Task 1: What is the meaning of symbols?

What do the below descriptions mean to you? How do they relate?

Record your ideas using a method of your choice.

Task 2: Complete the description

What should be added to either side of the balance scale or the equal sign to make both sides equal? Record your ideas using a method of your choice.

Task 1: Expressions and equations

Explore this video entitled Homework Zone: Mathematics to learn more about expressions and equations.

After exploring the video, answer the following questions using a method of your choice.

• What is the difference between a numerical and algebraic expression?
• What is the difference between an expression and an equation?

Task 2: Algebraic equations

$a+b=b+a$

The following is an algebraic equation. How do you know this?

What could the values be for $\mathrm{a }$ and $b$ ?

Are there multiple answers? How do you know?

When given an algebraic equation, the goal is to solve it by figuring out what the variable represents.

To do so, use the information that is provided by the rest of the equation.

If a variable is repeated, it must always be the same number. Therefore, both a’s are the same number (equal) and both b’s must be the same number (equal).

The equal sign shows us that there is the same number on both sides of the equation.

Task 3: Solving equations with a balance

Explore the video entitled Find the missing number in equations to learn more about how to balance an equation using a scale and manipulatives.

In the video, they use a blank square to represent the missing number.

This takes the place of a variable or a letter because both symbols mean that the value is unknown.

The teacher finds the value by using a balance scale and ensuring that there are the same number of circles on each side.

The missing number or variable is solved by figuring out how many more circles need to be added to become equal.

Whatever operation is done on one side of the equation, also needs to be done on the other.

Let’s do another example together.

$6+m=15-2$

Step 1. Solve the side of the equation that we can simplify: $15-2=13$

Step 2. Use a balance scale to model 6 + m = 13, where we have 6 balls on one side of the scale and 13 balls on the other side.

Step 3. Continue to add balls to the side with 6 until there is an equal amount on both sides. Count how many it takes to get to 13.

We then have a balanced scale with 13 balls on either side of the scale.

Step 4. m = 7 because 7 balls were added to the scale.

Step 5. Now to check our solution. Substitute 7 into the equation and solve.

$6+\left(7\right)=15-2$

$13=13,m=7$

If you would like, you can complete the next activity using TVO Mathify. You can also use your notebook or the following activity document.

Find the variables:

Check the below equations. Record how to balance the equations and find the variables.

$7+h=4-2+11$

$6\times 4=36-k$

$p-6=3\times 5$

$n+14=50÷2$

Task 4: Solving equations by isolating the variables

Balancing also means that whatever you do to one side of the equation you must do to the other side. The equation must always be balanced. So, whatever you add to or subtract from the left side, you must add to or subtract from the right side to keep things in balance. The same is for multiplying and dividing.

To solve for a variable, you need to isolate it–get it by itself–on one side of the equation. That way the equation will be: variable = all of the operations. Then, it will be easy to solve! You isolate the variable by balancing every operation. Let’s try an example:

$4p+7=19$

Step 1. First, get 4p alone by subtracting the 7. Subtract 7 on both sides of the equation.

$4p+7-7=19-7$

$4p=12$

Step 2. Get p alone by dividing by 4. We know that 4p means multiplication. Division is the opposite. Dividing by 4 will undo the multiplication because $4÷4=1.$

1p is the same as p by itself.

$4p÷4=12÷4$

$p=3$

Step 3. The answer is p = 3 because the variable is now completely alone on one side of the equation and has been balanced with the other side, which is now equal to 3.

Step 4. Check the answer by substituting the 3 into the equation.

$4\left(3\right)+7=19$

$12+7=19$

$19=19,p=3$

Now try a few on your own using this method. Record your ideas and solutions digitally, orally or in print. Don’t forget to verify or check your solutions!

• $6m+2=20$
• $24-4c+3=19$
• $9d-(3\times 2)=40-1$
• $5b-4=21$

Press the equations to access step-by-step solutions to check your work.

Task 5: Algebraic equations for pattern rules

Algebraic equations can be used to figure out a term in a pattern. The unknown term becomes the variable and the pattern rule is turned into an equation. For example, a pattern rule may be to double the term number and add 5. This pattern would be: 7, 9, 11, 13, etc…

Now let’s record this pattern rule as an expression with the term number represented as n. It would be: $2n+5.$

You can use this expression to solve for the 20th term, 50th term, 100th term of that pattern by substituting n with each term number. For example.

• 2(20) + 5 = 45. The 20th term would be 45.
• 2(50) + 5 = 105. The 50th term would be 105.
• 2(100) + 5 = 205. The 100th term would be 205.

You can also use this expression to figure out what the term number is, depending on the value. For example, 27 will occur in this pattern. What term number will 27 be?

First, write an equation for this problem: $2n+5=27.$

Then, solve for n. The goal is to get n by itself. You can balance the equation by making sure that every operation you do on one side, you also do to the other.

First, subtract 5.

$2n=27-5$

$2n=22$

Then, divide by 2 to get n alone.

$n=22÷2$

$n=11$

And check!

$2\left(11\right)+5=27$

$22+5=27$

$27=27,n=11$

Now, try it on your own. Record your ideas using a method of your choice.

Here is a pattern: 9, 12, 15, 18, 21…

• What is the pattern rule? Write it as an algebraic expression.
• What is the 25th term? The 50th term?
• 42 will occur in this pattern. What term number will 42 be?
• 186 will occur in this pattern. What term number will 186 be?

Record your thoughts with the chart, Algebraic Equations for Pattern Rules in your notebook or using the following activity document. You may also use a method of your choice.

Press the ‘Activity’ button to access Algebraic Equations for Pattern Rules.

Press ‘Answers’ to reveal the solutions to check your work.

Answers

Task 1: Pattern rules and algebraic equations

A baker has $10.00. Every day the baker makes $2.25 more selling mini cupcakes. Answer the following questions and record your ideas using a method of your choice.

• Write out the first 5 terms of this pattern.
• What is a pattern rule? Write it as an algebraic expression.
• How long until the baker saves $212.50? On what day will this occur?

Task 2: Creating and solving equations

Create an equation that has the solution $a=6.$

Create a different equation that has the solution $a=3.$

If possible, trade with a partner or someone at home to figure out which equation is

$a=6$

and which is

$a=3.$

How can you prove it? Record your response using a method of your choice.

Think about your learning

Reflect on and answer the following questions. If possible discuss with a partner or write a response using a method of your choice.

• Why is it important to balance an equation?
• What are some balancing strategies?
• What area do you still need to deepen your understanding in?

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