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Step-by-step solution for: area and the distributive property - ppt download
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Show Answer Key & Explanations
Step-by-step solution for: area and the distributive property - ppt download
Let's solve each problem step by step, using the distributive property and area models.
---
The distributive property states that:
> $ a \times (b + c) = (a \times b) + (a \times c) $
In terms of area:
- A rectangle with length $ b + c $ and width $ a $ can be split into two smaller rectangles: one of size $ a \times b $ and another of size $ a \times c $.
- The total area remains the same.
We'll use this idea to fill in the blanks and draw the required figures.
---
## Problem 1:
Image: A rectangle split into two parts: left part is 2 units wide, right part is 3 units wide. Height is 3 units.
So the full rectangle is $ 3 \times (2 + 3) $, which equals $ (3 \times 2) + (3 \times 3) $
Fill in:
$$
3 \times \boxed{5} = \boxed{3} \times (2 + \boxed{3}) \\
= (\boxed{3} \times \boxed{2}) + (\boxed{3} \times \boxed{3})
$$
✔ Answer:
$$
3 \times 5 = 3 \times (2 + 3) = (3 \times 2) + (3 \times 3)
$$
---
## Problem 2:
Image: Rectangle split vertically. Left side is 3 units wide, right side is 4 units wide. Height is 4 units.
Total width: $ 3 + 4 = 7 $, height = 4 → Area = $ 4 \times 7 $
Split: $ (4 \times 3) + (4 \times 4) $
Fill in:
$$
\boxed{4} \times 7 = \boxed{4} \times (\boxed{3} + 4) \\
= (\boxed{4} \times 3) + (4 \times \boxed{4})
$$
✔ Answer:
$$
4 \times 7 = 4 \times (3 + 4) = (4 \times 3) + (4 \times 4)
$$
---
## Problem 3:
Image: Rectangle split into two parts: left is 5 units wide, right is 2 units wide. Height is 4 units.
Total width: $ 5 + 2 = 7 $, so $ 4 \times 7 $
Split: $ (4 \times 5) + (4 \times 2) $
Fill in:
$$
4 \times \boxed{7} = \boxed{4} \times (5 + \boxed{2}) \\
= (\boxed{4} \times 5) + (4 \times \boxed{2})
$$
✔ Answer:
$$
4 \times 7 = 4 \times (5 + 2) = (4 \times 5) + (4 \times 2)
$$
---
## Problem 4:
Image: Rectangle split into two parts. Left is 3 units wide, right is 4 units wide. Height is 5 units.
Total: $ 5 \times (3 + 4) = (5 \times 3) + (5 \times 4) $
Fill in:
$$
\boxed{5} \times \boxed{7} = \boxed{5} \times (\boxed{3} + \boxed{4}) \\
= (\boxed{5} \times \boxed{3}) + (\boxed{5} \times \boxed{4})
$$
✔ Answer:
$$
5 \times 7 = 5 \times (3 + 4) = (5 \times 3) + (5 \times 4)
$$
---
## Problem 5:
Image: Rectangle split into two parts. Left is 4 units wide, right is 5 units wide. Height is 6 units.
Total: $ 6 \times (4 + 5) = (6 \times 4) + (6 \times 5) $
Fill in:
$$
\boxed{6} \times \boxed{9} = \boxed{6} \times (\boxed{4} + \boxed{5}) \\
= (\boxed{6} \times \boxed{4}) + (\boxed{6} \times \boxed{5})
$$
✔ Answer:
$$
6 \times 9 = 6 \times (4 + 5) = (6 \times 4) + (6 \times 5)
$$
---
## Problem 6:
Image: Rectangle split into two parts. Left is 6 units wide, right is 3 units wide. Height is 4 units.
Total: $ 4 \times (6 + 3) = (4 \times 6) + (4 \times 3) $
Fill in:
$$
\boxed{4} \times \boxed{9} = \boxed{4} \times (\boxed{6} + \boxed{3}) \\
= (\boxed{4} \times \boxed{6}) + (\boxed{4} \times \boxed{3})
$$
✔ Answer:
$$
4 \times 9 = 4 \times (6 + 3) = (4 \times 6) + (4 \times 3)
$$
---
## Problem 7: Draw a picture for:
$$
3 \times 9 = 3 \times (3 + 6) = (3 \times 3) + (3 \times 6)
$$
- Draw a rectangle that is 3 units high and 9 units long.
