Lesson 2Truth and Equations

Learning Goal

Let’s use equations to represent stories and see what it means to solve equations.

Learning Targets

I can match equations to real life situations they could represent.
I can replace a variable in an equation with a number that makes the equation true, and know that this number is called a solution to the equation.

Lesson Terms

coefficient
constant term
difference
factor
product
quotient
solution to an equation
sum
term
variable

Warm Up: Three Letters

Problem 1

The equation $a + b = c$ could be true or false.

If $a$ is 3, $b$ is 4, and $c$ is 5, is the equation true or false?
Find new values of $a$ , $b$ , and $c$ that make the equation true.
Find new values of $a$ , $b$ , and $c$ that make the equation false.

Problem 2

The equation $x \cdot y = z$ could be true or false.

If $x$ is 3, $y$ is 4, and $z$ is 12, is the equation true or false?
Find new values of $x$ , $y$ , and $z$ that make the equation true.
Find new values of $x$ , $y$ , and $z$ that make the equation false.

Activity 1: Storytime

Here are three situations and six equations. Which equation best represents each situation? If you get stuck, consider drawing a diagram.

$x + 5 = 20$

$x + 20 = 5$

$x = 20 + 5$

$5 \cdot 20 = x$

$5 x = 20$

$20 x = 5$

After Elena ran 5 miles on Friday, she had run a total of 20 miles for the week. She ran $x$ miles before Friday.
Andre’s school has 20 clubs, which is five times as many as his cousin’s school. His cousin’s school has $x$ clubs.
Jada volunteers at the animal shelter. She divided 5 cups of cat food equally to feed 20 cats. Each cat received $x$ cups of food.

Activity 2: Using Structure to Find Solutions

Here is a list of values followed by equations that each contain a variable. Think about what each equation means and find a solution in the list of values. If you get stuck, consider drawing a diagram. Be prepared to explain why your solution is correct.

List:

$\frac{1}{8}$
$\frac{5}{3}$
$\frac{4}{7}$
$\frac{3}{5}$
$\frac{3}{7}$
$\frac{7}{3}$
$0.01$
$0.1$
$0.5$

$1$
$2$
$8.5$
$9.5$
$16.7$
$20$
$400$
$600$
$1400$

$1000 - a = 400$
$12.6 = b + 4.1$
$8 c = 8$
$\frac{2}{3} \cdot d = \frac{10}{9}$
$10 e = 1$
$10 = 0.5 f$
$0.99 = 1 - g$
$h + \frac{3}{7} = 1$

Are you ready for more?

One solution to the equation $a + b + c = 10$ is $a = 2$ , $b = 5$ , $c = 3$ .

How many different whole-number solutions are there to the equation $a + b + c = 10$ ? Explain or show your reasoning.

Lesson Summary

An equation can be true or false. An example of a true equation is $7 + 1 = 4 \cdot 2$ . An example of a false equation is $7 + 1 = 9$ .

An equation can have a letter in it, for example, $u + 1 = 8$ . This equation is false if $u$ is 3, because $3 + 1$ does not equal 8. This equation is true if $u$ is 7, because $7 + 1 = 8$ .

A letter in an equation is called a variable. In $u + 1 = 8$ , the variable is $u$ . A number that can be used in place of the variable that makes the equation true is called a solution to the equation. In $u + 1 = 8$ , the solution is 7.

When a number is written next to a variable, the number and the variable are being multiplied. For example, $7 x = 21$ means the same thing as $7 \cdot x = 21$ . A number written next to a variable is called a coefficient. If no coefficient is written, the coefficient is 1. For example, in the equation $p + 3 = 5$ , the coefficient of $p$ is 1.