B. Number · Adding and Subtracting

Mental Math: Counting On, Counting Back and Doubles

B2.3 B2.1About 30 minutes

Learning goals

I can count on from the bigger number to add.
I can count back to subtract small numbers.
I know the doubles up to 10 + 10.
I can use a double to find a near double.
I can explain the strategy I used.

Words to know

mental math
Doing math in your head, without writing it all down.

strategy
A smart way or plan to solve a problem.

count on
Start at a number and count forward.

count back
Start at a number and count backward.

near double
A fact that is one more or one less than a double, like 6 + 7.

Math in your head

You don't always need blocks or paper. With smart strategies, you can add and subtract numbers up to 20 in your head. Today we learn three strategies.

Strategy 1: Count on

To add a small number, start at the bigger number and count on.

For 9 + 3: Say "9" in your head. Then count 3 more on your fingers: 10, 11, 12.

9 + 3: start at 9, jump 1 at a time: 10, 11, 12.

Careful!

Don't count the number you start on! Put it in your head and say the next number when you put up your first finger.

Worked example

Example 1: Count on from the bigger number

What is 2 + 14?

Which number is bigger? 14. Turn it around: 14 + 2. (Adding in any order is OK!)
Say "14" in your head.
Count on 2: 15, 16.

2 + 14 = 16. I counted on from the bigger number.

Strategy 2: Count back

To subtract a small number (1, 2 or 3), start at the bigger number and count back.

For 13 − 2: Say "13". Count back 2: 12, 11.

13 − 2: start at 13 and jump back 2 to land on 11.

Worked example

Example 2: Count back

There are 15 toonies in a jar. You take out 3. How many are left?

Taking out means subtract: 15 − 3.
3 is small, so count back. Start at 15.
Count back 3: 14, 13, 12.

12 toonies are left. 15 − 3 = 12.

When counting back is too long

Counting back works best for taking away 1, 2 or 3. For bigger numbers, like 12 − 9, it is easier to think addition: 9 + ? = 12. Count up from 9: 10, 11, 12. That's 3.

Strategy 3: Doubles and near doubles

You know doubles to 10. Now let's go all the way to 10 + 10!

Double	Sum
1 + 1	2
2 + 2	4
3 + 3	6
4 + 4	8
5 + 5	10
6 + 6	12
7 + 7	14
8 + 8	16
9 + 9	18
10 + 10	20

Doubles to 20. The sums are all the even numbers. Count by 2s: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20!

6 + 6: two ten frames with 6 in each. 5 + 5 = 10 and 1 + 1 = 2, so 6 + 6 = 12.

A near double is one away from a double. Use the double you know, then add or take away 1.

6 + 7: I know 6 + 6 = 12. 7 is one more than 6, so 6 + 7 = 13.
8 + 7: I know 8 + 8 = 16. 7 is one less than 8, so 8 + 7 = 15. (Or 7 + 7 = 14, plus 1 = 15.)

Worked example

Example 3: Near double

What is 5 + 6?

5 and 6 are next to each other when we count. That is a near double!
I know the double 5 + 5 = 10.
6 is one more than 5, so the answer is one more than 10: 11.

5 + 6 = 11. I used a double plus one.

Doubles help subtract too!

If 7 + 7 = 14, then 14 − 7 = 7. If 9 + 9 = 18, then 18 − 9 = 9. Same fact family!

Explain your strategy

After you solve a problem, say how you did it. For example: "I knew 8 + 8 is 16, so 8 + 9 is one more, 17." Explaining your thinking is a big part of math. There is often more than one good way!

Just for fun

Fun fact: a spider has 8 legs, 4 on each side. That is a double: 4 + 4 = 8. Spiders are walking doubles facts!

Hands-on activity

Double Trouble dice game

You need: one die (or number cards 1 to 10) and paper.

Roll the die. Double the number and say the fact. "I rolled 4. 4 + 4 = 8!"
Now say the near doubles: "4 + 5 = 9 and 4 + 3 = 7."
Take turns with a grown-up. Score 1 point for each correct fact.

Bonus round: Find doubles in your home. Egg cartons (6 + 6 = 12), your two hands, a pair of socks, a box of 2 rows of juice boxes.

Math talk: "Which double did you use to help you?"

Your turn

8 + 2 = ? Explain how you know.

3 + 15 = ?

11 − 2 = ?

7 + 7 = ?

7 + 8 = ? Use a double.

16 − 8 = ?

9 + 10 = ?

14 − 3 = ?

More practice: warm-up

12 + 1 = ?

13 − 1 = ?

5 + 5 = ?

16 + 2 = ? Count on.

Is 4 + 5 a double or a near double?

More practice: on your own

18 − 3 = ? Count back.

9 + 8 = ? Use a double.

12 − 6 = ? Use a double.

Ana has 13 toonies. She gets 2 more. How many toonies does she have now?

3 + 16 = ? Which number do you start from? Why?

Challenge

Find the mistake. For 7 + 3, Ben says "7, 8, 9" on his fingers and answers 9. What went wrong?

Show two different ways to solve 6 + 7.

Tia has 9 shells in one bucket and 9 shells in another bucket. She gives 2 shells to her friend. How many shells does Tia have now?

Check your understanding

Quiz

For 3 + 12, which number is best to start counting on from?

What is 9 + 9?

6 + 6 = 12. So what is 6 + 7?

What is 17 − 2?

Which is a near double?

What is 15 − 3?

8 + 8 = 16. So what is 8 + 9?

For parents and teachers

Big idea: B2.3 asks children to use mental math strategies for sums up to 20 and to explain the strategy. The goal is flexible thinking, not one "right" method.

Common mistakes:

Counting the starting number (for 9 + 3 saying "9, 10, 11" and answering 11). Model saying the first number in a "whisper voice", then counting the next numbers out loud on fingers.
Counting on from the smaller number. 2 + 14 by counting 14 more is slow and error-prone. Remind them they may turn it around.
Counting back for big subtractions. For 13 − 9, encourage "think addition" or counting up from 9.
Guessing near doubles. Ask "Which double is close? Is the answer one more or one less?"

How to help: In the car or at dinner, ask quick facts and always follow with "How did you figure it out?" Accept different good strategies.