These questions are for your own use to assess and reflect on your own understanding of some central ideas of this unit. After determining your own answers, click “Show Solution” to read more information about the question and the alternative responses. You may also find it valuable to discuss these questions with your colleagues.
Item 1:
In a lesson on multiplying fractions, several of Ms. Rose’s students came up with different area models to represent 2/5 x 3/7. Which of the following models should Ms. Rose accept as correct?
A. Ariel: I started with drawing 2/5 by dividing the rectangle into fifths and shading two parts. Then I divided that region into sevenths and shaded three parts. Since the overlapping region is 6 out of 14 squares, the answer is 6/14.
B. Brenda: I think we should start with 3/7 because the problem is two-fifths of 3/7. So, I first divide the rectangle into sevenths and shade three parts to make the 3/7. Then I take two-fifths of that shaded part by dividing into fifths and shading two parts. So the region I shaded twice includes the final parts, which is 6, out of 15 squares. So the answer is 6/15.
C. Carlita: I think you are both right about the part shaded twice being the final parts, but I think the total number of parts should be 35 and not 14 or 15. So the answer is 6/35.
D. James: I think the total number of parts should be 35. I ended up with 35 parts and 6 of them shaded, but they are a different shape than Carlita’s. My answer is 6/35.
ShowHide Solution
Answers A and B are incorrect. Ariel and Brenda each began by partitioning the area model correctly, according to their respective strategies. However, both then focused just on the part shaded initially and neglected to partition the full area with their second partitioning. This resulted in their responses having the correct numerator but an incorrect denominator. Ariel’s denominator implies that there is a total of 14 equal parts in the area model after partitioning, as opposed to 35. Similarly, Brenda’s denominator implies that there is a total of 15 equal parts in the area model after partitioning.
Answers C and D are both correct. In answer C, Carlita partitions the entire rectangular area into equal parts. For the problem of 2/5 x 3/7, she partitioned the rectangle horizontally into fifths, and vertically into sevenths. Two parts of the fifths and three parts of the sevenths are shaded and the overlapping parts show the results of the product 2 x 3 (i.e., 6 rectangles) in relation to the total number of rectangles, 5 x 7 (i.e., 35 rectangles). Thus, the result of multiplying 2/5 x 3/7 is 6/35.
In Answer D, James used a similar approach to Carlita but partitioned the rectangle vertically into fifths and horizontally into sevenths, to arrive at the same answer.
This task addresses multiplying non-unit fractions using area models. Area models are useful tools for students to use in learning to multiply fractions and can help students avoid the misconception that “multiplication always makes bigger”. As the incorrect solutions show, it is important that students keep track of the units in the problem so that both the numerator and the denominator in their answers are correct.
Item 2:
Mr. Wesley assigned his students to work in groups on some problems involving division of fractions. When he was walking around to check his students’ work, he noticed one of his students, Bonnie, wrote the following: 5/6 ÷ 1/2 = 6/10. What might be the best interpretation(s) of Bonnie’s work?
- Bonnie correctly applies the “invert and multiply” algorithm.
- Bonnie confuses division of fractions with multiplication of fractions.
- Bonnie knows that she needs to “invert and multiply” but does not apply the algorithm correctly.
- Bonnie does not use estimation to see that ⅚ is close to 1 and that there are 2 one-halfs in 1, so the answer should be close to 2. Since 6/10 is less than 1, it could not be correct.
ShowHide Solution
Answers A and B are not correct. Bonnie does not apply either the division or the multiplication algorithm correctly. If she had divided correctly, her answer would have been 10/6 or 5/3. If she had multiplied instead of divided, her answer would have been 5/12.
Answer C is correct. Bonnie’s solution follows the “invert and multiply” algorithm, but instead of inverting the second fraction (the multiplicand), she inverted the first fraction (the multiplier) so she answered that 6/5 x 1/2 = 6/10. The correct solution to the problem using the algorithm is 5/6 x 2/1 = 10/6 or 5/3.
Answer D is also correct. Students often fail to apply estimation to check their answers with fractions, even if they use estimation with integers. Estimating with fractions is an important skill that can help students avoid errors in solving problems with fractions.
This problem highlights why an exclusive focus on procedure when it comes to division of fractions can lead to errors in computation. There are a number of ways to incorrectly apply the “invert and multiply” algorithm, and students need opportunities to develop their conceptual understanding of the division of fractions to support their procedural understanding. With this, students would be able to better assess whether their results are reasonable and ultimately correct.
Item 3:
Item adapted from: 'Multiply or Divide?' from http://wested.mediacore.tv/media/multiply-or-divide-problems
Which of the following story problems can be solved by performing the operation 1 ¾ x ½?
- How many cups of sugar do you need to make a half batch of cookies if a full batch takes 1 ¾ cups of sugar?
- How many posters can you paint with 1 ¾ cans of paint if one poster takes ½ can of paint?
- How many pounds of birdseed do you need to fill a bird feeder if 1 ¾ pounds of birdseed fills the bird feeder ½ full?
- How many servings of lemonade can you make if you have 1 ¾ cups of lemonade and a serving is ½ cup?
- I only
- II only
- III only
- I and III only
- II and IV only
ShowHide Solution
Solution: Option A is correct. Problem I explains that a full batch of cookies requires 1 ¾ cups of sugar. As a result, in order to make a half of a batch, only half of the 1 ¾ cups of sugar is needed, or 1 ¾ x ½ = ⅞ cups.
Problem II describes a scenario in which ½ can of paint is required to paint one poster. With 1 ¾ cans of paint, you can identify how many posters you can paint by dividing the paint you have into sets of “½-cans”. You would have 3 ½ “½-cans”, which means that you could paint 3 ½ posters. This can be represented by the number sentence 1 ¾ ÷ ½ = 3 ½, as opposed to 1 ¾ x ½. Therefore, options B and E are incorrect, since they include problem II.
Problem III describes a scenario in which 1 ¾ pounds of birdseed fills half of a bird feeder. It therefore takes 2 sets of 1 ¾ pounds of birdseed, or 3 ½ pounds, to fill a bird feeder completely. This can be represented by the number sentence 1 ¾ x 2 = 3 ½. Therefore, options C and D are incorrect.
Problem IV describes a scenario that is similar to problem II. Each serving of lemonade measures ½ cup. You can divide the 1 ¾ cups of lemonade you have into “½-cups” to identify how many servings you can make. You would have 3 ½ “½-cups”, which means that you could make 3 ½ servings of lemonade. This can be represented by the number sentence 1 ¾ ÷ ½ = 3 ½, as opposed to 1 ¾ x ½.
This item highlights the complexity of division and multiplication meanings. It is worthwhile to link back to the operations with whole numbers (what should be an appropriate operation in the same word problems by replacing with whole numbers) and observe the structure of the operations when changing to fractions.
