Day 139 – Two Interesting Problems

Math 8Who Gets There Faster, Jim or Stan?

Today I worked on giving most of the instructions/prompts verbally. I thought it would be prudent to write down the numbers required though.  Maybe I should have tried it without the writing.

And a nice solution below…

While several groups clued in that distance/speed is time, lots of groups used equivalent ratios for their solution, which I was fine with.  The reasoning would be something like “Jim goes 9km in 2hrs, 13.5 km in 3hrs, and over 11km in 2.5 hours…”

The next problem was simply stated: I have a rectangular prism with side lengths 8, 9 and 12.  What is the furthest distance from one point to another?  One solution is shown below.  Once students were able to picture the vector they needed, many were able to find the correct answer. I liked this problem because it had a low floor – all groups were able to find some diagonals using the Pythagorean Theorem.

I was very proud of this lesson, both for what I had planned and on what students were doing. Virtually all the students were engaged and accountable. Even border kids who typically sit back a bit were interested in working on possible solutions.

Day 114 – Quantitative Energy

Physics 11 – Today we moved into more quantitative analysis of conservation of energy. I was pretty pleased with the previous class and the work done using bar graphs, and kids told me that they were feeling pretty good about. Gulp, that’s often a bad sign!

I could see today that several kids were not quite making the jump from bar graphs to using the conservation of energy equation.  Lots of the confusion stems from the students not knowing the beginning and end “situations.”  For example, consider the question below:

A 30.0kg gun is standing on a frictionless surface. The gun fires a 50.0g bullet with a muzzle velocity of 310m/s.  Calculate the kinetic energy of the bullet just after firing.

Several kids were very unsure of what the “beginning” and “end” are.  They were very used to have clearly drawn diagrams given to them.  This isn’t strange, research has shown that giving diagrams to students can actually impair their problem solving (I can’t find the reference just now, but I read it recently).

Day 65 – Bathtub Problem

Math 8 – Today students worked on the bathtub problem. It was totally solvable and as a result we saw a lot of success. The problem was difficult enough to stimulate discussion and reveal initial misconceptions though. Most groups first thought the solution would be 45 minutes. I think the logic was that if you add an hour and 30 minutes and then average/split it between two, you get 45 minutes. Listening to student conversations, there would inevitably be someone who would speak up and say something like, “That doesn’t make sense! If you use just hot water it only takes 30 minutes, it can’t take longer by adding even more water!”

Day 51 – Discounts

Math 8 – Today the class worked on the above problem. Lots of groups had problems with it, and couldn’t move off the idea that 30% + 45% + 25% = 100%. They couldn’t think of any other thing to do with the numbers.

There was also some nasty misconceptions with adding fractions! Lots of kids did 45/100 + 25/100 + 30/100 = 100/300.

By the third class of math 8 I finally found out what I had to do as a teacher to get the students on the right track. First, I had to make sure that the problem doesn’t turn into a money issue. If students equate the discount to saving money, there will be trouble. Keep the problem centered on saving gas, because everyone agrees that you can’t have a car drive without using fuel. Secondly, I had to really listen carefully to what the students say and really jump in when they say something like “45% of 70”. A ha! What operation is associated with the word “of”? That’s right, multiplication, not addition…

The neat thing about this problem, after I’ve done the good things needed to pull it off, is how engaged the students get. They can taste how close they are to a solution and want to solve it.  It was Friday afternoon, the dismissal bell had gone, and I had several kids sticking around to finish their work.

Day 47 – Specialize for Problem Solving

Math 8 – Today I am trying to get kids to extend their practice of using concrete examples and specialization for general math problem solving. The above picture was one example we did. Many students didn’t know how to approach this problem initially but they were able to re-frame it with a question they could do: they realized that 50% of 1480 is 740. This helps them figure out a method for doing the problem they really want an answer to.

Day 37 – Defining a Strategy

Math 8 – We’ve moved onto estimating square roots, but today we did a small assessment of problem solving. Students have the above handout to define a strategy for problem solving that they’ve been using. We had mostly inadvertently started using the rubric from Mason’s Thinking Mathematically.

My handout/assessment can be found here.

The problem presented was a lot easier than painted cubes and many kids had success. I was also very clear that I wanted to see kids work through the strategy. Try some specific examples, attack the problem with a try, check to see if it works…

Day 33 – Painted Cubes, High Fives

Math 8 – We continued with painted cubes today and several groups were able to start making generalizations. It was so great to see. There were lots of wins today. One student was so happy once she started to see that the cubes with 1 face painted increased with squares. She saw that if there was a 9x9x9 cube, the # of cubes with one face painted would be 7×7, and there would be 6 faces like that. Huge high-five!  Check out the whiteboard below.

Day 30 – Painted Cubes

Math 8 – Today the class started on the “Painted Cubes” problem. I figured this problem would be quite difficult for them, and I’m structuring the problem to focus on two things. First, I want students to use a problem solving strategy that helps them move from a specific case to a general case. Secondly, this problem is used to reinforce that squares, cubes, square roots and cube roots are not abstract ideas with weird symbols, but that they have physical meanings.

The students first have to understand what they know – the problem setup. Next, they have to be clear on what they’re looking for and what they want. To begin with this is the # of cubes that will have 3 faces painted, 2 faces painted, 1 face painted and 0 faces painted. What they’re doing today is solving a first specific example of this problem: a 3x3x3 cube. This is specialization. We’ll do a few more specializations, and then try a generalization.

In the top photo you can see a good example of the students checking their solution.  They knew they needed 27 cubes in total, and the solution in the bottom right corner didn’t allow them to have a cube with 0 faces painted.

The other photos show how some groups are trying to specialize with a 4x4x4 cube, and how to record their information in a way that makes sense.

Day 5 – Gauss 1 to 100

Math 8 – Today the kids tackled Gauss’ 1 to 100 problem.  How can a person add up the numbers from 1 to 100 in seconds?  What is the strategy?

This was a tough problem.  Two “hints” help the process along.  If kids write down all the numbers from 1 to 100, they can piece together  the strategy easier.  One thing I did was wait for kids to start grouping numbers, and then point out how grouping seems to help.

One interesting brain thing is that no one thought to add 1 + 100, 2 + 99, 3 + 98, for sums to 101. This includes myself. I think people are much more tuned and comfortable with 100, it’s less “outside the box.”

Next day we will do some formal work on integers and I will bring in the associative property of addition, using this problem as an example.