ASK: Is there more extra red at one end or extra blue at the other end? (more extra blue at one end than extra red at the other end) Line up the pencils correctly to check the prediction. Repeat with pencils that gradually become more similar in length. Use folding to compare lengths. Give student pairs a rectangular sheet of paper. SAY: Let s find out which side is longer, the top or the side. Have students discuss in pairs how to compare the lengths of the two sides. SAY: So far, we checked which pencil is longer by lining pencils up. We cannot line up sides of a sheet of paper, but we can place them one on top of the other. ASK: How can we place the top of the paper onto the side? (fold the sheet to bring the sides together) Demonstrate folding the sheet upward diagonally. Mark on the longer side where the shorter side ends as shown below: folded edge Activity 1 Which side is longer? Give students several different precut rectangles, such as those from BLM Rectangles, or ask students to cut out the rectangles from the BLM. Challenge them to predict which side is longer. Ask students to fold the rectangles to compare sides and use a colored marker or crayon to make a mark on the longer side, as shown above. (end of activity) (MP.2) Transitivity of length. Give three volunteers 1 pencil each, all of different lengths, without showing the pencils to the whole class. Ask the volunteer with the middle-sized pencil (Jin, for example) to demonstrate comparing the pencils first with one of the other two Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data H-3

4 volunteers (Ava and Kim, for example), then with the other. SAY: Jin s pencil is longer than Ava s pencil, and Kim s pencil is longer than Jin s pencil. Who has a longer pencil, Kim or Ava? (Kim) How do you know? (Kim s pencil is longer than Jin s, and Jin s pencil is longer than Ava s, so Kim s pencil is longer than Ava s and the longest of the three.) Have the other two volunteers compare the pencils to check. Introduce width. Explain that the distance across an object from side to side is called the width. Illustrate with a JUMP Math AP book: hold up the workbook and run your fingers along the front, first up and down, then side to side. ASK: Which way is the width? Is it this way (show from side to side)? Or is it this way (show from top to bottom)? Write width on the board. Explain that when you look at a book so that you can read it, the width is the distance across from side to side. Ask volunteers to show the width of various objects, such as the blackboard, a window, a door, a bookshelf, and so on. Compare widths. Place a JUMP Math AP book on the blackboard ledge. Explain that the workbook is not as wide as the blackboard. We say the blackboard is wider than the workbook. Write wider on the board, and explain that the word wider is used when comparing two widths. Ask students to find other objects in the classroom that are wider than their workbook, using their workbooks to check directly. Introduce narrower. ASK: What is the opposite of longer? (shorter) Explain that narrower is the opposite of wider. Write narrower on the board. Give each student three objects to compare, for example, an index card, a playing card, and a storybook. ASK: Is the book wider or narrower than the index card? (wider) Is the playing card wider or narrower than the index card? (narrower) Introduce height. Explain that the distance up and down an object is called the height. Write height on the board. Ask a volunteer to stand up, choosing a student who is of average height. ASK: Who thinks they are taller than (use the volunteer s name)? How can we check? After students share their ideas, ask a second volunteer to stand back to back with the first volunteer. Repeat the exercise with volunteers who think they are shorter. Finding the width of three-dimensional objects. Show students a toy car and ask them to identify the front of the car. Have volunteers trace a finger along the car from front to back, side to side, and top to bottom. ASK: Which way is the width of the car from front to back, side to side, or top to bottom? (from side to side) Draw a car on the board viewed from the side, so that the front is visible as shown below: H-4 Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data

5 ASK: Are we facing the front of the car or one of the sides? (one of the sides) Where is the front of the car? (the part on the right) Trace your finger along the car on the board from top to bottom, front to back, and side to side. Ask students to tell you (using thumbs up or down) if a distance you are tracing represents the width. Add lines to identify each dimension on the drawing (similar to those on AP Book 2.1 p. 176). Trace or color the line that represents the width. Repeat with other drawings of concrete objects, such as a toy train, a book, a chair, a bench, and so on. Have students also identify the height of each drawn object. Activity 2 I Spy. Play a few rounds of I Spy with students using the length, width, and height of classroom objects. Example: I spy something that is narrower than this sheet of paper. You might invite volunteers to lead a round of the game. (end of activity) Extensions 1. Obtain a red and a blue pencil that are close in length (the red pencil should be longer by less than 1 cm). SAY: I am going to show you something magical that we call an optical illusion. Hold the shorter blue pencil vertically and the longer red one horizontally. Ask students to predict which pencil is longer, then check their predictions. Explain that objects that sit vertically (from top to bottom) often look longer than objects that sit horizontally (from side to side). 2. Draw a rectangle on the board. Explain that geometric shapes, such as rectangles, do not have a front, so it is not clear which distance across is the width. Mathematicians have defined the width of rectangles to be the shorter side and the length to be the longer side. length width Give students some rectangles from BLM Rectangles. Ask them to trace the rectangles so that the width, the shorter side, goes from side to side (not up and down). Model an example (see below). Sample answers: A B C D E F 3. As a class, read Super, Super, Superwords by Bruce McMillan. The book has numerous illustrated examples of comparatives and superlatives. Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data H-5

