*From Patterns to Algebra*, I highly suggest you check it out before you read this post.)

The next 2 lessons in the book were similar to the first 2, but instead of just multiplying, now my students needed to think in 2 steps (can you say "nice connection to two-step equations"?) by first multiplying then adding. We started with the "Guess My Rule" game as before. The first day I used the rule

*x2+2*as suggested in the book. I had my students volunteer the random numbers for the input column and I gave them the output numbers one at a time. After 3 sets of input/output numbers I asked them to try to think of the rule. Many were stuck. One student (the same student I mentioned in part 1 as a matter of fact) said "This is hard!" (again). But once she figured it out, she again changed her tune and said "Oh, this is easy!" Most of my students just needed scaffolding to help them come up with the rule. I asked them to start by looking at just one pair of numbers - "What can you multiply the input number by so that you get close to but don't go over the output number? OK, then what do you have to add to get to the output number? Now see if that rule works for another pair of numbers in the table." Basically trial and error. Eventually most of them were able to figure this out.
The next day I started by giving them the first pair of input/output numbers and told them "I can think of 6 different rules for this, can you?" Loving a challenge, they all set out to work. That same girl I mentioned several times already had 4 correct rules and said she couldn't think of any more. That's when I broke into my rant about perseverance and productive struggle. With my prompting she went back to work and easily came up the last 2 rules. After we had all 6 rules written on the board, I added a new pair of inout/output numbers. I asked them "Now which rule is the only rule that works for both sets of numbers?" A few minutes later most had written down the correct rule on their whiteboards.

The third day we started out the same way with one pair of input/output numbers and me challenging them to find 4 rules. Once they had figured out the 4 rules, I gave them another input number and asked them to write down the output number using our rule. This was easy for them now that they had the rule, but then I gave them an output number and asked them to figure out what the input number was. Wow! You would have thought I'd asked them to explain Einstein's theory of relativity! We're in the middle of a unit on two step equations so I hinted at the connection. Love this! I love the way they're all engaged and challenged by this and they're starting to see the connections. At first, one student said "But there's no x!" so I had to explain to her that x represents the unknown number, which in this case is our input number. I saw the light bulb go on and she quickly wrote and solved the equation.

The third day we started out the same way with one pair of input/output numbers and me challenging them to find 4 rules. Once they had figured out the 4 rules, I gave them another input number and asked them to write down the output number using our rule. This was easy for them now that they had the rule, but then I gave them an output number and asked them to figure out what the input number was. Wow! You would have thought I'd asked them to explain Einstein's theory of relativity! We're in the middle of a unit on two step equations so I hinted at the connection. Love this! I love the way they're all engaged and challenged by this and they're starting to see the connections. At first, one student said "But there's no x!" so I had to explain to her that x represents the unknown number, which in this case is our input number. I saw the light bulb go on and she quickly wrote and solved the equation.

Next, I think I'll let them take turns being the rule makers as the book suggests. I didn't do that with the first lesson, but when I let them build their own patterns for each other to solve they really loved that. It gives them some ownership and, as I firmly believe, if you can teach something it really shows that you understand it. I plan on using this lesson for another day or two and then moving on to patterns with two steps. Stay tuned!

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