Minds On

Simplification

Simplify the following expression. Record your solution using a method of your choice.

3 × 6 + (5 - 4) + 6 ÷ 2

Simplify means to combine all the terms that you can.

Reduce the long equation to one single number by completing the operations.

Is there more than one answer? Why might someone get a different answer than you?

Action

Task 1: BEDMAS

When there are expressions with many different operations, there is an order that needs to be followed to find the correct answer.

The acronym “BEDMAS” can be used to guide you through the steps correctly.

Description

B – brackets. Any of the operations that are inside brackets are calculated first. In the equation $3 \times 6 + (5 - 4) + 6 \div 2$ , the first step would be to subtract five minus 4. Brackets are 2 curved lines, placed before and after the terms.

E – exponents. The next step would be to calculate any exponents, which is a number being multiplied by itself. You will not have to work with exponents this year. An example of an exponent is 10 with a superscript 2 beside it.

D and M – division and multiplication. After that, you divide or multiply based on whichever comes first in order from left to right. In the example, $3 \times 6 + 1 + 6 \div 2$ , step 2 would be calculating $3 \times 6$ and step 3 would be calculating $6 \div 2$ . The symbol for division is a horizontal line with one dot above and below the line. The symbol for multiplication is $\times$ .

A and S – addition and subtraction. The last step is to add or subtract, in order of the question from left to right. In the example, $18 + 1 + 3$ , there is only addition left. The symbol for addition is + and the symbol for subtraction is a horizontal line.

Press the ‘Activity’ button to access BEDMAS Order of Operations reference chart.

Activity

Compare your answer to the Minds On questions when following the order of operations vs. when answering the equation from left to right. Are the answers different? Why is it important to use BEDMAS?

3 × 6 + (5 - 4) + 6 ÷ 2

Task 2: Practicing the order of operations

Let’s practice a few questions involving order of operations together.

Before that, let's explore two step-by-step examples that involve Order of Operations.

Order of Operations
5 + 8 ÷ (1 + 1)	12 – 3 × 2 + (5 - 3)
Step 1. Brackets. 5 + 8 ÷ (1+1) 5 + 8 ÷ 2	Step 1. Brackets. 12 – 3 × 2 + (5 - 3) 12 – 3 × 2 + 2
Step 2. Exponents. There are no exponents.	Step 2. Exponents. There are no exponents.
Step 3. Division or multiplication. 5 + 8 ÷ 2 5 + 4	Step 3. Division or multiplication 12 – 3 × 2 + 2 12 – 6 + 2
Step 4. Addition or subtraction. 5 + 4 9	Step 4. Addition or subtraction. 12 – 6 + 2 6 + 2 = 8
Answer 5 + 8 ÷ (1 + 1) = 9	Answer. 12 – 3 × 2 + (5-3) = 8

Press the ‘Activity’ button to access Order of Operations.

Activity

Now, try some on your own. Answer the following questions using BEDMAS and other strategies or manipulatives to support your understanding. Record your ideas using a method of your choice.

For each equation/problem, select the missing number or phrase from the drop-down menu.

Task 3: Variables in expressions

Sometimes, long equations or expressions with multiple operations will include variables. Here is an example:

12 – 3 × h + 7 = 13

In this example, which part is the expression? Which part is the equation? Where is the variable?

Press each of the following terms to explore its definition.

Numbers, symbols, and operations grouped together that show a value. Does not include an equal sign.

Numbers, symbols, and operations that are equivalent or equal to something else. It will have an equal sign.

Symbol that represents a number or a value that is unknown.

When the variables are used for multiplication in a long equation with many operations, they may be represented differently.

Instead of including the multiplication symbol (×) because it can be mistaken for a letter variable, the variable and number that are being multiplied can be placed side by side.

The equation above may be represented as follows:

12 – 3h + 7 = 13

When you substitute a given value for the variable, you use brackets. If you are told that h=2, the equation would be recorded like this:

12 – 3(2) + 7 = 13

Rewrite the following expressions by substituting the given value for the variables. Record your thinking using a method of your choice.

Select the correct answer.

