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ASU MTE 494 - Fractions Proportional

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Solution Key 1 Part 1: Meaning of Fraction Fractions as reciprocal relationships of relative size 1. 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 1/7 of one candy bar. c) The candy bar is how many times as large as one piece? How do you know? The candy bar is 7 times as large as one piece. We know this because the whole was partitioned into 7 equal sized pieces. Saying that each piece is 1/7th of the whole candy bar means that the size of the whole bar is 7 times as much (large) as the size of each piece. 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? 13 pieces 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? We had to assume that the beam was cut up into 13 equal sized pieces. Without this assumption we could not conclude that the whole beam was cut into 13 pieces just by knowing that the length of one piece is 1/13 of the beam’s total length. c) Draw a diagram to represent the situation. d) The original wooden beam is 13 times as long as one of the pieces.Solution Key 2 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? _1/24th of the bag_. b) Do you know how many M&Ms each student receives? Explain. Without knowing how many M&M candies are in the bag, all we can say is that each student receives 1/24th of the number in the bag, not how many each one receives. c) Suppose you learn how many M&Ms one student received. What could you determine with this information? Explain your reasoning. Saying that each student receives 1/24th of the bag means that the total number in the bag is 24 times as much as the number that each student receives. This enables us to determine how many are in the bag if we know how many any student receives. By the equal splitting assumption in fractions, we would also know that every student received that same amount (number) of M&Ms. 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? 1/11th of the pack of gum. 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. By the equal splitting assumption, this tells you that each of the 11 students received less than half of one piece of gum. This in turn enables us to conclude that the pack contains fewer than 11 half-pieces of gum (since a pack must contain 11 times as much as each student receives). We can thus conclude that a pack contains fewer than 6 pieces of gum. 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 help us 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. If B is partitioned into n parts of equal size, then the size of B must be n times as large as the size of each part (since B is made up of the n parts). Thus, by our meaning of fraction we say that the size of each part is “1/nth” as large as the size of B.Solution Key 3 b) How large is B compared to the size of each part? Explain your reasoning. B is n times as large as each part, since B is comprised of n equal sized parts. 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 piece in relation to the C? Explain your reasoning. Each piece is said to be 1/mth as large as C, since this is what it means to say that C is m times as large as the size of each piece. b) Why it is critical to know that the pieces are all the same size? Explain your reasoning using diagrams to support your explanations. If the pieces are not assumed to be all of the same size, then the most that can be said about the whole amount C is that it is comprised of some number of pieces. We cannot conclude anything about the relative size relationship between the whole and the parts. Example: The diagram shows an amount partitioned into 9 pieces, not all of which have equal size. Notice that the whole is not 9 times as large as the size of each of the pieces, although it is comprised of 9 pieces. The assumption of equipartitioning is built in to the very meaning of fractions developed here. A student whose meaning of “1/9” is limited to “one of nine pieces that make up the whole” would call each piece “1/9th of the whole” and would not necessarily be thinking of relative size relationships between the whole amount and the constituent pieces. This sequence of activities develops a meaning of fractions as reciprocal relationships of relative size. The reciprocity is embedded in the fact that thinking of amount A being 1/nth as large as amount B is equivalent to thinking of amount B being n times as large as A. Fractions do involve partitioning amounts into some number of parts, but the partitions must be equipartitions AND the focus must be on thinking of the size of the parts in relation to that of the whole and vice versa.Solution Key 4 Part 2: Meaning of Multiplication Note: 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. 5 * 120 represents 5 copies of 120oz cans. It represents a multiplicity of 120oz cans, the quantification of which is 5. It also represents a multiplicity of ounces of paint (assuming all the cans are full to capacity), whose quantification is five 120 oz or 600 oz. …


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