Crooked Paths. Based on Activity 20.3, p To help students understand that length is an attribute that need not be in a straight line.

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1 Expanded Lesson for CW Ch 20 p. 1 Crooked Paths Based on Activity 20.3, p. 379 Grade Level: Late kindergarten or early first grade. Mathematics Goals To help students understand that length is an attribute that need not be in a straight line. For example, a distance around an object or a non-straight path has length just as does a straight object. To provide the opportunity to use an intermediary object as a basis of comparison. Thinking About the Students Students have made comparisons of straight objects or paths and have learned the meaning of longer and shorter in that context. Students have not used units or rulers to measure lengths. Materials and Preparation This will be a station activity. Set up three identical stations around the room. Each station consists of two crooked paths made of masking tape. Try to make them about the same in each station. Path A is a zigzag of 4 straight line segments that total about 9 feet in all. Path B is more S-shaped and is about 7 feet long. Make path B look longer by spreading it out more. Make one recording sheet for every two students. (See Blackline Master 69, Crooked Paths )

2 Expanded Lesson for CW Ch 20 p. 2 Rope, string, or yarn that is at least 10 feet long for each station. LESSON BEFORE Begin with a simpler version of the task: Show the class pairs of straight objects such as a meter stick and teacher, pencil and crayon, two lines drawn on the board. For each pair ask, Which is longer? Which is shorter? How can we tell? On the board draw a half circle and beneath it a line segment about as long as the diameter. How can we tell which of these is longer? Solicit ideas. Be sure they hear the idea that the curve is longer than the segment and that some students provide good reasons. For example, if you had to walk on these it would take longer to walk the curved path.

3 Expanded Lesson for CW Ch 20 p. 3 The Task Gather students around one station. Tell them, One path might be longer or they might be the same. Your task is to decide. Establish Expectations: Show students the worksheets and explain how to use them. Explain that they are to circle the path that is longer or circle both paths if they think they are the same. Then they draw a picture to show how they decided. Have children work in pairs. Ask students which path they think is longer or do they think they are the same. Before you begin work, put an X on the picture of the path that you think is longer. This is your guess. Have a few students share their guesses and their reasoning. Show students that there is string but that they can use their bodies, or blocks, or whatever materials they want to help them decide. DURING Monitor station activity but do not interfere. Be sure students are completing worksheets to the best of their abilities. If a pair seems unable to make a decision, ask, If a toy car was going to go along these paths, which path would it travel the longest on? OR Could you use some blocks from over in the block corner to help? Challenge task for capable students: Make a long row of blocks in a straight line that is just as long as the curvy path. AFTER (When all students have completed the station.) Remind students of the task of comparing the two paths. Return their worksheets so that they can use them to talk about what they did.

4 Expanded Lesson for CW Ch 20 p. 4 Ask, How many thought the zigzag path was longer? (Count and record on the board next to a zigzag.) How many thought the curvy path was longer? (Count and record.) How many thought they were about the same? (Count and record.) Ask students if the guesses they made were the same as the result that they figured out. Were they surprised? Why? Select pairs to explain what they did. Begin with timid students who are correct. Then ask students with different results. Get as many different ideas and methods as possible. If students disagree about which path is longer, have students explain their reasoning in a way that might convince those that disagree. Give students the opportunity to change their minds but ask, What made you change your minds? ASSESSMENT NOTES Look for students who understand how length can exist on a curved path (correctly compare or make an appropriate attempt). This can be a checklist item based on the discussion or observations. If students used the string, did they exhibit correct understanding of intermediate unit? (Not a requirement of the activity.) For students who used units, did they use like-sized units or make appropriate use of materials?

5 Name Name Crooked Paths Circle the longer path. If they are the same, circle both. How we decided: (Draw pictures) Crooked paths 69 Field Experience Guide: Resources for Teachers of Elementary and Middle School Mathematics Allyn and Bacon