C. Algebra · Patterns, Equations, Inequalities, Coding and Modelling

Words, Expressions and Pictures: Showing Equivalent Relationships

C2.1 A1About 40 minutes

Learning goals

I can write an algebraic expression for a phrase in words.
I can describe an expression in words and show it with a picture.
I can show that two expressions are equivalent using a picture or by testing numbers.
I can explain why some expressions that look alike are not equivalent.

Words to know

variable
A letter or symbol that stands for a number that is unknown or can change.

algebraic expression
A math phrase with numbers, variables and operations, but no equal sign, like 3 × n + 4.

equivalent expressions
Expressions that are always equal, no matter what number the variable stands for, like n + n + n and 3 × n.

brackets
( ) Brackets show which part to do first. 2 × (n + 3) means find n + 3, then double it.

visual representation
A picture of a relationship, like bags and counters, a bar model or a balance.

Remember from Grade 4

In Grade 4 you learned that a variable is a letter or symbol that stands for a number. An expression like c + 5 names an amount and has no equal sign. An equation like c + 5 = 12 has an equal sign.

In Grade 5 you will move back and forth between words, expressions and pictures, and you will discover that different-looking expressions can be equivalent: always worth the same.

Let's start with a story

The school is collecting cans for the food bank. Mr. Ahmed's class fills some bags. Each bag holds the same number of cans, but nobody has counted them yet. Let's call the number of cans in one bag n.

The class has 3 full bags and 4 loose cans.

Three paper bags labelled n, and 4 loose counters.

We can describe the cans three ways:

Words: 4 more than 3 times the number in a bag.
Expression: 3 × n + 4
Picture: 3 bags and 4 loose counters (above).

All three show the same amount. Translating means changing from one to another.

Words to expressions

Words	Expression
5 more than a number	n + 5
7 less than a number	n − 7
7 decreased by a number	7 − n
triple a number (3 times a number)	3 × n
a number divided by 4	n ÷ 4
twice a number, plus 3	2 × n + 3
4 less than double a number	2 × n − 4

Common phrases and their expressions. Read carefully: 7 less than a number is n − 7, but 7 decreased by a number is 7 − n.

Writing multiplication

You may also see 3 × n written as 3n. The number goes in front, and the × sign is left out. Both mean the same thing. In this unit we will usually keep the × sign so it's easy to read.

In an expression like 3 × n + 4, multiply first, then add. If you want to add first, use brackets: 3 × (n + 4).

Equivalent expressions

Two expressions are equivalent if they are always equal, whatever number the variable is.

n + n + n and 3 × n are equivalent: three n's added together is 3 groups of n.
n + 5 and 5 + n are equivalent: you can add in any order.

Pictures help us see equivalence. Here are two groups, each with a bag and 3 counters. Now put all the bags together and all the counters together.

Left: two dashed groups, each with one bag labelled n and 3 counters. Right: the same things regrouped as 2 bags and 6 counters. Both sides show the same amount.

Two groups of (n + 3) is the same as 2 bags and 6 counters, so 2 × (n + 3) = 2 × n + 6. This is the splitting rule (distributive property) from Grade 4, now with a variable!

Worked example

Example 1: Words to an expression

Sam has b books. His sister has 3 times as many books as Sam, plus 2 more. Write an expression for his sister's books.

3 times as many as Sam is 3 × b.
Plus 2 more adds 2.

3 × b + 2 (or 3b + 2).

Worked example

Example 2: An expression to words and a picture

Describe 2 × n − 1 in words, and draw it.

2 × n is double a number or twice a number.
− 1 is 1 less.
Picture: draw 2 bags, each with n counters, then cross out 1 counter.

Words: 1 less than twice a number. Picture: 2 bags with 1 counter taken away.

Worked example

Example 3: Prove two expressions are equivalent

Show that 2 × (n + 3) and 2 × n + 6 are equivalent.

Use the picture above: 2 groups of (bag + 3 counters) is 2 bags and 6 counters.
Test a number. Let n = 4. 2 × (4 + 3) = 2 × 7 = 14. 2 × 4 + 6 = 8 + 6 = 14 ✔
Test another. Let n = 10. 2 × 13 = 26. 20 + 6 = 26 ✔

They are equivalent. The picture shows why it works for every number, and the tests agree.

