Sequences for Mental Math Addition and Subtraction

The following strategies can be introduced to students with a sequence of similar problems. Select a strategy for students to invent and practice. Write a sequence of problems that can be solved with the strategy. Display one at a time for students to solve, let students solve it, and discuss the strategy that they used. If no one uses the strategy. Try to think of a simpler problem where the strategy could be more easily used. If that fails then you might want to say, "Here is the way that I solved it." Then follow up with another and ask what they think about the strategy.

The order of the following sequences is not the only order in which they could be presented. As knowledge varies from person to person the place to start varies also. Each strategy varies in complexity relative to the values of the numbers. A spiral approach would seem to be an effective way to proceed.

Making leaps/jumps of ten(s)

Leaps of ten(s) and compensation,

Leaps of ten(s) and adjusting, or moving to the next nearest ten and adjust (15 + 9 = 20 + 4

15 + 9 = (15 + 10) - 1
If no one uses the strategy of ten. Give the next problem(s) with ten (15 + 10). When addition of ten is used, return to nine.
15 + 9
15 + 19
28 + 19
28 + 32
39 + 21
28 + 44 = 28 + 40 + 4 = 68 + 4=72
63 + 10
43 + 10
123 + 10
143 + 100
143 + 107
138 + 20
138 + 23
138 + 123

Making jumps of ten backward

150 - 39 = 62 - 10 62 - 30 62 - 34 178 - 10 178 - 30 178 - 39

Using known facts

(2 + 2 = 4, so 2 + 3 = 5) or (8 + 6 = 14, so 8 + 7 = 14 + 1) or (9 + 6 = 15, so 8 + 6 = 14)

Counting on, counting backward, adding on or removing

Because addition and subtraction are related so are adding on and removing.

62 - 4 makes more sense to remove 4 or work backwards from 62 to 58. But with 62 - 54 it makes more sense to add on from 54, 6 + 2 = 8. Usually when numbers are closer together it is better to add. When they are far apart it's better to work backward. Structure your sequences with that in mind to get students to learn when to add or remove.

33 - 4 (remove or work back) 33 - 7 42 - 37 (add on) 33 - 28

It is important students learn that subtraction problems can be solved with addition strategies.

343 - 192 (I start with192 and jump 8 to 200. Find the difference of 200 and 343 and add 143 and 8 = 151)
175 - 139 = (Write all solutions to encourage thinking.)
175 - 19

Doubles

Read Madeline "…In an old house in Paris, that was covered with vines, lived twelve little girls in two straight lines…"

How many people in each line if in pairs? What if 14,…

Illustrate skip counting number line with pictures of kids in line two by two. Below appropriate kids put 1 + 1, 2 + 2, 3 + 3 and below that put 2, 4, 6, 8.

What else comes in twos? Mittens, shoes… Shoes off and at door. Each kid draws their family's shoes at the door.

Double your number. Board game where double a roll on a die to move. It's like 10 shoes, five left and five right, like 5 pairs or ten shoes.

Doubles, near doubles with addition, doubles with plus and minus

5 + 5 5 + 6 5 + 4 7 + 7 7 + 6

When students understand this then give them problems with 2 more or less to develop Compensation (8 + 6, same as 7 + 7 if subtract 1 from 8 and add it to 7)

For bigger numbers start with a double and give them three problems to use compensation then switch to another double if all okay.

25 + 25 25 + 26 25 + 28 25 + 24

Doubles and near doubles with subtraction

52 - 25 = 25 + 2 50 - 25 52 - 25 70 - 35 72 - 35 40 - 20 40 - 21 40 - 19

Constant difference

Compensation 49 + 51 = 50 + 50 doesn't work with subtraction. Think of subtraction as the difference. 52 - 34 add 6 to each, 58 - 40, explain with number line as constant difference, 18 can slide in either direction. To understand subtraction as the distance between two points it can be modeled on a number line. Grade 2 +. This is a difficult strategy for children to understand. 1436 - 188, add 12, 1448 - 200, 175 - 139 174 - 138 173 - 137

Canceling out common amounts

Is really the same as constant difference. 120 - 109, cancel 100 from both and left with 20 - 9, 11 Can show with a double number line that 100's are the same and 20 is 11 more than 9. 20345 - 10012=10333

Splitting

28 + 44 = 20 + 40 + 8 + 4 = 60 + 12=72 Children usually split (decompose) numbers by place values. Not real efficient. However, more so than counting.

Moving to the next friendly ten by counting back

143 - 3 143 - 23 143 - 24 (move 3 to 140 then 20 to 120 then 1 to 119) 164 - 25 182 - 43

Moving to the next friendly number.

98 + 37 = 100 + 35 27 + 49 = 26 + 50 36 + 118 = 34 + 120 227 + 164 = 230 + 161 38 + 6 38 + 26 47 + 24 50 + 21 47 + 4 47 + 34 50 + 31 100 + 27 98 + 29 96 + 29 95 + 30 300 + 35 298 + 37

Swapping digits

293 + 919 Swapped digits 213 + 999, then used compensation 1000 + 212 34 + 19 39 + 14 71 + 26 76 + 21 449 + 192 142 + 499 442 + 199

Commutative property of addition (8 + 7 = 15, so 7 + 8 = 15)