Math 8 – Students had their last classroom day to work on the project. This was a nice way to end the unit, with the students being assessed on whether or not they could formulate a cohesive understanding of integers and integer operations. As well, it becomes a formative task for them because some students will have learned new things from doing this project and can now re-test a learning objective for integers.
Math 8 – We started class with doing a bit of Peer Instruction (PI) on order of operations. Lots of pedagogy here: choosing random students to answer questions, peer instruction, spaced practice, etc…
This student did a wonderful job of coming up to the front of the class and giving a good solution to the problem presented
The first vote had about 50% correct answers, and this increased to about 88% after the second vote.
The rest of the class was spent on an integer project (more on that later).
Math 8 – Today we put some formal words and notes around the order of operations, and started some practice on questions that are more difficult than ones that students would have seen in elementary school.
Above is a worked example that the class solved. I randomly chose students to complete each step.
Math 8 – Today we worked on the Four 4’s problem to check and practice our understanding of order of operations. The photo above reveals one of the most common mistakes: working from left to right without first doing division before addition/subtraction.
Today was a good example of how things can slow down progress. Getting kids to properly build a Table of Contents has not been smooth. So while this happens to be a math class, students are also learning fundamental organizational (life?) skills.
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.
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:
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.