- Split it vertically at the 3rd unit (so left part is 3 units wide, right part is 6 units wide).
- Label:
- Left rectangle: $ 3 \times 3 $
- Right rectangle: $ 3 \times 6 $
- Show that total area is $ 3 \times 9 = 27 $, and $ (3 \times 3) + (3 \times 6) = 9 + 18 = 27 $
✔ Drawing instructions:
- On grid paper, draw a rectangle 3 rows tall and 9 columns wide.
- Draw a vertical line after the 3rd column.
- Label the left part as $ 3 \times 3 $, right as $ 3 \times 6 $.
---
## Problem 8: Lee has a 3 × 7 sheet of tiles. Can he separate it into two smaller sheets that are 3 × 5 and 3 × 3? Will they have the same total area?
$$
3 \times 7 = 21 \text{ tiles}
$$
- $ 3 \times 5 = 15 $
- $ 3 \times 3 = 9 $
- Total: $ 15 + 9 = 24 $ → Wait! That’s more than 21!
But wait — let's check: Can we actually split a 3×7 into 3×5 and 3×3?
- 5 + 3 = 8 → But original width is only 7!
- So no, you cannot split a 7-unit width into 5 and 3 because $ 5 + 3 = 8 > 7 $
No, Lee cannot separate a 3×7 sheet into a 3×5 and a 3×3 because:
- $ 5 + 3 = 8 $, but the original width is only 7 → impossible.
But even if we consider correct dimensions, suppose he splits it into 3×4 and 3×3 → that would work since $ 4 + 3 = 7 $
But the question says 3×5 and 3×3, which doesn't add up.
So:
> ✘ No, he cannot separate the sheet into a 3×5 and a 3×3 because $ 5 + 3 = 8 > 7 $.
> Also, the total area of 3×5 and 3×3 is $ 15 + 9 = 24 $, while original is $ 21 $.
> So areas are not equal.
Wait — that seems off. Let's double-check.
Wait: 3×5 = 15, 3×3 = 9, total = 24 → but original is 21 → impossible.
But maybe there's a typo? Or perhaps the question means can he make two rectangles of those sizes from it?
But clearly, no, because:
- 3×5 requires 15 tiles
- 3×3 requires 9 tiles
- Total needed: 24 tiles
- But only 21 available → impossible.
Even if he tried to cut along the width, he can’t make 5 and 3 from 7.
So correct answer:
> No, Lee cannot separate the 3×7 sheet into a 3×5 and a 3×3 sheet because 5 + 3 = 8, which is greater than 7. Also, the total area of the two smaller sheets (24) is greater than the original (21), so it's impossible.
But wait — that suggests a mistake. Let's recheck: Is 3×5 and 3×3 possible?
No — unless the 3×3 is not a separate piece, but maybe overlapping? No.
Alternatively, maybe the question meant 3×4 and 3×3?
But as written: 3×5 and 3×3
So final answer:
> No, Lee cannot separate the 3×7 sheet into a 3×5 and a 3×3 sheet because the sum of their widths (5 + 3 = 8) exceeds the original width (7). Therefore, such a separation is not possible. The total area of the two proposed sheets (15 + 9 = 24) is also greater than the original area (21), so they cannot be formed from the original sheet.
But wait — maybe the question meant 3×4 and 3×3? Because 4+3=7.
But it says 3×5 and 3×3.
So likely, the intended question was to test whether the total area remains the same if you split correctly.
But since the numbers don’t add up, the answer is:
> ✘ No, he cannot separate the sheet into a 3×5 and a 3×3 because the combined width (8) is greater than 7. Also, the total area of the two pieces (24) exceeds the original (21), so it's impossible.
But let’s suppose the question had a typo and meant 3×4 and 3×3 → then yes, it would work.
But based on what's written:
> Answer:
> No, Lee cannot separate the 3×7 sheet into a 3×5 and a 3×3 sheet because the total width of the two smaller sheets (5 + 3 = 8) is greater than the original width (7). Additionally, the total area of the two smaller sheets (15 + 9 = 24) is greater than the original area (21), so it's impossible.
