Yesterday I read this awesome post by Dylan Kane. Today I posed the same question to my students.
Before I get into the details I have to say - I'm so excited to have Algebra 1 classes this year. There are always so many great ideas for algebra and I haven't been able to use them in the past. I am currently teaching two double block courses of Fundamentals of Algebra 1. They are for students who arrive in high school below grade level in math. The majority of the students have an IEP and I co-teach with my awesome special ed partner (this is our fourth year teaching together which means we know each other well enough to finish each others sentences). So these kids need some number sense and problem solving foundations solidified as we work on mastering the concepts of Algebra 1. 90 minutes every day is a blessing (but I didn't realize exactly how much prep it would be - I haven't taught Algebra 1 in 5 years so I'm mostly starting from scratch). Enough background, on to today:
We start each Monday with a mental math problem. Since most of my kids are allowed calculators even on state tests I'm modifying these to be "estimate first, then do a calculation on paper/calculator, then compare." I hope this will reinforce thinking while typing rather than blindly trusting the calculator.
Just like Dylan, I got some kids who said 31 because there are two numbers so you must do something with just those numbers. I also had a student in each class who said the situation was preposterous and several who agreed. In the first class (red ink) I was able to describe the situation as checking your email every half hour. In the second class they took offense and couldn't move on so I changed it to "the phone battery" and then they were able to move forward. Another student made the point that it depends on a lot of things (and then sidetracked to ask if his battery life is effected by the fact the side of his phone broke off) so I acknowledged his excellent point and emphasized "what do you expect?"
Listening to kids react to each other's estimates is really interesting. We're working on respectful disagreement. There were many noises when 0.50 went up but nothing specific. In the other class, however, when a student said 1% another student responded by saying "Percent?" With some prompting the students directed their questions at each other rather than at me and I got them to state that answers should be in terms of hours. They even questioned the next two responses to make them include units!
The first class (red ink) had several strategies which you can see don't make much sense. The last one was the 7/3*93 and another student very intelligently asked "Where did the 7 come from?" When the student responded I took that opportunity to write "3 hours, 7% used" and then pose the question "What percent would be used after 6 hours?" The rest of that table was a combination of my posing questions and students offering suggestions.
The second class (black ink) was unwilling to share any strategies. While I was wandering the room as they thought, I noticed a student who wrote "7% goes away after 3 hours" and so I asked him to share what he wrote. First he shared "I don't know the answer" but then he shared that other relationship (I'm fascinated that he knew enough to discover that relationship but lacked the confidence to even make a guess). I posed the 6 hours again and then this class headed off in a different direction. They came up with the multiplicative relationship right away (the other class was additive all the way down the table). One student decided it would be about 17 hours because 15 hours would be 56% (28% doubled) and 18 hours would be 112%. Another student - who I'd asked earlier in the class to face forward and stop talking (about the game tonight) - asked me “Can I respond to that?” Then he turned around and asked the other student if 112% was possible. My co-teacher and I still needed to run interference as they weren't listening so much as waiting their turn to talk, but it was exciting to see the beginning of dialogue. They figured out that student 1 chose 17 hours because he knew 112% was too high, but also that the doubling technique only works if you double both the hours and the percents. Then I asked the class to go back to their previous observation. We eventually got to “What times 7 equals 100?” After each kid told me a guess and how close their guess was, I asked “Can you get closer?” Soon I had students frantically typing into calculators and excitedly yelling out their next best solution. (Yes, I know that 9th graders should know to divide in this situation but we started with mental math and I wasn’t about to interrupt that enthusiasm.)
I really enjoyed this problem. It led to some fascinating conversations on units, proportionality and reasoning. Not to mention all the practice we had with good math discussion skills!