Grade by Grade
C. Algebra · Patterns, Variables, Coding and Modelling

Mystery Symbols: Using Variables

C2.1About 35 minutes
Learning goals
  • I can tell when a symbol, like ☐, ? or a letter, is standing in for a number.
  • I can describe what the symbol stands for in a problem.
  • I can find the number a symbol stands for.
  • I can tell the difference between a symbol that stands for a number and a symbol like +, − or =.

Words to know

symbol
A mark or picture that stands for something, like +, =, ☐ or ⭐.
variable
A symbol that stands for a number. The number might be unknown (a mystery), or it might be able to change.
unknown
A number we do not know yet and need to find.
equation
A number sentence with an equal sign, like 4 + 5 = 9 or 4 + ☐ = 9.

A mystery in math!

Omar had some hockey cards. His cousin gave him 5 more. Now Omar has 12 cards. How many cards did Omar have at the start?

We do not know that number yet. We can write a symbol to hold its place:

☐ + 5 = 12

The box ☐ stands for the number of cards Omar had at the start. A symbol that stands for a number is called a variable.

Symbols all around us

Math uses lots of symbols. Some tell you what to do. Others stand for a number.

  • + means add. − means subtract. = means is the same amount as. ¢ means cents. These are not variables.
  • ☐, ?, ⭐, △ or a letter like n, when they hold the place of a number, are variables.
Number sentenceIs there a variable?What it stands for
7 + 2 = 9NoAll the numbers are known.
7 + ☐ = 9Yes, ☐The number added to 7 to make 9 (it is 2).
? − 3 = 10Yes, ?The starting number (it is 13).
n + n = 8Yes, nA number added to itself to make 8 (it is 4).
A variable stands for a number. Every n in the same number sentence stands for the same number.
Worked example

Describe the variable

Chloe has 8 mittens. Some are on the shelf and 3 are in her backpack. The number sentence is:

☐ + 3 = 8

What does ☐ stand for? What number is it?

  1. Read the story again. There are mittens on the shelf and in the backpack.
  2. 3 is the backpack mittens. 8 is all the mittens.
  3. So ☐ stands for the number of mittens on the shelf.
  4. Find it: what plus 3 makes 8? Count on from 3: 4, 5, 6, 7, 8. That is 5 counts.
  5. Check: 5 + 3 = 8. ✔

☐ stands for the mittens on the shelf. ☐ = 5.

☐ + 38
The balance is level, so both sides are the same amount. ☐ must be 5.

Same symbol, same number

If the same symbol shows up twice in one number sentence, it stands for the same number both times.

⭐ + ⭐ = 10 → ⭐ is 5, because 5 + 5 = 10.

If there are two different symbols, they can be different numbers:

🔺 + 🔵 = 10 → 🔺 could be 6 and 🔵 could be 4. Or 🔺 could be 1 and 🔵 could be 9. There are many answers! Here the variables can change.

Same symbolDifferent symbols+= 10Both stars must be thesame number:5 + 5 = 10+= 10Many answers work:6 + 4 · 1 + 93 + 7 · 5 + 5
When the same symbol shows up twice, it is the same number both times. Different symbols can be different numbers, so there can be many answers.
Worked example

A variable that can change

A bus has 10 seats. Some are filled with children and some are empty.

c + e = 10

What do c and e stand for? Find 3 different pairs that work.

  1. c stands for the number of seats with children. e stands for the number of empty seats.
  2. At each stop, children get on and off, so c and e can change. But the total is always 10.
  3. Try c = 4. Then e = 6, because 4 + 6 = 10.
  4. Try c = 7. Then e = 3. Try c = 10. Then e = 0 (the bus is full!).

c is the seats with children and e is the empty seats. Some pairs: 4 and 6, 7 and 3, 10 and 0. Any two numbers that make 10 work.

Ask yourself

When you see a symbol, ask: Does it tell me what to do (+, −, =)? Or is it standing in for a number? If it stands in for a number, it is a variable. Then ask: What number could it be?

Just for fun

Why did the ☐ go to math class?

It was tired of feeling empty! 📦

Worked example

Write a number sentence with a variable

Write a number sentence for this story. Use ? for the unknown.

There were 15 geese on the pond. Some flew away. Now there are 9 geese.

  1. Start: 15 geese.
  2. Some flew away: that is the unknown, so write ?. Flying away means subtract.
  3. Now there are 9.
  4. Number sentence: 15 − ? = 9.
  5. Find it: 15 − 6 = 9, so ? = 6.

15 − ? = 9. The ? stands for the number of geese that flew away. ? = 6.

Your turn

1

Which symbol is a variable? 6 + ☐ = 11

2

Find the number: 6 + ☐ = 11

3

Find the number: ? − 4 = 6

4

Find the number: ⭐ + ⭐ = 12

5

Is = a variable?

6

Leah read some pages on Monday and 7 pages on Tuesday. She read 20 pages in all. ☐ + 7 = 20. What does ☐ stand for?

Hands-on activity

Mystery cup

You need: a paper cup, about 20 small objects (beans, buttons or cereal), paper and a pencil.

  1. A grown-up secretly puts some objects under the cup and some beside it.
  2. They tell you the total, like There are 14 in all.
  3. Count the objects you can see (say 9). Write a number sentence: ☐ + 9 = 14.
  4. Say what ☐ stands for: the beans under the cup.
  5. Figure out ☐, then lift the cup to check!
  6. Switch roles. Try a version with a minus: I had 14 and hid some. 9 are left.

More practice: warm-up

1

Find the number: 3 + ☐ = 10

2

Find the number: ? + 2 = 9

3

Find the number: 10 − △ = 8

4

Which of these is not a variable when it is in a number sentence: ☐, ?, +

5

Find the number: n + n = 4

More practice: on your own

1

Find the number: ☐ + 20 = 50

2

Find the number: 18 − ? = 10

3

Write a number sentence with a variable: I had some stickers. I got 6 more. Now I have 15. What number is the variable?

4

🔺 + 🔵 = 8. Give two different answers.

5

Tom says in ⭐ + ⭐ = 14, one star could be 10 and the other could be 4. Is he right?

Challenge

1

Find both numbers: ☐ + ☐ = 16 and ☐ + △ = 10

2

A pack of juice boxes has j boxes. Two packs have 12 boxes in all: j + j = 12. How many boxes are in one pack? Explain.

3

Write a story that matches ? − 5 = 20. Tell what ? stands for and find it.

Check your understanding

Quiz

1.

In 9 + ☐ = 15, what is the ☐?

2.

What number is ☐? 9 + ☐ = 15

3.

What is ⭐? ⭐ + ⭐ = 18

4.

Which number sentence matches? I had some apples. I ate 2. Now I have 5.

5.

Which symbol is not a variable?

6.

🔺 + 🔵 = 10. Which could be true?

For parents and teachers

A variable is a symbol (box, question mark, shape or letter) that stands for a number. This is an early step toward algebra. In Grade 2, children identify variables, say what they stand for in a story, and find the value.

Common mistakes

  • Treating the box as "the answer goes here" and adding all the numbers (writing 24 in 9 + ☐ = 15). Ask: What does the box stand for in the story?
  • Giving the same symbol two different values (in ⭐ + ⭐ = 14). Remind: same symbol, same number.
  • Calling +, − or = a variable.

How to help

  • Play mystery cup with snacks.
  • Make up short stories with an unknown and have your child write the number sentence.
  • Use letters that match the story (c for cookies, p for pages). Children enjoy this!