Suggestons to create sequences to teach strategies to solve multi digit addition and subtraction

Introduction

This page includes instructional suggestions and sequences to help develop people's ablity to solve addition and subtraction problems mentally. And sequences to introduce different strategies, which everyone is capable of learning and using.

Prior to teaching or learning these strategies it is helpful for learners to have had experiences to develop their spacial abilities. Abilities to visualize the relationships of numbers. For example, relationships of different values, as represented on a hundred chart or number line as well as addition and subtraction with arrow math on a hundred chart. Also beneficial, is numerous experiences with single digit addition and subtraction and undeerstanding the different ways addition and subtration are represented. Additionally instruction which has included deconstruction and reconstruction of values to add and subtract is helpful. For example:

• 7 + 8;
  • can be deconstructed to (5 + 2) + 8;
  • rearranged as 5 + (8 +2);
  • then 5 + (10); 15
• Or 7 + 8;
  • can be deconstructed to (5 + 2) + (5 + 3);
  • rearranged as (5 + 5) + (2 + 3);
  • the (10) + (5); 15

Strategies like these can be taught with a general instructional sequence and a list of problems as is described by the following sequences:

General instructional method

Planning:

• Select a strategy for learners to invent and practice (jumps of ten; using tens & adjusting; making jumps of ten backward; using known facts, counting on, back, adding on or removing; subtraction with addition; doubles; doubles, near doubles with addition, doubles with plus & minus; doubles & near doubles with subraction, sliding differences; canceling out common amounts; decompose; removing friendly numbers by counting back; adjust to the next friendly number; swapping digits; & commutative property of addition). The list seems daunting at first, but with experience they become second nature as a consequence of number literacy.
• Create a list of sample problems. Suggestions for making and using a sample list of problems:
  • A sample list should be selected, made, and motified depending on the ability of the learner who is to learn or pratice the strategy. Problems in a sequence should be incrementally challengeing, but not so hard the learner is unable to be successful using the strategy.
• Display one problem at a time.
• Ask. Solve it.
• Provide adequate wait time for them to solve it.
• Share different solution methods.
• As you share, write all solutions to encourage critical thinking.
• Discuss all the different strategies used.
• If no one uses the selected strategy, think of a simpler problem where the strategy could be more easily used. If that fails then you might say:
  Here is a way I have seen others solve it.
• Follow up with another problem.
• Ask. Did you use the strategy?
• Continue until they are proficient.
• Conclude by asking. What do you think about the strategy?

Periodically review:

To help learners remember and use the different strategies, a spiral approach that periodicaly returns to each presented strategy unexpectedly is helpful. Periodically randomly presenting a problem from each strategy to review.

The problems in the following strategies are provided as samples, which portions can be used across all grades. The order of the problems are not the only order in which strategies can be presented. Depending on the student's experiences with mathematics and their use of a strategy, the list may vary from the orders presented here.

Conclusions:

Decide where to start, what problem to follow each problem with, and when to stop if the learners are not ready to move through a complete list.

Jumps of ten

Background: counting by tens.

• 10, 20, 30, 40, ...
• 15 + 10; 15, 25, 35, 45, ...
• 4, 14, 24, 34, 44, ...
• 26, 36, 46, 56, ...

Samples:

Randomly select a number and have students call out ten more:

• 5; 15
• 24; 34

• 8 + 10
• 15 + 10
• 22 + 10
• 34 + 10
• 46 + 10
• 78 + 10
• 14 + 20
• 26 + 30
• 24 + 80
• 112 + 10
• 234 + 10
• 23.5 + 10

Using tens & adjusting

Background: counting by tens, making tens & left overs

Samples:

• 15 + 9; 15 +(5 + 4); (15 + 5) + 4; 20 + 4; 24
• 15 + 9; 15 + (10 - 1); (15 + 10) - 1; 25 - 1; 24
• 28 + 44; 28 + (40 + 4); (28 + 40) + 4; 68 + 4; 72
• If no one uses the strategy of ten. Give problems from the jumps of ten list. When they are back on track, return to problems with nine.

• 15 + 9
• 15 + 19
• 28 + 19
• 28 + 32
• 39 + 21
• 28 + 44
• 63 + 10
• 43 + 10
• 123 + 10
• 143 + 100
• 143 + 107
• 138 + 20
• 138 + 23
• 138 + 123

Making jumps of ten backward

Background: counting by tens backwards, reducing or adding to tens with left overs. If no one uses the strategy of ten. Give problems from the jumps of ten list. When they are back on track, return to problems with nine.

