Day 15 – Multiplying Negative Numbers

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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.

 

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Day 13 – First Look at SBG

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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

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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:

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Day 9 – Adding Integers Again

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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 5 – Gauss 1 to 100

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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.

Day 3 – Intro to Problem Solving

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Math 8 – Students had their first taste of this year’s Thinking Classroom culture and active learning problem solving. After I introduced the outline for the year and talked about what we’ll be doing, the kids started off on some problems.

Student engagement was very high. We got through two problems using randomized groups and working on whiteboards and windows with dry erase markers.  At the end of one class I asked one student if they liked math.  The student said that they used to, but then got demoralized by bad test results.  However, the student added that he liked math now. I guess that goes in the win column.

Thanks to the math teachers at Killarney (Tanya, Sue, Sonya) for helping me down this path to a thinking classroom.