Use Area Models for Distribution - 3rd Grade Math

1. What is the area using distribution: $7(2+1)$?

   $21$. $7(2+1)=7(3)=21$ or $7\cdot^2+7\cdot^1=14+7=21$.

2. Identify the correct distributed form of $8(4+3)$.

   $8\cdot 4+8\cdot 3$. Distribute $8$ to both $4$ and $3$.

3. Identify the missing term: $6(4+2)=6\times 4+\underline{\ \ }$.

   $6\times 2$. Distribute $6$ to both $4$ and $2$.

4. Find and correct the error: $4(2+7)=4\cdot 2+7$.

   Correct: $4(2+7)=4\cdot 2+4\cdot 7$. Must distribute 4 to both 2 and 7.

5. A rectangle has height $4$ and widths $3$ and $6$. What is the total area?

   $36$ square units. $4 \times (3 + 6) = 4 \times 9 = 36$.

6. What property does the area model show in $a(b+c)=ab+ac$?

   The distributive property. Shows that multiplying a sum equals the sum of products.

7. What does $ac$ represent in an area model with height $a$ and right width $c$?

   Area of the right smaller rectangle. Product of height and right width gives right area.

8. What does $ab$ represent in an area model split into widths $b$ and $c$ with height $a$?

   Area of the $a$ by $b$ part. The left section has width $b$ and height $a$.

9. What is the area using distribution: $5(8+2)$?

   $50$. $5(8+2)=5(10)=50$ or $5\cdot^8+5\cdot^2=40+10=50$.

10. Identify the correct area statement for a rectangle of height $2$ and total width $9+1$.

    $2(9+1)=2\times 9+2\times 1$. Shows distributive property for this rectangle.

11. What is the value of $7(2+6)$ using the area model sum $7\times 2+7\times 6$?

    $56$. $7 \times 2 + 7 \times 6 = 14 + 42 = 56$.

12. What is the value of the missing factor: $6(3+4)=6\cdot 3+6\cdot \underline{\hspace{1cm}}$?

    $4$. Distribute $6$ to both addends: $6\cdot^3+6\cdot^4$.

13. A rectangle has height $9$ and total width $5+2$. What is its area?

    $63$ square units. $9 \times (5 + 2) = 9 \times 7 = 63$.

14. Find the total area: a rectangle is $2$ units by $6+3$ units. What is the area?

    $18$ square units. $2(6+3)=2(9)=18$ square units.

15. What is the total area if $a=6$, $b=2$, and $c=5$ for a rectangle with sides $a$ and $b+c$?

    $42$ square units. $a(b+c)=6(2+5)=6(7)=42$ square units.

16. What is the area of a rectangle with side lengths $a$ and $b+c$?

    $a(b+c)$ square units. Multiply height by total width to find area.

17. What is the area using distribution: $3(5+2)$?

    $21$. $3(5+2)=3(7)=21$ or $3\cdot^5+3\cdot^2=15+6=21$.

18. What is the area of a rectangle with side lengths $a$ and $b+c$ written as a sum of two products?

    $ab+ac$. The rectangle splits into two parts with areas $ab$ and $ac$.

19. Identify the missing term: $a(b+c)=ab+\underline{\hspace{1cm}}$.

    $ac$. Distribute $a$ to both $b$ and $c$.

20. What is $8(10+1)$ written using the distributive property as a sum?

    $8\cdot 10+8\cdot 1$. Multiply 8 by each addend: $8\cdot 10$ and $8\cdot 1$.

21. What does $ab$ represent in an area model with height $a$ and left width $b$?

    Area of the left smaller rectangle. Product of height and left width gives left area.

22. What does $ac$ represent in an area model split into widths $b$ and $c$ with height $a$?

    Area of the $a$ by $c$ part. The right section has width $c$ and height $a$.

23. Which equation matches a rectangle tiled into two parts: $a\times b$ tiles and $a\times c$ tiles?

    $a(b+c)=ab+ac$. Total tiles equal sum of tiles in each section.

24. What is the missing side length: if $a(b+c)=a\cdot 9$ and $b=4$, what is $c$?

    $c=5$. Since $b+c=9$ and $b=4$, then $c=9-4=5$.

25. Identify the missing term: $a(b+c)=\underline{\hspace{1cm}}+ac$.

    $ab$. Distribute $a$ to both terms in parentheses.

26. Which expression matches an area model with height $a$ and widths $b$ and $c$ added together?

    $ab+ac$. Same as splitting a rectangle into two parts.

