Solving Cluster Problems
Submitted by Angel Greenley
greenleya@billings.k12.mt.us
Adapted from TERC Investigations – Arrays and Shares Unit
Materials Needed: Cluster Problem Worksheets
Lesson:
1. Cluster problems are sets of 3 or more problems that use known facts
from the first problems to find an answer to a more difficult problem.
You want to encourage students to look for patterns and connections
between the problems. The main focus of cluster problems is to
help build number sense and fluency with numbers.
2. Display the following set of problems on the board
7x8=
7x6=
7 x 10 =
7 x 16 =
3. Have volunteers answer the first three problems. Then ask, “How can
we use information from any of the first 3 problems to find the
answer to the 4th problem?” Take all possible solutions from students.
Here are some possibilities:
Solution 1:



I added 6 + 10 and got 16. (Using problem 2 & 3)
Then I added 42 + 70 = 112. (The products for problems 2 & 3)
So I found the answer 112.
Solution 2:

I used the first answer from 7 x 8. Since 16 is a double of 8; I
doubled the product of 56 and knew it was 112.
4. Display the 2nd set of cluster problems.
4X5
2 x 15
4 x 10
4 x 15
Since some of these are not basic facts (2 x 15), you may have students
explain how they arrived at that answer. Possible solutions for finding 4 x
15 include:
Solution 1:
I used 4 x 5 and 4 x 10. Since 5 + 10 equals 15, I can add the products
of these two problems. So 4 x 15 equals 20 + 40 = 60.
Solution 2:
I knew that 4 is double of 2. Since 2 x 15 = 30; then I know that 4 x
15 is double that, so it is 60.
5. Continue using sample problems until students are comfortable with
the many different strategies.
These can be used as a regular lesson and then followed up as a morning
opener each day to keep the strategies fresh.
Multiplication Cluster Problems
Solve each cluster of problems. Look for ways the problems in each cluster are
related.
10 x 123
20 x 123
2 x 123
22 x 123
10 x 18
5 x 18
50 x 18
2 x 18
20 x 18
40 x 18
45 x 18
47 x 18
400 x 9
500 x 9
90 x 9
8x9
2x9
498 x 9
2 x 72
10 x 72
5 x 72
20 x 72
200 x 72
210 x 72
215 x 72
Multiplication Cluster Problems
Solve each cluster of problems. Look for ways the problems in each cluster are
related.
20 x 4
25 x 4
2x4
23 x 4
3 x 25
10 x 25
12 x 25
13 x 25
5 x 50
10 x 50
7 x 50
17 x 50
3x5
200 x 5
233 x 10
233 x 5
4 x 25
10 x 25
40 x 25
50 x 25
6 x 25
46 x 25
500 x 4
1000 x 4
2500 x 4
2507 x 4
Multiplication Cluster Problems
Solve each cluster of problems. Look for ways the problems in each cluster are
related.
8 x 25
30 x 25
40 x 25
38 x 25
4 x 40
70 x 40
75 x 40
74 x 40
25 x 4
200 x 4
250 x 4
275 x 4
277 x 4
25 x 5
200 x 5
2000 x 5
2025 x 5
Challenging Multiplication Cluster Problems
Solve each cluster of problems. Look for ways the problems in each cluster are
related.
7 x 40
25 x 40
250 x 40
275 x 40
282 x 40
3 x 50
20 x 50
200 x 50
197 x 50
0.5 x 50
2.5 x 5
25 x 5
2.5 x 50
12 x 24
25 x 12
0.5 x 24
12.5 x 24
Cluster Problems
Make an estimate for the first problem in each cluster. Then solve the problems
in the cluster.
Estimate for 34 x 50 _________________________________
3 x 50
10 x 50
34 x 25
30 x 50
34 x 50
Estimate for 62 x 23 ____________________________________
60 x 20
62 x 10
62 x 3
62 x 23
Estimate for 36 x 43 ______________________________________
10 x 43
30 x 40
5 x 43
10 x 36
36 x 43
Estimate for 37 x 27 ____________________________________
10 x 27
30 x 20
37 x 2
40 x 27
37 x 27
Cluster Problems
Make an estimate for the first problem in each cluster. Then solve the problems
in the cluster.
Estimate for 36 x 20 _________________________________
6 x 20
3 x 20
30 x 20
36 x 20
Estimate for 26 x 30 ____________________________________
6x3
20 x 3
26 x 3
26 x 30
Estimate for 43 x 35 ______________________________________
30 x 3
40 x 5
40 x 30
43 x 35
Estimate for 29 x 15 ____________________________________
10 x 15
9 x 15
20 x 15
29 x 15