5) Illustrative Mathematics: 2 Real-World Problems using Comparing
5.NF.B.5 A
5.NF.B.5 A
![]() I can mentally compare the size of a product to the size of one of the factors by thinking about the other factor in the problem.
Illustrative Mathematics You can click above to go to the site, or look below. Running a Mile Curt and Ian both ran a mile. Curt's time was 8/9 Ian's time. Who ran faster? Explain and draw a picture. Commentary There is a subtlety worth noting: we are given information about the boys' times but asked about their speeds. Since the distance they run is the same, this isn't difficult to reason through. The two solutions reflect different competencies described in 5.NF.5. The first solution uses the idea that multiplying by a fraction less than 1 results in a smaller value. The second actually uses the meaning of multiplying by 8/9 to explain why multiplying by that fraction will result in a smaller value. Solutions Solution #1: Scaling by a number less than 1 To find Curt's time, you would multiply Ian's time by 8/9. Since we are multiplying Ian's time by a number less than 1, Curt's time will be less than Ian's time. The picture shows Ian's time multiplied by 1 above the number line and Ian's time multiplied by 8/9 below the number line. Since they both ran the same distance but Curt ran it in less time, he must have been running faster. Solution #2: Using the meaning of fraction multiplication Curt's time is 8/9× Ian's time. That means that if you divide Ian's time into 9 equal time intervals and take 8 of those intervals, you will have Curt's time. So Curt's time to run a mile is less than Ian's and he must be going faster. |
![]() Fundraising
Cai, Mark, and Jen were raising money for a school trip.
Solutions Solution #1: Drawing a picture It is relatively easy to see that both Cai and Jen raised more money than Mark. The question is whether Cai made more than, the same as, or less than Jen. Drawing a picture can help. The picture shows that Cai raised 2 1/2 as much as Mark by representing the amount that Cai made as a strip that is 2 1/2 times as long as a strip that represents Mark's amount. Conveniently, Mark's strip is already divided into two equal pieces. Since His amount is 2/3 Jen's amount, we just need to add one more of those pieces to represent the total amount that Jen raised. Cai raised 5/2 the amount that Mark raised. Jen raised 3/2 the amount that Mark raised. Mark raised 2/2 the amount that Mark raised. Cai raised the most, Jen raised the second most, and Mark raised the least. Solution #2: No picture While more abstract, one can reason through this without a picture. Mark raised 2/3 as much as Jen, so Jen raised 3/2 as much as Mark. Cai raised 5/2 the amount that Mark raised. Jen raised 3/2 the amount that Mark raised. Mark raised 2/2 the amount that Mark raised. Cai raised the most, Jen raised the second most, and Mark raised the least. |