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Guided Practice
Algebraic expressions can be written using words or symbols.

For example, "subtract y from 5" can be written as 5 – y.

If Raul is 12 years old, and his sister is 2 years older, you can write 2 + r to represent Raul's sister's age.

"Five groups of t toys each" can be written as 5 × t or 5t.

If you take the 5t toys and divide them into g equal groups, you will have 5t ÷ g, or 5t/g toys in each group.

To evaluate an algebraic expression, first substitute the number for the variable. Then perform the operation.

For example, evaluate the expression x + 9 for x = 1.

StepResult
Substitute the number for the variable.x + 9 → 1 + 9
Perform the operation.1 + 9 → 10    

Complete the algebraic expression to represent the area of the shaded section.

12(a + 20) – ×

Write an algebraic expression to represent the total amount. Use a correct operation: (+, -, x, or /).

Write an algebraic expression to represent the length of the segment.

Write an algebraic expression to represent the amount of water in each container if the same amount is poured into each, with none left in the original container.

Write an algebraic expression to represent the situation.

"3 more than q"

Simone has 21 candies that she will share with f friends. Write an algebraic expression to represent the amount each friend will get.

Zack has 21 boxes of shoes with f shoes in each box. Write an algebraic expression to represent how many shoes he has in all.

Evaluate the expression a + 8 for a = 39.

+ 8 =

Evaluate the expression t × 1.5 for t = 6.

× 1.5 =

Evaluate the expression 28 ÷ j for j = 28.

28 ÷ =

Evaluate the expression 100 – w for w = 93.

100 – =

Many math expressions have special names. When you add two or more numbers, the expression is called a sum. A multiplication expression is called a product. The expressions multiplied in a product are called factors. The expressions added together in a sum are called addends. A division expression is called a quotient.

sum
6 + 2
a + 5
c + x + 3
product
5 × 5
a • 1 • 2
2p
quotient
3/9
10 ÷ m

The product 2p has two factors: and .

Rewrite the quotient 3/9.

3/9 = ÷

Identify the sum in the expression. Write the sum without parentheses.

r • 5 • (3 + x) • 0

Sum:

Identify the sum in the expression. Write the sum without parentheses.

10 ÷b/2• (4 + q)

Sum:

This expression has two factors. Write the factors without parentheses.

(4 + s) • (m + 1)

Factors: and

This expression has two factors. Write the factors without parentheses.

(d + t) • (5 + 6.09)

Factors: and

This expression has two factors. Write the factors without parentheses.

12(g +1/3+ 8)

Factors: and

Think of this expression as having two addends. Write the addends without parentheses.

(4m) + (2.5 – 1.5)

Addends: and

Think of this expression as having two addends. Write the addends without parentheses.

7w + ( 1 –1/8)

Addends: and

This expression has two addends. Write the addends without parentheses.

2z + bg

Addends: and

This expression has two factors. Write the factors without parentheses.

(6 + m) • (0.5 + m)

Factors: and

Rewrite the quotient 5/11.

5/11 = ÷

Rewrite the quotient 8/9.

8/9 = ÷

A coefficient is usually a number that comes in front of a variable and is multiplied.

The coefficient of 3m is .

When a variable is by itself, without a number in front of it, its coefficient is 1.

The coefficient of the expression y is .

A fraction like m/6 is equal to 1/6 • m. So, the coefficient of the expression m/6 can be 1/6.

The coefficient of n/9 is .

The coefficient of 3x2 is .

You can write some expressions with variables in different ways and they stay the same. These are equivalent expressions. For example, the addition expression y + y + y is equivalent to the multiplication expression 3 • y, or 3y.

You can substitute any number for y to see that the expressions are equivalent:

(2.5) + (2.5) + (2.5) = 3 • 2.5

Write the multiplication expression equivalent to p + p + p.

p + p + p =

Write the power expression equivalent to m • m.

m • m =

Write the one-value expression equivalent to
0 • b.

0 • b =

Write the one-variable expression equivalent to
c • 1.

c • 1 =

Write the one-variable expression equivalent to
q + 0.

q + 0 =

Write an expression equivalent to (m + 5) • 1.

