Cooking with the Whole Cup
Task
Travis was attempting to make muffins to take to a neighbor that had just moved in down the street. The recipe that he was working with required $\frac34$ cup of sugar and $\frac18$ cup of butter.

Travis accidentally put a whole cup of butter in the mix.

What is the ratio of sugar to butter in the original recipe? What amount of sugar does Travis need to put into the mix to have the same ratio of sugar to butter that the original recipe calls for?

If Travis wants to keep the ratios the same as they are in the original recipe, how will the amounts of all the other ingredients for this new mixture compare to the amounts for a single batch of muffins?

The original recipe called for $\frac38$ cup of blueberries. What is the ratio of blueberries to butter in the recipe? How many cups of blueberries are needed in the new enlarged mixture?


This got Travis wondering how he could remedy similar mistakes if he were to dump in a single cup of some of the other ingredients. Assume he wants to keep the ratios the same.

How many cups of sugar are needed if a single cup of blueberries is used in the mix?

How many cups of butter are needed if a single cup of sugar is used in the mix?

How many cups of blueberries are needed for each cup of sugar?

IM Commentary
While the task as written does not explicitly use the term "unit rate," most of the work students will do amounts to finding unit rates. A recipe context works especially well since there are so many different pairwise ratios to consider.
This task can be modified as needed; depending on the choice of numbers, students are likely to use different strategies which the teacher can then use to help students understand the connection between, for example, making a table and strategically scaling a ratio.
The choice of numbers in this task is already somewhat strategic: in part (a), the scale factor is a whole number and in part (b), the scale factors are fractions. Because of this difference, students will likely approach the parts of the task in different ways. The teacher can select and sequence a discussion of the different approaches to highlight the structure of the mathematics and allow for connections to proportional relationships.
This task was submitted by Travis Lemon for the first IMP task writing contest 2011/12/122011/12/18.
Solutions
Solution: Solution


The ratio of cups of sugar to cups of butter is $\frac34 : \frac18$. If we multiply both numbers in the ratio by 8, we get an equivalent ratio that involves 1 cup of butter. $$8\times \frac34 = 6$$ and $$8\times \frac18 = 1$$
In other words, $\frac34 : \frac18$ is equivalent to $6 : 1$, and so six cups of sugar is needed if there is one cup of butter.

In the previous part we saw that we have 8 times as much butter, so all the ingredients need to be increased by a factor of 8. That is, the quantity of each ingredient in the original recipe needs to be multiplied by 8 in order for all the ratios to be the same in the new mixture.

The ratio of cups of blueberries to cups of butter is $\frac38 : \frac18$ in the original recipe, so Travis will need to add $8\times\frac38=3$ cups of blueberries to his new mixture.



The ratio of cups of sugar to cups of blueberries is $\frac34 : \frac38$. If we multiply both numbers in the ratio by $\frac83$, we get an equivalent ratio.
$\frac83 \times \frac34 = 2$ and $\frac83 \times \frac38 = 1$.
Since $\frac34 : \frac38$ is equivalent to $2 : 1$, two cups of sugar is needed if there is one cup of blueberries.

The ratio of cups of butter to cups of sugar is $\frac18 : \frac34$. If we multiply both numbers in the ratio by $\frac43$, we get an equivalent ratio.
$\frac43 \times \frac18 = \frac16$ and $\frac43 \times \frac34 = 1$.
In other words, $\frac18 : \frac34$ is equivalent to $\frac16 : 1$, and $\frac16$ cup of butter is needed if there is one cup of sugar.

The ratio of cups of blueberries to cups of sugar is $\frac38 : \frac34$. If we multiply both numbers in the ratio by $\frac43$, we get an equivalent ratio.
$\frac43 \times \frac38 = \frac12$ and $ \frac43 \times \frac34= 1$.
Since $\frac38 : \frac34$ is equivalent to $\frac12 : 1$, Travis would need $\frac12$ cup of blueberries if there is one cup of sugar.

