Dividing Fractions

My 6th graders have been working with dividing fractions for the last two weeks. We explore these four ways, in this order:

1. Number line

2. Rectangles — I wrote about this here.

3. Dividing by one

4. Common denominator

It's completely intentional that we work with the number line and rectangles first. I want my kids to see the answer and that it should match their intuition and understanding.

Teaching Absolute Value

I know Common Core does not have absolute value in grade 8, but I'm teaching it anyway because we're still doing "algebra 1" this year. (A year ago Raymond Johnson looked into the inclusion of this topic in the different grade levels.) My 8th graders know that the absolute value of 5 is 5, and the absolute value of -5 is also 5. Some recall that it's the distance from 0 on the number line. We begin by solving a few of these: abs(x) = 4, abs(a) = 0, abs(w) = -3. A few trip up on the last one but recover quickly and move along.

Me: What is the distance between these two points?

Class: Eight.

M: How did you get eight?

C: Subtract.

M: What about this one? The points are at -3 and 9.

C: The distance is twelve.

M: This one?

C: Thirty-two.

M: Good. Distance is always positive... How did you find the distance between the points again? What operation did you use?

C: Subtraction!

M: Then I'm going to add an equation below each number line showing subtraction. Is that okay?

M: So, the distance between 5 and 13 is 8. Then, what is the distance between 13 and 5?

C: Eight

M: Woah! It's the same? Meaning I can write the equation either way?

Kids agree that subtraction can be commutative when it's inside the absolute value bars because we’re just measuring distance. The distance from Johnny to Julie is the same distance as from Julie to Johnny. I'm not going to argue.

Me: Given two points, you can tell me the distance between them. So now I’m going to give you just one of the two points but tell you the distance between them, and you find the missing point x.

C: x is ten.

M: Yea, ten works. Let's try to read this open sentence. How would you say it?

C: x minus six... The absolute value of x minus six is four.

M: Hmmm. Oh, you say the words 'absolute value' because they're there. Let's try again without saying those words. Use the word 'distance' instead.

C: The distance of x minus six equals four.

M: Let me show you again the first one that I'd asked you. I remember just asking you, 'What is the distance between 5 and 13?' What did I not say even though it's there?

C: Minus.

M: Right. Let's not add stuff we don't need. You know naturally that finding distance implies subtraction. So, say the equation again.

C: The distance between x and six is four.

M: Or you could say...? Can we switch the points around?

C: The distance between six and some point x is four.

M: Alright. Is 10 the only answer for x? We are trying to find a point on the line that makes the equation true. So, let's use the number line to solve this. Because we know 6 is one of the points, let's locate it. We need to find the other point that would be a distance of 4 away from 6. So, it could be to the right of 6, or to the left of 6. Where does this put us at?

M: Oh, why isn't the point -6? I see a 'minus six' in the equation.

C: Remember, that minus is for subtracting. We need it there to find distance.

M: I remember. We need it.

We do a few more of these. Enough to bore us, need something new.

M: Let's try this.

C: No subtraction sign.

M: And you said we needed it. Then create it. Make it happen without changing the problem of course.

C: Change it to minus minus...

M: What does this problem say now?

C: The distance between x and negative eight equals five.

We do a few more of these. Enough to bore us, need something new again.

M: What about this?

M: Nothing terribly exciting. The other point(s) that we find is now worth 2x, so we just need to solve for x.

Then we do a batch of these:

Hey, what about these, where there's more stuff stuck around the absolute value quantity. Oh, we just need to first isolate the absolute value, then it's business as usual.

We spend the next whole class using Desmos to explore the shifts/changes to the parent absolute value function. Students need to write down their predictions first before graphing. One student was very excited when she got the V-shape to turn upside down.We discuss some real-life scenarios that may involve absolute values: margins of error, ranges of measurements: distances, scores, speed, temperatures, pH levels, elevations, etc.

Not proud to admit that I spent a lot of hours in college playing pool instead of studying, but never once did I associate the path of the ball as an absolute value function. Consider me odd if I always thought of angles instead.

