I’m having trouble remembering exactly all that happened this week. I know we had some good lessons and I think it’s perhaps nothing sticks out as a negative experience; that’s good! We started exploring inequalities in Pre-Algebra, fractions actually went better than anticipated in Math Skills, and Robotics…well you can see how that’s going through my weekly episodes of Middle School Robotics 🙂
Reflecting on the 2-step equations project the students did last week, a good amount of them have done fairly well. It was a test grade but I treated it as a learning experience as well; I was willing to work with students, they could work together, and it was a much better alternative to just having them solve a bunch of 2-step equations. Check out the video I did last week where I explain the project and provide resources for it. A few students really didn’t do well since as they worked, they had never checked in with me so I wasn’t able to catch any misconceptions while they worked. They will be able to reassess though.
It was kind of funny how at least one person in each Pre-Algebra class made a joke about getting rid of inequality when I introduced the unit. Absolutely loved THIS LESSON I found on how to introduce inequalities and basically copied it verbatim. Great way to tap into the concept first, then talk about mathematical notation and graphing. Here’s the slideshow I made to help guide our discussion and remind of some of the questions to ask. It went much better than last year since last year I just told them how to do it but we didn’t really talk about them in the context of ‘is this a solution to this situation or not.’ Some got a little bit tripped up when solving 1-step inequalities (here’s the slideshow for that), especially since the ones we did in class were working with whole numbers but their Khan Academy assignment ended up pulling in some fractions.
So after trying to teach multiplying mixed numbers with circle models and complicated area models like this:
2/3 * 4/5 = 8/15
I opted for a simpler area model which seemed to click with a lot of them.
They know they can just convert to improper and multiply or they could draw this. A lot of them said it cleared up the process and they were able to work with numbers that were a lot smaller (since converting to improper will sometimes give you a bit a hefty numerator).
When I moved to division, I tried something different. Before I showed them the traditional algorithm of multiplying by the inverse, I had them find a common denominator and dividing the numerators. If you have 10/12 and you’re dividing by 2/12, you can sort of just think of 10/2, which is 5. I was hesitant to do this because then I didn’t want them thinking they had to find common denominators to multiply fractions (it would work but you’ll just end up with much higher numbers with has more potential for mistakes). And in reality, you can really just ‘divide across’ as well but then you end up with a fraction over a fraction, which I know would be really confusing for some…and still result in having to divide two fractions.
I worked to really help them see the connection between what we did when we found the common denominator and divided and how that related to just multiplying by the reciprocal. My favorite example for showing multiplication of the reciprocal is examining is:
Students intuitively just do this in their head when I ask them what 20 ÷ 1/2. While many will still not know exactly WHY this works, I’d much rather them be able to at least say:
I’m turning this division problem into a multiplication problem by multiplying by the reciprocal of the divisor
You just cross multiply! YOU LIKE CROSS THEM AND MULTIPLY THEM AND JUST PUT THE NUMBERS WHEREVER YOU WANT!
Some of them were already anticipating that we would just have to turn mixed numbers into improper and do that same thing. That’s good.