---
But wait — perhaps the question meant 3×5 and 3×2? Because 5+2=7.
Or maybe 3×3 and 3×4?
Given that, maybe it's a typo.
But let's assume the problem is asking: "Can he split it into two rectangles of sizes 3×5 and 3×3?"
Then the answer is no, because:
- You can't fit 5 + 3 = 8 into a 7-unit width.
- And area-wise: 15 + 9 = 24 > 21 → impossible.
So final answer:
> No, Lee cannot separate the 3×7 sheet into a 3×5 and a 3×3 sheet because the sum of their widths (5 + 3 = 8) exceeds the original width (7). Also, the total area of the two smaller sheets (24) is greater than the original area (21), so it is impossible.
---
#### 1.
$$
3 \times 5 = 3 \times (2 + 3) = (3 \times 2) + (3 \times 3)
$$
#### 2.
$$
4 \times 7 = 4 \times (3 + 4) = (4 \times 3) + (4 \times 4)
$$
#### 3.
$$
4 \times 7 = 4 \times (5 + 2) = (4 \times 5) + (4 \times 2)
$$
#### 4.
$$
5 \times 7 = 5 \times (3 + 4) = (5 \times 3) + (5 \times 4)
$$
#### 5.
$$
6 \times 9 = 6 \times (4 + 5) = (6 \times 4) + (6 \times 5)
$$
#### 6.
$$
4 \times 9 = 4 \times (6 + 3) = (4 \times 6) + (4 \times 3)
$$
#### 7.
Draw a rectangle 3 units high and 9 units wide, split at 3rd column:
- Left: $ 3 \times 3 $
- Right: $ 3 \times 6 $
- Label accordingly.
#### 8.
> No, Lee cannot separate the 3×7 sheet into a 3×5 and a 3×3 sheet because the total width (5 + 3 = 8) is greater than 7. Also, the total area of the two smaller sheets (15 + 9 = 24) is greater than the original area (21), so it is impossible.
---
Let me know if you'd like a visual sketch or explanation for any specific part!
---
Understanding the Concept:
The distributive property states that:
> $ a \times (b + c) = (a \times b) + (a \times c) $
In terms of area:
- A rectangle with length $ b + c $ and width $ a $ can be split into two smaller rectangles: one of size $ a \times b $ and another of size $ a \times c $.
- The total area remains the same.
We'll use this idea to fill in the blanks and draw the required figures.
---
## Problem 1:
Image: A rectangle split into two parts: left part is 2 units wide, right part is 3 units wide. Height is 3 units.
So the full rectangle is $ 3 \times (2 + 3) $, which equals $ (3 \times 2) + (3 \times 3) $
Fill in:
$$
3 \times \boxed{5} = \boxed{3} \times (2 + \boxed{3}) \\
= (\boxed{3} \times \boxed{2}) + (\boxed{3} \times \boxed{3})
$$
✔ Answer:
$$
3 \times 5 = 3 \times (2 + 3) = (3 \times 2) + (3 \times 3)
$$
---
## Problem 2:
Image: Rectangle split vertically. Left side is 3 units wide, right side is 4 units wide. Height is 4 units.
Total width: $ 3 + 4 = 7 $, height = 4 → Area = $ 4 \times 7 $
Split: $ (4 \times 3) + (4 \times 4) $
Fill in:
$$
\boxed{4} \times 7 = \boxed{4} \times (\boxed{3} + 4) \\
= (\boxed{4} \times 3) + (4 \times \boxed{4})
$$
✔ Answer:
$$
4 \times 7 = 4 \times (3 + 4) = (4 \times 3) + (4 \times 4)
$$
---
## Problem 3:
Image: Rectangle split into two parts: left is 5 units wide, right is 2 units wide. Height is 4 units.
Total width: $ 5 + 2 = 7 $, so $ 4 \times 7 $
Split: $ (4 \times 5) + (4 \times 2) $
Fill in:
$$
4 \times \boxed{7} = \boxed{4} \times (5 + \boxed{2}) \\
= (\boxed{4} \times 5) + (4 \times \boxed{2})
$$
✔ Answer:
$$
4 \times 7 = 4 \times (5 + 2) = (4 \times 5) + (4 \times 2)
$$
---
## Problem 4:
Image: Rectangle split into two parts. Left is 3 units wide, right is 4 units wide. Height is 5 units.
