Estimation 180... Mr. Stadel's height compared to a bus & clearance height of garage, then angle or percentage of pie eaten. A few students had answers that didn't make sense but changed them when I pointed something out about the picture. For example, for this picture:
one student had a guess of 80%. I mentioned that we all knew the amount eaten should be more than 50%, which was enough for that student to realize that it should also be less than 75%.
After our estimating, we moved on to solving systems of linear equation by substitution. Well, we did it more than one way. Here was the first example:
My students make sense of problems by looking at data. It's a concrete starting point for them. So they looked data which wasn't going up very quickly. I suggested going up by more than 1 student at a time.
We got to our solution numerically pretty quickly. Great. How does this connect graphically? They graphed by hand; I graphed in Desmos:
We talked about what the break even point meant and why it is important to know the break even point before an event. Then we tackled the problem algebraically. We started by creating equations to model the situation, then solved the system:
There we go - solved numerically, graphically and algebraically. They worked through this at their own pace - those who were done quickly moved on to the next question. Some didn't see the advantage of solving this algebraically but it quickly became apparent with question 3:
None of my students found the solution numerically. They could estimate graphically (if graphing by hand), but algebra allowed them to find the exact solution.
That's it - 2 questions today, but lots of good mathematical connections going on. This is how we are ending cycle 2. We will work on review questions tomorrow. There is a big field trip on Friday for those students taking history so the test will be on Monday.