Exam 7: Computing With Fractions

Three children took a hike, carrying one heavy backpack. One child carried the backpack for 3/8 of the hike, and a second carried it for 1/6 of the hike. For what part of the hike did the third child carry the backpack? Which child carried the backpack for the greatest part of the hike?

What calculation would solve this story problem? A woman paid $4.80 for 3/4 of a pound of candy. How much does the candy cost, per pound?

Tuna costs 70¢ a can at Store X and 80¢ a can at Store Y. How much will eight cans of tuna cost if one-quarter of the cans are bought at Store X and the rest at Store Y?

Jody mows 1/8 of the yard then takes a break. After the break, Jody mows another 1/3 of the yard then stops for lunch. After lunch, Jody mows another 1/6 of the yard and then goes swimming. What part of the yard does Jody still have to mow?

Consider this number line: Points R and S represent numbers between 0 and 1 as indicated. Which of the remaining points best represent the following? A) $R \times S$ B) $R + S$ C) $R - S$ D) $1 \div R$

Tell what 8 $\frac { 7 } { 8 }$ ÷ 3 means A) with the repeated subtraction or measurement view. B) with the partitive or equal-sharing view.

Which point of A-E would BEST locate the point for 0.89 × n on the number line (n is positive)?

A farmer keeps 40% of his land UN-cultivated. Two-thirds of the cultivated portion is planted with corn. What fraction of the entire land is planted with corn? Make a drawing and show your work clearly.

Draw a picture that represents the following: A) 4/3 × 2/5 B) 2/5 × 4/3

Following are three students' solutions to 9/16 of 48. A) For each student, first evaluate the student's mathematical reasoning. Check whether the steps are mathematically correct or flawed, or indicate that you cannot tell. B) Next, mark each student's work with "doesn't appear to understand," "may or may not understand," or "shows good understanding of" multiplication of fractions. Jessica: 9/16 is 1/16 more than1/2. Half of 48 is 24, and 1/16 of 48 is 3, so 27. A) Mathematical steps: choose one. - Jessica's steps are mathematically correct. - Jessica's steps are mathematically flawed. - I cannot tell if Jessica's steps are mathematically correct or flawed. B) Understanding multiplication of fractions: choose one. - Jessica doesn't appear to understand multiplication of fractions. - Jessica may or may not understand multiplication of fractions. - Jessica shows good understanding of multiplication of fractions. C) If Jessica's steps are mathematically correct, use her way of thinking to solve 3/8 × 32. If they are not, explain how Jessica's reasoning is flawed. Justin: A) Mathematical steps: choose one. - Justin's steps are mathematically correct. - Justin's steps are mathematically flawed. - I cannot tell if Justin's steps are mathematically correct or flawed. B) Understanding multiplication of fractions: choose one. - Justin doesn't appear to understand multiplication of fractions. - Justin may or may not understand multiplication of fractions. - Justin shows good understanding of multiplication of fractions. Stacy: 9/16 of 48? 9 × 16 = 144; 144 ÷ 48 = 3; and 9 × 3 = 27. So 9/16 of 48 is 27. A) Mathematical steps: choose one. - Stacy's steps are mathematically correct. - Stacy's steps are mathematically flawed. - I cannot tell if Stacy's steps are mathematically correct or flawed. B) Understanding multiplication of fractions: choose one. - Stacy doesn't appear to understand multiplication of fractions. - Stacy may or may not understand multiplication of fractions. - Stacy shows good understanding of multiplication of fractions.

Use a drawing to help explain why $\frac { 3 } { 4 } \times \frac { 5 } { 7 }$ is equal to $\frac { 3 \times 5 } { 4 \times 7 }$ . Be explicit.

One meaning of 45/59 × 35/17 is, "How many ones is 45/59 of 35/17 of 1?"

Four identical pizzas are shared among three people. Show two ways that the pizzas could be cut.

Why is the division of fractions confusing to people? Give two reasons.

Is one meaning of 1/4 ÷ 1/3 the following: how many 1/3s of 1 are in 1/4 of 1? If it is one meaning, how many 1/3s or how much of 1/3 of 1 are/is in 1/4 of 1?

Use these circles to show each division. For each part tell what question is being asked. Tell what your answer is in each case, and show how you obtained it. A) 7/8 ÷ 1/4 B) 1/4 ÷ 3/8

A recipe calls for a 1/2 cup of sugar. You have 2/3 of a cup of sugar. If you use all of your sugar, how many recipes can you make? Solve by means of a diagram. Referent unit for the 1/2: _____ Referent unit for the 2/3: _____ Referent unit for the answer: ــــــــــــــــــــــــــــــــــ

One day, Joe's old car used 5/8 of a quart of oil. The next day, it used only 2/3 as much oil. On day three, the old car used 1 $\frac { 1 } { 4 }$ Quarts of oil! But on day four, it used only 60% as much as it did on day three. How many quarts of oil did the car use on day four? Which computation will answer the question?

A) Show 1 ÷ 3/8 with a rectangular region as the whole. Write your answer as a mixed number. B) Clearly explain how the fraction part of your answer relates to your drawing.

For this number line, A) the point for p + q would be closest to which letter? B) the point for p × q would be closest to which letter?