# Rational Problem Solving Tim is mixing 1 L of juice concentrate with 5L of water to make juice for his 10 guests.

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Students should be able to compare numbers and manipulate the values to derive other forms of the numbers to make comparing less inhibiting and more accessible.

Students will also use ratios and proportional reasoning to solve problems in various contexts.

Student 1: I think Mix V will be the most orangey because it has 5 cups of concentrate, and no other juice has that much, so it has to be the most orangey. Student 2: Well, the denominator is the total, so the total would be 3 cups. But shouldn't we be able to use the fraction $\frac$? Student 1: Because when we compared all of the ratios, they were equivalent: 2 to 3 is equivalent to 4 to 6, is equivalent to 10 to 15.

Student 2: I think Mix S and Mix T will be the same because they each have only one more cup of concentrate than juice, so they should taste the same, shouldn't they? Student 2: Find out what percent of each is concentrate? In Mix S, though, there are 3 cups of concentrate AND 2 cups of water, so there's really 5 cups of ingredients in the juice. If we write them all as fractions we can see that even better. Note that this does not necessarily imply that "hours spent on homework" = 2 or that "hours spent in school" = 7.

: A recipe calls for milk, flour and sugar in a ratio of 4:6:3 (this is how recipes are often given in large institutions, such as hospitals).

How much flour and milk would be needed with 1 cup of sugar?

Student 1: Won't it depend on the brand of juice we use, too? Teacher: So if we wanted to know what fraction is concentrate, what fraction would we use? Again, we are trying to figure out which one is the most orangey. Teacher: So, we can see from these two strategies that Mix T is the most orangey. Teacher: Let's move on to the next part of the problem. For each mix, you need to now figure out how many batches are needed to be made to serve all of the campers. Student 4: I think we need to first figure out how many cups are in each batch of juice. Student 4: So we can scale the recipe up to make enough to feed the campers. Go ahead and give it a try and let me know how it goes. You are right that you need 100 cups, but that is 100 cups of juice. We did that same process for the rest of them, too, like our table shows.

Teacher: You are right; that could be a factor, and that could be something we explore at a different time, but for today, we are going to assume that all the juices are the same brand, so that won't be a factor. Cam and Scott are in charge of making juice for the 200 students at camp. Student 2: $\frac$, because 5 is the total amount of cups in the juice and 2 is the number of cups of concentrate. The teacher continues to circulate around the room, listening to conversations that different groups are having. Which mix does your group think is the most orangey? We set up ratios of concentrate to water, and compared the decimal values. Student 3: We found out what percent of the juice was concentrate. Student 3: Can't we just multiply 200 by $\frac$ to get 100? That doesn't say how many cups of each of the 2 ingredients you need. By doing it our way, we won't have as much left over, because they rounded the number of batches, and we set up proportions so we have a closer number of cups that would be needed. To finish the lesson, here is your final task: Which of the following will taste most orangey: 2 cups of concentrate and 3 cups of water; 4 cups of concentrate and 6 cups of water; or 10 cups of concentrate and 15 cups of water?

(This is how recipes are often given in large institutions, such as hospitals.) How much flour and milk would be needed with 1 cup of sugar? Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. Teacher: How many of you have made orange juice from a can before? Teacher: When you made juice or helped, what did you have to do? So, we knew we needed 100 cups total, and there were 5 total cups in the batch made.

Student 3: Make sure we didn't add too much water or it wouldn't taste very good. Student 2: Make sure we add enough water or it would be too sour and strong tasting. Student 4: We used proportional reasoning to solve it. Since we needed two cups of concentrate, then we can scale that up.

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