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# Dimensional Analysis

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## Real World Applications – Algebra I

### Topic

How can we make enough chocolate chip cookies for an exact amount of people with only one recipe?

### Student Exploration

In this individual activity, you’re going to calculate how much of everything we need in order to make enough cookies for a lot of people. For the sake of this activity, we’re going to use Algebra for you to figure out how much of each ingredient we need to feed everyone in your class, classes (if you have more than one class), the whole grade level, and your whole school.

First, let’s dig into what the recipe’s telling us. How many cookies does this recipe give us? Will it be enough for a classroom of students and teachers, assuming each person will eat two cookies?

Let’s say that some of the other teachers at your school want you to make cookies for their classes too. We want to quadruple the recipe. How much of each ingredient will you need, if you now need to bake cookies for 150 students (or more, depending on the school you attend)?

If there are 450 students in the whole school, what is the scale factor, or the multiplier, for each ingredient?

Well, if we are trying to give cookies for 450 people, that’s 900 cookies that we need to bake. That means we need a multiplier of 19 to make sure that everyone will get at least one. That means we’ll need 19 cups of butter, 19 cups of white sugar, 19 cups of brown sugar, 38 eggs, 38 teaspoons of vanilla extract, 57 cups of flour, 19 teaspoons of baking soda, 38 teaspoons of hot water, and 9.5 teaspoons of salt, 38 cups of chocolate chips, and 19 cups of chopped walnuts. That’s a lot of ingredients!

One thing most people do NOT want to do is to measure and count 19 cups of any ingredient. There’s an easier way! By using conversions!

First, let’s look at how much flour we need. 1 cup of flour is 8 ounces, and 1 pound of flour is 16 ounces. If each bag of flour is 5lbs, then we can get about 10 cups of flour per bag. Since we need 57 cups of flour, we should have 6.5lb bags of flour. Measuring 7 cups of flour is better than measuring 57 cups of flour! See below on how we used conversions to find how many cups of flour are in one 5lb bag:

$& \underline{1 \ cup \ of \ flour} \qquad \times \qquad \underline{16 \ ounces} \qquad \times \qquad \underline{5 \ pounds \ of \ flour} \quad = \quad 10 \ cups \ of \ flour \ per \ bag\\& \quad 8 \ ounces \qquad \qquad \quad \ 1 \ pound \ of \ flour \qquad \qquad \ \ 1 \ bag \ of \ flour$

The ounces units “cancel” in both the numerator and the denominator, and so the units of “pounds of flour,” so the units left are “cups of flour.” Using arithmetic, we multiply the numerators and divide by the denominator.

We can also use proportions to find out how many bags of flour we need for our recipe. From our calculations above, we know that 10 cups of flour are in one bag. What if we have 57 cups of flour? See below:

$& \underline{10 \ cups \ of \ flour} \ = \ \underline{57 \ cups \ of \ flour}\\& \ 1 \ bag \ of \ flour \qquad \ x \ bags \ of \ flour$

Using the method described in the concept, we now have $10x = 57$. Divide both sides by 10, so $x = 5.7$. This means that we’ll need 5.7 bags of flour. But, of course, when we’re making 450 cookies, we will realistically get 6 bags of flour, use 5 whole bags for the recipe and measure 7 cups out of the sixth bag.

Applying our knowledge of mixed numbers, we can also represent the amount of flour needed as $\frac{57}{10}$ bags of flour, or $\frac{57}{10}$ bags of flour. Which representation do you think would be more useful for measuring and baking?

Try doing the same for the other ingredients. Here’s some helpful information, but you might not need all of it. Some of the conversions are here because big bags of some ingredients are given by weight (i.e. in pounds), instead of measurement. You might also want to apply your understanding of mixed numbers to clearly figure out fractions of certain ingredients.

For butter: $8 \ tbsp =\frac{1}{4} \ lb, \frac{1}{4} \ lb=\frac{1}{2}\ cup, 2 \ cups = 1 \ lb$. Butter is usually sold in packages of 4 sticks of 8tbsp each.

For white sugar: $1 \ cup = 0.5 \ lb$. White sugar is usually sold in 5lb bags.

For brown sugar: $1 \ cup = 0.44 \ lb$. Brown sugar is usually sold in 2lb bags.

For eggs: Eggs are usually sold in packages of 12 or 18.

For vanilla extract: $1 \ oz = 8 \ tsp$, $1 \ fluid \ ounce = 6 \ tsp$. Small bottles of vanilla extract are usually sold in 8 fluid ounce bottles.

For baking soda: $1 \ tsp = 0.1667 \ oz$. Baking soda is usually sold in 8oz and 1lb boxes.

Now that we’ve figured out how to make a really big batch of cookies, how would we figure out how to make a really small batch of cookies? The recipe on the website given yields 4 dozen cookies, or 48 cookies. What if we want one dozen cookies?

If we want one dozen cookies, we would divide all of the measurements by 4.

What if we wanted to only make 10 cookies with no leftovers? How could we alter the recipe?

Let’s use a conversion, or a scale factor. We know that the recipe gives us 48 cookies, but we only want 10 cookies. We can divide 10 by 48 and simplify the fraction, since it’s best to use measurements for cooking with fractions. $\frac{10}{48}= \frac{5}{24}$. Now that we have our factor, we can multiply every measurement by this fraction to find our new measurements.

$\left(\frac{5}{24}\right) (1 \ cup \ butter) = \frac{5}{24} \ cups \ of \ butter$

$\left(\frac{5}{24}\right) (1 \ cup \ sugar) = \frac{5}{24} \ cups \ of \ sugar$

$\left(\frac{5}{24}\right) (1 \ cup \ brown \ sugar) = \frac{5}{24} \ cup \ brown \ sugar$

$\left(\frac{5}{24}\right) (2 \ eggs) = \left(\frac{5}{24}\right) \left(\frac{2}{1}\right) = \frac{10}{24} = \frac{5}{12} \ eggs$

As you continue finding the other ingredients’ measurements, what do you notice?

How can wet get $\frac{5}{24}$ cup of butter? How can we get $\frac{5}{12}$ of one egg?

What unit fractions are close to $\frac{5}{24}$? What unit fractions is close to $\frac{5}{12}$? Do you think we can use $\frac{6}{24}$, or $\frac{1}{4}$ of a cup for some of these ingredients? Do you think we can use $\frac{6}{12}$ or $\frac{1}{2}$ of an egg instead of $\frac{5}{12}$ of an egg?

What do you think the benefits are from changing the amount of some of the ingredients? What are some of the drawbacks?

Do you think it would be easier to make a dozen cookies instead?

### Extension Investigation

What if we don’t have all of the measuring tools we need? What if you only had certain measuring tools?

Try finding a favorite family recipe and calculate how much of each ingredient you will need to make enough for your entire school. You might need the internet to help you with conversions with different measurements in weight and volume.