Section A Section A Checkpoint

Problem 1

Here are the first two steps of a pattern of squares that look like + symbols. At each step, 4 squares are added.

pattern of gridded rectangles.
pattern of gridded rectangles. Step 1, 5 squares total, 1 square in middle, 1 square adjacent to each side of middle square. Step 2, 9 squares total, 1 square in middle, 2 squares extending from each side of middle square.

  1. Draw the next figure in the pattern.
  2. How many squares will be needed in step 7? Explain or show your reasoning.
  3. Will there be a figure that uses exactly 40 squares? Explain or show your reasoning.

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Solution

  1. pattern

  2. 29. Sample response: There are 5 squares in step 1. Adding 4 squares 6 times takes us to step 7. 5+(6×4)=5+24=295 + (6 \times 4) = 5 + 24 = 29.
  3. No. Sample response: The number of squares will always be an odd number because we start with an odd number and always add 4, an even number, each time.
Show Sample Response
Sample Response

  1. pattern

  2. 29. Sample response: There are 5 squares in step 1. Adding 4 squares 6 times takes us to step 7. 5+(6×4)=5+24=295 + (6 \times 4) = 5 + 24 = 29.
  3. No. Sample response: The number of squares will always be an odd number because we start with an odd number and always add 4, an even number, each time.

Problem 2

The rule for a pattern is “start with 12, keep adding 12.”

start with 12,
keep adding 12
  1. Complete the table with the first 10 numbers of the pattern.
  2. What do you notice about the numbers in the pattern? Explain or show why it happens.
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Solution
  1. 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
  2. Sample response: The digit in the ones place changes the same way it does if the pattern is add 2. You are adding 2 ones each time when you add 12. It also makes all the numbers even. The digits in the tens place start increasing by 1, but increase by 2 after 5 numbers. This happens because you add two 5 times and make a ten. So every 5th number you are increasing by 2 tens.
Show Sample Response
Sample Response
  1. 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
  2. Sample response: The digit in the ones place changes the same way it does if the pattern is add 2. You are adding 2 ones each time when you add 12. It also makes all the numbers even. The digits in the tens place start increasing by 1, but increase by 2 after 5 numbers. This happens because you add two 5 times and make a ten. So every 5th number you are increasing by 2 tens.