So for this last week before winter break I switched up the warmups from estimations. Mr. Rodinsky was getting fed up with students coming into class with the correct estimate on their paper with no reasoning written. Some see it as a game and I tried to make the analogy that it is like telling someone the ending to a movie before they see it.
I found this visual pattern prompt on twitter, and it's posted under my Favorite Problems link at the top of this page. Here it is again:
After reading about Dylan Kane's ideas about Visual Patterns as an instructional routine, I set the timer for 3 minutes of independent think time where they could copy the patterns down if they wanted. Then 3 minutes to tell their elbow partner what they noticed about the pattern. Then we shared out.
Also, I purposely meant to introduce this pattern the day before introducing the border problem number talk, which I will discuss in the following blog post.
Many students shared that they saw the grey squares as the figure number, squared. So for figure 2, 2^2 is 4, and for figure 3, 3^2=9 the number of grey squares. Students also used what they knew about linear patterns to see that the number of red squares was growing by 4. Some reasoned that the figure before must be 4 so the rule was y=4x+4. Some students used that for question 3 and set 4x+4 equal to 64 to get 4x=60, and x equals 15 or figure 15 having 64 red tiles. 1st period didn't come up with this, but one eager student told me how he solved it first thing in the morning.
For figure 20 in question 1, students reasoned there were 400 grey squares because 20^2 is 400. They said it would have 84 red tiles, because 4(20)+4 is 84. Students reasoned that you could add those two together to get the total squares or you could take the figure number and add 2 to it, then square it.
The highlight was one student saying the rule was x^2+4x+4 also. I assumed and asked him if he had learned something outside of my class about dealing with (x+2)^2? The student replied no. The x^2 is the grey tiles, the 4x is the red growing by 4, and the plus 4 is the red corner squares. My jaw dropped. If you didn't figure out, I assumed the student was going to say the dreaded "FOIL" method (I'm a big fan of area model or generic rectangle first). I was incorrect in my assumption. The student also showed how to see which figure had 121 tiles by undoing the squaring on both sides of the expression to get 11=x+2 and getting x=9 for question 2.
So, for a 10 to 15 minute warm-up, it provided rich discussions and laid some foundation for the border problem the next day.