Samples:

• 62 - 34;
  • 62 - 30 - 4;
  • (62 - 30) - 4;
  • 32 - 4; 28
• 178 - 39;
  • 178 - 30 - 9;
  • (178 - 30) - 9;
  • 148 - 9;
  • (148 - 8) - 1;
  • 140 - 1; 139

• 40 - 10
• 40 - 20
• 62 - 10
• 62 - 30
• 62 - 34
• 150 - 30
• 178 - 10
• 178 - 30
• 178 - 39

Using known facts

Background: some basic facts

Samples:

• Know: 2 + 2 = 4, ∴ 2 + 3 = 5
• Know: 7 + 7 = 14, ∴ 8 + 6 = 14
• Know: 8 + 6 = 14, ∴ 8 + 7 = 14 + 1
• Know: 9 + 6 = 15, ∴ 8 + 6 = 14)
• Know: 8 + 2 + 5 = 15, ∴ 8 + 7 = 15

Initally help students learn the addition facts they struggle to remember, which an include most of the facts with an addend above five.

• 4 + 7
• 8 + 6
• 14 + 6
• 7 + 7
• 19 + 6
• 18 + 4
• 23 + 5
• 24 + 7
• 36 + 7

Counting on, counting back, adding on or removing

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

Samples:

• 62 - 4
  • Makes more sense to remove 4, or work backwards, from 62 to 58.
• 62 - 54
  • This makes more sense to add on from 54, 6 + 2 = 8.
• 33 - 4;
  • Remove or work back 33 - 3; 30 - 1; 29
• 33 - 7;
  • Remove or work back 33 - 3; 30 - 4; 26
• 42 - 37;
  • Add on 37 + 3; + 2; 5
• 33 - 28;
  • Add on 28 + 2; + 3; 5

Suggestions:

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 help learn when to add or remove.

• 31 - 3
• 31 - 24
• 42 - 35
• 32 - 28
• 67 - 12
• 67 - 54

Subtraction with addition

Background: It is important to learn subtraction problems can be solved with addition strategies.

Samples:

• 343 - 192
• Start with 192 and jump 8 to 200.
• Find the difference of 200 and 343; 143
• Then add 143 and 8; 151
• 175 - 139
  • Start with 139 and jump to 140
  • Remember 1;
  • Jump to 150; remember 10
  • Add 10 to 1; 11 and
  • Lastly jump to 175; 11 + 25; 36

• 243 - 43
• 243 - 239
• 243 - 192
• 156 - 40
• 166 - 32
• 986 - 943
• 346 - 193
• 258 - 136
• 894 - 144

Doubles

Strategy to introduce doubles to young children:

• 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.
• Ask. What else comes in twos? Mittens, shoes …
• Ask. What happens when the shoes come off and at door?
• Tell. Draws your 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.

Randomly say numbers and ask for their double.

• 3, 4, 2, 1, 5, 8, 5, 7, 2, 9, 11, 5, 6, 7, 2, 9, 3, 8, 2, 5, 3, 4, 10
• 13, 8, 6, 9, 4, 22, 12

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

Background: doubles.

Samples:

• 3 + 3; double
• 3 + 4; use double plus one
• 4 + 4; double
• 4 + 5; use double plus one
• 5 + 5; double
• 5 + 6; use double plus one
• 5 + 4; use double plus one
• 7 + 7; double
• 7 + 6; use double plus one
• 8 + 8; double
• 8 + 9 use double plus one
• 9 + 9 double
• 8 + 6 near double; 7 + 7
• 12 + 14 near double; 13 + 13
• 12 + 10 near double; 11 + 11
• 25 + 27 near double 26 + 26

• When students understand these, give them problems with 2 more or less to develop near doubles;
• 8 + 6, near doubles to 7 + 7 if subtract 1 from 8 and add it to 7

Mix them up:

• 8 + 8
• 8 + 7
• 8 + 9
• 8 + 6
• 12 + 12
• 13 + 12
• 12 + 14
• 12 + 10
• 25 + 25
• 25 + 26
• 25 + 27
• 25 + 24
• 23 + 25
• 18 + 16
• 18 + 17
• 18 + 14

Doubles and near doubles with subtraction

Background: See doubles and near doubles above.

Samples:

• 30 - 15; think double 15 is 30; ∴ 30 - 15 is 15
• 52 - 25; think 52 is double 25; ∴ 52 - 25 is 25 + 2

• 50 - 25
• 52 - 25
• 70 - 35
• 72 - 35
• 40 - 20
• 40 - 21
• 40 - 19

Sliding differences

Adding 51 + 49 by adjusting each to 50 + 50, doesn't work the same way with subtraction.

Subtraction is the difference. 51 - 49. Visualize a number line. The difference between the numbers has to stay the same. Therefore, to keep them the same, a value of adjustment can be added or subtracted to both. For example: to make it easier for some to see, add one to both, or sliding both. 51 and 49; slide one to 52 - 50; which is easier to see a difference of 2.

Another: 52 - 34. Could add 6 to each; 58 - 40, which is easier to see a difference of 18. You can use a number line to model how the differeence between 52 and 34 stays the same (18) if the numbers are slid from 52 t0 58 and 34 to 40. can slide in either direction.

This strategy is Not for below third grade. This is a difficult strategy for children to understand. However for problems such as: 1436 - 188, adding 12, makes a much easier problem 1448 - 200,

• 175 - 139
• 174 - 138
• 173 - 137

Canceling out common amounts

Is the same as sliding differences. 120 - 109, cancel 100 from both makes 20 - 9; 11.

