Can you help me understand this Mathematics question?

Discussion Question

1. Explicitly describe (in several sentences) how you visualize the change in the growth pattern that is displayed below. Try to explain it as you would expect your students to explain it to you and their classmates.
2. Use this description to generate an algebraic expression/equation that will give the number of squares, n, at any stage, s.

Peer Response (2 points): Demonstrate your understanding of their strategy and expression/equation to find the number of squares at any other size not already used by one of your peers. Additionally, respond to their description and expression/equation.

Peer Response 1

As discussed in last week’s article on patterns, first I would ask the students to give their view about the patterns. I know some of the answers that would be given would be “There are 3 shapes of different sizes”. Another anticipated answer would be the squares are increasing by 3. Another anticipated answer may be 1 square is added on the top line, 1 square is added on the line going downward, and 1 square is added on the line at the bottom. I would explain to the students that they are all correct based on their explanations of what they observed from the changes in each pattern. For better understanding of the subject matter, I would introduce them to a simpler way to have an accurate result at all times. First, I would ask them to spot the differences between figure 1 and figure 2. Then, I would have them use a red crayon to color the squares that are different between figure 1 and 2. Then, I would ask them to shade the squares that are different between figure 2 and figure 3. I would place a model on the board so that they could see the shading. Finally, I would let shade 3 squares in figure 1 in order for them to see the initial value. After shading 3 boxes from figure 1 our initial value would be 2. We would use that answer to generate our equation.

The algebraic expression that I would use for the increase in size is the following: n=3s+2. In order to ascertain the answer and determine the next pattern, I would substitute the value of 4 for s in the equation. The answer for n would be 14 which shows the increase is 3 from the subsequent pattern. Finally, I would also use the value of 5 for s and the answer for n would be 17. This proves that my equation is correct because I constantly have an increase of 3.

Peer Response 2

The first thing I observed in this growth pattern is that in each successive step the total number of squares increases at a rate of 3 squares per step. To help visualize the change and better understand the growth pattern, I decided to work backward and create what the pattern would look like at Step 0. I visualized Step 0 as two vertical squares, one on top of the other. Therefore the constant in all my steps is + 2.

From this image, I decided how the three squares being added in each step were being placed, one off to the left horizontally, one block being placed in between squares vertically, and a final square horizontally to the bottom right. Given this pattern, no matter what step number we are at we can use a single equation to determine the total number of squares in the pattern, where three additional squares per step are added to a constant of two squares.

There are certainly other ways to interpret this relationship, and equivalent expressions that could be used; however, the equation that supports my reasoning is:

n = 3s + 2

Can you help me understand this Mathematics question?

Discussion Question

1. Explicitly describe (in several sentences) how you visualize the change in the growth pattern that is displayed below. Try to explain it as you would expect your students to explain it to you and their classmates.
2. Use this description to generate an algebraic expression/equation that will give the number of squares, n, at any stage, s.

Peer Response (2 points): Demonstrate your understanding of their strategy and expression/equation to find the number of squares at any other size not already used by one of your peers. Additionally, respond to their description and expression/equation.

Peer Response 1

As discussed in last week’s article on patterns, first I would ask the students to give their view about the patterns. I know some of the answers that would be given would be “There are 3 shapes of different sizes”. Another anticipated answer would be the squares are increasing by 3. Another anticipated answer may be 1 square is added on the top line, 1 square is added on the line going downward, and 1 square is added on the line at the bottom. I would explain to the students that they are all correct based on their explanations of what they observed from the changes in each pattern. For better understanding of the subject matter, I would introduce them to a simpler way to have an accurate result at all times. First, I would ask them to spot the differences between figure 1 and figure 2. Then, I would have them use a red crayon to color the squares that are different between figure 1 and 2. Then, I would ask them to shade the squares that are different between figure 2 and figure 3. I would place a model on the board so that they could see the shading. Finally, I would let shade 3 squares in figure 1 in order for them to see the initial value. After shading 3 boxes from figure 1 our initial value would be 2. We would use that answer to generate our equation.

The algebraic expression that I would use for the increase in size is the following: n=3s+2. In order to ascertain the answer and determine the next pattern, I would substitute the value of 4 for s in the equation. The answer for n would be 14 which shows the increase is 3 from the subsequent pattern. Finally, I would also use the value of 5 for s and the answer for n would be 17. This proves that my equation is correct because I constantly have an increase of 3.

Peer Response 2

The first thing I observed in this growth pattern is that in each successive step the total number of squares increases at a rate of 3 squares per step. To help visualize the change and better understand the growth pattern, I decided to work backward and create what the pattern would look like at Step 0. I visualized Step 0 as two vertical squares, one on top of the other. Therefore the constant in all my steps is + 2.

From this image, I decided how the three squares being added in each step were being placed, one off to the left horizontally, one block being placed in between squares vertically, and a final square horizontally to the bottom right. Given this pattern, no matter what step number we are at we can use a single equation to determine the total number of squares in the pattern, where three additional squares per step are added to a constant of two squares.

There are certainly other ways to interpret this relationship, and equivalent expressions that could be used; however, the equation that supports my reasoning is:

n = 3s + 2