Repeating patterns are often introduced to children using colored blocks, chips, or some other object.  It is important to expose students to different objects in repeating patterns so they are able to translate between different representations of the same type of pattern. This helps them understand patterns much like using different variables helps one understand algebra.  Eventually they learn to write out patterns on paper using letters (for example: ABABAB…).

Watch this video as a sample of how the study of patterns begins in elementary school (~ 2 minutes):

Watch the following video for a more general understanding of patterns and type of patterns (~4 minutes):

https://youngmathematicians.edc.org/math-topic/patterns-and-algebra/

2.1 Examples

Example 2.1.1

Assume the colors represent colored blocks.

a) Consider the following pattern (showing two repetitions):
red, red, green, red, red, green,…
Suppose this pattern continued. What color will the 25th block be? Explain how you got your answer.

b) Consider the following pattern (showing two repetitions):
red, green, blue, red, green, blue,…
Suppose this pattern continued to 25 blocks. How many red blocks would there be in the pattern? Explain how you got your answer.

c) Consider the following pattern (showing two repetitions):

yellow, green, green, yellow, green, green,…

Suppose this pattern continued until there were 9 yellow blocks total and then stopped.  How many green blocks would be in the pattern? Explain how you got your answer.

How can we figure out answers to Example 2.1.1 without writing out all the blocks?  Discuss.

Observe the quotient and remainder are important concepts in answering repeating pattern questions, especially when the numbers get too large to list out the objects.

The division algorithm for whole numbers states that for any dividend n and divisor [latex]d>0[/latex] there is a unique quotient q and unique remainder r with [latex]0\le r\lt d[/latex], such that [latex]n=q\cdot d + r[/latex]

Illustration (division algorithm):  Consider [latex]265\cdot 3[/latex].  Our result is 88 ⅓ or 88 with remainder 1 so the following equality holds [latex]265=88\cdot 3 + 1[/latex].  In this problem n = 256 is the dividend, d = 3 is the divisor, q = 88 is the quotient and r = 1 is the remainder.

Recall Example 2.1.1

 a) Consider the following pattern: red, red, green, red, red, green,…

Suppose this pattern continued. What color will the 25th block be?

Using division to solve this problem, determine the following values and explain which one gives you your answer and why.  Write the equation  with your numbers filled in to check equality.

Dividend (n):

Divisor (d):

Quotient (q):

Remainder (r):

[latex]n=q\cdot d + r[/latex]:

b) Consider the following pattern: red, green, blue, red, green, blue,…

Suppose this pattern continued to 25 blocks. How many red blocks would there be in the pattern?

Using division to solve this problem, determine the following values and explain which one gives you your answer and why.  Write the equation  with your numbers filled in to check equality.

Dividend (n):

Divisor (d):

Quotient (q):

Remainder (r):

[latex]n=q\cdot d + r[/latex]:

Example 2.1.2

1) How does the quotient relate to repeating pattern problems (in general). That is, what does the quotient represent in repeating pattern problems?

2) How does the remainder relate to repeating pattern problems (in general). That is, what does the remainder represent in repeating pattern problems?

Example 2.1.3

Create a repeating pattern and write three questions about it that you could ask children to promote their algebraic reasoning. Make the numbers big enough to motivate the use of the division algorithm as opposed to writing out all the blocks.

Then partner with someone in the class and try to answer each other’s questions.

Pattern:

Question 1:

Question 2:

Question 3:

Example 2.1.4

Beginning with 2, the digits in the ones place of even numbers are 2, 4, 6, 8 or 0. What digit is in the ones’ place of the 358th even number?  Explain how you got your answer.

Example 2.1.5

A ribbon design consists of an elephant, a lion, a giraffe and a zebra in that order. Each figure on the ribbon is 5 centimeters wide. If you cut a 1468 cm long piece of the ribbon that starts with an elephant, what will be the last figure on your piece of ribbon? Explain how you got your answer.