Crafty Combinations

This activity encourages students to look for clues that could help them select a strategy (in this case, combining numbers) to solve a problem more easily. Students are often encouraged to look for combinations of numbers that add to 10 or multiples of 10 when they first use combining strategies, but this activity helps students to recognise that the strategy can be used with other numbers as well. The strategy is applied to addition and subtraction of whole numbers and then to fractions and decimal fractions.
For questions 1–4 and 6, students need to be advanced additive thinkers (stage 6) or ready to make the transition to this stage. Students who are not familiar with fractions and decimal fractions may find question 5 an interesting introduction to them.
Before an independent group starts work on the activity, say: Sometimes when you need to add several numbers together, you can combine some of the numbers first and then add the answers together.  Why might you do this? Then work through the first example together. Some students might like to record by drawing curved lines to connect the pairs of numbers and writing the answer along the line.
Note that combining involves the idea of average or “middle”. For example, in question 3, the numbers are around a middle of 26: 21, 24, 25, 26, 27, 27, 29.
Use the same introductory question with a guided teaching group. Then work through the introduction and question 1 as a practical activity. Get the students in the group to make the following numbers by linking cubes together:

Write 3 + 5 + 2 + 6 + 7 + 4 on the board and then say: You might notice that some of these numbers combine to make exact 9s. Which ones are they? Get the students to join the cubes physically to make 3 groups of 9.
Show the students a hundreds board. Say: I want to work out what the first row of the hundreds board adds up to, but I want to find a quicker way than adding each one in order. I can see that 1 + 10 and 2 + 9 both add up to 11. How might that be useful?

After the students have found the other groups of 11, reinforce the idea of combining with this diagram:

Give each student a photocopy of the combinations copymaster so that they can do their own recording of their linking for question 1 and the rest of the questions.
With question 5, the students might find it useful to make the fractional numbers with a foam fraction kit.
Question 6 is designed to promote algebraic thinking. It asks students to think about when combining would be a useful strategy. Encourage the students to look carefully at the numbers in problems before they rush into solving them so that they have time to notice the clues that might help them to solve the problem with minimal difficulty and can choose a strategy that suits the problem. (One clue is the clustering of numbers around a middle number, as described above for question 3.)
To encourage this, ask questions such as:
Look at the problem carefully before you start to solve it. Which strategies might you use? Which do you think would be best? Why?
Now that you’ve done the problem, were you happy with your choice of strategy? If you were doing it again, what would you do differently?

Extension

Have the students investigate the use of combining strategies to add consecutive numbers for:
• the numbers 1 to 20
• all the numbers on a hundreds board
• multiples of 5 up to 100 (5, 10, 15, 20 …)
• any sequence of consecutive numbers, such as 23 + 24 + 25 + 26 + 27 + 28,or 20 + 22 + 24 + 26.
 

Ask:
Which numbers will you combine to help you add all these numbers quickly?
Does it make a difference if there is an odd or an even number of numbers to add?
What instructions would you give someone who wanted to quickly add a sequence of consecutive numbers?

 

Answers to Activity

1. 55. (1 + 10 = 11, 2 + 9 = 11, 3 + 8 = 11, 4 + 7 = 11, 5 + 6 = 11. 5 x 11 = 55)
2. Yes. (12 + 10 = 22, 11 + 11 = 22, 13 + 9 = 22, 14 + 8 = 22. 4 x 22 = 88)
3. $179. (21 + 29 = 50, 26 + 24 = 50, 27 + 27 = 54. 25 + 50 + 50 + 54 = 179)
4. 2 000. 800 + 200 = 1 000, 14 + 986 = 1 000, 555 + 445 = 1 000, and – 999 – 1 = – 1 000. 1 000 + 1 000 + 1 000 – 1 000 = 2 000
5. a. 14. (One way is: 4 lots of  1/2 = 2. 1 + 1 = 2, 2 + 2 = 4, 3 + 3 = 6.
2 + 2 + 4 + 6 = 14. Another way is to combine pairs of numbers to make 4:
+ 3 = 4, 1 + 3 = 4, 1 + 2 = 4. 3 x 4 + 2 = 14)
b. 8. (Look for decimals that add up to 2. 0.1 + 1.9 = 2, 1.2 + 0.8 = 2, 0.5 + 1.5 = 2, 1.6 + 0.4 = 2. 2 + 2 + 2 + 2 = 8)
c. 6. Look for fractions that add up to 1.
3/4 + 1/4 = 1,  2/5 + 3/5 = 1, 2/3 + 1/3 = 1, 4/7 + 3/7 = 1.1 + 1 + 1 + 1 = 4. 10 – 4 = 6.
6. a. They all contain numbers that combine to give whole numbers that are easy to add or subtract.
b. Not always. Sometimes the numbers will not add up to whole numbers that are easy to add or subtract.