Question

activity b

1. what is the relationship between the length and the width of this \family\ of rectangles?

figure 1 figure 2 figure 3 figure 4
(images of four rectangles labeled figure 1 to figure 4)

1. use the relationship between the width and length to complete the table and the graph.

(table with columns width x, sketch, length y; rows for x=1,2,3,4,5,x)
(graph with axes)

1. based on your graph, if you were to continue the pattern, what ordered pair represents the length of a rectangle with a width of zero?
2. does the ordered pair make sense in this situation? why or why not?
3. write an equation that could be used to determine y, the length of the rectangle, given x, the width of a rectangle, in this family.

Explanation:

Step1: Analyze Figure 1

Figure 1: Width (x) = 1, Length (y) = 3 (since it's 3 units long horizontally).

Step2: Analyze Figure 2

Figure 2: Width (x) = 2, Length (y) = 4? Wait, no, wait. Wait Figure 1: Let's count the number of squares. Figure 1: 1 row, 3 columns? Wait no, looking at the figures:

Figure 1: It's a rectangle with width (vertical) 1, length (horizontal) 3? Wait no, maybe width is the number of rows, length is the number of columns.

Wait Figure 1: Rows (width) = 1, Columns (length) = 3.

Figure 2: Rows (width) = 2, Columns (length) = 4? Wait no, Figure 2: 2 rows, 4 columns? Wait no, the second figure: 2 rows, 4 columns? Wait no, looking at the drawing:

Figure 1: 1 row, 3 columns (so width=1, length=3)

Figure 2: 2 rows, 4 columns? Wait no, maybe I got it reversed. Wait the "width" is the vertical side, "length" is horizontal.

Wait Figure 1: vertical (width) =1, horizontal (length)=3

Figure 2: vertical (width)=2, horizontal (length)=4? Wait no, Figure 2: 2 rows (vertical), 4 columns (horizontal)? Wait no, the second figure: 2 rows, 4 columns? Wait the third figure: 3 rows, 5 columns? Fourth: 4 rows, 6 columns.

Ah! Now I see the pattern:

Width (x, number of rows) and Length (y, number of columns):

Figure 1: x=1, y=3

Figure 2: x=2, y=4? Wait no, 1→3, 2→4? No, 1→3, 2→4? Wait 3-1=2, 4-2=2? No, 1+2=3, 2+2=4, 3+2=5, 4+2=6. Yes! So the pattern is y = x + 2.

Check:

x=1: y=1+2=3 ✔️ (Figure 1)

x=2: y=2+2=4 ✔️ (Figure 2: 2 rows, 4 columns? Wait no, Figure 2: 2 rows, 4 columns? Wait the second figure: 2 rows, 4 columns? Wait the third figure: 3 rows, 5 columns (3+2=5) ✔️, fourth: 4 rows, 6 columns (4+2=6) ✔️. Yes! So the relationship is length (y) = width (x) + 2.

So for question 1: The length (y) is equal to the width (x) plus 2, so y = x + 2.

Now question 2: Complete the table.

Width (x) | Length (y)
--- | ---
1 | 1+2=3
2 | 2+2=4
3 | 3+2=5
4 | 4+2=6
5 | 5+2=7
x | x + 2

So the table:

For x=1: y=3

x=2: y=4

x=3: y=5

x=4: y=6

x=5: y=7

x: y = x + 2

Now question 3: Ordered pair when width (x)=0: (0, 0 + 2) = (0, 2)

Question 4: Does (0,2) make sense? A rectangle with width 0 (no height) can't exist, so no, because a rectangle needs positive width and length.

Question 5: Equation: y = x + 2

Now let's go step by step for each part.

Part 1: Relationship between length and width.

Step1: Identify values from figures.

Figure 1: Width (x)=1, Length (y)=3

Figure 2: x=2, y=4

Figure 3: x=3, y=5

Figure 4: x=4, y=6

Step2: Find the pattern.

Calculate the difference: 3-1=2, 4-2=2, 5-3=2, 6-4=2. So y = x + 2.

Part 2: Complete the table.

Step1: For x=1:

y = 1 + 2 = 3

Step2: For x=2:

y = 2 + 2 = 4

Step3: For x=3:

y = 3 + 2 = 5

Step4: For x=4:

y = 4 + 2 = 6

Step5: For x=5:

y = 5 + 2 = 7

Step6: For general x:

y = x + 2

Part 3: Ordered pair when x=0.

Step1: Use the equation y = x + 2.

Substitute x=0: y = 0 + 2 = 2. So ordered pair (0, 2).

Part 4: Does (0,2) make sense?

Step1: A rectangle has width (x) and length (y) as positive dimensions (number of rows/columns must be positive integers).

x=0 means 0 rows (no height), so it's not a valid rectangle. Thus, the ordered pair does not make sense.

Part 5: Equation.

Step1: From the pattern, y = x + 2.

Answer:

1. Relationship:

The length \( y \) is 2 more than the width \( x \), so \( y = x + 2 \).

2. Completed Table:

Width \( x \)	Length \( y \)
2	4
3	5
4	6
5	7
\( x \)	\( x + 2 \)

3. Ordered Pair:

\( (0, 2) \)

4. Sensibility:

No. A rectangle with width \( x = 0 \) (0 rows) is not a valid rectangle (dimensions must be positive).

5. Equation:

\( y = x + 2 \)