This can be modeled with a double number line to show that the difference between the numbers stay the same as the numbers are slid along the number line. Which, is the same as canceling the 100's from each to leave 20 - 9; 11 or the difference between 20 and 9 is 11. The same difference between 120 and 109.

20342 - 10012; 10330

Decompose, compose and create mental algorithms

Children usually decompose numbers left to right by place values.

28 + 44 is decomposed into

• 20 + 40; 60
• 8 + 4; 12
• 60 + 12; 72

Learners will solve problems by decomposing by place values. It takes a lot of experience before the are able to understand and operationalize addition and subtraction with two or more digits, each with different place values. Eventually, they will understand to add 23 and 34, theymust operate on all numbers within their place values:

• 23 + 34; decompose
  • (20 + 3) + (30 + 4); regroup
  • (20 + 30) + (3 + 4); add the tens,
  • 50 + (3 + 4); add the ones
  • 50 + 7; then combine the tens and the ones; 57

To help learners move to a more efficient strategy, you may need to encourage them not to decompose both numbers, but to select one to decompose and one to add on from.

• 23 + 34; decompose 23 and don't decompose 34
  • (20 + 3) + 34; commutative property
  • (34 + 20) + 3; add
  • 54 + 3; 57
• 78 + 26; decompose 26 and don't decompose 78
  • 78 + (20 + 6)
  • (78 + 20) + 6 add
  • 98 + 6; 104
• 183 + 58; decompose 58 and don't decompose 123
  • 183 + (50 + 8); commutative property
  • (183 + 50) + 8; add [think (183 + 20 + 30); (203 + 30)
  • 233 + 8; add [think 233 + 7 + 1]
  • 241

In today's world with calculators and spread sheets the strategies presented here along with mental algorithms created with the use of decomposition, compositioni, and working from left to right can proved probably all the a person needs for everyday calculations. The beauty of the traditional algorithm isn't in using it when other strategies are more efficient it is in its elegancy of working in all situations, particularly in advanced mathematics.

Removing to friendly numbers by counting back

Background: friendly numbers are tens and fives

Sample:

Subtract parts by removing amounts to work with friendly numbers.

• 143 - 24;
  • think remove move 3 to 140
  • 143 - 3; leaves
• 140 - 21;
  • then think to remove 20
  • 140 - 20; leaves
• 120 - 1; 119

• 164 - 25
• 182 - 43

Adjust to the next friendly number

Background: friendly numbers are tens and fives

Samples:

• 47 + 4;
  • adjust to 50
  • 50 + 1; 51
• 38 + 6; adjust to 40
  • 40 + 4; 44
• 98 + 37; adjust to 100
  • 100 + 35; 135
• 27 + 49; adjust to 50
  • 26 + 50; 76
• 36 + 118; adjust to 120
  • 34 + 120; 154
• 227 + 164; adjust to 230
  • 230 + 161; 391
• 47 + 24; adjust to 50
  • 50 + 21; 71
• 47 + 34; adjust to 50
  • 50 + 31; 81
• 98 + 29; adjust to 100
  • 100 + 27; 127 Or adjust to 30; 97 + 30; 127
• 96 + 29; adjust to 30
  • 95 + 30; 125
• 298 + 37 adjust to 300
  • 300 + 35; 335
• 442 + 199 adjust to 200
  • 441 + 200; 641

• 48 + 4
• 36 + 6
• 88 + 7
• 99 + 6
• 47 + 33
• 42 + 58
• 46 + 33
• 118 + 83

Swapping digits

Background: This is one that probably will need modeling.

Samples:

• 293 + 919; swap digits in the same place value (tens)
  • 213 + 999; then use adjust (add 1 to 999 and subtract one from 213)
  • 212 + 1000 = 1 212; add
• 34 + 19 swap digits (ones)
  • 39 + 14; then use adjust (add 1 to 39 and subtract one from 14)
  • 39 +1 + 13 = 53; add
  • Hard to argue using tens and adjusting is not easier (34 + 20) - 1; 53
    Good to prove learning is working if it is mentioned. ☺
• 71 + 26 swap digits
  • 76 + 21; then adjust
  • 76 + 1 + 20; add
  • 97
• 449 + 192 swap digits
  • 499 + 142; then adjust
  • 500 + 141; 641

• 53 + 29
• 82 + 78
• 84 + 78
• 129 + 93
• 449 + 291

Commutative property of addition

Background: Commutative propery of addition. Commutative property of addition is the order in which a pair of addends is added does not affect the sum. For all real numbers a and b, a + b = b + a

Samples:

• 8 + 7 = 15, ∴ 7 + 8 = 15
• 9 + 8 = 17, ∴ 8 + 9 = 17
• 8 + 6 = 14, ∴ 6 + 8 = 14

Dr. Robert Sweetland's notes
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