Patterns in Arithmetic

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Objective

Students will be able to identify and create the themes of a pattern in a sequence of equations or terms.

Big Idea

Students are given a chance to explore and extend number patterns.

Start Up

15 minutes

Partner Pattern Hunting

30 minutes

Students have 30 minutes to develop and conjecture about the direction of this pattern. I avoid using standard scientific and graphing calculators here. Students need a large display to record this pattern. I encourage them to use a spreadsheet problem like excel. Then can enter their equations and extend the pattern. The fun is defining the pattern. This introduces a wonderfully subjective element to mathematics. Mathematics is about truth, but truth is always about perception. Here students are shattering the myth that math “always has an answer.” Instead the way in which we define the patterns we see are based on our choices we make as we hunt for order and structure. The very way in which we define order and structure might vary as well.

If students finish early, they can attempt a spice rack problem. These are problems I keep in the room for students who are ready to move on.

 I offer these three:

Mild (easy):

Write the next 5 equations in this pattern. Then try the 6th equation. What changes?

1 x 8 + 1 = 9

12 x 8 + 2 = 98

123 x 8 + 3 = 987

123 x 8 + 4 = 9876

Medium (medium):

Write the next 3 equations in this pattern. Why might the pattern ‘end’ in the next 3 equations?

76, 923 x 1 = 076923

76923 x 10 = 769230

76923 x 9 = 692307

Spicy (tough)

Write the next 4 equations in this pattern. Why might the pattern ‘end’ in the next 4 equations?

76, 923 x 2 = 153846

76923 x 7 = 538461

76923 x 5 = 384615

Summary

15 minutes

I start the summary by framing the intent of the lesson. The major take away is that students should consider this type of pattern hunting for their last project of the year. This is a topic that can be pursued if they don’t want to or can’t find a way to connect their science and math.

I drive this conversation by gathering student work from their pattern hunting session. I either put their work under a document camera or have them record their observations on chart paper and then ask them to talk about their findings to the class. I like to discuss the opening pattern and only get to the spice rack problems if there is time. If students are really excited about sharing the spice rack problem but ran out of time, we can have them show the next day and repeat a similar type of lesson that focuses on the pattern they solved. There is certainly no reason to rush here. We are encouraging them to hunt. The world is deliberate. A master hunter is patient and calm.

I tell them to “think of the lion hunting its prey. Think about how still it must be until it finds its prey.”

With these patterns we must be patient. We must wait for a pattern to emerge. We certainly want to hear ideas from the group and won’t waste their hard work by rushing past their observations. The first pattern develops nicely as we expand it. Students using excel might show something like this:

 If students struggled with using the spreadsheet, I demonstrate or have a student demonstrate how valuable a spreadsheet is in pattern hunting. 

Students notice the transformation of the pattern from the digits in each product doing a simple climb from 1 to 9 and then a simple descent back to 1 to something new:

The pattern changes to 12345679 and then has a 00 as if to make room for a digit for 10 and then descends 98765 and then hops down to 0000. Then in the next step all seems lost. If climbs as 12345679 again, but then climbs 0120988000000. I take this chance to encourage perseverance in math. If they keep going a new pattern takes hold. The 12345679 holds and a new mini climb and descent grows. First 0123210 and then 01234320 in the next number. The pattern isn’t the same, but there is something else developing. And that is the magic of pattern hunting.