6 MD2-2 Measuring Length Pages Standards: 2.MD.A.1 Goals: Students will identify how to correctly measure the length of an object by lining up identical units in the same position, with no gaps or overlaps. Prior Knowledge Required: Count beyond 20 Compare lengths directly by placing objects side by side and aligning properly Vocabulary: length, longer, measure, unit, unit of measurement Materials: 1 cm connecting cubes (at least 21 for each student) pencil that is an exact number of centimeters long 3 cm and 5 cm paper clips (at least 5 small and 8 large for each student) strips of paper scissors masking tape variety of items to serve as units of measurement (e.g., pattern blocks, play coins, 2 cm connecting cubes) finger paint precut cardboard rectangles several same-size elastics (see Extension 1) newspapers and cardstock paper or construction paper (see Extension 2) (MP.5) Why we need measurement. Hold up a pencil and and ask a volunteer to hold up a pencil, too. SAY: I want to know whose pencil is longer. How can we check? (line up the pencils) Invite the volunteer to check. Then explain that you want to check which pencil is longer, the one you have in your hand or the one you have at home. ASK: How can I check? (bring the pencil to school and check) Can you check which pencil is longer without bringing it to school? Can you check which is longer, your pencil, or the pencil of a friend in China? How might you check? Students may suggest measuring both pencils and comparing the measurements. Introduce measure. SAY: When we use a number to show how long a pencil is, we are measuring its length. Measuring is different from comparing a pencil s length to the length of another pencil. We can say that a ruler is longer than a book, but that is not very helpful since lots of things are longer than a book. (You might brainstorm a list together to illustrate the point.) When we say that a ruler is 6 inches long, we are giving more information. H-6 Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data

7 Introduce unit of measurement. Write unit of measurement on the board. Explain that a unit of measurement is something that we can use over and over again to measure with. Obtain a pencil that is an exact number of centimeters long. Place a row of connecting cubes below the pencil as shown, or show the picture below: SAY: For example, if we place connecting cubes in a row below a pencil, we can count the cubes. ASK: How many cubes are in the row? (8) SAY: This pencil is as long as 8 cubes, so we say that it is 8 cubes long. We can measure the length of other objects in cubes. Using the same unit to measure different objects allows us to compare objects, even when we can t place the objects side by side. Explain that many objects can be used as a unit of measurement. An example is connecting cubes. Give each student at least 21 small connecting cubes and ask them to make a chain of cubes that is as long as their AP book is wide. ASK: How many cubes wide is the book? (21) Measuring using centimeter cubes. Remind students that to compare lengths properly, they need to align the ends of the objects. Use a pencil that is an exact number of centimeters long and a chain of connecting cubes to demonstrate properly aligning the end of the pencil with one end of the chain. Have students make a chain of cubes for each pencil or crayon in Questions 1 8 on AP Book 2.1 p Exercises: Make a chain of cubes that is as long as each pencil or crayon. How many cubes long is each pencil? Answers: 1. 5 cubes, 2. 5 cubes, 3. 7 cubes, 4. 6 cubes, 5. 4 cubes, 6. 5 cubes, 7. 6 cubes, 8. 7 cubes Units of measurement need to be placed the same way. Have a volunteer use cubes to measure a pencil that is an exact number of centimeters long. Write on the board: The pencil is 8 cubes long. Demonstrate measuring the same pencil incorrectly by arranging connecting cubes as shown below): Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data H-7

8 (MP.6) SAY: I think this pencil is 7 cubes long. Is that correct? (no; thumbs down) Have students show thumbs up or down to represent yes or no, respectively. How is this different from the way we measured earlier? (the cubes are not in a chain; some cubes are turned so they take more space) Emphasize that the units need to be placed exactly the same way for measuring. Explain that you can also use paper clips to measure length. Show measuring the pencil incorrectly in another way by arranging paper clips as shown below: ASK: Is this a good way to measure with paper clips? (no; thumbs down) Why not? (some units are lying down and some are standing up) Have a volunteer place the paper clips correctly. If the pencil is not an exact number of paper clips long, point that out and say that the pencil is a little more (or a little less) than 4 paper clips long. Units of measurement must be the same length. Give students a variety of small and large paper clips. Show them how to make a chain of paper clips and how to keep the chain straight. Demonstrate marking the length of a sample chain of 3 paper clips on a strip of paper, and cut the strip along the mark. Tell students to make their own chain using any 5 paper clips and then cut a strip of paper that is as long as their chain. Make sure that at least one student uses only small paper clips and another uses only large paper clips for their chains. Invite many volunteers to tape their paper strips on the board. Compare the length of the strips directly, particularly those made using only one size of paper clip. SAY: All these strips are 5 paper clips long, but they are not all the same length. Why not? (everyone used different paper clips) Show or draw a pencil that is 4 large paper clips long and a pencil that is 5 small paper clips long. Tape or place the paper clips alongside each pencil as shown: (MP.3) ASK: Which pencil is longer? How can you tell? (the one that measures 4 large paper clips; because that pencil extends beyond the other one) Should 5 paper clips be longer than 4 paper clips? Explain. (only if the paper clips are the same size) H-8 Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data

