C. Algebra · Patterns, Equations, Inequalities, Coding and ModellingVariables in Expressions and Equations
C2.1 A1About 35 minutes
Learning goals- I can use a letter or symbol as a variable.
- I can tell the difference between an expression and an equation.
- I can write an expression or equation for a story or a word sentence.
- I can find the value of an expression when I know the value of the variable.
Words to know
variable
A letter or symbol that stands for a number we don't know yet, or a number that can change.
expression
Numbers, variables and operation signs together, with no equal sign. Example: n + 5.
equation
A number sentence with an equal sign. Both sides have the same value. Example: n + 5 = 12.
value
How much something is worth. If n = 7, the value of n + 5 is 12.
substitute
Put a number in place of a variable.
Remember from Grade 3
In Grade 3 you learned that a variable is a letter or symbol that stands for a number. It can stand for an unknown number (n + 4 = 11, so n = 7) or for an amount that changes (2 × b wheels on b bikes). You also met expressions (no equal sign, like 2 × b) and equations (with an equal sign, like n + 4 = 11). In Grade 4 you will use variables in both: tell them apart, write them for stories, and find their values.
Let's start with a story
Maya has some hockey cards in a box. She doesn't know how many. Her cousin gives her 5 more.
We can't write the total as a number yet, but we can write it with a variable. Let c be the number of cards in the box. The total is c + 5.
Later Maya counts: she has 12 cards in all. Now we can write c + 5 = 12.
Expression or equation?
- c + 5 is an expression. It is like a math phrase. It names an amount: 5 more than c. There is no equal sign.
- c + 5 = 12 is an equation. It is like a math sentence. It says two things have the same value. It has an equal sign.
An expression can use any operation: n + 8, 20 − k, 4 × b, m ÷ 2.
Symbols can be variables
A variable does not have to be a letter. Any symbol can stand for a number: ☐ + 3 = 10, ★ × 4 = 20, ? − 6 = 9. In Grade 4 you will mostly see letters, because they are quick to write.
In one problem, the same symbol always means the same number. In a + a = 14, both a's are 7.
| Words | Expression |
|---|
| 5 more than a number | n + 5 |
| a number take away 3 | n − 3 |
| 8 less than a number | n − 8 |
| 4 groups of a number | 4 × n |
| a number shared equally among 2 | n ÷ 2 |
| double a number | 2 × n |
Careful with "less than"
8 less than a number means start with the number and take 8 away: n − 8, not 8 − n. If n is 20, 8 less than 20 is 12, and 20 − 8 = 12. ✔
Finding the value of an expression
If you know what the variable stands for, you can substitute (put in) that number and work out the value.
If b = 6, then 4 × b = 4 × 6 = 24.
Worked exampleExample 1: Expression or equation?
Sort these: (a) 3 × p (b) 3 × p = 18 (c) y − 7 (d) 25 = 10 + k
- Look for an equal sign.
- (a) and (c) have no equal sign. They are expressions.
- (b) and (d) have an equal sign. They are equations. (The equal sign can be on either side!)
Expressions: 3 × p and y − 7. Equations: 3 × p = 18 and 25 = 10 + k.
Worked exampleExample 2: Write an expression from a story
A pack of juice boxes has 6 juice boxes. Write an expression for the number of juice boxes in p packs.
- Each pack has 6. For p packs, we have p groups of 6.
- Groups of means multiply.
6 × p (or p × 6). If p = 3 packs, that is 6 × 3 = 18 juice boxes.
Worked exampleExample 3: Write an equation from a story
Liam had some money. He spent $9 on a book. Now he has $16. Write an equation.
- Pick a variable for the unknown: let m be the money Liam had at first.
- He spent $9, so we take 9 away: m − 9.
- What is left is $16.
m − 9 = 16. (In the next lesson you will solve it: m = 25.)
Worked exampleExample 4: Find the value
Find the value of each expression when k = 8: (a) k + 15 (b) 30 − k (c) 5 × k (d) k ÷ 2
- Substitute 8 for k each time.
- (a) 8 + 15 = 23
- (b) 30 − 8 = 22
- (c) 5 × 8 = 40
- (d) 8 ÷ 2 = 4
Writing multiplication with variables
Write 4 × b with the times sign for now. When you are older, you will see it written as 4b, which means the same thing.
