## Day 17 – Equivalence

Math 8 – The above picture shows the 2nd vote (peer instruction).  On the first vote the choices were evenly split between A, B, and C. Obviously we spent more time on this question.  Pretty good evidence that this topic needs to be explicitly dealt with, we shouldn’t assume that kids know what the equals sign means.  This vote was after I asked the class to consider the difference between 4×8 (an expression) and 4×8=32 (an equation).

Above shows the class collaborating and discussing a voting question.

In general, we had good Flow today…

## Day 16 – Mistake Game

Physics 11 – Today was the physics’ students first shot at the mistake game. The above picture shows a group’s work on a ranking question.

As well, I handed back last day’s quiz on graphing, best fit lines, slopes and models. I asked students to look at three other quizzes to compare mistakes. They then had to write down the three most common mistakes and hand this sheet in to me. This is my attempt to stop them from repeating the same old mistakes time and time again…

## Day 15 – Multiplying Negative Numbers

Math 8 – This is the third time we’ve used patterns and tables to explain or reason around a rule that we use with integer arithmetic.   Today’s class the students did most of the work though (gradual release of responsibility).  They were given the following prompt, “In groups, come up with a table that you could use to convince someone how to multiply a negative number by a negative number.” Most groups managed to succeed.  The most common problem was when the group started by using the rule that we’re supposed to find. In other words, the “correct” table would start with a positive x negative because we had previously determined that this resulted in a negative number.  However, some students started with a negative x negative, saying that the product is positive.  This only works if you already know the rule.

The day also included a few more practice questions on adding integers, and their first practice with multiplying integers.  I’m trying hard to utilize spaced practice.

Finally, the students signed up for Moodle and enrolled in my Math 8 course.  The Moodle course will hold mini-lessons/notes on the topics along with auto graded quizzes for practice.

## Day 13 – First Look at SBG

Math 8 – Students had their first short quiz today and I introduced them to SBG. Some kids got what I was putting down but I think many are still unsure of what I’m talking about. However, this will sort it out over the coming weeks as we do more assessments.

For a breakdown of my grading scheme, take a look at my physicsoflearning blog post on it.

## Day 11 – Finding Patterns and Reasons

Math 8 – Today the kids practiced adding and subtracting integers by working on suduko type puzzles together.  Next we took at look at multiplying integers. We reasoned that a negative multiplied by a positive must be a negative, by modeling expressions like 4 x (-2), which corresponds to 4 groups of negative two.  Things get tricky when you consider (-3) x 4, because exactly what does it mean to have negative three groups?

There are ways to model negative groups but I find them completely unsatisfactory. I believe that models in science and math are meant to represent a feature of the world such that it is easier to understand.  Some of the modeling in mathematics clearly does not do this.  Adding a bunch of matched pairs of positive and negatives such that we can subtract a negative where one didn’t used to exist is the opposite of easy. And certainly very few people would come up with this model or use.

Instead we used the communitive property of multiplication to show that (-3) x 4 = 4 x (-3) and then easily modeled the latter.  At this point I asked the classes to come up with a rule for when a negative is multiplied by a positive.  With examples on the boards and having discussed it for a bit, I was surprised that many kids couldn’t pick up the pattern – that the product is always negative. I think we were dealing with mental fatigue at this point.

We also took at the pattern generated from a table:

## Day 10 – Buggy Whiteboards

Physics 11 – I did a lot of talking again today, which was expected but still a bit disappointing.  It was the first board meeting I had with one class, and they were very quiet and unsure of what to say, ask or critique.

The top picture shows some good analysis and you can see the influence of math in the notation. We later talked about better symbols to use, and why a decimal is better than a fraction in this context.  The bottom picture is a screen grab from the consensus we came to for making whiteboards and models.  This class didn’t have anyone generalize the model, so I had to go over that part.

Next the students will have some practice moving through different representations (words, graphs, motion maps), which will highlight how they all apply to the same model.

## Day 9 – Adding Integers Again

Math 8 – All my grade 8’s had their timetables changed so I was essentially starting with a new class today. About 2/3 of my students were new to me.  Having already done a constructivist review lesson with adding integers, today I took a faster more direct route because the first half of the class was spent going over class business (books, course outline, expectations, etc).

I quickly reviewed two methods for adding and subtracting integers, using modeling and a number line.  From what I saw on Wednesday, almost all kids are comfortable with this.

The situation that is hardest for them to understand and the hardest for me to explain is when you subtract a negative number. For this, I had students fill in the 6 – n table shown above.  Students were confident that the pattern was correct and would hold for subtraction.  From this, we reasoned then that 6 – (-1) = 7, 6 – (-2) = 8, etc.

I get kids to explicitly state their reasoning each time.  It’s repetitive but I want to reinforce that answers come from reasons, not tricks.

From the data table and the equations we developed, I asked students to write down a rule on how to subtract a negative. Most students give rules #1 and #2. I ask around until we get to rule #3 (we had already covered the definition of an “additive inverse”). I then asked the students to give the pros and cons of each rule and we generally agree that rules 1 and 2 don’t give any explanation and they can also be confusing.

Most kids do have a rule for this situation. They say you turn one negative around and put it on top of the other negative, which makes it a positive. Excuse me while I gag…

Next day I will offer an analogy for adding and subtracting integers:a hot air balloon where positives are helium and negatives are weights.  It’s the best analogy I’ve come across that fits with subtracting a negative (if you remove a weight, the balloon rises / gets more positive / does the same thing as if you added  helium).  I might make a Scratch game/simulation for this.

One last note… I had students return some Plickers voting cards since they weren’t going to be in my class anymore. One student was walking away when she stopped, turned around and said, “I really liked your class. It’s the most fun math class I’ve ever had.”

So that goes in the win column.

## Day 8 – Buggy Whiteboarding

Physics 11 – Today was the students’ first shot at whiteboarding models that they’ve developed.  The photo above is interesting because during their presentation, this group realized that the direction on their graph was wrong.

The group below has a valuable statement on possible errors in data:

And the group below were the only ones that got to point of getting a mathematical model of the buggy motion. They had some good things to say with their presentation, more than what is shown here.