9 Next show pencils that are both 8 paper clips long, but use paper clips of different sizes as shown below: SAY: Both pencils are 8 paper clips long, but they are not the same length. Explain that when we measure, we need to use units that are the same length. There must be no gaps or overlaps between units of measurement. Show another incorrect way to measure a pencil by leaving gaps as shown below: ASK: Is this pencil 3 paper clips long? (no, thumbs down) What is wrong with the measurement? (the paper clips are spread out) Explain that there should be no spaces between the units. Add 2 more paper clips that cover the gaps but create overlaps as shown below: ASK: Is this pencil 5 paper clips long? (no, thumbs down) What is wrong with the measurement? (the paper clips overlap each other) Invite a volunteer to place the paper clips correctly. Have students show thumbs up if the paper clips are arranged correctly. (MP.6) Identifying correct measurements. Show different correct and incorrect ways to measure items, such as paper clips, cubes (1 cm and 2 cm), pattern blocks, and play coins. Have students show you thumbs up or down to represent arrangements that are correct or incorrect, respectively. Invite volunteers to correct the arrangements. Activity Using fingerprints as units of measurement. Distribute paint and cardboard rectangles of various lengths. Have students use fingerprints (from the same finger) to measure the length of the rectangles. Make the rectangles wide enough so that students can write or finger-paint the number of fingerprints as well. (end of activity) Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data H-9

10 Extensions 1. Features of good units of measurement. Explain to students that paper clips are good units of measurement because it is easy to find many that are the same size. SAY: You can use many small paper clips or many big paper clips to measure. SAY: Each small paper clip is the same length. Units of measurement also need to stay the same length. Paper clips stay the same length because they are rigid. Let s find out if elastics are good units of measurement. Use many elastics of the same size to demonstrate how their length changes when you stretch them. ASK: Are elastics good units of measurement? Why do you think so? (no, they stretch to different lengths). Invite students to identify classroom objects that would make good units of measurement. As a class, discuss the units students found as well as any good units that they may have missed. Have each student draw and label an appropriate measuring unit and add it to a class poster titled Good Measuring Units. (MP.6) 2. The need for standard units. SAY: A long time ago, people did not use paper clips or cubes or even centimeters to measure lengths. They used other things. For example, in ancient Egypt, they used cubits. A cubit is the length from the tip of your middle finger to the outside of your elbow. Show this length on your arm. Have students do the following to illustrate the disadvantages of non-standard units of measurement and the need for standard units. Have students work in groups of 4 to make a table. Have each student use their own cubit to make one leg that is 2 cubits long. Students can roll old newspapers to make the legs and use cardstock paper or construction paper to make the tabletop. (Rolling newspapers diagonally works well; the ends are thinner and easier to cut off.) Then have students lay a pencil on their table. ASK: Does the pencil roll off? Why? (yes, the table is not level) Are the legs the same height? Why not? (no, the length of each cubit is different) Explain that the ancient Egyptians recognized the need for everyone to use the same, or a standard, cubit. ASK: Whose arm do you think they used to determine the length of a standard cubit? The king s! Discuss how people recorded and shared the king s cubit so that everyone would know what it was. (They used a wooden stick with lines scratched in it to show the length of the king s cubit, just as we use rulers.) H-10 Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data

11 MD2-3 Measuring in Centimeters Pages Standards: 2.MD.A.1 Goals: Students will measure pictures of objects in centimeters using centimeter cubes and then a centimeter ruler. Prior Knowledge Required: Count beyond 20 Compare lengths directly by placing objects side by side and aligning properly Measure length using non-standard units Vocabulary: centimeter (cm), length, longer, measure, ruler, unit of measurement Materials: 1 cm connecting cubes (at least 30 for each student) 30 cm rulers BLM Measuring in Centimeters (p. H-32) BLM Concrete Rulers (p. H-33) BLM Measuring with a Ruler (p. H-34) Measuring by counting cubes in a long chain. Give each student at least 15 small connecting cubes, and tell them to make a chain using the cubes. Explain to students that they will use the chain to measure pictures. Help students recall how to measure correctly one end of the picture should align with one end of the chain. Tell students to note the place on the chain where the picture ends and then count the cubes from the end of the chain to the place they noted. Demonstrate by using a picture on the board or an object that is an exact number of centimeters long. Have students use their chains to measure the pictures on BLM Measuring in Centimeters and then write the measurements in cubes. (1. 13 cubes, cubes, 3. 7 cubes, 4. 5 cubes, 5. 6 cubes) Using a concrete ruler. Tell students that it is often inconvenient to use a chain of cubes to take measurements. SAY: I want to try using a picture of cubes instead. Show students a concrete ruler from BLM Concrete Rulers and then distribute concrete rulers to them. ASK: Are the units placed correctly on the ruler? Are the units placed so there are no spaces or overlaps? Are the units the same size? (yes, thumbs up to all) Have students check that the units on the concrete ruler and the connecting cubes are exactly the same size. ASK: How is the picture of the row of cubes different from an actual row of cubes? (e.g., picture is bendable, picture is flat, cubes have color) Make sure students notice that the picture of the row of cubes does not start at the end of the strip. Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data H-11

