It’s the end of the first week back. I got my new group of Robotics students, we began with some divisibility and exponents in 6th Grade Math, and we’re exploring multiplying rational numbers in Pre-Algebra.

#### Robotics

Robotics is just a semester course so I have a new group this semester. Last semester was nine 7th-8th grade boys. This semester is eleven 6th grade boys, a 7th grade girl, and and a 7th grade boy. Really different dynamic, but almost all of them I’ve had in math class before. The semester feels like it’s running a bit on autopilot since not much is changing. We got started right away with the CS First curriculum, which is meant for after school programs but works as an intro to my semester course. I feel kind of bad because just about all the instruction is video based but it works so well with the various levels students are at. They’re learning the basics of the Scratch blockly programming software to begin thinking sequentially and programmatically. They watch about 10 minutes of video and then they create something on their own. I have a lot of time where I’m just watching them watch the videos and work, waiting for someone who has a questions, so I’ve decided that I would work to advance my own skills in Scratch while they work on their projects. One student remarked “this isn’t even like a class, we’re just playing on Scratch the whole time! It’s awesome!” Ha, he doesn’t even realize he’s learning 🙂

#### 6 Grade Math

6th grade math went fine this week; nothing really sticks out as awesome or terrible. I’m pretty excited because I just bought some linking cubes for an activity on teaching prime numbers. We haven’t done a lot of manipulative work in there and I know they’re craving it. I’ll share more on the activity and I plan on making a video of how it goes once I sit down with it and work out all the details. I actually found myself excited to do the activity myself so that’s a good sign.

#### Pre-Algebra…& The Fractions

I had a moment today in my 6th period where I was walking through an example and trying to explain multiplying mixed numbers with a number line and I thought ‘wow, I’m confusing myself right now…this is crashing and burning.” My students were gracious but man, I did such a terrible job. I’ll walk through my examples and why there were terrible. Let me preface with the fact that I was trying to just avoid telling them what the rules were and having them see more conceptually what’s happening to (hopefully) illuminate the rules.

Students were to attempt this problem on their own before we started the lesson:

I had students share ideas without saying if it was right or wrong, just asking why they did what they did.

I multiplied the whole numbers together, then the numerators together, then the denominators together…because…that’s the rule.

I converted to improper then multiplied the numerators together and denominators together…because it’s easier with improper fractions.

I think I’m supposed to flip the fractions around and turn it into a division problem…or something?

Well I know my answer is reasonable because 2.5 x 1 is going to be 2.5 so my answer is probably going to be around there.

For two of my classes, I stopped the conversation there and said, ‘before I tell you which way is right, lets see if we can understand what we’re actually doing when multiplying fractions.’ My last class (the 6th period one), I thought, ‘perhaps I can try and show them this example on a number line.’ It sort of when downhill from there.

I proceeded through the following train of thought:

- 2 1/2 one time would be 2 1/2 (no problems there).
- 2 1/2 twice would be 5 (still OK).
- I then tried to show them 1/4 of 2 1/2 on a number line. I broke each half up into 4, which made 8ths.
- I then counted up all the 8ths in 2 1/2 (20 eighths).
- I then said we need 1/4 of the 20/8, which would leave us with 5/8.
- We then would add 5/8 to 2 1/2, but we need common denominators there.
- Let’s convert to improper fractions to do that easily.

Ugh, I’m grossed out just by writing it all out. What stinks is that in my first class that I taught this lesson, one student asked how a particular problem could be done and it was a MUCH BETTER example that worked out visually so well, but I forgot the problem (later on I found out it was just 1/4 * 3/4…much smarter to just show it visually with proper fractions first and then move to mixed).

We went through a simpler example (1/2 * 5) to show that we’re still just doing repeated addition even with fractions and I gave them a couple more mixed number multiplication problems. By the end of it, I was a bit at a loss for how to help them realize it conceptually that I just started saying ‘you don’t need common denominators because…’ and ‘just convert to improper and multiply the numerators and denominators.’ It stunk because if I was just going to do that, I could’ve saved about 20 minutes of class time.

It all seemed like a good idea when I was writing the lesson plan :/

Let’s hope division will be a better story…I think it will. Students don’t struggle with division right?

My students would rather change fractions to decimals to do the math. I think multiplication and division of fractions is hard for most students no matter how you present it.

Some of mine do the same. I know some of the Algebra 1 students are really struggling with some of the work since they can’t work with and manipulate fractions without trying to turn them into decimals. Thanks for reading Lee!

Conceptually splitting into parts via the commutative property makes this a bit simpler

. I.e. 5/2 x -5/4 = -1 × (5 x 5) × 1/2 × 1/4.

The remaining fractions are much simpler to model and then this flows towards why we we just multiply numerator and denominators.

That makes sense. I had never seen it done like that. Thanks for sharing Ben! 🙂

[…] venturing back into fractions. I wrote at length last year about this being one of the most challenging units. I plan on having them work with both number lines and […]