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 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 84 – Drawing Conclusions

WP_20160122_15_00_04_Pro.jpg Science 9 – This is a result from one group’s experiment with yeast budding. Their experimental variable was water pH, and it looks like they may have made a mistake. This turns out to be excellent for everyone because it will give the class a chance to apply reasoning to an experimental results.

E: pH 3
F: pH 5
G: pH 7
H: pH 11

If maximum gas was at pH 7, then clearly the pH 3 solution should have less gas than the pH 5. It’s hard to tell from the picture but balloon E was the 2nd fullest.  A few students were able to apply CER (Claim, Evidence, Reasoning) to find this mistake.