12 (MP.3, MP.6) Demonstrate measuring an object incorrectly by aligning the end of the object with the end of the strip (instead of aligning it with the end of the picture of the cubes), as shown below. ASK: Is this a good way to measure? (no, thumbs down) Ask students to explain your mistake (MP.3) Letting rulers do the counting. Show students a concrete ruler. Discuss the meaning of the numbers above the cubes on the ruler. ASK: What is drawn above the picture of cubes? (a number line) What number do you see on the tick that matches the beginning of the row of cubes? (0) Draw a line 5 cm long on the board and show how to measure the line using the cubes on the ruler. Write 5 cubes long beside the line. Then place the number line side of the ruler below the line you drew, as shown SAY: Look, this way the line is 6 units long! Is that correct? (no) Have students explain your mistake. Emphasize that the line is 5 spaces between the tick marks long. ASK: How should I place the number line so that I do not need to count spaces? (align with 0) Demonstrate doing so and show how the number line does the counting. (MP.6) Have students measure the pictures on BLM Measuring in Centimeters again, this time using the number line. Emphasize the importance of starting at 0. Point out that since the distance between each pair of marks on the number line is exactly one cube long, it is like measuring the pictures in cubes. Beside each measurement in cubes, have students put a check mark if the answer is the same. Afterward, discuss if students got the same answers using the number line and the cubes. If not, ask why. PROMPT: Did you forget to line up one end of the object with 0? Did you count the cubes correctly? Were there so many cubes that you got lost in the counting? Which way of measuring was easier? Which way was less work? (using the number line) Why? (because the number line does the counting for us) Relating the length of small connecting cubes to spaces on a standard ruler. ASK: Do people usually use pictures of cubes to measure length? (no) What do they use? (rulers) Explain that a ruler is a tool to measure small lengths. Write ruler on the board. H-12 Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data

13 Hand out a standard centimeter ruler and at least 30 small connecting cubes to each student. Have students look at the ruler. ASK: How is a ruler like a number line? (both have numbers in counting order; there are equal spaces between the markings) Have students line up the cubes on the ruler so that they fit in the markings. ASK: What do you notice? (the spaces between the markings are the same length as the cubes; the cubes fit exactly between the markings) Explain that the length of each small cube is called a centimeter. People often use centimeters to make rulers. Write centimeter on the board. Centimeter and cm. Have a volunteer find and circle the letters c and m in the word centimeter. Tell students that people often write just cm for centimeter. Write cm on the board. Have students find the label cm on their ruler. Have students practice measuring by completing BLM Measuring with a Ruler. (1. 4 cm, 2. 5 cm, 3. 6 cm, 4. 3 cm, 5. 7 cm, 6. 2 cm, 7. 2 cm, 8. 3 cm) Activity Have students find classroom objects that are about 1 cm long, wide, or high. (end of activity) Extension Have students use a 30 cm ruler to measure objects that are between 20 cm and 30 cm long, such as a sheet of paper. Have students work in pairs, with each partner measuring the same objects, one at a time, and then comparing their results. Have them look for any discrepancies. Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data H-13

14 MD2-4 Length and Subtraction Pages Standards: 2.MD.A.1 Goals: Students will measure pictures of lines and objects in centimeters using centimeter cubes and a centimeter ruler. Prior Knowledge Required: Count beyond 20 Subtract using a number line Measure lines and objects that are exact numbers of centimeters long Compare lengths directly by placing objects side by side and aligning properly Measure length using non-standard units Vocabulary: centimeter (cm), length, measure, ruler, unit of measurement Materials: a ruler from BLM Concrete Rulers (p. H-33) 1 cm connecting cubes (at least 10) measuring tape BLM Subtraction Using a Measuring Tape (p. H-35; see Extension) Review rulers and centimeters. Remind students that a ruler is a tool to measure length. Although we can measure length using different units, some units are very common people use them in many countries. One of the most common units is the centimeter, which is the same length as a small connecting cube. When we count units on a centimeter ruler, we count centimeters. Counting centimeters as jumps to find the distance on a ruler. Draw a centimeter ruler on the board and add two arrows as shown below: cm SAY: To find how far apart the arrows are, we can jump from one number on the ruler to the next number, and so on, and count the jumps. Draw the jumps as done on AP book 2.1 p Trace the arrows with a finger. ASK: How long is each jump? (1 cm) How many jumps do you need to make from 0 to 3? (3 jumps) How far apart are the arrows? (3 cm) H-14 Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data

15 Draw the arrows in several different positions on the ruler, including situations when the first arrow is not at zero. Have students show the number of fingers equal to the number of jumps needed to get from one arrow to the other. Record the answer on the board: The arrows are cm apart. Have a volunteer fill in the number for each pair of arrows. Progress to a longer ruler and longer distances. Point out that when students count jumps on the centimeter ruler, they are actually counting centimeters. Show the concrete ruler from BLM Concrete Rulers and emphasize that each jump on a number line is exactly the same as a small cube, so it is exactly 1 centimeter. Measuring length of lines and objects by counting centimeters. Draw a line 5 cm long and place a concrete ruler above it as shown: ASK: How many cubes long is this line? (5 cubes) How many centimeters long is this line? (5 cm) SAY: Instead of counting cubes, we can just count jumps (or centimeters) on the ruler. Move the ruler below the line so that the line starts at 2 and ends at 7. Show how to count centimeters on the ruler to find the length. Draw several different lines (not starting at zero) above the ruler and have students find each length by counting centimeters. Use lines that are shorter than 10 cm so that students can signal the answer by holding up the correct number of fingers. Write on the board: The line is cm long. Have a volunteer fill in the answer. Repeat the exercise using pictures of objects instead of lines. (MP.4) Counting jumps and subtracting on a number line. Help students recall counting jumps when subtracting on a number line. For example, for 8 5, have students sketch a number line between 5 and 8, and count the jumps as shown. Have a volunteer write the subtraction = 3 Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data H-15

