MTE 494Meanings of fraction, multiplication, measure, and division This document contains a sequence of instructional activities designed to foster the development of a well-connected and conceptually coherent understanding of fractions, multiplication, measure, and division.1 The sequence is in four parts (pp. 2-14). Pages 15-16 each elaborate the understandings/meanings aimed for in Parts 1, 2, and 3. Part 1 will be started in class (Jan. 21) and will be followed by a discussion. Complete Parts1-2 for homework and submit on _01/28_. Submission date(s) for Parts 3-4 yet to be determined.1 Created by Pat Thompson and Alan O’Bryan. Arizona State University.1Name: ___________________________________Part 1: Meaning of Fractions — Fractions as reciprocal relationships of relative size1. Imagine that you take a candy bar and partition it into seven equal pieces.a) Draw a diagram to represent this situation.b) Each piece is ______ of one candy bar.c) The candy bar is how many times as large as one piece? How do you know?2. Suppose that a wooden beam is cut up into several pieces. Suppose that the length of one of the pieces is 1/13 of the length of the original beam.a) How many pieces was the beam cut into?b) In part (a) you had to make an assumption about the situation in order to answer the question (even if you didn’t realize you were making one!). What assumption did you have to make?c) Draw a diagram to represent the situation.d) The original wooden beam is ____ times as long as one of the pieces.3. Suppose a large bag of M&Ms is opened and shared equally among a class of 24 students.a) What fraction of the M&Ms will each student will receive? _______________________. b) Do you know how many M&Ms each student receives? Explain.2c) Suppose you learn how many M&Ms one student received. What could you determine with this information? Explain your reasoning.4. Imagine that a pack of bubble gum is split equally among a group of 11 friends.a) What fraction of the bubble gum will each friend receive? _____________________________________.b) Suppose you learn that one friend ended up with less than half of one piece of gum. What does this tell you? Explain your reasoning.We want to generalize our thinking from the earlier exercises so that we can understand the idea of a fraction in all situations we might encounter. The next two sets of questions will helpus do this.5. Some amount (call it B) is partitioned into n equal parts. a) How large is each part in relation to B? Explain your reasoning.b) How large is B compared to the size of each part? Explain your reasoning.6. Some amount (call it C) is partitioned into equal pieces. C is m times as large as the size of each piece.a) How large is each part in relation to the C? Explain your reasoning.b) Why it is critical to know that the pieces are all the same size? Explain your reasoning using diagrams to support your explanations.3Name: ___________________________________Part 2: Meaning of MultiplicationNote: Remember that the meaning of multiplication, such as in u * v, is to make u copies of something of size v. The product (u*v) is the amount made by u copies of v.1. Suppose that you have a collection of paint cans (containing paint) that you will be using to paint your house. Each can is able to hold up to 120 ounces of paint.a) Use the meaning of multiplication to explain how to think about what is being represented by the expression 5 * 120. Then draw a diagram to visually represent your explanation.b) Given the same context, use the meaning of multiplication to discuss how to think about the expressions (13178) * 120 and (1/9) * 120.2. To help offset budget cuts, Campo Verde held a supply drive. Each student was asked to bring in three reams (packages) of printer paper. Suppose 657 students did so.a) Using the meaning of multiplication in this context, explain how 657 * 3 represents the number of reams these students brought in.b) The commutative property of multiplication states that we can write 657*3 as 3 * 657 and still get the same number. Using the meaning of multiplication, explain how 657*3 reflectsa different way of thinking about the situation of students bringing reams of paper.c) How many reams of paper did Campo Verde students collect in the supply drive? 4d) The number of reams of paper brought in is _____ times as large as the number of studentswho collected them. How do you know?e) The number of reams of paper that each student collected is how many times as large as the number of students who collected them? How do you know?f) The number of reams each student collected is _____ times as large as the total amount of paper brought in. What is your reasoning?g) The number of students who collected paper is _____ times as large as the number of reams they collected. What is your reasoning?3. Suppose that Juan walks home from school, and that the distance between his home and school is 13 blocks. Suppose also that it takes Juan 81 steps to travel one block.a) What does 13 * 81 represent?b) Using the meaning of multiplication in this context, explain why 13 * 81 represents what you say it does in part (a).c) The total number of steps he walks to get home is how many times as large as the number of steps he walks per block? How do you know?d) The number of steps Juan walks to get home is _____ times as large as the number of blocks he walks to get home. How do you know?f) The number of steps Juan walks per block is _____ times as large as the total number of steps he walks to get home. What is your reasoning?g) The number of blocks Juan walks to get home is how many times as large as the total number of steps he walks to get home? What is your reasoning?h) Look at parts (a) through (g) again and reflect on the following questions (think about these, don’t write anything – these are good questions to discuss with a classmate). How 5would your answers change if Juan’s house was 13.6 blocks from school instead of 13 blocks? (Make sure to think through your reasoning, don’t just say, “I would replace all ofthe 13s with 13.6.” Think through it as if it were a new problem.) What if it took Juan 81.1steps to walk each block instead of 81 steps?We want to generalize our thinking from the earlier exercises so that we have a powerful, flexible way of thinking about multiplication in all situations we might encounter. The next setof questions will help us to generalize our thinking. Remember that