This week, since it had been a while since we worked on the patterns lessons due to my week off before break coupled with the 10-day Christmas break, I started out with just reviewing by using the Guess My Rule, first having them come up with as many rules as they could given just one pair of input and output numbers (2 and 10 in the example below), then giving them a second pair of input-output numbers (6 and 26) to have them decide which is the only rule that works for both pairs of input-output numbers. Once they determined the correct rule (x4+2), I first gave them another input number (7) and they had to calculate the output, then I switched it up and gave them an output number (18) and they had to figure out what the input number was. This is always more challenging for them and even though I demonstrated a few weeks ago how to write a two-step equation to find the missing input number, they still mainly rely on guess-and-check. I tried to point out to them that they can use the input-output pairs they already know to narrow down what numbers to use when they guess-and-check. For instance, if the output number is 18, that falls between the output numbers of 10 and 26 that we already know, so the input number must also fall between the 2 and 6. Some students got this strategy, while others didn't.
The next day we did the "secret pattern" lesson from the book. I gave out slips of paper to each student with a secret rule and they had to create their own patterns with the color tiles and 2-sided chips to go along with the rules they were assigned. Here are some of their patterns:
(One thing I realized after the fact is that I should cut the position number strips apart so they could spread them out more and line up the position numbers under the actual patterns for each position.)
Once I had checked everyone's patterns to ensure they had set them up correctly, I had them rotate to someone else's pattern to first try to write the rule for the pattern, then build the 4th position in the pattern.
Most of my students have gotten pretty good at finding the rules. As a class, they have decided that the part of the pattern that is the same in every position is the constant and should be written as the part that is added. They're really good at drawing the 0 position because they are good at finding that constant. They find the multiplier by looking at the first position's tiles that are not the constant. I've demonstrated that once they find the multiplier in the first position (for example, 1 set of 5 red tiles as in the last pattern shown above) sometimes it's easier to see that the second position is 2 sets of 5 red tiles is they make a slight separation between each set of 5 red tiles. I also try to emphasize that they can use the multiplier and the position number to find out how many red tiles there will be at any other position (and that "total tiles = position number x multiplier + constant"). We've practiced describing what position 12 or 15 or 100 would look like - how many yellow circles and how many red tiles will there be. They're now good at knowing that there will always be 2 yellow circles at any position and that they really just need to calculate how many red tiles by multiplying the position number I give them by the multiplier they discovered in the first position.
Something that I pointed out after they were all done with writing the rule and building the 4th position is (as the book suggests) that even though some of their rules were the same, their patterns didn't look anything alike. And we also observed how rules that were opposite, such as x5+3 and x3+5 look different even though they have the same numbers.
Next, going off script a little from the book, I had my students relating the patterns to the tables. First I gave them a pattern to determine the rule. I had them write out the equations to find the total tiles in each position ("total tiles = position number x multiplier + constant"). I explained that using an organized list would make it easier, so I reminded them to start off with position 1, then 2, etc. Next, I gave them a table and had them go through a Guess My Rule exercise. Finally, I split the screen on the SmartBoard and had them look at the numbers from both activities to see if they saw anything in common. Some of them spotted the fact that position 1 had a total of 8 tiles and that if the input was 1 the output was 8, so as a class we decided that the input numbers were the same as the position number in the pattern and that the output numbers were the same as the total tiles in the pattern. Now we have a connection between the patterns and the tables in Guess My Rule.
The next lessons coming up will help get them more prepared for our linear equations unit. We will be observing how changing the multiplier and keeping the same constant, or changing the constant and keeping the same multiplier effect how our patterns act. Stay tuned for more updates...