Category: Math 7

Circles and Mindmaps

5/30/2015

Alana Gilliam @MrsAGilliam tweeted this great mindmap...

My students were just finishing our circle explorations at the time, and I haven't been using mindmaps this year, so I asked what they could read from this one.

1) List all the things about circles that you can find in this picture.

2) Is there anything missing that should be in there?

3) Is there anything that you don't understand?

A chord was only thing they thought was missing. And some of them didn't understand the circles filled in with yellow and surrounded by yellow, but others did. And they were very excited to realize how the colors helped (and they noticed the one spot where the colors didn't quite match. We figured that was when the idea started.)

I have tucked this away for another discussion next year. Thanks Alana.

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An Alternate Way to Measure Circumference and Find Pi

5/24/2015

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I can't remember when I first saw this... perhaps at a California Math Council conference session. As an alternative to wrapping string around a circle or a shape, students can draw a line on a big sheet of paper and roll objects along it, carefully marking the start and end points. Then they can mark out the diameter along the line. With the measurements in hand, you can calculate and compare the ratios, and/or mark out three diameters along the circumference for a great visual of "pi is 3 and a little bit more."

I did this as a Pi Day activity, and my students clearly benefited from revisiting circumference with strings again later in the year.

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Area of a Circle Redux

5/24/2015

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My 7th graders had already been introduced to the circumference of a circle and the area of a circle in 6th grade. By me. But we all know how these things go, so I did lesson with them on the circumference of a circle on Pi Day, and this week reviewed the area formula with a shortened version of Fawn Nguyen's area of a circle lesson. We had also looked at the volume and surface area of a rectangular prism using nets that they cut out and assembled. They seemed to have a solid idea of why multiplying the area of the base times the height worked:

"Because you have to figure out how much in one section and then multiply by the number of sections."
"It's the top layer times how many layers."

So far so good. Next I filled a coffee mug with cubes and asked them to estimate how many cubes in the cup, without touching the cubes inside. 5 minute exercise: count the cubes in the top layer, stack cubes up next to the cup, multiply... we dumped out the cubes and counted. The student who got the closest even thought to take one layer off to account for the thickness of the bottom of the mug. So I figured we were all set for tomorrow's task.

I set out 12 cylinders - cans, jars, salt boxes - and grouped them into threes and fours. In each group, I asked them to decide which cylinder was biggest by volume. We had an interesting discussion about how extra height might cancel out extra weight and vice versa. Their thinking seemed to be largely linear:
"This can is just a little bit wider, but the skinny one is way taller."
[Turning the tuna can sideways and measuring it against the soda can] "I am trying to picture how many times this will go into that one."

Then I asked them how they might find the volume of the cans so that we could compare them. Now I will be honest. My goal for this activity is not deeply conceptual. I want them to practice using the volume formula for a cylinder, and to care enough to want to be accurate and to be willing to compare their work if two students disagree. We already get the layers of a cylinder or prism, area of the base times the height, right?

Blank stares.
Me:    "How did you guys get the number of cubes in the coffee cup yesterday?"
Them:    "Oh, we counted the number of cubes in one layer and multiplied by the height."
Me:    "So how could we use that here?" Nothing.
Me:    "What were you finding when you counted the number of cubes in the bottom of the cup."
Them:    "How many cubes were in one level of cubes."
Me:    "And what is that?"    Blank faces again.

I am afraid I was not wise enough to abandon ship and plan to circle around the next day. I dragged the thing out another painful five minutes or so, AND THEN abandoned ship and tried to regroup.

There was a disconnect between the cubes on the bottom of the coffee cup and the actual area of the circle at the base. They knew that the area of the circle would be different than the number of cubes because of the spaces between the cubes, so that seemed to prevent them from using one to substitute for the other, which makes perfect sense when I think about it their way.

The next day, I handed out this sheet of paper:

I asked them to find the radius and diameter of one of the circles. What kind of units do we measure that in?

Next I asked how they might measure the circumference of the circle. The formula was rusty from the earlier lesson, and I realized some students almost equated circumference with area, since both "had an inside." I also asked them what kind of units we use to measure circumference. It was almost split between "units" and "square units", with a few "don't know"s. As soon as

someone suggested string I passed the string around. As they found the distances, I asked them to compare the distance to the one the formula gave them, and they seemed to be comfortable with the two as related. We also talked about whether those were centimeters or square centimeters, and a number of students were truly surprised to realized we were not measuring area. But it seemed to make sense.

