Minds On

Solving a problem

A group of students were presented with the following word problem:

A gymnast, boxer, and acrobat have a combined age of 60. How old is each person?

g + b + a = 60

The students found it was impossible to solve this question without more information.

Brainstorm

Let’s think!

Brainstorm what information you would need to solve this question.

Record your ideas digitally, orally, or in print.

Action

Working with unknowns

A group of students were presented with the same word problem from the Minds On section:

A gymnast, boxer, and acrobat have a combined age of 60. How old is each person?

g + b + a = 60

This time, the teacher added this information:

The acrobat is 12 years older than the gymnast and the boxer is 3 years younger than the acrobat.

The students decided this question was solvable because they could relate the ages to each other to create a single unknown (variable).

Gathering like terms

Here is what the students did to solve the word problem.

Let gymnast = g ,
boxer = b
and acrobat = a .

The students thought about how they could relate the acrobat’s age and boxer’s age to the gymnast. This is what they did:

Acrobat = g   +   12  
and boxer = g   +   12 - 3 ,  
so boxer = g +   9  

This is logical since the acrobat is g + 12
and the boxer is 3 years younger than the acrobat.

Next, the students wrote the algebraic equation:

g +   b +   a   =   60

g   +   ( g + 12 )   +   ( g   +   9 )   =   60

How many “ g  ”s do we have in total? Is there a way to group the “ g  ”s together? What if we added the “ g  ”s together and placed them in the equation?

How would this appear? Record your ideas digitally, orally, or in print.

When you are ready, press ‘Answer’ to reveal the solution.

We have 3 “ g  ”s in total so we can group the like terms together like this:

3 g   +   21   =   60

The 21 was possible by adding 9 + 12 together.

This is called gathering like terms.

Test Your Skills!

Solve it

Solve the equation and calculate everyone’s age. Confirm your answer using a method of your choice.

Record your solution and strategies using a method of your choice.

Binomials and monomials

Explore the following definitions:

A mathematical expression consisting of two terms connected by a plus sign or minus sign.

A mathematical expression consisting of a single term.

Practice 1

The following are four mathematical expressions. Which of the following would be a monomial and which would be binomial?

Is this expression binomial or monomial?

Is this expression binomial or monomial?

Is this expression binomial or monomial?

Is this expression binomial or monomial?

Simplify the expression

Can you simplify the following expression?

3 x   +   3 x

Record your solution using a method of your choice.

When you are ready, press ‘Answer’ to reveal the solution.

We could group the like terms! Now, this expression becomes 6 x .

3 x   +   3 x =   6 x

Practice 2

Try to simplify a few expressions on your own. Select the correct response for the following three questions.

Word problem

Examine the following diagram of a rectangular garden:

A rectangle. The length is 3 x plus 1 and the width is 2 x subtract 4.

The rectangular garden has a length of
3 x   +   1
and a width of
2 x   –   4 .

Using this information, show how you could calculate the perimeter of the garden.

Record your response digitally, orally, or in print.

Consolidation

Task 1

Use the information in the following word problem to create an equation. Your equation should include a monomial or binomial. Once you have your equation, solve it. Record your work digitally, orally, or in print.

A student is trying to find out the number of goals scored by a soccer team. The striker currently has scored the most goals on the team. The mid-fielder has the next highest total and has 8 fewer goals than the striker. The defender has the third highest total and has 2 fewer goals than the mid-fielder. The top 3 scorers on the team have a combined 27 goals.

How can you write the equation so that it can be solved? Is it a monomial or a binomial? How many goals does each player have?

Task 2

Use what you have learned about adding and subtracting monomials and binomials to solve the following riddle. If the symbols are too unfamiliar, try replacing them with letters.

Record your work digitally, orally, or in print.

Let’s try another!

Examine the following riddle and then determine each variable.

x + y + z = − 1

x × x + x × x = 8

y - 1 = x + z

y + z ÷ z = 1

x = ?

y = ?

z = ?

Reflection

As you read the following descriptions, select the one that best describes your current understanding of the learning in this activity. Press the corresponding button once you have made your choice.

I feel...

Now, expand on your ideas by recording your thoughts using a voice recorder, speech-to-text,or writing tool.

When you review your notes on this learning activity later, reflect on whether you would select a different description based on your further review of the material in this learning activity.

Press ‘Discover More’ to extend your skills. 

Now it’s your turn! Create a logic puzzle using symbols, or letters. Provide a solution key.