Solving absolute value inequalities start similarly enough. Kids know from graphing inequalities that there's "shading" involved. They also know the difference between open and closed points. So I just have the kids use their thumb and forefinger to indicate the distance between the two points, then if the inequality says less than [or equal to], then it's natural to pinch their fingers closer together, indicating that the region inside the points need to be shaded.

The textbook will tell kids to set up the "two cases" to solve these inequalities (same thing with equations). 

Then, kids are asked to graph the solution. But if kids learned to solve using the number line itself, then there would be nothing to memorize because they learned distance way back when they started learning to crawl. And since the graph shows the solution, then writing what that solution is is easy because it matches the graph. Like below, x lives on the green line between -8 and -2, being ≥ -8 and ≤ -2.

For greater than inequalities, the student would naturally spread his fingers apart to indicate shading outside of the points.

I don't know why there seems to be a lot of rules when learning to solve absolute value equations — which inequality sign for when it's and or when it's or.

The Question Was Mine, but the Answer Was All His

The algebra kids have been doing a lot of graphing inequalities on the number line, and the geometry kids have been doing a bunch of a-squared-plus-b-squared-equals-c-squared for their Viewmongus lesson.

So I'm looking at my geometry kids and thinking: Hmmm... I wonder if you know how to graph an irrational number on the number line.

I ask them to locate sqrt(7) on the number line. Two students pull out their calculators. The rest are scribbling and erasing, doodling too I suppose. They know sqrt(7) is somewhere between 2 and 3.

Five minutes pass quietly.

Another five minutes, their faces grow longer. Gabe brings his journal up to me. He writes:

You could create a right triangle with a vertex at 0 and the side being sqrt(7) going along the number line. The other vertex would be the point where the sqrt(7) is located.

I nod my head but ask, "Want to help me understand what you wrote with a picture?"

He comes back with this sketch. I tell him, "I see. Something to do with right triangles. Okay. But you know that I still don't see how you arrive at sqrt(7) right there, right? I think you're on to something, Gabemeister."

Enough individual think/stuck time has passed, I put them into random groups of three. I walk around, they're talking about the problem, but I'm not seeing much on the whiteboards. Maia [bottom right] thinks she's funny. Gabe gets himself a compass.

I'm watching the clock. Jack wonders if he could just make a number line for irrational numbers only. You know, 0 here, then sqrt(1) here, then sqrt(2) here, then oh never mind. Gabe looks busy with his compass. His group mates are observing him carefully and listening to him mumble.

Time to nudge them along. I ask, "This question that I'm asking you of where is sqrt(7) or sqrt(any irrational number) on the number line came from my watching you in the last activity. What were you doing in last two days?"

They burst out:

Aspect ratio thingies.

The diagonal of the TV!

Mathalicious!

Right triangles!

Pythagorean theorems — but this [number line] is not a right triangle!

Viewmongus!

The diagonal of the TV is the number line!

I say nothing more and continue to walk around. Now I'm seeing more on their whiteboards.

All the groups are working with right triangles, but I'm unable to take more pictures of their work as I'm now with Gabe's group and listening to him explain how he thinks he has the answer.

He goes over his drawing above step-by-step by drawing another similar one.

I'm retelling his story using Geometer's Sketchpad.

Step 1: Draw an arc of radius 3 units.

Step 2: Draw an arc of radius 4 units.

Step 3: Draw the line y = -3.

Step 4: Draw a perpendicular line (pink) to the red line where it intersects blue arc.

Step 5: Sharing this final step right now with you makes me tear up.

Tomorrow Gabe will get to share his way of finding sqrt(7) on the number line with his class.

Then maybe I'll ask the class to try finding sqrt(11), sqrt(10), sqrt(bring-it-on-Mrs.Win-we-got-this). Maybe they'll have fun doing this.

I'd posed a question that wasn't lined up with the curriculum. It just came to me as I was watching my kids do math. But the question lined up with my gut, and my heart went along and said, Do this because it'll remind you of why you love this job.