B. Number · Operations: Adding, Subtracting and Equal GroupsMath Rules: Any Order, Zero and Grouping
B2.1 A1About 30 minutes
Learning goals- I can add numbers in any order and get the same sum.
- I can explain what happens when I add or subtract 0.
- I can group numbers in a smart way to add three or more numbers.
- I can explain why subtracting in a different order does not work.
Words to know
property
A math rule that is always true, like "you can add in any order".
sum
The answer when you add.
difference
The answer when you subtract.
friendly numbers
Numbers that are easy to work with, like 10, 20 or 30.
partners of 10
Two numbers that add up to 10, like 3 and 7, or 4 and 6.
Let's start with a question
Which is easier to work out in your head?
Most people think 9 + 2 is easier: start at 9 and count on 2. Here is a secret: they have the same answer, 11. Today you will learn some math rules that are always true. Mathematicians call them properties. They make adding faster.
Rule 1: You can add in any order
When you add two numbers, you can switch them around. The sum stays the same.
4 + 7 = 11 and 7 + 4 = 11
So it is smart to start with the bigger number and count on.
Worked exampleExample 1: Start with the bigger number
- Switch the order: 3 + 16 is the same as 16 + 3.
- Start at 16 and count on 3: 17, 18, 19.
19. 3 + 16 = 16 + 3 = 19.
Rule 2: Adding or subtracting 0
Zero means nothing. If you add nothing, the number stays the same. If you take away nothing, the number stays the same.
- 15 + 0 = 15
- 0 + 32 = 32
- 18 − 0 = 18
And if you take away all of something, nothing is left:
Zero is sneaky
18 − 0 is 18, not 0. You did not take any away, so you still have all 18.
18 − 18 is 0. You took all 18 away, so none are left.
Rule 3: You can group numbers any way
When you add three or more numbers, you can add them in any order and group any two first. The total stays the same.
Look for pairs that are easy, like partners of 10 (1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5) or doubles.
| Partners of 10 | |
|---|
| 1 + 9 | 6 + 4 |
| 2 + 8 | 7 + 3 |
| 3 + 7 | 8 + 2 |
| 4 + 6 | 9 + 1 |
| 5 + 5 | 10 + 0 |
Worked exampleExample 2: Look for a ten
- Look for partners of 10. 6 and 4 make 10!
- Group them first: 6 + 4 = 10.
- Then add the 5 that is left: 10 + 5 = 15.
15. (6 + 4) + 5 = 10 + 5 = 15.
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6 red leaves and 4 red leaves make 10. Add 5 more leaves: 15.Worked exampleExample 3: Look for a double
- Look for a double. 7 and 7 make 14.
- Then add 2: 14 + 2 = 16.
16. 7 + 7 + 2 = 14 + 2 = 16.
Worked exampleExample 4: Four numbers
At a skating party, the snack table has 8 juice boxes, 5 milk boxes, 2 water bottles and 5 apple juices. How many drinks are there?
- Write it: 8 + 5 + 2 + 5.
- Group partners of 10: 8 + 2 = 10.
- Group the double: 5 + 5 = 10.
- Add the tens: 10 + 10 = 20.
Rule 4: Subtraction is different!
You cannot switch the numbers when you subtract.
12 − 5 = 7, but 5 − 12 is not 7. You cannot take 12 away from 5 cookies, because you only have 5!
When you subtract, the order matters. Always start with the whole (the amount you have).
| Rule | Adding | Subtracting |
|---|
| Switch the order? | Yes: 4 + 9 = 9 + 4 | No: 9 − 4 is not the same as 4 − 9 |
| With 0 | 13 + 0 = 13 | 13 − 0 = 13 |
| Same number | 6 + 6 = 12 (a double) | 6 − 6 = 0 |
Prove it!
Mathematicians like to show that a rule is true. You can prove that 4 + 9 = 9 + 4 by building 4 cubes and 9 cubes, then turning the line around. You still have 13 cubes.