Instructional Note: For part (b), I have encouraged students to think about unit fractions as an intermediate step to developing an understanding of how to multiply by fractions. With the emphasis on unit fractions in the CCSSM, I decided to use this approach this year and have found success. Students see the value of scaling to a unit fraction and then going from there.
For example, if a student realizes that $\frac38$ needs to become 1 to answer part (b.i), she can first take $\frac13$ of the amount to create a unit fraction of $\frac18$ and then multiply this by 8 to create 1.
The composite result of these calculations is equivalent to multiplying by $\frac83$. Students often find that the two calculations (taking $\frac13$ of the amount to create a unit fraction of $\frac18$ and then multiply this by 8) made independently are more mentally accessible, which makes them a nice intermediate step in understanding the composite calculation of multiplying by the reciprocal.
Solution: Using tables


The ratio of cups of sugar to cups of butter is $\frac34 : \frac18$. If we set up a table, we can successively double the amounts:
cups of sugar $\frac34$ $\frac64$ $\frac{12}{4}$ $\frac{24}{4}=6$ cups of butter $\frac18$ $\frac28$ $\frac48$ $\frac88=1$ So six cups of sugar is needed if there is one cup of butter.

In the previous part, we had to double the quantities three times: $2\cdot2\cdot2=8$. So Travis needs 8 times as much butter as the original recipe required. If we want to keep all the ingredients in the same ratio, Travis needs to multiply the amount of each ingredient by 8.

The ratio of cups of blueberries to cups of butter is $\frac38 : \frac18$ in the original recipe, so Travis will need to add $8\cdot \frac38=3$ cups of blueberries.



It is much harder to solve this problem using a table because the scale factor is no longer a whole number. Students who solved the first part using a table may need guidance from their classmates or the teacher to see that multiplying both numbers in the ratio by the reciprocal of the amount of blueberries will give an equivalent ratio with 1 cup of blueberries. Here is where the teacher should highlight the importance of being able to find a unit rate.
The ratio of cups of sugar to cups of blueberries is $\frac34 : \frac38$. If we multiply both numbers in the ratio by $\frac83$, we get an equivalent ratio.
$\frac34 \times \frac83 = 2$ and $\frac38 \times \frac83 = 1$.
So two cups of sugar is needed if there is one cup of blueberries.

The ratio of cups of butter to cups of sugar is $\frac18 : \frac34$. If we multiply both numbers in the ratio by $\frac43$, we get an equivalent ratio.
$\frac18 \times \frac43 = \frac16$ and $\frac34 \times \frac43 = 1$.
So $\frac16$ cup of butter is needed if there is one cup of sugar.

The ratio of cups of blueberries to cups of sugar is $\frac38 : \frac34$. If we multiply both numbers in the ratio by $\frac43$, we get an equivalent ratio.
$\frac38 \times \frac43 = \frac12$ and $\frac34 \times \frac43 = 1$.
So $\frac12$ cup of blueberries is needed if there is one cup of sugar.

Cooking with the Whole Cup
Travis was attempting to make muffins to take to a neighbor that had just moved in down the street. The recipe that he was working with required $\frac34$ cup of sugar and $\frac18$ cup of butter.

Travis accidentally put a whole cup of butter in the mix.

What is the ratio of sugar to butter in the original recipe? What amount of sugar does Travis need to put into the mix to have the same ratio of sugar to butter that the original recipe calls for?

If Travis wants to keep the ratios the same as they are in the original recipe, how will the amounts of all the other ingredients for this new mixture compare to the amounts for a single batch of muffins?

The original recipe called for $\frac38$ cup of blueberries. What is the ratio of blueberries to butter in the recipe? How many cups of blueberries are needed in the new enlarged mixture?


This got Travis wondering how he could remedy similar mistakes if he were to dump in a single cup of some of the other ingredients. Assume he wants to keep the ratios the same.

How many cups of sugar are needed if a single cup of blueberries is used in the mix?

How many cups of butter are needed if a single cup of sugar is used in the mix?

How many cups of blueberries are needed for each cup of sugar?
