Nicole Hawkins
CCLM^2 Project, Summer 2012
This material was developed for the Leadership for the Common Core in Mathematics (CCLM^2) project
at the University of Wisconsin-Milwaukee.
Part 1 Grade Level: 6 Domain: Ratios and Proportional Relationships (6.RP) Cluster: Understand ratio concepts and use ratio reasoning to solve problems. Standard: 2. Understand the concept of a unit rate a/b associated with a ratio a:b with b 0, and use rate language in the context of a ratio relationship. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger." Part 2 Explanation 6.RP.2 is intricately related to 6.RP.1 and they should be taught together within the same context. In 6th grade, the progression document explains, the standards do not require that students set up equations to solve ratio relationships. As students explore proportional relationships, they will draw on their knowledge of the multiplication table to create tables of pairs of equivalent ratios. During this process, the idea of unit rate will naturally emerge. As students discover the multiplicative relationship between equivalent ratios, they will move into being able to calculate them. Unit rate is really calculating the ratio that is “for each one.” Unit rate is finding the number of units of one quantity (A) per 1 unit of the other quantity (B) for a ratio A:B. It can be said as _____ units of A for each one unit of B. Students need to become proficient in using correct ratio language in order to understand the meaning of the ratio and specifically the meaning of a unit rate. Along those lines, the students need to be very specific in their labels and in keeping the ratio in context of how it was presented. (See example 2) Finally, tie unit rate to graphing the ratios on a coordinate plane as the place where the line hits the one on the x or the y axis. This shows the relationship between the original ratio and the unit rate. Students need to see unit rate as a specific kind of ratio to understand its purpose. Examples Example 1: Joe found a recipe in his grandmother’s recipe box for lasagna. It calls for 16 ounces of cream cheese to make enough lasagna to serve 8 people. Joe only needs enough for himself so he decides to cut the recipe down. How much cream cheese will he need to have to make a mini-­‐lasagna for himself? 16 ounces of cream cheese : 8 servings Ounces of cream cheese 16 8 4 2 Servings 8 4 2 1 Answer: Joe will need 2 ounces of cream cheese for each serving of lasagna. Example 2: Julie is training for a marathon. She ran 7 miles in 56 minutes. In this problem, the unit rate could either be 8 minutes per mile or 1/8 of a mile per minute. Students need to know what the problem is asking before they can solve. The most common answer would be the 8 minutes per mile as then Julie could solve how long it would take her to run the marathon. However, it is possible that the unit rate per minute could be used if she wants to go for a training run one day but perhaps she only has 30 minutes to run. She could figure out how far to go in order to still be able to get home in time. Part 2 We use the Prentice Hall textbook series. However, we are looking in the next year or two to update our text series. Because of this, I wanted to look at the Big Ideas texts as well as our current texts. In the 6th grade textbook, there is one lesson on unit rate. The text explains unit rate in a couple of different ways, but it is very heavy on the numerical equation part of ratios and unit rates. The first example shows finding the unit rate by dividing the numerator by the denominator because “A unit rate has the denominator of 1.” The second example shows a picture model of finding the calories in one serving. However, instead of building a table showing the relationship between number of servings and number of calories, the example again just shows the division strategy. Moving on, the book gives an example of using unit rate to solve problems. The way given to solve the problem is to multiply both terms by the same number. Lastly, unit price is given as a separate instructional section which is given as a way to compare costs. This instruction is not incorrect, but it leaves a lot to be desired in terms of deeper understanding. As we look at the common core, keeping in mind the 8 Math Practices, this instruction is lacking. The equation method would most likely produce student understanding limited to rote application of steps. Students would not understand what unit rate really means and would only be able to apply it to a limited number of situations. It would not become a tool that they could use in multiple situations. If students are given the opportunity to explore ratios and eventually discover unit rate, they will have done work with tables and equivalent ratios and graphing. Then, they will see how unit rate fits into the bigger picture of ratios. When a student is asked to persevere through a problem that may be unfamiliar to them, they need to see how the tools they have learned fit in the mathematical world so they can determine the appropriate application of them. Additionally, when a student is asked to defend their thinking in solving a problem, it is not sufficient to say they multiplied both terms by the same number. That does not show any depth of understanding. As we go on to Grade 7, there is again one lesson on unit rate. It is almost identical to the lesson in the Grade 6 book. The only additional steps that students have to do is convert something like minutes and seconds into just seconds in order to find unit rate. There are no additional methods of solving for unit rate. I then looked into the 6th grade book of Big Ideas Math, hoping to see many different examples of ratios and rates and their graphs and how the unit rate fits into this, but was disappointed to see that the instruction was very similar to the Prentice Hall text. There were more examples, but the equation method was the go-­‐to to solve the problems.