1!“During grades 3–5, students should build their understanding of fractions as parts of a whole ... They will need to see and explore a variety of models of fractions … By using an area model in which part of a region is shaded, students can see how fractions are related to a unit whole, compare fractional parts of a whole, and find equivalent fractions.” !"!—NCTM, Principles and Standards for School Mathematics, 2000"2!“Seeing” Fractions!Luis Saldanha, Arizona State University.!Adapted from Providing a foundation for teaching mathematics in the middle grades, pp. 223-250, Sowder, J. T. & Schappelle, B.P. (Eds.) (1995). SUNY Press. Albany: NY.!3!What do you see?!Do you see a woman’s face?!Do you see a saxophonist?!Do you see both?!4!What do you see?!Do you see a face?!Do you see the word “Liar”?!Do you see both?!5!Of what do you see 3/5?!Of what do you see 5/3?!• Do you see 5/3 of something?!• Do you see 3/5 of something?!6!5/3 of 3/5 is how much of 1? Explain in terms of the diagram.!• Do you see 2/3 of 3/5?!2/3 of 3/5 is how much of 1? Explain in terms of the diagram.!• Do you see 5/3 of 3/5?!7!What does “1 ÷ 3/5” mean?!• Do you see 1 ÷ 3/5?!1 ÷ 3/5 = 1 2/3. Explain this in terms of the diagram.!“1” in “1 ÷ 3/5 = 1 2/3” refers to 1 what?!“2/3” in “1 ÷ 3/5 = 1 2/3” refers to 2/3 of what?!8!“5/3” in “1 ÷ 3/5 = 5/3” refers to 5/3 of what? !1 ÷ 3/5 = 5/3. Explain this in terms of the diagram.!9!Of what do you see 2/7?!• Do you see 2/7 of something?!Of what do you see 7/2?!• Do you see 7/2 of something?!10!7/2 of 2/7 is how much of 1? Explain in terms of the diagram.!• Do you see 7/2 of 2/7?!• Do you see 3/2 of 2/7?!3/2 of 2/7 is how much of 1? Explain in terms of the diagram.!11!“Three-fourths of a schoolʼs budget is allocated for athletics. Two-fifths of the athletics budget is allocated for softball”. "What fraction of the school budget is allocated for softball?"What does it mean that one amount is three-fourths (3/4ths) of another amount?!12!“Two-fifths of the athletics budget is allocated for softball” means that the softball budget is two times as much as 1/5 of the athletics budget. "Compare the softball budget with the total budget. #It is 6 times as much as 1/20 of total budget (each little rectangle is 1/20 of total budget because total is 20 times as large as one of them). #So the softball budget is 6/20ths as much as the total school budget "13!Teacher: One package weighs 0.55 kilograms. And Adam used 35 hundredths of that one package. How many did he use? In order to find out how many he used, I’ve got to subtract."Interviewer: How would you explain your solution to a student?"Teacher: …I’d have them read the problem to me and my question would be, “Do you have to add, subtract, multiply, or divide?” "… and from the last question, “How much did he use?” they know that they have to take away. That’s what they’re looking at, is the last question. So then I’d say to them, “How much did he start out with?” Fifty-five kilograms. “How much did he use up?” Thirty-five. And then everything’s set up all right, so then we subtract."A package of printing paper weighs 0.55 kg. Adam used 0.35 of a package for his research paper. How many kilograms did he use?!14!An alternative approach: Reason about the quantities in the situation:"Imagine breaking up the package of paper into 100 equal parts. Then each part weighs one 1/100th of 0.55kg. Or equivalently, "0.55 kg/100 = 5.5g:"15!Now, 0.35 is 35/100ths, which is 35 times as much as 1/100th. So, 35/100ths of this package weighs 35 times as much as 1/100th of it. "That is 35 times as much as 5.5g.
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