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# Equations that Describe Patterns

## Describe numerical sequences by finding a rule.

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Cheez Its
Teacher Contributed

## Real World Applications – Algebra I

### Topic

How are algebraic relationships represented in the nutrition facts of snacks?

### Student Exploration

Below is a copy of the nutrition facts for a box of Cheez Its (Four Cheese), one of the most popular snack foods.

There are a lot of different ways that we can represent the information in this picture.

Let’s say that I’m addicted to Cheez Its and I can never really have just one serving size, which is 36 crackers. We can represent this relationship in a table.

# of serving size 0 1 2 3 4
# of crackers total 0 36 72 108 144

According to this table, 144 crackers eaten is equivalent to 4 servings! That’s a lot!

If we were to calculate the number of crackers in 10 servings, we can write this as an expression and figure it out. If x\begin{align*}x\end{align*} represents the number of serving sizes, the expression is 36x\begin{align*}36x\end{align*}. So, in 10 servings, I could eat 360 crackers. (I might also have a stomach ache too.)

We can also represent this relationship as an equation. If y represents the number of crackers, our equation would be y=36x\begin{align*}y = 36x\end{align*}. If I wanted to eat 200 crackers we can figure out how many serving sizes that would equate to. 200=36x\begin{align*}200 = 36x\end{align*}. We would divide both sides by 36, and find that x=5.6\begin{align*}x = 5.6\end{align*}, or a little over 5 and a half serving sizes.

When you get older, you will start to see that a lot of adults care about their calorie intake. According to the nutrition facts, one serving size has about 140 calories. People usually like to base what they eat on a 2000 calorie diet. Taking this fact into account, what is the maximum number of cheez its that I can eat in a day? Let’s create another equation to represent this relationship.

Let x\begin{align*}x\end{align*} represent the number of cheez its I eat, and let y\begin{align*}y\end{align*} represent the number of calories. Our equation is xy=36140\begin{align*}\frac{x}{y} = \frac{36}{140}\end{align*}. This special equation is also called a proportion. If we use multiplication to eliminate the denominators, our new equation is (140x)36=y\begin{align*}\frac{(140x)}{36} = y\end{align*}. Since we can’t exceed our 2000 calorie diet, our inequality is (140x)362000\begin{align*}\frac{(140x)}{36} \le 2000\end{align*}. What value of x\begin{align*}x\end{align*} would make this statement true?

We can use just two steps to figure this out! First, multiply both sides by 36 to eliminate the denominator. We now have 140x72000\begin{align*}140x \le 72000\end{align*}. Next, we divide both sides by 140. So, x514.3\begin{align*}x \le 514.3\end{align*}. This means that I can’t eat more than 514 crackers in one day, assuming that’ll be the ONLY food I eat all day.

Now we have the maximum amount of crackers I can have in one day, how many servings is this?

51436=14\begin{align*}\frac{514}{36} = 14\end{align*} servings.

Note: It is not suggested that a person eat 514 Cheez Its!

### Extension Investigation

Try looking at the nutrition facts label on your favorite food, and figure out what the maximum amount you can eat of that particular food, based on a 2000 calorie diet.

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