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.

Activity 3: Card Sort: What’s in an Expression?

Find cards that match each mathematical term. Some cards may match more than one mathematical term. Parts of the expressions may be underlined to clarify what part of the expression should be matched to the mathematical term. Explain or show your reasoning to your partner with each match you make.
When you and your partner are in agreement about all the card matches, create three parts of an expression on the three blank cards for the mathematical terms of your choosing.

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.