Exploring equations

Let's explore the following two equations.

Equation #1

The description represents the following equation: 4 × h = 48

Each hamburger is represented with the letter ‘h’ and there are 4 hamburgers.

• What do you think the letter ‘h’ means in this equation/description?
• What other symbols could you use other than the letter ‘h’?
• What does the equation demonstrate?

Equation #2

The description represents the following equation: 1,000 ÷ j = 5

There are 5 cups of choice, each labelled with the letter ‘j.’ There is an empty jug beside the cups.

• What do you think the letter ‘j’ means in this equation/description?
• What other symbols could you use other than the letter ‘j’?
• What does the equation demonstrate?

What is a variable?

A variable is shown as a symbol that represents a number that is unknown or can change.

In the equation 4 × h = 48, h is the variable, meaning that we don’t know what number it is.

The h may represent the price of the burger.

There are 4 burgers at some unknown price, which multiplies to $48 in total.

The price of each burger is not known.

Explore this video on variables.

Equations with variables

An equation includes an equal sign with information or numbers on either side of that equal sign.

a – b = 20.

a and b represent two numbers. Subtracting b from a will equal 20.

This equation could be representing different scenarios.

• One scenario could be that someone had a certain amount of money (a) and spent some of that money (b). Now, they have $20 left.
• Or this equation could be representing a scenario where someone has a certain distance that they want to drive (a) and have driven some of that distance already (b). They have 20km left to drive.

Task 1: Create your own scenarios

Create scenarios for each of the following equations.

Record your ideas using a method of your choice.

• 25 – y = 17
• 9 + g = h
• 40 × 7 = z
• a + j = 73

Expressions

An expression is only one side of an equation.

It includes the variables and operations but does not have an equal sign as the answer is not known.

An expression is an unsolved equation.

We can solve algebraic expressions, by substituting or “plugging in” different numbers into the variables in an expression.

For example,

2 + n

If n = 7, the expression becomes 2 + 7 and can be solved.

Task 2: Substituting variables in expressions

Select the correct answer.

Substituting variables in expressions when the variable is unknown

How can you solve these expressions if the variable is unknown, or the value is not given to you?

Explore the video to help to answer that question:

An input/output table can help you solve expressions where the variable is unknown by plugging in multiple possibilities for the variable.

In an input/output table, what does the input represent? What does the output represent?

Let’s practice creating an input/output table for the following equation: 7 + n – 1.

Step 1. Create the table

Step 2. Put some different values for n into the input column.

Step 3. Replace the n in the equation with each number in the input column to determine the output.

Task 3: Solving the equations when the variables are unknown

A baker made 24 cookies. The baker tasted a few while baking to make sure they were delicious. How many cookies does the baker have left?

Develop an algebraic expression. Create an input/output table and record at least 5 values to provide possible solutions for your expression.

Record your ideas using a method of your choice.

Press 'Possible solutions for the expression' to access the input/output table.

Possible Solutions for the Expression

Monomials

Sometimes a variable might be written beside a number. For example: 4n, 8j, or 10p.

This is a kind of algebraic expression where there is one term or number with the variable, called a monomial.

It means that there are 4 terms with the value of n or n + n + n + n, 8 terms with the value of j or j + j + j + j + j + j + j + j, or 10 terms with the value of p or p + p + p + p + p + p + p + p + p + p.

In other words, the term is being multiplied by the variable.

Simplify the expressions below. Record your responses using a method of your choice.

• g + g + g
• h + h + h + h + h + h
• m + m

Monomials may also be included in expressions with more operations. For example: 4n + 3n, 8j − j, or 12p − 10p. When the same variable is repeated multiple times in an expression, it always represents the same value.

That means we can simplify these expressions by combining the number in the monomial and leaving the variable. As we learned above, 4n + 3n would be the same as adding n 4 times and 3 times, n + n + n + n + n + n + n. This can be simplified to 7n.

You can use algebra tiles, cubes or another manipulative to help you simplify these expressions. The variable (1n) is represented by one bar. 4n or n + n + n + n would be represented by 4 algebra tiles.

Therefore, we can use them to help us understand how many of a variable is in a monomial and easily add those to another monomial.

Task 1: Develop your own scenario

Create your own real-life scenario that involves a variable and an operation.

Create an input/output table with five possible values.

Record your idea using a method of your choice.

Task 2: Think about your learning

Answer the following questions using a method of your choice:

• Why do we need to know the value of a variable in a real-life situation?
• How does using real-life situations help your understanding of variables?

Reflection

As you read through these descriptions, which sentence best describes how you are feeling about your understanding of this learning activity? Press the button that is beside this sentence.

Now, record your ideas using a voice recorder, speech-to-text, or writing tool.

Connect with a TVO Mathify tutor