16 SAY: We can use subtraction to find how far apart arrows are or how long an object is. Above a ruler, draw a line that starts at 5 and ends at 8, as shown below. ASK: How long is the line? (3 cm) cm Above the ruler, draw another line that starts at 2 and ends at 6. SAY: Write the subtraction to find the length of the line. (6 2 = 4; the line is 4 cm long) Repeat with other lines. Include lines that start at zero and longer lines. Distance from zero. Draw two arrows, one at 0 and another at 7, as shown below: cm ASK: How far apart are the arrows? (7 cm) SAY: The second arrow is 7 cm away from 0. Draw an arrow at 5. ASK: How far from 0 is the arrow? (5 cm) ASK: Do you need to count jumps or subtract to find how far away from zero an arrow is? (no) How else can you do it? (just look at the ruler) How do you know? (the second arrow points at the answer) SAY: We can count jumps to find a distance or we can let the ruler do the counting for us. Extension Explain that a measuring tape is like a very long ruler, rolled into a tight roll. Show students a measuring tape. Have students use BLM Subtraction Using a Measuring Tape to subtract two-digit numbers to find the length. Answers: = 5, 5 cm; = 6, 6 cm; = 4, 4 cm; = 4, 4 cm; = 7 cm, 7 cm; = 12 cm, 12 cm; = 15, 15 cm H-16 Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data

17 MD2-5 Measuring to the Closest Centimeter Pages Standards: 2.MD.A.1 Goals: Students will measure length to the closest centimeter. Prior Knowledge Required: Count numbers beyond 20 in order Subtract using a number line Measure lines and objects that are exact numbers of centimeters long Compare lengths directly Measure length using non-standard units Vocabulary: about, centimeter (cm), closer to, closest, exactly, length, measure, ruler, unit of measurement Materials 2 cm connecting cubes or 5 cm paper clips (10 15 for each student) a pencil, a marker, a pencil case, and an eraser for each student 30 cm rulers 10 pattern block squares for each student measuring tape several small curved objects (e.g., cup, water bottle) Review measuring in different units. Help students recall what they know about units of measurement. ASK: What different units have you used to measure the length of objects? (connecting cubes, paper clips, centimeters) How do you measure objects correctly? (use units that are the same length; place units in the same position; leave no gaps or overlaps between units) Give each student large connecting cubes or paper clips and ask them to make a chain. Remind students how to measure length using a chain, before having them find a classroom object that is an exact number of units long. ASK: How long is the object? Length and closer to. Draw two rows of connecting cubes separated by a line that is between 5 and 6 connecting cubes long, as shown: ASK: How long is the line? Is it longer or shorter than 5 cubes? (longer) Is it longer or shorter than 6 cubes? (shorter) SAY: We can see that the line is not exactly 5 cubes long and not Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data H-17

18 exactly 6 cubes long the line is between 5 and 6 cubes long. I would like to know if the line is closer to 5 cubes long or closer to 6 cubes long. Write the words closer to on the board. Highlight the distance between the end of the line and the last cube above and below it, as shown below: Explain that the line is closer to 5 cubes, so we say the line is about 5 cubes long. Write about on the board. Repeat with lines of different lengths. Record the length each time on the board: The line is about cubes long. Have a volunteer fill in the number. Measuring to the closest unit. Pick an object that is not an exact number of connecting cubes (or paper clips, depending on the units that students are using) long and demonstrate measuring it with large cubes. ASK: Which measurement is the length closest to, 4 cubes or 5 cubes? (4 cubes) ASK: Which cube is the pencil closest to? Show what you mean by tracing the horizontal distance from the end of the fourth cube to the end of the pencil; then trace the distance from the end of the pencil to the end of the fifth cube. For the exercise, provide students with large connecting cubes to measure a variety of objects. Have them work in pairs to check each other s answers by measuring the same objects and checking that the answers are the same. (MP.6) Exercises: Use cubes to measure the object. About how many cubes long is the object? a) pencil b) marker c) pencil case d) eraser Sample answers: a) 7 cubes, b) 5 cubes, c) 10 cubes, d) 1 cube Review using a ruler to measure. ASK: What do people often use to measure an object? (ruler) Hold up a centimeter rule and ASK: What are the units on this ruler? (centimeters) Remind students to align the zero mark on a ruler with the end of the object they are measuring. ASK: How does the ruler count the centimeters for you? (the end of the object points to the answer on the ruler) Measuring to the closest centimeter. Draw a ruler on the board and draw a line that ends between two centimeter marks. ASK: How many centimeters long is the line? Have students signal the answer by raising the correct number of fingers. Repeat several times by drawing H-18 Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data

