I am currently piloting an Algebra 1 textbook from Big Ideas. I've been teaching math for 12 years (or so) and I spent a couple of decades before that reinforcing the idea that I know everything. So, when I started the year, I used the textbook the same way I always had. I teach the way I feel most comfortable and then use the book for problem sets. This week I decided to learn something new.
I read the suggested methodology in the book and took the kids through an exploration in the prescribed manner. The lesson was regarding graphing linear equations using intercepts. I started the kids off with a warm up: How many pairs of numbers have a sum of 7? I was surprised and delighted to see the responses.
First of all, all 30 freshmen were immediately engaged. I didn't see anyone not discussing the problem. Clearly they were intrigued. This was 8am Friday morning!
Then I started wandering around the class to see and hear what they were discussing. I noticed one student had simply written infinaty on his paper (the well known alternative spelling to infinity). At first I was not sure what he meant. Then another student called me over and asked whether he could use negative numbers as well as positive numbers. The light bulb started to illuminate over my head. I hadn't specified that you could only use whole numbers. With no restrictions, some of the kids became quite innovative.
Another student asked whether they could use fractions and yet another asked about decimals. I was so excited. In my classroom, kids are seated in groups of 4, so no matter who had the idea, others became aware of the possibilities very quickly.
I got the attention of the whole class and took some time to discuss the idea that based on my loosely worded question,you could have an infinite number of pairs. Then I asked students to limit themselves to non-fraction and non-decimal positive numbers.
I asked how many pairs again. Several students volunteered that you could have 4 pairs, 0+7, 1+6, 2+5 and 3+4. So then I asked the class whether 4+3 would be considered another pair. One girl loudly exclaimed, "mind blown."
After this warm up activity, we proceeded with the Exploration in the Student Journal where we guessed what pairs of numbers would satisfy the equation 3x + 2y = 6? Eventually one girl figured out that you could easily find a pair if you set x or y to zero.
All in all it was a very satisfying lesson. Thanks, Big Ideas!