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 1 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 2 What do you see Do you see a woman s face Do you see a saxophonist Do you see both 3 What do you see Do you see a face Do you see the word Liar Do you see both 4 Do you see 3 5 of something Of what do you see 3 5 Do you see 5 3 of something Of what do you see 5 3 5 Do you see 5 3 of 3 5 5 3 of 3 5 is how much of 1 Explain in terms of the diagram Do you see 2 3 of 3 5 6 2 3 of 3 5 is how much of 1 Explain in terms of the diagram Do you see 1 3 5 What does 1 3 5 mean 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 7 1 3 5 5 3 Explain this in terms of the diagram 5 3 in 1 3 5 5 3 refers to 5 3 of what 8 Do you see 2 7 of something Of what do you see 2 7 Do you see 7 2 of something Of what do you see 7 2 9 Do you see 7 2 of 2 7 7 2 of 2 7 is how much of 1 Explain in terms of the diagram Do you see 3 2 of 2 7 10 3 2 of 2 7 is how much of 1 Explain in terms of the diagram Three fourths of a school s budget has been allocated for athletics Two fifths of the athletics budget has been allocated for softball What does it mean that one amount is three fourths 3 4 of another amount This amount is 1 4 of total school budget Athletics budget is 3 4 of school budget means that total is 4 times as much as this amount because it is 3 times as much as 1 4 of school budget 11 Two fifths of the athletics budget has been allocated for softball means that the softball budget is two times as much as 1 5 of athletics budget This amount is 1 5 of athletics budget because athletics budget is 5 times as much as this amount Softbal budget is 2 5 of athletics budget because it is two times as much as 1 5 of 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 20 or 6 one twentieths of the school budget 12 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 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 Thirtyfive And then everything s set up all right so then we subtract 13 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 14 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 15
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