19 lines slightly shorter and slightly longer than an exact number of centimeters. Then SAY: When a line is exactly halfway between two units, we use the longer measurement. Draw or show the pictures below and then SAY: The line is about 6 cm long cm Have students use a centimeter ruler to measure the objects in the exercise. Have them work in pairs to check each other s answers. (MP.6) Exercises: Use a ruler to measure the object. About how many centimeters long is the object? a) pencil b) marker c) pencil case d) eraser Sample answers: a) about 14 cm, b) about 10 cm, c) about 21 cm, d) about 2 cm Extensions 1. Have students make a row of 10 pattern block squares and measure its length to the closest centimeter. Answer: 10 pattern block squares are about 25 cm long. 2. Teach students to measure round or curved objects, such as a cup or a water bottle, using a measuring tape. Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data H-19

20 MD2-6 Estimating in Centimeters Page 188 Standards: 2.MD.A.1, 2.MD.A.3 Goals: Students will use finger width to estimate length in centimeters. They will compare their estimates to measurements made using a centimeter ruler. Prior Knowledge Required: Count numbers beyond 20 in order Measure length to the closest centimeter using a ruler Vocabulary: about, centimeter (cm), closer to, estimate, exactly, length, measure, predict, ruler, unit of measurement Materials 30 cm rulers small objects (e.g., small paper clips, play coins) finger paint cardboard rectangles a pencil case a pencil, a lunch bag, an eraser, and an index card a stick or meter stick 1 m sparkly gift wrap ribbon for each student BLM Table Template (p. H-36) BLM Lights (p. H-37; see Extension) NOTE: If you plan to do Activity 1, divide it into two parts. Have students complete the first part before recess, lunch, or another break in the day to allow time for fingerprints to dry. Using finger width as a benchmark for 1 cm. Give each student a ruler. Have them place each finger in turn, including the thumb, between the grid marks on the ruler. ASK: Which finger is closest to 1 cm wide? Which finger is farthest from 1 cm wide? Students will need this information for Activity 1. SAY: One of your fingers is not exactly 1 cm wide but it is pretty close. So, you can use that finger to measure in centimeters when you need an answer that is close to an exact answer. Measuring short objects using finger widths. SAY: We cannot use many copies of the same finger to measure something because we have only two copies of the finger we need. Your pointer, middle finger, and ring finger are all about the same width, so for small objects that are about 3 fingers long or less, you can use 2 or 3 fingers placed close beside each other. Have students use 2 or 3 fingers placed close beside each other to estimate the length of a small object, such as a small paper clip or a penny. H-20 Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data

21 Using only one unit to measure. Draw a line 10 cm long on the board. ASK: Are 3 fingers enough to measure this? (no) Explain that a different technique is needed. Measure the line using both index fingers by placing one index finger on the line and alternating with the other one, as shown below (put chalk or ink on your finger to make fingerprints). Record the length. Then have a volunteer measure the line using his or her index fingers. Explain that when you think and then guess about something, you predict. Predict means to guess based on what you know. Have the class predict which measurement will be closer to the measurement in centimeters (a student s finger is closer to 1 cm in width, so the estimate using a student finger is better). Measure the line using a ruler. Then place your index finger and the volunteer s index finger between the markings on a ruler to see whose finger is closer to 1 cm wide. Activity 1 Using fingerprints to measure. Distribute paint and cardboard rectangles of various lengths. Have students use fingerprints (from the same finger) to measure the length of the rectangles. Make the rectangles wide enough so that students can write or finger-paint the number of fingerprints as well. Have students measure each length twice: using the finger closest to 1 cm wide and then using the finger farthest from 1 cm wide. 8 6 Wait until the fingerprints have dried to continue the activity. Have students circle the measurement that they think will be closest to the actual measurement in centimeters. Then have them measure the length of the rectangle using a ruler. ASK: Was your prediction correct? How close was that fingerprint measurement to the actual centimeter measurement? (end of activity) Introduce estimate. SAY: A guess based on what you know is called an estimate. The more you know, the closer an estimate will be to the correct answer. Write estimate on the board. Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data H-21

22 Emphasize that an estimate is not just a guess; it is a guess based on information. To estimate means to think and then guess. Explain that students can use finger width as an estimate for 1 cm. Have students estimate the length of their pencil cases. Recording information in a table. Explain that when we need to record a lot of information that is similar, such as lengths of objects, it makes sense to use a table. Draw a table as shown. SAY: This table has 2 columns (trace down each column with your finger). Object Estimate Provide students with BLM Table Template. Have them write the labels in the first 2 columns of the top row. Explain to students that they will learn how to fill in the table. SAY: In the first column, you will write the name (or draw a picture) of an object you will measure (trace down the column with your finger). In the second column, you will write your estimate of its length. SAY: This is a row (trace across the row with your finger). Demonstrate how to fill in the first row for a pencil case. In the first column, write pencil case. Have students do the same. Ask a volunteer to use his or her finger width to measure the pencil case. In the second column, write about cm. Ask the volunteer to fill in the missing number. Have students do the same using their estimate of its length. Fill in the first column of the table with the names of the objects in the exercises, and have students do the same in their table. Then point to a cell in the second column of the table. ASK: For which object will you write the length here? Repeat by pointing to each cell in the second column. Tell students to place the eraser on the cell that shows its length. For the exercise, have students use the BLM to record their answers. Exercises: Measure the longest side of the object. Use the finger that is closest to 1 cm wide. a) pencil b) lunch bag c) JUMP Math AP book d) eraser e) index card Sample answers: a) about 18 cm, b) about 24 cm, c) about 28 cm, d) about 4 cm, e) about 12 cm Measuring to check the estimate. Explain to students that they will check their estimate for each object and record the information. Point out that the table on the BLM has 3 columns. Add a column to the table on the board, and label it Measurement. Object Estimate Measurement Have students label the third column on their copy of the BLM. Point to a cell in the third column of the table. ASK: For which object will you write the length here? Repeat by pointing to each H-22 Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data