Task 4: Solving for a variable

When the goal is to find the value of a variable, there is more than one strategy that can be used.

Guess-and-check

Using what you know from the information in the question, you can make a reasonable guess about the value.

Test your answer by substituting for the variable and comparing your answer to the answer of the equation.

Adjust your guess based on how different your answer is from the equation’s answer.

Use the guess-and-check method or another strategy to answer the following questions and record your ideas using a method of your choice.

The first has been done for you.

m ÷ 4 – 2 = 10

m must be a lot bigger than 10 because I need to divide it by 4. I’m going to start with 40 first.

40 ÷ 4 – 2 = 10

10 – 2 = 10

8 ≠ 10, hmm too small. I’m going to try 44.

44 ÷ 4 – 2 = 10

11 – 2 = 10

9 ≠ 10, hmm still too small – but getting closer! I’m going to try 48

48 ÷ 4 – 2 = 10

12 – 2 = 10

10 = 10, m = 10

3 + 6n = 21
9g – (8 + 5) = 23
A librarian bought 10 books and 7 pencils and erasers. Each book costs $12.50. Each eraser costs $0.30 more than each pencil. The total price was $150.00. How much does each pencil cost?

Reflect on the following questions:

How did you decide on what number to use for your first guess? Did you use the same strategy each time?
What are the benefits of the guess-and-check strategy?
What are the cons of this strategy?
Is there more than one possible answer for the variable?

Reverse Flow Chart

You can also solve these problems by creating a flow chart.

A flow chart breaks down the steps used to solve the problem in order.

To check, reverse each stage and complete the flow chart backwards to arrive at the value of the variable. For example: 4 + 6s = 22

This is a flow chart for this question: Did you notice the expression has been re-ordered to follow order of operations? 6 × s would be done first, because multiplication comes before addition according to BEDMAS, so it’s first in the flow chart.

Solve for s by undoing each of these steps. Start at 22 and reverse each operation. Notice that when we go the opposite way in the flow chart, we use the opposite order of operations. We start with the addition/subtraction first and end with multiplication/division.

Use the reverse flow chart method, guess and check or another strategy to answer the following questions. Record your ideas using a method of your choice.

48 ÷ 4 + i = 14
3j + 5 – 1 = 25
(6 + 7) – 3r = 4

Reflect on the following questions:

What are the benefits of this strategy?
What are the cons of this strategy?
Is there more than one possible answer for the variable?

Task 5: Real-world problem

A tourist went to the carnival with friends. The cost for entry was $20/person. Each person bought 50 tokens for $15/person. In total, the friends spent $175. How many people were in the tourist’s group at the carnival? Create an algebraic equation to explain the question and then solve using any strategy. Record your ideas using a method of your choice.

Press ‘Answer’ to reveal a possible solution to check your work.

Answer

$20p + $15p = $175

First, I would group like terms:

35p = 175

Then I would divide both sides by 35 to isolate p:

p = 5

Therefore, there were 5 people in Tourist’s group at the carnival.

Consolidation

Task 1: Creating an equation with BEDMAS

Think of a real-life scenario that will lead to an equation that meets the following criteria and record your idea using a method of your choice.

The equation should have:

at least one variable
one addition/subtraction operation
one multiplication/division operation

Record your scenario using a method of your choice.

Student Success

Think-Pair-Share

If possible, trade scenarios and use the other scenario to create an equation and solve using the strategy of your choice.

Task 2: Think about your learning

Reflect on and answer the following questions using a method of your choice:

Why is it important to use BEDMAS?
What is an advantage of using algebraic equations and expressions to write about real-life scenarios?
How do you select which strategy to use when solving an algebraic expression or equation?

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.

I feel...

I have an excellent understanding of the concepts I have a very good understanding of the concepts I have some understanding but need to review some more my understanding of the concepts needs a lot more work

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

Connect with a TVO Mathify tutor

Think of TVO Mathify as your own personalized math coach, here to support your learning at home. Press ‘TVO Mathify’ to connect with an Ontario Certified Teacher math tutor of your choice. You will need a TVO Mathify login to access this resource.