In this unit students are encouraged to abandon counting methods for adding and subtracting in favour of more complex but more efficient mental strategies using known doubles facts to derive the answers to addition problems.
be able to recall instantly doubles from 1 + 1 to 9 + 9
use known doubles facts to work out addition problems mentally using part/whole reasoning
In this unit students who currently use "counting on" or "counting down" methods to solve addition and subtraction problems are encouraged to use doubles and "part/whole" mental methods to speed up mental computation.
Examples of "counting on" or "counting down" methods are
- The students work out 16 – 13 by starting at 13 and count 14, 15, 16; they know the answer is 3.
- The students work out 29 + 4 by counting 30, 31, 32, 33; they know the answer is 33.
Examples of "part/whole" methods for doubles are:
- The students work out 8 + 6 by removing 2 from the 8 to leave 6, then 6 + 6 =12 from a known doubles fact, and then add the removed 2 to 12 to give 14.
- The students work out 26 + 25 by adding 25 + 25 to give 50 and adding 1 to give 51.
It is desirable for students to move from counting methods to part/whole methods as counting methods are too slow for larger numbers. For example, students who attempt to work out 35 + 36 by "counting on" will soon lose their way.
The transformation from a counting method to the more abstract part/whole methods may prove difficult for students. The idea of breaking numbers into suitable parts to help solve the problem is not initially obvious and may take considerable effort for the students to understand the principle. For example students might model this problem on counters: Jean has 7 red sweets and buys 8 green sweets at the shop. How many has she got altogether?
A green sweet is removed to leave 7 in one pile and 7 in the other. Students, because they know their doubles, now say 7 + 7 = 14. Critically they must now add the removed sweet to 14 to give 15.
While this process often makes sense for students when materials are present they may experience difficulty when the materials are removed and they are asked to imagine the process. For example, to work out 9 cakes plus 8 cakes the students are asked to "see" without materials, that 9 contains 8 and 1, then to understand they use the known double 8 + 8 = 16, then they add the 1 to give 17.
The step to being able to ignore materials and working out addition problems by imaging is an important step along the way to being able to quickly and reliably use a "near doubles " strategy. Persistence on the part of the teacher is needed to help students make this transformation from needing materials to be able to image the answer.
- Check that the students know their doubles facts before starting. If the students do not know all the facts identify which ones they do know work from these ones.
- Pose addition problems that are near doubles, that is to say the two numbers to be added are almost equal. Using counters encourage the students to solve these problems by removing or adding counters to make the piles equal, then adding or removing to finish the problem.
Sarah has 6 sweets and Matua has 8 sweets. (Students model this with counters.)
Sarah complains it is not fair because Matua has got more sweets.
Matua gives a third student some of his sweets so Sarah and Matua are now equal (Students act this out with counters).
How many sweets does Matua give away? (2)
How many sweets do Sarah and Matua have now? (6+6=12)
How many sweets are there altogether (12+2=14)
Another strategy for solving the problem is to make Sarah’s and Matua’s piles equal (7+7=14). It usually takes time for the students to understand that Matua needs to give Sarah one sweet. The usual initial response is for the students to say that Matua should give two to Sarah.
- Pose another doubles problem. This time encourage the students to work out answers without materials by using imaging the counters or other mental processes.
Over the next 2-4 days the students work with a partner or in a small teaching group to solve problems involving near doubles. As the students solve the problems they are encouraged to share their strategies with others.
- Pose number stories for the students to solve using, for example, 6 + 7, 5 + 7, 9 + 8, 8 + 6.
- Encourage the students to use one or both of the strategies discussed on the previous day:
- Students use materials to solve them using the technique of removing some counters to make the two piles equal, using a known doubles fact then adding back the removed counters.
- Students use materials so that each pile has the same amount.
- As the students become confident with the solution of the problems using materials encourage them to adjust the numbers mentally.
June has $18 and Mere has $22. Mere kindly agrees to give June some of her money so they will have equal amounts. How much money does Mere give June? How much money do they have altogether?
- Increase the size of the numbers as the students become confident with making mental adjustments with the small numbers. Pose number stories using numbers such as: 21 + 19, 28 + 32, 49 + 51, 23 + 27, 48 + 22, 79 + 11.
- Encourage the students to adjust the numbers to make whole numbers of tens. (The numbers have been specially selected so that after adjusting to make the numbers equal the answer can be found by adding the relevant number of tens.)
- For the "experts" pose number stories using these numbers: 102 + 98, 97 + 303, 298 + 102, 499 + 1001
At the end of each session ask volunteers to explain their working and thinking to the rest of the class.