To solve these puzzles, we need to find two numbers for each "X" that satisfy two conditions:
1.
Product (Top): The two numbers multiplied together equal the top number.
2.
Sum (Bottom): The two numbers added together equal the bottom number.
Here is the step-by-step solution for each problem:
1. Top: 9, Bottom: -6
* We need factors of 9: $(1, 9)$ or $(-1, -9)$ or $(3, 3)$ or $(-3, -3)$.
* Check sums:
* $1 + 9 = 10$
* $-1 + (-9) = -10$
* $3 + 3 = 6$
* $-3 + (-3) = -6$ (This matches!)
*
Answer: -3, -3
2. Top: 4, Bottom: 4
* We need factors of 4: $(1, 4)$, $(2, 2)$, etc.
* Check sums:
* $1 + 4 = 5$
* $2 + 2 = 4$ (This matches!)
*
Answer: 2, 2
3. Top: -30, Bottom: -13
* Since the product is negative, one number is positive and one is negative.
* Factors of 30: $1 \times 30$, $2 \times 15$, $3 \times 10$, $5 \times 6$.
* We need a sum of -13, so the larger number must be negative.
* $-15 + 2 = -13$ (This matches!)
*
Answer: -15, 2
4. Top: -84, Bottom: 5
* Product is negative (one positive, one negative). Sum is positive (the positive number is larger).
* Factors of 84: $7 \times 12 = 84$.
* Check signs: $12 - 7 = 5$.
*
Answer: 12, -7
5. Top: -24, Bottom: -5
* Product is negative. Sum is negative (the negative number is larger).
* Factors of 24: $3 \times 8 = 24$, $4 \times 6 = 24$.
* Check sums:
* $-8 + 3 = -5$ (This matches!)
*
Answer: -8, 3
6. Top: 6, Bottom: -5
* Product is positive (same signs). Sum is negative (both are negative).
* Factors of 6: $1 \times 6$, $2 \times 3$.
* Check sums with negatives:
* $-1 + (-6) = -7$
* $-2 + (-3) = -5$ (This matches!)
*
Answer: -2, -3
7. Top: -15, Bottom: -14
* Product is negative. Sum is negative.
* Factors of 15: $1 \times 15$, $3 \times 5$.
* Check sums:
* $-15 + 1 = -14$ (This matches!)
*
Answer: -15, 1
8. Top: -75, Bottom: -10
* Product is negative. Sum is negative.
* Factors of 75: $1 \times 75$, $3 \times 25$, $5 \times 15$.
* Check sums:
* $-15 + 5 = -10$ (This matches!)
*
Answer: -15, 5
9. Top: 12, Bottom: 7
* Product is positive. Sum is positive (both are positive).
* Factors of 12: $1 \times 12$, $2 \times 6$, $3 \times 4$.
* Check sums:
* $3 + 4 = 7$ (This matches!)
*
Answer: 3, 4
Final Answer:
1. -3, -3
2. 2, 2
3. -15, 2
4. 12, -7
5. -8, 3
6. -2, -3
7. -15, 1
8. -15, 5
9. 3, 4
Parent Tip: Review the logic above to help your child master the concept of diamond factoring worksheet.