Worked example

Example 4: Look-alikes that are NOT equivalent

Are 2 × n + 3 and 2 × (n + 3) equivalent?

Test n = 5. 2 × 5 + 3 = 10 + 3 = 13.
2 × (5 + 3) = 2 × 8 = 16.
13 is not 16. One test that doesn't match is enough to show they are not equivalent.
Pictures: 2 × n + 3 is 2 bags and 3 counters. 2 × (n + 3) is 2 bags and 6 counters.

Not equivalent. The brackets make a big difference!

Worked example

Example 5: A shape relationship

A square has side length s cm. Write two equivalent expressions for its perimeter (the distance around it).

Add the four sides: s + s + s + s.
Four sides the same length is 4 × s.
Check with s = 5: 5 + 5 + 5 + 5 = 20 and 4 × 5 = 20 ✔

s + s + s + s = 4 × s. In words: the perimeter of a square is 4 times its side length.

Just for fun

Why are variables great at hide-and-seek?

Because nobody knows what number they are hiding until you open the bag!

Hands-on activity

Mystery cups

You need: 4 paper cups or small containers, about 30 coins, buttons or dried beans, paper and a pencil.

Your grown-up secretly puts the same number of items under each cup. Call that number n.
Your grown-up sets out 2 cups and 6 loose items. Write an expression (2 × n + 6) and say it in words.
Now move things around: make 2 groups, each with 1 cup and 3 loose items. Write the new expression: 2 × (n + 3). Did the total amount change? (No! They are equivalent.)
Lift the cups and count to check that both expressions give the same total.
Your turn: set up cups and items for your grown-up and say an expression in words. They build it.

What to look for: your child explains that moving the items into groups doesn't change the total, so the two expressions are equivalent.

Grown-up note: keep coins and beans away from babies and toddlers.

Your turn

Write an expression: 8 more than a number n.

Write an expression: a number n divided by 5.

Write an expression: 6 less than 4 times a number n.

Describe 5 × k + 1 in words.

A picture shows 4 bags (n in each) and 2 loose counters. Write the expression.

Are n + n and 2 × n equivalent? Explain.

Are n + 4 and 4 × n equivalent? Test n = 1.

More practice: warm-up

Write an expression: 10 more than t.

Write an expression: twice a number m.

Write an expression: 3 less than p.

Is x + 9 the same as 9 + x?

Describe n − 2 in words.

More practice: on your own

Tickets to the science centre cost $12 each, plus $5 for parking. Write an expression for the cost for t tickets.

Write an expression: 15 decreased by a number y. Then write one for a number y decreased by 15.

Show that 3 × (n + 2) and 3 × n + 6 are equivalent with a picture and one test.

Which two are equivalent? A: 5 × n B: n + n + n + n + n C: n + 5

A rectangle is w cm wide and 8 cm long. Write two equivalent expressions for its perimeter.

Challenge

Which expression is equivalent to 4 × n + 8? A: 4 × (n + 2) B: 4 × (n + 8) C: 8 × n + 4

Jada says n + 3 and 3 × n are equivalent because they are both 6 when n = 3. Is she right?

Write an expression in words, as an expression, and draw it: I had some stickers. I doubled them, then gave away 5.

Check your understanding

Quiz

Which expression means 6 more than a number n?

Which phrase matches 3 × n − 2?

A picture shows 2 bags of n counters and 5 loose counters. Which expression matches?

Which is equivalent to n + n + n + n?

Which is equivalent to 2 × (n + 5)?

How can you show that two expressions are not equivalent?

For parents and teachers

Big idea: Grade 5 students translate among words, algebraic expressions and visual representations that describe equivalent relationships (C2.1). Children should see that n + n + n and 3 × n, or 2 × (n + 3) and 2 × n + 6, are always equal, and be able to explain why with a picture.

Common difficulties:

Order in subtraction phrases. "7 less than n" is n − 7, not 7 − n. Test with a real number: 7 less than 10 is 3, and 10 − 7 = 3.
Ignoring brackets. Compare 2 × n + 3 and 2 × (n + 3) with cups and coins.
Thinking one matching test proves equivalence. n + 3 and 3 × n match only when n = 3.

How to help: Use cups (or envelopes) with hidden coins. Build an expression, regroup the pieces, and ask "Did the total change?" Social-emotional learning (A1): encourage your child to explain their thinking out loud and listen to yours; talking about math builds confidence.