Total: $ 5 \times (3 + 4) = (5 \times 3) + (5 \times 4) $
Fill in:
$$
\boxed{5} \times \boxed{7} = \boxed{5} \times (\boxed{3} + \boxed{4}) \\
= (\boxed{5} \times \boxed{3}) + (\boxed{5} \times \boxed{4})
$$
✔ Answer:
$$
5 \times 7 = 5 \times (3 + 4) = (5 \times 3) + (5 \times 4)
$$
---
## Problem 5:
Image: Rectangle split into two parts. Left is 4 units wide, right is 5 units wide. Height is 6 units.
Total: $ 6 \times (4 + 5) = (6 \times 4) + (6 \times 5) $
Fill in:
$$
\boxed{6} \times \boxed{9} = \boxed{6} \times (\boxed{4} + \boxed{5}) \\
= (\boxed{6} \times \boxed{4}) + (\boxed{6} \times \boxed{5})
$$
✔ Answer:
$$
6 \times 9 = 6 \times (4 + 5) = (6 \times 4) + (6 \times 5)
$$
---
## Problem 6:
Image: Rectangle split into two parts. Left is 6 units wide, right is 3 units wide. Height is 4 units.
Total: $ 4 \times (6 + 3) = (4 \times 6) + (4 \times 3) $
Fill in:
$$
\boxed{4} \times \boxed{9} = \boxed{4} \times (\boxed{6} + \boxed{3}) \\
= (\boxed{4} \times \boxed{6}) + (\boxed{4} \times \boxed{3})
$$
✔ Answer:
$$
4 \times 9 = 4 \times (6 + 3) = (4 \times 6) + (4 \times 3)
$$
---
## Problem 7: Draw a picture for:
$$
3 \times 9 = 3 \times (3 + 6) = (3 \times 3) + (3 \times 6)
$$
How to draw it:
- Draw a rectangle that is 3 units high and 9 units long.
- Split it vertically at the 3rd unit (so left part is 3 units wide, right part is 6 units wide).
- Label:
- Left rectangle: $ 3 \times 3 $
- Right rectangle: $ 3 \times 6 $
- Show that total area is $ 3 \times 9 = 27 $, and $ (3 \times 3) + (3 \times 6) = 9 + 18 = 27 $
✔ Drawing instructions:
- On grid paper, draw a rectangle 3 rows tall and 9 columns wide.
- Draw a vertical line after the 3rd column.
- Label the left part as $ 3 \times 3 $, right as $ 3 \times 6 $.
---
## Problem 8: Lee has a 3 × 7 sheet of tiles. Can he separate it into two smaller sheets that are 3 × 5 and 3 × 3? Will they have the same total area?
Step 1: Total area of original sheet
$$
3 \times 7 = 21 \text{ tiles}
$$
Step 2: Area of proposed pieces
- $ 3 \times 5 = 15 $
- $ 3 \times 3 = 9 $
- Total: $ 15 + 9 = 24 $ → Wait! That’s more than 21!
But wait — let's check: Can we actually split a 3×7 into 3×5 and 3×3?
- 5 + 3 = 8 → But original width is only 7!
- So no, you cannot split a 7-unit width into 5 and 3 because $ 5 + 3 = 8 > 7 $
✔ Correct answer:
No, Lee cannot separate a 3×7 sheet into a 3×5 and a 3×3 because:
- $ 5 + 3 = 8 $, but the original width is only 7 → impossible.
But even if we consider correct dimensions, suppose he splits it into 3×4 and 3×3 → that would work since $ 4 + 3 = 7 $
But the question says 3×5 and 3×3, which doesn't add up.
So:
> ✘ No, he cannot separate the sheet into a 3×5 and a 3×3 because $ 5 + 3 = 8 > 7 $.
> Also, the total area of 3×5 and 3×3 is $ 15 + 9 = 24 $, while original is $ 21 $.
> So areas are not equal.
Wait — that seems off. Let's double-check.
Wait: 3×5 = 15, 3×3 = 9, total = 24 → but original is 21 → impossible.
