Let's solve the
Diamond Problems shown in the image. These are common math exercises that help students practice multiplication and addition, especially with integers and fractions.
Each diamond has four sections:
- Top:
Product of the two numbers on the left and right.
- Bottom:
Sum of the two numbers on the left and right.
- Left and Right: Two numbers that multiply to give the top number and add to give the bottom number.
The pattern is:
-
Top = (Left) × (Right)
-
Bottom = (Left) + (Right)
We are given either two numbers or one number and need to find the missing ones.
---
a.
```
?
/ \
3 4
\ /
?
```
Given: Left = 3, Right = 4
-
Top = 3 × 4 =
12
-
Bottom = 3 + 4 =
7
✔ So:
- Top:
12
- Bottom:
7
---
b.
```
?
/ \
-2 -3
\ /
?
```
Given: Left = -2, Right = -3
-
Top = (-2) × (-3) =
6
-
Bottom = (-2) + (-3) =
-5
✔ So:
- Top:
6
- Bottom:
-5
---
c.
```
12
/ \
? 7
\ /
?
```
Given: Top = 12, Right = 7
We need to find the
Left number such that:
- Left × 7 = 12 → Left = 12 ÷ 7 =
12/7
Then check sum:
- Bottom = Left + Right = 12/7 + 7 = 12/7 + 49/7 =
61/7
So:
- Left:
12/7
- Bottom:
61/7
✔ But let’s double-check if this makes sense.
Wait — maybe we should try to see if there’s an integer solution?
But 12 ÷ 7 = 12/7 → not an integer, but that’s okay.
So:
- Left =
12/7
- Bottom =
61/7
Alternatively, could it be the other way around? No — the right is already given as 7.
So yes, this is correct.
But wait — perhaps we made a mistake.
Wait: The
top is 12, and
right is 7. So:
Let’s call the left number
x.
Then:
- x × 7 = 12 → x = 12/7
- Then sum = x + 7 = 12/7 + 49/7 = 61/7
✔ So:
- Left:
12/7
- Bottom:
61/7
---
d.
```
?
/ \
4 1/2
\ /
?
```
Given: Left = 4, Right = 1/2
-
Top = 4 × (1/2) =
2
-
Bottom = 4 + 1/2 =
4.5 or
9/2
✔ So:
- Top:
2
- Bottom:
9/2 or
4½
---
e.
```
?
/ \
8 -6
\ /
?
```
Given: Left = 8, Right = -6
-
Top = 8 × (-6) =
-48
-
Bottom = 8 + (-6) =
2
✔ So:
- Top:
-48
- Bottom:
2
---
Final Answers:
#### a.
```
12
/ \
3 4
\ /
7
```
#### b.
```
6
/ \
-2 -3
\ /
-5
```
#### c.
```
12
/ \
12/7 7
\ /
61/7
```
(Or write as mixed numbers: 12/7 ≈ 1 5/7, 61/7 ≈ 8 5/7)
#### d.
```
2
/ \
4 1/2
\ /
9/2
```
#### e.
```
-48
/ \
8 -6
\ /
2
```
---
Summary of Completed Diamonds:
| Problem | Top | Bottom |
|--------|-----|--------|
| a | 12 | 7 |
| b | 6 | -5 |
| c | 12 | 61/7 |
| d | 2 | 9/2 |
| e | -48 | 2 |
---
✔ Pattern: For each diamond:
-
Top = Left × Right
-
Bottom = Left + Right
You use this rule to fill in the missing values.
Let me know if you'd like these drawn out or explained further!
Parent Tip: Review the logic above to help your child master the concept of diamond problem worksheet.
