Question

bluebonnet learning k-5 math
name __ date __

1. what two equations can this rectangle represent?

rectangle with squares
a. ____
b. ____
c. using whole numbers, what other rectangle(s) can be made with this same area?
d. is 9 prime or composite? how do you know?

1. is 2 prime or composite? ____

explain how you know.
____
____
a. draw the rectangle(s) you can make using two squares.
b. list the factors of 2. ____

Explanation:

Question 1

Part a and b (Equations for the rectangle)

Assuming the rectangle has 9 squares (since later questions involve 9, and the diagram shows a row of 9 squares, maybe 1 row of 9 or 3 rows of 3? Wait, the diagram: let's count the squares. The first diagram: let's see, the rectangle has, say, length 9 and width 1, or maybe 3x3? Wait, no, the first part: "What two equations can this rectangle represent?" Let's assume the rectangle has area 9 (since question 1d is about 9). So if it's a 1x9 rectangle, the area equation is \( 1 \times 9 = 9 \). If it's a 3x3 rectangle, \( 3 \times 3 = 9 \). So:

Step 1: Identify the number of squares

Count the squares in the rectangle. From the diagram, let's say there are 9 squares (since 1 row with 9 squares, or 3 rows of 3). So the area is 9.

Step 2: Write the multiplication equations

For a rectangle with length \( l \) and width \( w \), area \( A = l \times w \). If it's 1 row of 9: \( 1 \times 9 = 9 \). If it's 3 rows of 3: \( 3 \times 3 = 9 \).

So a. \( 1 \times 9 = 9 \)
b. \( 3 \times 3 = 9 \) (or other factor pairs of 9, like \( 9 \times 1 = 9 \))

Part c (Other rectangles with same area)

The area is 9. The factor pairs of 9 are (1,9), (3,3), (9,1). We already considered 1x9 and 3x3. Wait, but 9 is \( 3 \times 3 \), so the only whole - number rectangles (with length and width as whole numbers) are 1 row of 9, 9 rows of 1, and 3 rows of 3. But 1x9 and 9x1 are similar (just rotated). So the other rectangle (besides 3x3 if we first took 1x9) would be 3x3, or if we first took 3x3, then 1x9. Wait, the area is 9. The factors of 9 are 1, 3, 9. So the possible rectangles are:

• Length = 1, Width = 9 (area \( 1\times9 = 9 \))
• Length = 3, Width = 3 (area \( 3\times3 = 9 \))
• Length = 9, Width = 1 (area \( 9\times1 = 9 \))

If the original rectangle was 1x9, then the other rectangle is 3x3. If the original was 3x3, the other is 1x9.

Part d (Is 9 prime or composite?)

Step 1: Recall definitions

A prime number has exactly two distinct positive divisors: 1 and itself. A composite number has more than two distinct positive divisors.

Step 2: Find divisors of 9

The divisors of 9 are 1, 3, 9. Since 9 has three distinct positive divisors (1, 3, 9), it is composite.

Question 2

Is 2 prime or composite?

Step 1: Recall definitions

Prime: exactly two distinct positive divisors (1 and itself). Composite: more than two.

Step 2: Find divisors of 2

The divisors of 2 are 1 and 2. Since it has exactly two distinct positive divisors, 2 is prime.

Part a (Draw rectangle with two squares)

Two squares can be arranged in a rectangle with length 2 and width 1 (or length 1 and width 2, which is the same when rotated). So draw a rectangle with two squares side - by - side, forming a 1x2 or 2x1 rectangle.

Part b (List factors of 2)

The factors of a number are the positive integers that divide it without leaving a remainder. For 2, the numbers that divide 2 are 1 and 2. So the factors of 2 are 1 and 2.

Final Answers

Question 1

a. \( \boldsymbol{1\times9 = 9} \) (or \( \boldsymbol{9\times1 = 9} \) or \( \boldsymbol{3\times3 = 9} \) depending on the rectangle)
b. \( \boldsymbol{3\times3 = 9} \) (or other factor pair of 9)
c. Rectangle with dimensions \( \boldsymbol{3\times3} \) (or \( \boldsymbol{9\times1} \)) (depending on the original rectangle)
d. 9 is \(\boldsymbol{composite}\) because it has divisors 1, 3, 9 (more than two distinct positive divisors).

Question 2

Is 2 prime or composite? \(\boldsymbol{Prime}\)

2 has exactly two distinct positive divisors, 1 and 2.
a. Draw a rectangle with two squares placed side - by - side (1 unit wide and 2 units long, or vice - versa).
b. Factors of 2: \(\boldsymbol{1, 2}\)

Answer:

2 has exactly two distinct positive divisors, 1 and 2.
a. Draw a rectangle with two squares placed side - by - side (1 unit wide and 2 units long, or vice - versa).
b. Factors of 2: \(\boldsymbol{1, 2}\)