Next I asked them to arrange as many cubes as they could in one of the blank circles and count them. We collected numbers and then I asked them to estimate the number of squares in the circle with the grid. Would that be more or less or equal to the number of cubes they just counted? Why? [I realize now I first could have asked them to tell me what they thought the number of cubes told me about the circle, and what units that number represented.]

"The squares will be more because there are no spaces in-between." Pretty much unanimous.
What are we measuring about the circle when we count the squares? "Area." Okay good, this seems to be working. What value for the area does the formula give you? They calculate that and it seems natural to everyone that we got a number that is close.

Did I fix it? The class before lunch seemed to have lots of aha's, the one after lunch not so much. In fact, they way over-counted the partial squares by taking every sliver, no matter the size, as half a square. "We're tired." But when I ask again how to calculate the volume of the cylinders, many more students are on board.

Overall, I am pleased. I think the activity clarified some misconceptions, and this experience is a reminder to me that there can be gaps in the students' understanding that are invisible to me unless I ask about their thinking. This is a lesson that I have traditionally tried to simply illustrate and explain very well, then have them practice the formula. Did it work? When they got started actually measuring and calculating, most of them could do it on their own. Next week we get to see which ones are larger. I hope they still care. Next year it will be smoother.

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Sequences

5/21/2015

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I have been using patterns like the ones on Fawn Nguyen's site to help students explore functions. I've learned a lot from the geometric grouping strategies that Fawn and her students use, and I've had some great conversations with my own students this year. The students can see the pattern growing, can work with it in a way that makes sense to them, and they learn to express those relationships using algebra. They are a great way to learn about the role functions play, and where equations come from.

I wanted to be able to transfer that understanding to functions in table form. It seems to be a similar kind of thinking, but every year I get a significant fraction of the class who would come to this conclusion whenever they work on their own:

And it tended to be a very sticky misconception. I always ended up feeling like I was just telling them what to do, and they were not seeing any reason behind it.

I realized that when I look at this table, I see a sequence in the y values that continues in both directions, and I can imagine other values that are part of that sequence and how to get them. That seemed to be something that the students didn't see or know how to reason with.

So I started to formulate some questions about sequences, asking them to make predictions about specific terms in sequences, and discussing strategies that were useful and why.

I introduced the word "term" and we began with simple questions like:

What is the next term in the sequence?
What is the term in the sequence above the arrow? How did you find it?
Is there a shorter way that you could find it?
What would be the 20th term in the sequence? How did you find it?

I was amazed at how quickly the students picked up on the idea, and were able to do their own reasoning and explaining about finding missing terms. I wanted them to be able to reason backwards as well, and to think about gaps in the sequence:

Here is my recording of some student thinking:

We extended the concept by numbering the terms, introducing larger gaps, and gradually moving to a table format. All along the way, I also included sequences that were decreasing as well, and ones that involved negative numbers:

In discussing how to find the larger numbers, and searching for more generalized methods, we were able to talk about how the number of the term (I was not always strict with using the word "index") was related to the value of the term. They didn't come up with the idea of a zeroth term on their own, but when I did introduce it, it made a lot of sense, and they immediately saw the value.
Here is some student thinking as we were beginning to write functions for the sequences:

While we were exploring sequences, I also introduced in parallel the idea that a function could generate a sequence:

In the end, of course there were still students who got confused when we were working with vertical tables and graphing points, but the sequences were the go-to tool for discussions at that point, and brought a lot of "aha"s much more quickly than in previous years. I will definitely be keeping sequences as part of my math talks next year.

Below are some of the worksheets I used this year, due for some reworking this summer.

Questions I still have:
1) Does it matter if the indices (0,1,2,3,4,5,..) are above the sequence or below it? Does one or the other help the transition to vertical tables?
2) How can I connect the students' understanding here to the continuity of a function, and the line on a graph?
3) How can I better relate this way of looking at a table (as a sequence with indices) to looking at a table as a set of points?

02_06__7_sequences_for_lesson.doc
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02_06_w6a__7_sequences 1.doc
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02_06_w6b__7_sequences_2.doc
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02_06_w6c__7_sequences_3.doc
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02_06_w6d_7_sequences_4.docx
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Circles and Mindmaps

An Alternate Way to Measure Circumference and Find Pi

Area of a Circle Redux

Sequences

Archives

Categories

All original content copyright Doug McKenzie, 2014-2021

Circles and Mindmaps

An Alternate Way to Measure Circumference and Find Pi

Area of a Circle Redux

Sequences

﻿﻿Archives﻿﻿

Categories

All original content copyright Doug McKenzie, 2014-2021

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