23 cell in the third column. Tell students to place a paper clip in the cell that should show the actual length of their lunch bag. Demonstrate how to fill in the third column for the pencil case. Ask a volunteer to measure the pencil case. In the third column, write cm. Ask the volunteer to fill in the missing number. Have students do the same. Have students use a centimeter ruler to measure each object from the exercise and record the measurement in the table. Then have volunteers fill in the measurements in the table on the board. Point to different cells in the second and third columns of the table and have students identify what each cell shows (for example, estimate for the length of a pencil case, measurement for the length of a pencil). Estimating centimeters without using fingers. Have students estimate how many centimeters long, tall, or wide objects are and then check using a ruler. Choose objects that are less than 30 cm long. To estimate, students should make educated guesses based on their experiences with centimeters thus far. Have students record their work on BLM Table Template. Activity 2 (MP.2) On a stick or on the back of a meter stick make markings of 10 cm, 20 cm, 30 cm, and so on from the end of the stick. Label the marks clearly. Explain that the markings show the distance from the end of the stick. Use the stick to play an estimation game with students. Ask them to tell you to stop when your finger has reached 15 cm from the end of the stick. Slide a finger slowly along the stick and then mark the place where students tell you to stop. Have a volunteer measure the distance and check the guess. Repeat with other distances. Students can play the game in pairs with one partner sliding a finger along the stick and the partner telling when to stop. Have partners measure the distance from the end. Have partners switch roles after each round. (end of activity) Extension (MP.1) Give students BLM Lights. Tell students that they will decorate a picture of a house with ribbon. The ribbon will represent the outdoor lights that people use to decorate their house for a holiday. Explain that they will glue the ribbon on the thick lines of the house (and the door and window) but that you are not sure how much ribbon will be needed. To find the length needed for each feature, have students estimate the length of each thick line and then measure it. Have them add the lengths. Then give students a length of gift wrap ribbon and have them measure and cut the length needed for each feature. Answer: 77 cm of ribbon Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data H-23

24 MD2-7 Estimating in Meters Page 189 Standards: 2.MD.A.1, 2.MD.A.3 Goals: Students will estimate measurements in meters. They will compare their estimates to measurements made using a meter stick. Prior Knowledge Required: Count numbers in order Measure length to the closest centimeter using a ruler Measure length to the closest unit Know that length can be measured in different units Vocabulary: about, centimeter (cm), closer to, estimate, exactly, length, measure, meter (m), meter stick, predict, ruler, unit of measurement Materials meter sticks several objects close to 1 m in length, including a baseball bat and a scarf or a piece of yarn 1 m length of string for each student newspapers masking tape scissors Introduce meters. SAY: I would like to measure the length of the classroom. Do you think it would make sense to use connecting cubes? Why? Do you think we have enough cubes? How else could I measure the length in a faster and easier way? Explain that measuring a long distance, such as the classroom, requires hundreds of cubes, paper clips, or centimeters. SAY: We use meters to measure long distances. A meter is about the same length as a meter stick. Write meter on the board. (MP.2) Show students a meter stick and ask them to think of other objects that are about the same length or height as a meter (e.g., a baseball bat, a child s golf club, a 4-year-old child, a bus wheel). If available, show students some of these objects, including a baseball bat and a piece of yarn or a scarf 1 m long. If the meter stick has a small space between the end and the start of the number line, point that out, and explain that a meter is slightly shorter than the meter stick. Meter and m. Have students recall the short form for centimeter (cm). SAY: You know that people often write just cm for centimeter. They also often write just m for meter. H-24 Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data

