Question

learning goal 2.4: students will rewrite the sum of two composite numbers with a common factor using the distributive property
complete the missing values to rewrite each expression using the distributive property.

1. 64 + 42 = 2(__ + __)
2. 182 + 351 = ____(14 + 27)

rewrite an equivalent expression for each sum using the distributive property with a common factor.

1. 84 + 70
2. 108 + 72

Explanation:

Problem 1:

Step 1: Divide 64 by 2

To find the first missing number, we divide 64 by 2. $64\div2 = 32$.

Step 2: Divide 42 by 2

To find the second missing number, we divide 42 by 2. $42\div2 = 21$.
So, $64 + 42 = 2(32 + 21)$.

Problem 2:

Step 1: Find the common factor

We know that $182 = 13\times14$ and $351 = 13\times27$. So we divide 182 (or 351) by 14 (or 27) to find the missing factor. $182\div14 = 13$ (or $351\div27 = 13$).
So, $182 + 351 = 13(14 + 27)$.

Problem 3:

Step 1: Find the GCD of 84 and 70

The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84. The factors of 70 are 1, 2, 5, 7, 10, 14, 35, 70. The greatest common divisor (GCD) is 14.

Step 2: Divide each term by 14

$84\div14 = 6$ and $70\div14 = 5$. Using the distributive property $a\times c + b\times c=(a + b)\times c$, we get $84 + 70 = 14(6 + 5)$.

Problem 4:

Step 1: Find the GCD of 108 and 72

The factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108. The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. The GCD is 36.

Step 2: Divide each term by 36

$108\div36 = 3$ and $72\div36 = 2$. Using the distributive property, we get $108 + 72 = 36(3 + 2)$ (we could also use other common factors like 12: $108\div12 = 9$, $72\div12 = 6$, so $12(9 + 6)$; or 18: $18(6 + 4)$; or 24: $24(4.5 + 3)$ but since we usually use whole numbers, 36 is a good choice as it's the GCD).

Answer:

1. $64 + 42 = 2(\boldsymbol{32} + \boldsymbol{21})$
2. $182 + 351 = \boldsymbol{13}(14 + 27)$
3. $84 + 70 = \boldsymbol{14(6 + 5)}$ (or other valid forms with common factors like $2(42 + 35)$, $7(12 + 10)$)
4. $108 + 72 = \boldsymbol{36(3 + 2)}$ (or other valid forms with common factors like $12(9 + 6)$, $18(6 + 4)$, $24(4.5 + 3)$ but whole - number factor forms are preferred)