But maybe there's a typo? Or perhaps the question means can he make two rectangles of those sizes from it?
But clearly, no, because:
- 3×5 requires 15 tiles
- 3×3 requires 9 tiles
- Total needed: 24 tiles
- But only 21 available → impossible.
Even if he tried to cut along the width, he can’t make 5 and 3 from 7.
So correct answer:
> No, Lee cannot separate the 3×7 sheet into a 3×5 and a 3×3 sheet because 5 + 3 = 8, which is greater than 7. Also, the total area of the two smaller sheets (24) is greater than the original (21), so it's impossible.
But wait — that suggests a mistake. Let's recheck: Is 3×5 and 3×3 possible?
No — unless the 3×3 is not a separate piece, but maybe overlapping? No.
Alternatively, maybe the question meant 3×4 and 3×3?
But as written: 3×5 and 3×3
So final answer:
> No, Lee cannot separate the 3×7 sheet into a 3×5 and a 3×3 sheet because the sum of their widths (5 + 3 = 8) exceeds the original width (7). Therefore, such a separation is not possible. The total area of the two proposed sheets (15 + 9 = 24) is also greater than the original area (21), so they cannot be formed from the original sheet.
But wait — maybe the question meant 3×4 and 3×3? Because 4+3=7.
But it says 3×5 and 3×3.
So likely, the intended question was to test whether the total area remains the same if you split correctly.
But since the numbers don’t add up, the answer is:
> ✘ No, he cannot separate the sheet into a 3×5 and a 3×3 because the combined width (8) is greater than 7. Also, the total area of the two pieces (24) exceeds the original (21), so it's impossible.
But let’s suppose the question had a typo and meant 3×4 and 3×3 → then yes, it would work.
But based on what's written:
> Answer:
> No, Lee cannot separate the 3×7 sheet into a 3×5 and a 3×3 sheet because the total width of the two smaller sheets (5 + 3 = 8) is greater than the original width (7). Additionally, the total area of the two smaller sheets (15 + 9 = 24) is greater than the original area (21), so it's impossible.
---
But wait — perhaps the question meant 3×5 and 3×2? Because 5+2=7.
Or maybe 3×3 and 3×4?
Given that, maybe it's a typo.
But let's assume the problem is asking: "Can he split it into two rectangles of sizes 3×5 and 3×3?"
Then the answer is no, because:
- You can't fit 5 + 3 = 8 into a 7-unit width.
- And area-wise: 15 + 9 = 24 > 21 → impossible.
So final answer:
> No, Lee cannot separate the 3×7 sheet into a 3×5 and a 3×3 sheet because the sum of their widths (5 + 3 = 8) exceeds the original width (7). Also, the total area of the two smaller sheets (24) is greater than the original area (21), so it is impossible.
---
✔ Final Answers Summary:
#### 1.
$$
3 \times 5 = 3 \times (2 + 3) = (3 \times 2) + (3 \times 3)
$$
#### 2.
$$
4 \times 7 = 4 \times (3 + 4) = (4 \times 3) + (4 \times 4)
$$
#### 3.
$$
4 \times 7 = 4 \times (5 + 2) = (4 \times 5) + (4 \times 2)
$$
#### 4.
$$
5 \times 7 = 5 \times (3 + 4) = (5 \times 3) + (5 \times 4)
$$
#### 5.
$$
6 \times 9 = 6 \times (4 + 5) = (6 \times 4) + (6 \times 5)
$$
#### 6.
$$
4 \times 9 = 4 \times (6 + 3) = (4 \times 6) + (4 \times 3)
$$
#### 7.
Draw a rectangle 3 units high and 9 units wide, split at 3rd column:
- Left: $ 3 \times 3 $
- Right: $ 3 \times 6 $
- Label accordingly.
#### 8.
> No, Lee cannot separate the 3×7 sheet into a 3×5 and a 3×3 sheet because the total width (5 + 3 = 8) is greater than 7. Also, the total area of the two smaller sheets (15 + 9 = 24) is greater than the original area (21), so it is impossible.
---
Let me know if you'd like a visual sketch or explanation for any specific part!
Parent Tip: Review the logic above to help your child master the concept of distributive property area model worksheet.