Just for funZero is the laziest number in math. Add it, take it away, and the answer doesn't change one bit: 18 + 0 = 18. Nice work, zero. Very relaxing.
Hands-on activityRule detectives
You need: 20 small things (pennies, beads or cereal pieces) and 3 cups.
- Put 3 things in one cup, 8 in another and 7 in another.
- Pour two cups together first and count. Then add the third cup. Write the total.
- Put them back. Now pour together a different two cups first. Is the total the same? (It should be 18 every time.)
- Which two cups were easiest to put together first? (Most people choose 3 and 7, because they make 10.)
- Try to take 8 things away from a cup that has 3. What happens? Talk about why subtraction order matters.
Math talk: "How could you prove to a friend that you can add in any order?"
Your turn
1If 8 + 9 = 17, what is 9 + 8?
17. You can add in any order.
25. Adding zero does not change the number.
0. If you take away all of them, none are left.
4Solve 2 + 9 + 8. Which two numbers did you add first?
19. Add the partners of 10 first: 2 + 8 = 10. Then 10 + 9 = 19.
5Solve 4 + 13. Which number would you start with?
17. Start with the bigger number, 13, and count on 4: 14, 15, 16, 17.
6Is 7 − 3 the same as 3 − 7? Why?
No. 7 − 3 = 4. You cannot take 7 away from 3 things. In subtraction, order matters.
16. Double 5 + 5 = 10, then 10 + 6 = 16.
More practice: warm-up
3Fill in the blank: 3 + 6 = 6 + __
3. Switching the order keeps the sum the same.
4Which two numbers are partners of 10: 3, 5, 7?
17. 1 + 9 = 10, then 10 + 7 = 17.
More practice: on your own
20. Group partners of 10: 4 + 6 = 10 and 8 + 2 = 10. Then 10 + 10 = 20.
2True or false: 15 + 4 = 4 + 15
True. Both sides make 19. You can add in any order.
3Tom says 16 − 0 = 0. Is he right?
No. 16 − 0 = 16. Taking away nothing leaves all 16.
24. Double 9 + 9 = 18. Then 18 + 6 = 24. (Or 9 + 6 = 15, then 15 + 9 = 24.)
5A family packs 3 granola bars, 4 apples and 7 carrots for a hike. How many snacks? Show a smart way.
14 snacks. Add 3 + 7 = 10 first. Then 10 + 4 = 14.
Challenge
1Solve 6 + 7 + 3 + 4 in your head. Explain how you grouped the numbers.
20. 6 + 4 = 10 and 7 + 3 = 10. 10 + 10 = 20.
2Fill in the blank: 8 + 5 + 2 = 10 + __
5. 8 + 2 = 10, and the 5 is left over. Both sides make 15.
3Jada says, "I can switch the numbers in 10 − 4 to get 4 − 10. It is the same." Use a story to show why she is wrong.
Example: "I have 10 stickers and give away 4. I have 6 left." With 4 − 10, you would have only 4 stickers and try to give away 10, which you cannot do. Subtraction order matters.
Quiz
1. Which is the same as 6 + 9?
4. What is a smart first step for 7 + 4 + 3?
Big idea: These are the properties of addition the curriculum asks Grade 2 students to use (B2.1): the commutative property (add in any order), the identity property (adding or subtracting 0), and the associative property (group numbers any way when adding). Children do not need to learn these formal names. They need to use the rules to make calculations easier, and to know that subtraction does not work in any order.
Common mistakes:
- 18 − 0 = 0. Act it out: "You have 18 crackers and I take none. How many do you have?"
- Always adding left to right. Encourage scanning for partners of 10 and doubles first.
- Switching the numbers in subtraction (5 − 12 = 7). Use a story: you can't give away more than you have (with whole numbers, in Grade 2).
How to help: When adding several prices, scores or snacks, ask "Which numbers would you put together first? Why?" Ask your child to prove a rule with objects; explaining and proving is part of the mathematical processes (strand A).