Patti BakerWinnipeg, CA,

November 18, 2018

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Learn to use counting on, make 10, doubles, near doubles, landmark numbers, and compensation.

- Grade 3
- Grade 4
- Grade 5

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**Strategies for Addition**

Please note all of the strategies below require frequent practice as well as experience and knowledge in rote counting, representing, decomposing and reconstructing numbers.

In this lesson you will learn to identify and express the following addition strategies:

**Counting On****Doubles****Near Doubles****Make 10 or Landmark Numbers****Compensation**

**Counting On: **

This strategy is one of the first a student can learn to add two addends together to determine their sum. For example: 2 + 4 is quite the conundrum for a student that has not developed automaticity for this fact.

First the student should be taught to identify the greater addend, that is to say: the largest number. In this example it is 4. We identify and start this strategy at the number 4 in order to reduce the amount of counting an individual must perform. So begin at 4 and count aloud, from four, two times. Early users to the strategy often use their fingers, counters, or physical manipulatives to aid them in counting on the correct number of times.

It should sound like: “4, 5, 6. The answer is 6.”

**Doubles:**

This strategy is a natural extension of skip counting skills. Once an individual can skip count using 1’s, 2’s, 4’s, and 5’s. If a child can count by 3’s and 6’s even better!

The strategy can be used as follows: 4 + 4

This seemingly simple calculation can be tricky. Using counting on it would take five steps: 4, 5, 6, 7, 8! but using doubles it is simply one step: 4, 8.

**Near Doubles:**

This strategy allows the user to add uneven addends using the doubles strategy. The notion of adding one or subtracting one is brought into use to enable the addition of odd numbers, or unlike terms.

4 + 5

In this strategy, as before the user would identify that five is almost four. 4 + 5 is the same as 4 + (4+1). So 4, 8 and one more is 9.

This also works for other addends and can be used to make a subtraction operation as follows: 7 + 8 is the same as 8 + (8 -1) because 8 -1 is 7. It is important for brackets to be introduced here so that students can identify how certain numbers can be expressed as equations.

For example 10 is the same as: ( 9 + 1) or (12 - 2)

**Make 10 & Landmark Numbers**

Make 10 is often referred to Landmark Numbers when used with addends that sum to amounts larger than 10.

This strategy/rote learning is very much based on rote practice, much like learning the times tables. The idea is an individual will recite, practice and explore making sums to ten using physical objects, their fingers, rhymes, and games to master the complements to 10.

At the end of constant, consistent rote learning activities (perhaps 25 minutes a day for two weeks) an individual will master and identify:

0+10 is 10

1+9 is 10

2+8 is 10

3+7 is 10

4+6 is 10

5+5 is 10

6+4 is 10

7+3 is 10

8+2 is 10

9+1 is 10

10+0 is 10

Once the individual has mastered this knowledge via demonstrations of using this skill, they will be able to apply it at higher and higher levels in conjunction with counting by tens. This is referred to as Landmark Numbers.

Landmark numbers is very much the same as the make 10 strategy. The only major difference is the individual, having mastered the make 10 strategy will immediately move on to counting by 10's to 100. Rather than identify 10 as the total to sum to, the individual will identify the nearest ten (20, 30, 40) etc.

Having succeeded in using Make 10, this individual will be ready for Landmark Numbers.

Landmark Numbers is an addition strategy that works as follows:

32 + 47 = ?

First, the individual will count the tens either using manipulatives, their fingers (via the counting on strategy), or doubles, near doubles.

They should immediately identify that 30 + 40 is seventy. Once this occurs, the individual using the strategy should record this sum. Next, they should add up the ones: 2 + 7, knowing that 2 + 7 is 9 the individual should write another Nine. 7 tens and nine more is 70 + 9, which is 79.

Notice that in the beginning this strategy should not be taught with numbers that require regrouping in the ones position. That is an added layer of complexity to the problem that often requires some understanding of compensation. An individual should practice this skill extensively before moving on to regrouping.

**Compensation**

This addition strategy is very important for understanding many multiplication and division strategies. It is also a key step along the developmental path toward the standard algorithm for addition.

Students are expected to compensate to a given landmark using the previous strategy in a slightly different way. For example: 9 + 2 is the same as 9 + (1 + 1). It is important to introduce brackets here as it allows students to still clearly identify how the 1 + 1 is 2, but also to very much appreciate that 9 + 1 + 1 is 10 + 1 and so makes 11. We simply **borrow **a one from the two, leaving one behind. This makes the nine into a ten and leaves an extra one.

It is important to note that 14 + 8 can be seen as: 10 + 4 + (6 + 2). We know from our make 10 facts that 4 + 6 make 10 and we also know how to count by 10’s. Thus, we can examine 14 + 8 as: 10 + 10 + 2, or simplify it to 22.

I will provide four more examples:

28 + 9, using compensation this question can quite comfortably written as: 28 + (2 + 7) or somewhat more awkwardly as: (27 + 1) + 9. It is in my experience, preferred, that students perform operations with the smaller addend in order to have the least affect on the over all sum should a mistake happen. An inaccuracy of 2 or 3, is much more different, than being off by 20 or 30, and this also leads to rich mathematical conversations.

36 + 47

36 + 47 can be summed as follows: (30 + 3 + 3) + (40 + 7) or any other number of ways, but this seems most efficient.

Then it would be consolidated or added up in steps as follows:

30 + 3 + 40 + 10. (The 7 + 3 were used to make a 10)

then 30 + 40 + 10 is 80, + 3 more is 83.

54 + 26

54 + 26 can be summed as follows: (50 + 4) + (20 + 6). This allows us to combine the 4 + 6 which make 10 according to our make 10 strategy. 10 + 50 + 20 is 80. So the answer is 80.

12 + 79

Last but not least is yet another regrouping problem. It can be deconstructed as follows: (10 + 1 + 1) + (70 + 9). We borrow the one from the 12 and it allows us to make 79 into 80. Thus we have 80 plus 10 plus one. Or 80 + 10 + 1 = 91.

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