25 Explain that 1 m is 100 cm long. Write on the board: 1 meter = 100 centimeters 1 m = 100 cm Invite a volunteer to circle the common letter(s) in the whole word and the abbreviation for each unit (m and cm, respectively). (MP.2) Activity 1 Finding benchmarks for a meter. Demonstrate how to check that a length of string is 1 m long (line up the string with a meter stick and check that the string starts at 0 and ends at 100 cm). Give each student a 1 m length of string. Have them verify that the string is 1 m long. Then have students use the string to find classroom objects that are about 1 m in length, width, or height, and objects that are more than and less than 1 m in length, width, or height. Suggest that students include distances, such as the distance around the seat of a chair or the distance from the floor to a door knob. Afterward, have students share the objects that are about 1 m long. (end of activity) Using meters to measure. SAY: Predict how many meters long the blackboard is. Demonstrate how to measure the blackboard correctly using a meter stick. If the meter stick is slightly longer than 1 m, show making a mark at exactly 100 cm (not at the end of the stick). SAY: You make a mark at exactly 100 cm because that is how long a meter is. Then demonstrate measuring incorrectly by holding the stick diagonally and making a mark at 100 cm. Have a volunteer make the mark correctly by holding the stick straight along the board. (One way to ensure the stick is straight is to align it with the blackboard ledge.) Review measuring to the closest unit. Point out that the blackboard is longer than, say, 3 m, but shorter than, say, 4 m. Compare the distance from the last marking. ASK: Is the leftover distance close to 1 m or much smaller than 1 m? Is the blackboard about 3 m long or about 4 m long? Activity 2 Making a meter stick. Have students roll up a newspaper tightly and tape it with masking tape. Model rolling up newspaper diagonally so that the ends are only one layer thick and easier to cut. Help students cut the newspaper roll so that it is exactly 1 m long by lining up the roll with the 0 and 100 cm marks on a meter stick. As a class, use multiple meter sticks to determine how many meter sticks long the classroom is from side to side and from front to back. At the point when an additional whole meter stick will not fit in the remaining space, show students how to estimate the remaining distance by placing another meter stick so that the last stick in the row and the additional stick overlap as shown. gap overlap Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data H-25

26 Which is larger, the gap or the overlap? If the gap is larger than the overlap, the total distance is closer to, say, 7 m. If the overlap is larger than the gap, the total distance is closer to 6 m. (end of activity) Measuring with only one meter stick. ASK: Is the hallway wider than the classroom? Is it wider than the classroom is long? Ask students to predict the width of the hallway. Then have students use their meter sticks to measure the width of the hallway. Have them use small pieces of masking tape to mark the end of each stick measurement. Students can verify their measurements by using multiple meter sticks to measure. (MP.2) Using a benchmark to predict and then check. On the board, list the lengths of items that students have already measured (for example, blackboard: about 3 m wide). Pick an item not on the list (for example, a window). ASK: What item on the list is closest to the width of the window? Is the item larger or smaller than the window? SAY: The window is smaller than the blackboard, but larger than the door. The door is about 1 m wide and the board is about 3 m wide. What is a good estimate for the window? (about 2 m) Have students check the prediction. Repeat with several more items. Use units that are about the same size to find approximate measurements. SAY: Remind me about what makes a good unit of measurement. (It is easy to find many units the same size.) SAY: I want to tell a friend how long the school hallway is. She does not need to know exactly how big it is; she just wants an idea. I plan to take big steps all the way down the hall to see how many steps it takes. Do you think my steps are good units? Can I take many steps that are all the same size? Will my steps be at least close to the same size? Even though steps are not all exactly the same size, they will all be close to the same size if I try to take big steps all the time. This will not tell me exactly how long the hallway is, but it will give me a good idea. Sometimes, that is all you need. Large steps are about a meter long. Mark three long lines 1 m apart on the floor. Have students stand with their heels on one line, facing the second line, and take a step so that their toes touch the second line. Have students practice taking a step in this way several times. For multiple steps, show students how to lead off with the same foot each time: take one giant step (positions 1 and 2 below); bring the back foot forward and place it in front of and to the side of the first foot, with the heel directly in front of the front toe (position 3); bring the back foot forward beside the front foot (position 4), then take another giant step (position 5) Step 1 Step 2 Have students take 5 steps that they think will be about 1 m each, and have a partner check using a meter stick. Emphasize that it is much harder to take 5 big steps all in a row than 5 steps walking normally. H-26 Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data

27 (MP.5) SAY: I want to know about how many meters long the hallway is. Have students take giant steps along the hallway to help you find out. Remind them to take identical 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 hallway. ASK: Did we get the same answer both ways? Were our answers close? Which way was faster? Explain that when the answer does not have to be exact, we can save time using giant steps instead of meter sticks. Discuss situations where it is important to know exact measurements. Examples: competing in a race, ordering glass to fix a broken window, making a ruler to sell, making paper to put in a book, and making legs for a table. Estimating distances using a giant step. Have students predict the distance in meters to, say, the school library. Have them measure the distance in giant steps to check predictions. Allow students to adjust their prediction after measuring part of the distance. Repeat with another location at school. Have students think about whether the new location is farther or closer than the previous one. ASK: Should the estimate be larger or smaller than the distance just measured? Measure to check the estimates, allowing students to adjust their estimate after measuring part of the distance. Extensions (MP.3) 1. SAY: Let s estimate how many students with outstretched arms we will need to go all the way across the room. Will this give an exact measurement of how wide the classroom is? Why? (no, because the outstretched arms of students are not all the same length even though they are close) Will we need more or fewer students than meter sticks? What do you predict? Are outstretched arms longer or shorter than a meter stick? (longer, so we will need fewer students) Have students use outstretched arms to measure the width of the classroom as shown. Then have students measure the width using meter sticks and compare the actual width with their prediction. ASK: Was your prediction correct? Explain. 2. When students are not in the room, draw two lines on the board, both 1 m long, with arrows at the ends as shown. Ask students to predict which line is longer. (Ignore the arrows.) Then have a volunteer check by comparing both lines to a self-made meter stick. Teacher s Guide for AP Book 2.1 Unit 7 Measurement and Data H-27