Creating Expressions to Estimate Cost, Part 1

1.1.2 Creating Expressions to Estimate Cost, Part 1

1.1.2 • Creating Expressions to Estimate Cost, Part 1

Activity

Imagine your class is having a pizza party.

Work with your group to plan what to order and to estimate what the party would cost.

A delicious pizza slice with melted cheese stretching, topped with pepperoni, mushrooms, olives, and green peppers, alongside fresh vegetables on a wooden surface.

What is your favorite type of pizza to order?

Compare your answer:

Your answer may vary, but here are some samples:

A supreme pizza that has pepperoni, sausage, bell peppers, mushrooms, and olives.
A deep-dish cheese pizza.
A thin crust pizza with hamburger meat.
A Hawaiian pizza that has Canadian bacon and pineapple on it.
Any type of pizza is my favorite!

How many toppings are on your favorite type of pizza?

Compare your answer:

Your answer may vary.

What type of crust does your favorite pizza have?

Compare your answer:

Your answer may vary, but here are some samples:

Pan crust
Deep-dish
Stuffed crust
Thin and crispy
Hand-tossed

For questions 4 – 9, use the following group instructions.

Work with a group to research the cost of ordering pizza from a local restaurant. Then, determine what to order for the class pizza party and determine how much it would cost. Answer the following questions regarding your group's plan.

What is the cost estimate for your group's planned class party?

Compare your answer:

Your answer may vary.

Write down one expression that show how your group’s cost estimate was calculated.

Compare your answer:

Your answer may vary, but here are some samples:

$10 x (3 (1.50) + 21.99) + 2.50$

In your expression, which quantities, if any, might change on the day of the party?

Compare your answer:

Your answer may vary, but here are some samples:

The delivery cost might change.
The number of toppings we select may change.
The total number of pizzas ordered may change if someone is absent.

Rewrite your expression, replacing the quantities that might change with letters.

Compare your answer:

Your answer may vary, but here is an example.

$(1.50 t + 21.99) p + d$

What do the letters represent?

Compare your answer:

Your answer may vary, but here are some samples:

$t =$ number of toppings
$p =$ number of pizzas ordered
$d =$ delivery charge

How would you convince the class to go with your group’s plan?

Compare your answer:

Your answer may vary, but here is an example.

Ordering 10 extra large pizzas is cheaper than ordering 30 small pizzas. If three people share a pizza, they can have up to three toppings or they can each get their favorite topping on their portion of the pizza. And, if someone wants more pizza, then two people can share one pizza instead of three people.

Are you ready for more?

Extending Your Thinking

A local pizzeria sells gourmet pizzas for the prices posted below. Remember the sizes are determined by the diameter of each pizza. Compare the cost per square inch of the sizes.

Pizza size	10” Small	12” Medium	14” Large	16” XLarge
Price	$12.99	$15.99	$19.99	$21.99

This table lists various pizza sizes and their respective prices.

How did you determine the cost per square inch of each pizza?

Compare your answer:

First, determine the total area of the pizza using the area formula for a circle, ${π r}^{2}$ . Then, divide the cost of the pizza by the area measurement. The expression could be $\frac{c}{{π r}^{2}}$ , if $c = c o s t$ .

Which size pizza is the better deal?

Compare your answer:

Xlarge

Which size pizza is the most expensive per square inch?

Compare your answer:

small

What is the difference in price per square inch between the most and least expensive sizes?

Compare your answer:

.06

Self Check

A ninth-grade class is ordering pizza for lunch. There are

n

students in the class. Each slice of pizza costs $3 and the delivery fee is $8. Which expression represents the total cost if each student gets 2 slices of pizza?

$3n + 8$
$(2)3n + 8$
$(2)3n$
$3n$

Additional Resources

Translate an English Phrase to an Algebraic Expression

You can use the following operation symbols to translate English phrases into algebraic expressions.

Look closely at these phrases using the four operations:

The SUM of $a$ and $b$

$a + b$

The DIFFERENCE of $a$ and $b$

$a - b$

The PRODUCT of $a$ and $b$

$a \cdot b = a b = (a) (b)$

The QUOTIENT of $a$ and $b$

$\frac{a}{b} = a \div b$

This table lists basic mathematical operations (sum, difference, product, quotient) for variables 'a' and 'b', showing their verbal descriptions and corresponding algebraic notations.

Classroom Connection

Log into student.desmos.com using the information provided by your teacher to complete the activity.

Identify other phrases that represent the same algebraic concepts by categorizing the following list according to the four operations.

$b$ added to $a$	$a$ divided by $b$	$b$ subtracted from $a$
The product of $a$ times $b$	$a$ sets of $b$	The sum of $a$ and $b$
$a$ plus $b$	The ratio of $a$ and $b$	$a$ decreased by $b$
$b$ less than $a$	$a$ increased by $b$	$b$ divided into $a$
The quotient of $a$ and $b$	$b$ more than $a$	The total of $a$ and $b$
$a$ times $b$	$a$ minus $b$	The difference of $a$ and $b$

This table provides various verbal phrases representing common mathematical operations (addition, subtraction, multiplication, and division) between variables 'a' and 'b'.

The SUM of $a$ and $b$ $a + b$	The DIFFERENCE of $a$ and $b$ $a - b$	The PRODUCT of $a$ and $b$ $a \cdot b = a b = (a) (b)$	The QUOTIENT of $a$ and $b$ $\frac{a}{b} = a \div b$
$a$ plus $b$	$a$ minus $b$	$a$ times $b$	$a$ divided by $b$
$b$ added to $a$	$b$ subtracted from $a$	The product of $a$ times $b$	$b$ divided into $a$
$b$ more than $a$	$b$ less than $a$	$a$ sets of $b$	The ratio of $a$ and $b$
$a$ increased by $b$	$a$ decreased by $b$		The quotient of $a$ and $b$
The sum of $a$ and $b$	The difference of $a$ and $b$
The total of $a$ and $b$

This table illustrates common verbal expressions and their corresponding symbolic representations for the four basic arithmetic operations: sum, difference, product, and quotient, using variables 'a' and 'b'.

Each phrase tells us to operate on two numbers. Look for the words "of" and "and" to find the numbers.

Later in this course, we’ll apply our skills in algebra to solving applications. The first step will be to translate an English phrase to an algebraic expression.

Example

The length of a rectangle is 14 less than the width. Let $w$ represent the width of the rectangle. Write an expression for the length of the rectangle.

Step 1 - Write a phrase about the length of the rectangle.

14 less than the width

Step 2 - Substitute $w$ for "the width."

14 less than $w$

Step 3 - Rewrite less than as subtracted from.

14 subtracted from $w$

Step 4 - Translate the phrase into algebra.

$w - 14$

Try it

Translate an English Phrase to an Algebraic Expression

Translate the following into an algebraic expression:

June has dimes and quarters in her purse. The number of dimes is seven less than four times the number of quarters. Let $q$ represent the number of quarters. Write an expression for the number of dimes.

Compare your answer:

Here is how to translate this into an algebraic expression:

Step 1 - Write a phrase about the number of dimes.

seven less than four times the number of quarters

Step 2 - Substitute $q$ for the number of quarters.

7 less than 4 times $q$

Step 3 - Translate $4$ times $q$ .

7 less than $4 q$

Step 4 - Translate the phrase into algebra.

$4 q - 7$