Sums and Products

Understanding that the term “sum” describes an addition relationship and the term “product” describes a multiplication relationship is vital with this problem. Students need to see that although the numbers used are the same, the relationship between them makes for a different result.
Have the students clarify the problem by restating it in their own words. Let them attempt two or three examples on their own and then bring them together to discuss the strategies they used. Most will be using a guess-and-check strategy. Ask them if they can find a more efficient way. If necessary, suggest that they explore the number of possible addends that can make the sum and the number of whole-number factors
that can make the product.
A systematic approach would be to draw up a table comparing addends and factors. For question 1b, it would look like this:

The students may be able to see that while there are four possible additions for this problem, there can only be two multiplications. It would therefore be more efficient to find the factors that make the product and then use this list to find the addends that sum to the correct total.
So in question 1f, if the students listed the factors for 35 as 1 x 35 = 35 and 5 x 7 = 35 and then looked to see which numbers were used again to make a sum of 12, they would quickly discover that 5 + 7 is the answer.

You can use question 2c as a challenge to see who can make up 10 sum-and-product questions for their group to solve. The students must know the answers before they ask their classmate to solve the problem, and they must be able to explain how the problems work.

Question 3 is there to challenge the incorrect view held by most students that multiplication always results in products that are greater than the factors used. To find a sum that is larger than the product, the students need to use a factor that is equal to or smaller than 1.
Make sure that your students understand the identity element for multiplication. They should be able to explain clearly that when a number is multiplied by 1, the result stays the same.

Answers to Activity

1. a. 1 and 4
b. 2 and 5
c. 4 and 3
d. 2 and 6
e. 2 and 8
f. 7 and 5
2. a. Answers will need to be two of the following:
6 + 4. 6 + 4 = 10, 6 x 4 = 24;
3 + 8. 3 + 8 = 11, 3 x 8 = 24;
2 + 12. 2 + 12 = 14, 2 x 12 = 24;
1 + 24. 1 + 24 = 25, 1 x 24 = 24.
b. Answers will need to be two of the following:
5 + 6. 5 + 6 = 11, 5 x 6 = 30;
3 + 10. 3 + 10 = 13, 3 x 10 = 30;
2 + 15. 2 + 15 = 17, 2 x 15 = 30;
1 + 30. 1 + 30 = 31, 1 x 30 = 30.
c. Answers will vary.
3. Answers will vary. For example, any number multiplied by 1 will fit:
3 and 1 because 3 + 1 = 4 and 3 x 1 = 3;
78 and 1 because 78 + 1 = 79 and 78 x 1 = 78