Just for funWhy are variables so mysterious?
They never tell you their number... until you solve them!
Hands-on activityMystery bag expressions
You need: a paper bag or envelope, about 30 small things (pasta pieces, buttons or paper clips), paper and a pencil.
- Your grown-up secretly puts some items in the bag. Call that number b.
- Your grown-up puts 4 more items next to the bag. Write an expression for the total: b + 4.
- Now your grown-up tells you the total, for example 15. Write the equation: b + 4 = 15. Guess what b is, then open the bag and count to check.
- Play again with new actions: take 2 items out of the bag (b − 2); put 3 bags with the same number in each (3 × b).
- Switch roles: you fill the bag, and your grown-up writes the expressions.
What to look for: your child writes an expression (no equal sign) before the total is known, and an equation (with an equal sign) after.
Grown-up note: keep small items like these away from babies and toddlers.
Your turn
1Is 12 − t an expression or an equation? How do you know?
An expression. It has no equal sign.
2Is w + 9 = 30 an expression or an equation?
An equation. It has an equal sign.
3Write an expression for 7 more than a number.
n + 7 (any letter is fine).
4Write an expression for the number of legs on d dogs.
4 × d. Each dog has 4 legs.
5Find the value of 3 × r when r = 9.
6Write an equation: A number plus 15 equals 40.
7In ★ + ★ = 18, what number is ★?
9. The same symbol stands for the same number, and 9 + 9 = 18.
More practice: warm-up
1Circle the equation: a + 2 or a + 2 = 9
2Write an expression for a number take away 5.
3If y = 4, what is y + 10?
4What symbol does every equation have?
5If m = 20, what is m ÷ 2?
More practice: on your own
1Write an expression: a box of crayons has 24 crayons, and you lose some, l.
24 − l. Start with 24 and take away the l lost crayons.
2Write an equation: Some students were on the bus. 6 got off. Now 19 are on the bus.
s − 6 = 19 (s = students at first).
3Find the value of 50 − h when h = 18.
4Ali writes 9 less than a number as 9 − n. Is he right? Explain.
No. 9 less than a number starts with the number and takes 9 away: n − 9. For example, 9 less than 20 is 11, and 20 − 9 = 11, but 9 − 20 does not make sense here.
5Movie tickets cost $8 each. Write an expression for the cost of t tickets. What is the cost when t = 5?
8 × t. When t = 5: 8 × 5 = $40.
Challenge
1Write a story that matches the equation 4 × p = 28.
Sample answer: Some pencils come in packs. 4 friends each have the same number of pencils, p. Together they have 28 pencils. (p = 7.) Any story with 4 equal groups making 28 works.
2Find the value of n + n + 3 when n = 12.
27. Substitute 12 for both n's: 12 + 12 + 3 = 27.
3A snack bag holds g grapes. Lena eats 5 grapes from it, then shares the rest equally with her friend. Write an expression for how many grapes each person gets.
(g − 5) ÷ 2. First take away the 5 she ate, then divide by 2. (The brackets mean do this part first.) For example, if g = 25: 25 − 5 = 20, and 20 ÷ 2 = 10 grapes each.
Quiz
2. Which expression means 6 less than a number?
3. What is the value of 7 × a when a = 5?
4. Which equation matches: Some birds were on a tree. 8 more came. Now there are 21.
5. Which expression shows the number of wheels on t tricycles?
6. In ▲ + ▲ + ▲ = 15, what is ▲?
Big idea: The Ontario curriculum (C2.1) asks Grade 4 students to identify and use symbols as variables in expressions and equations. A key idea is the difference between an expression (a math phrase, like n + 5) and an equation (a math sentence with an equal sign, like n + 5 = 12).
Common difficulties:
- Writing "less than" backwards (6 − n for 6 less than n). Try it with a real number to check.
- Thinking a letter stands for its place in the alphabet (a = 1, b = 2). Remind your child that the letter is just a name for a number.
- Thinking different letters must be different numbers, or the same letter can be two numbers. In one problem, the same letter always means the same number.
How to help: Turn everyday moments into expressions: "Each egg carton holds 12 eggs. How many eggs in c cartons?" (12 × c). Let your child invent stories for you to write as equations. Praise clear explanations (Strand A: communicating and representing).