(m + 5) • 1 = +

Write a multiplication expression equivalent to
(s + 1)3.

(s + 1)3 =

Write the single-term expression equivalent to
(m + 1/2 + 10) • 0.

(m + 1/2 + 10) • 0 =

Write the addition expression equivalent to 2x.

2x = +

=

You can enter equivalent expressions in the tool on the left. A statement which shows that two expressions are equivalent is called an equation.

Try these equivalent expressions. Enter them in the tool and press go. Notice that the model is not drawn to scale.

  • 3y = 60
  • 60 = 3q
  • a = 20
  • 2m + 15 = 35
  • 200 = 10 + 4z
  • g + 10 = 90

Let's explore how to set up and solve equations in order to solve problems.

Suppose you buy 6 team jerseys for your whole family and spend $126. If each jersey costs the same amount, you can set up and solve an equation with one variable to figure out how much each jersey costs.

6 jerseys × j dollars = 126 total dollars

Enter this relationship as the equation 6j = 126. What is the cost of each jersey? How do you know?

Each of the 7 people at a party won the same number of prizes. If they won 98 prizes in all, how many prizes did each person win?

You can set up and solve an equation with one variable to figure out how many prizes each person won.

7 people × p prizes = 98 total prizes

Enter this relationship as the equation 7p = 98. How many prizes did each person win? How do you know?

Suppose you jog at a fast 13 miles per hour. How long will it take for you to jog 78 miles?

You can set up and solve an equation with one variable to figure out the time it will take to run 78 miles.

13 miles per hour × h hours = 78 miles

Enter this relationship as the equation 78 = 13h. How long will it take? How do you know?

You can draw on the model to reason about problems. This model shows 5x = 180.

You can see that x = 36. To determine the value of x, you can divide: 180 ÷ 5 = 36. Why does this work?

Suppose that you bought 6 team jerseys, and you spent $24 on a hat, and the total was $126. Let's model this new situation with an equation.

6 jerseys × j dollars + 24 dollars = 126 dollars

You can set up and solve an equation to solve this problem: 6j + 24 = 126.

Now how much does each jersey cost? Explain what operations you can use to figure it out. How does this compare with the previous problem about buying jerseys?

Let's study the model for 6j + 24 = 126. Draw on the model to show the reasoning below.

Both the model and the equation show you that 6j + 24 is equal to 126. But what is 6j equal to? It must be equal to 126 – 24, or 102. You can even enter a new equation to show this: 6j = 102.

From there, we can see that j is equal to 102 ÷ 6, or 17. So each jersey cost $17.

We can check: 6(17) + 24 = 102 + 24 = 126.

Enter the equation 65 = 33 + 4g. Draw on the model to show the reasoning below.

Both the model and the equation show you that 33 + 4g is equal to 65. But what is 4g equal to? It must be equal to 65 – 33, or 32. You can even enter a new equation to show this: 32 = 4g.

From there, we can see that g is equal to 32 ÷ 4, or 8.

We can check: 65 = 33 + 4(8) = 33 + 32 = 65.

Enter the equation 355 = 7n + 40. Draw on the model to show the reasoning below.

Both the model and the equation show you that 7n + 40 is equal to 355. But what is 7n equal to? It must be equal to 355 – 40, or 315. You can even enter a new equation to show this: 315 = 7n.

From there, we can see that n is equal to 315 ÷ 7, or 45.

We can check: 355 = 7(45) + 40 = 315 + 40 = 355.

Try to solve these equations. Use the model to help you.

  • 8x + 25 = 321
  • 29 + 2r = 87
  • 191 = 10s + 61
  • 215 = 15 + 8y

Your turn. Evaluate the expression k + 2 for k = 6.

k + 2 →

Evaluate each expression for k = 9.

k + 2 →

9 – k →

k • k  →

k ÷ 3 →

Evaluate each expression for d = 13.

(3 • d)   – d ÷ 1 →

(d ÷ 1)   + 3 • d →

(d ÷ 1) + (3 • d) →

(3 • d) – (d ÷ 1) →