27. Find the total area: a rectangle is $5$ units by $7+1$ units. What is the area?

    $40$ square units. $5(7+1)=5(8)=40$ square units.

28. What is the area using distribution: $4(6+3)$?

    $36$. $4(6+3)=4(9)=36$ or $4\cdot^6+4\cdot^3=24+12=36$.

29. What is the missing expression: $a(b+c)=\underline{\hspace{1cm}}+ac$?

    $ab$. Distributing $a$ to both $b$ and $c$ gives $ab+ac$.

30. Which expression matches the area model with height $a$ and widths $b$ and $c$: $a(b+c)$ or $a+b+c$?

    $a(b+c)$. Area equals height times total width.

31. What is the value of $6(7+3)$ using $6\times 7+6\times 3$?

    $60$. $6 \times 7 + 6 \times 3 = 42 + 18 = 60$.

32. Find $a(b+c)$ when $a=3$, $b=8$, and $c=1$.

    $27$. $3 \times (8 + 1) = 3 \times 9 = 27$.

33. What is $5(9+6)$ written using the distributive property as a sum?

    $5\cdot 9+5\cdot 6$. Multiply 5 by each addend: $5\cdot 9$ and $5\cdot 6$.

34. What is the value of $5(3+4)$ using $5\times 3+5\times 4$?

    $35$. $5 \times 3 + 5 \times 4 = 15 + 20 = 35$.

35. What is $7(3+5)$ written using the distributive property as a sum?

    $7\cdot 3+7\cdot 5$. Multiply 7 by each addend: $7\cdot 3$ and $7\cdot 5$.

36. Find the area using distribution: $7(5+2)$.

    $7\cdot 5+7\cdot 2=49$. $7\cdot 5=35$ and $7\cdot 2=14$, sum is $49$.

37. What is the area of a rectangle with side lengths $a$ and $b+c$ written as one product?

    $a(b+c)$. The total width is $b+c$, multiplied by height $a$.

38. What is the value of $8(5+1)$ using $8\times 5+8\times 1$?

    $48$. $8 \times 5 + 8 \times 1 = 40 + 8 = 48$.

39. What is the value of the missing factor: $9(1+5)=9\cdot \underline{\hspace{1cm}}+9\cdot 5$?

    $1$. Distribute $9$ to both addends: $9\cdot^1+9\cdot^5$.

40. What is the value of $3(10+2)$ using the distributive property?

    $36$. $3 \times 10 + 3 \times 2 = 30 + 6 = 36$.

41. What unit is used to count area when using tiling with $1\times 1$ squares?

    Square units. Each tile is one square unit.

42. What is $6(4+2)$ written using the distributive property as a sum?

    $6\cdot 4+6\cdot 2$. Multiply 6 by each addend: $6\cdot 4$ and $6\cdot 2$.

43. Find $ab+ac$ when $a=4$, $b=5$, and $c=2$.

    $28$. $4 \times 5 + 4 \times 2 = 20 + 8 = 28$.

44. Find the area using distribution: $9(4+3)$.

    $9\cdot 4+9\cdot 3=63$. $9\cdot 4=36$ and $9\cdot 3=27$, sum is $63$.

45. Find the total area: one rectangle is $3\times 8$ and the other is $3\times 2$.

    $3\cdot 8+3\cdot 2=30$. Add areas: $24+6=30$ square units.

46. What property does the area model show when $a(b+c)=ab+ac$?

    The distributive property of multiplication over addition. Shows that multiplying a number by a sum equals the sum of the products.

47. What is the sum of the two sub-areas when a rectangle is split into widths $b$ and $c$ with height $a$?

    $ab+ac$ square units. Add the areas of both sub-rectangles.

48. Find the total area: one rectangle is $4\times 6$ and the other is $4\times 3$.

    $4\cdot 6+4\cdot 3=36$. Add areas: $24+12=36$ square units.

49. What does $b+c$ represent on an area model rectangle split into two parts labeled $b$ and $c$?

    The total width. Sum of the two width segments.

50. What equation shows splitting width $b+c$ into $b$ and $c$ for a rectangle of height $a$?

    $a(b+c)=ab+ac$. Shows area of whole equals sum of parts.

51. What is the missing expression: $a(b+c)=ab+\underline{\hspace{1cm}}$?

    $ac$. Distributing $a$ to both $b$ and $c$ gives $ab+ac$.

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