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Algebra 1

7.3.2 Writing Equations for Patterns with Squares

Algebra 17.3.2 Writing Equations for Patterns with Squares

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Activity

Three steps of a growing pattern with squares. In Step 1, there is a square in the center, with four more squares at each vertex of the center square. The overall image of step 1 looks like a big X consisting of small squares. Step 2 replaces the center square with four squares arranged in two rows of two squares. There are still 4 squares at the corners. Step 3 places nine squares arranged in three rows of three squares at the center of the figure. There are still 4 squares at the corners.

1. If the pattern continues, what will we see in Step 5 and Step 18?

a. Sketch or describe the figure in each of these steps.

b. How many small squares are in each of these steps? Explain how you know.

2. Write an equation to represent the relationship between the step number nn and the number of squares yy. Be prepared to explain how each part of your equation relates to the pattern. (If you get stuck, try making a table.)

3. Sketch the first three steps of a pattern that can be represented by the equation y=n21y=n21.

Are you ready for more?

Extending Your Thinking

1.

For the original step pattern in the statement, y=n2+4y=n2+4, write an equation to represent the relationship between the step number, nn, and the perimeter, PP.

2.

For the step pattern you created in Part 3 of the activity, y=n21y=n21, write an equation to represent the relationship between the step number, nn, and the perimeter, PP.

3.

Are these linear functions?

Self Check

Three steps of a growing pattern. Step 1: Total of two squares, stacked one atop the other. Step 2: Total of five squares, two squares on the bottom row and the second row and one square on the top row. Step 3: Total of ten squares, three squares on the bottom row, row 2, and row 3, and one square on the top row.

Which equation represents the relationship between the step number n and the number of squares y ?

  1. y = n ( n + 1 )
  2. y = n 2 + 1
  3. y = 2 n + 1
  4. y = n + 1

Additional Resources

Patterns Represented by Quadratics

See Steps 1 – 3 of a pattern of squares below. Write an equation representing the relationship between the step number nn and the number of squares yy.

A pattern of squares. Step 1 has a column of three squares. Step 2 has two columns with six squares in each column. Step 3 has three columns with 9 squares in each column.

Make a table for the number of columns and rows of each step and the number of squares:

Step # of columns # of rows # of squares
1 1 3 3
2 2 6 12
3 3 9 27
nn nn 3n3n 3n23n2

First, count how many columns there are in each step (the length):

For each, there are nn columns.

Next, count how many rows there are in each step (the height):

For each, there are 3n3n rows.

The area of this figure is then l×hl×h or n×3nn×3n, which becomes y=3n2y=3n2.

Try it

Try It: Patterns Represented by Quadratics

See Steps 1 – 3 of a pattern of squares below. Write an equation representing the relationship between the step number nn and the number of squares yy.

A pattern with squares. Step 1 has zero squares. Step 2 has a column of 2 squares and then a column with 1 square. Step 3 has 2 columns with three squares each and then a column of two squares.

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