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Area of a Circle

Whether it’s polygons in geometry, or under a curve in calculus, I have a favorite way to explore area in class: cut up shapes made of paper and glue them back together in a new way. This time, I’m applying the idea to visually prove the formula for the area of a circle.

If it isn’t clear, the parallelogram-ish shape was originally the same as the circle at the top of the page, but its sectors have been cut apart and glued into the alternating up-and-down pattern that’s shown.

I printed eight circles to a page, and cut them into four sections so that each student could have two of them. There have been past years where I’ve had students draw and cut out their own circles to do this activity, which has the nice side effect of showing that students with different sized circles still get the same result. However, I decided this year that it was more important to get students into the activity quickly, so I gave them the template to use. Also, the sized circles I used (the radius is 1.35 in) seem ideal for fitting in a composition notebook.

You can download files for the circles here.

A few pointers on how to approach this lesson:

  • I wrote clear instructions on my board to only cut apart one circle. If you don’t emphasize this, you’re likely to have a kid who has to glue an extra circle back together unnecessarily.
  • Encourage students to get on with the task quickly. I find that while cutting and pasting shouldn’t take very long, students can drag things out if given the opportunity, and sometimes feel like they’re doing work even if the pair of scissors in their hand. Even the students who showed up five minutes early were told to get going the moment they entered the room, which set the tone for the following students.
  • Also encourage students to be precise with their gluing. Some of my kids had strange looking shapes that either curved down the page, or had large gaps between each sector and couldn’t fit the whole thing on their page.
  • Students will finish the gluing part of of their notes at very different times. I was prepared by having the notes finished in my notebook, so students could copy them as they needed. I also allowed them to take a picture of the notes on their phone to copy from, so they wouldn’t have to wait for another student to finish. Also, because students were finishing at different times, I had other work for students to go on with when they were done with the notes too.
  • I waited until most students had finished gluing before we discussed the meaning of the activity. I tried to prompt the students themselves to recognize what’s going on here so they could explain their understanding to the rest of the class; this worked to varying degrees in my different classes.

These notes include the formula for the area of a sector, but our justification of it is not included on this page. This post from a few years ago outlines how I like to introduce that concept.

 
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Perimeters of Polygons INB Page

I mentioned in my last post that students sometimes describe perimeter as the “outside of a shape” (as opposed to area being the “inside”.) Though it’s an easy answer to give, and an easy answer for a teacher to accept as being correct, I wanted to put my students’ understanding of perimeter on a more sure footing. I wanted to give students a clear definition, showing that when we use the perimeter, we’re talking about a length.

At the same time, the math involved in the perimeter of polygons is pretty simple: its the sum of the lengths of the sides. There’s a reason it’s introduced at third grade (in the Oklahoma standards, at least.) But while adding a set of numbers is pretty easy for a high school student, I wanted to layer in more challenge.

My students need to figure out the missing side lengths of the polygon before they can make that simple calculation. Or, they’ll get the perimeter and have to figure out something else. Given this unit has already covered theorems related to quadrilaterals, and the previous unit was on the Pythagorean theorem, special right triangles and trigonometry, I had lots of options of for how to make students determine the information they need.

Also, students complain all the time about word problems, which tells me they probably need more exposure to and practice with word problems. So I gave my students ten problems, for which they had to draw a diagram (as I hadn’t given them one) and show all their mathematical working.

Downloads are available here.

 
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Areas of Polygons Cut and Paste Activity

One of the biggest challenges in teaching math is allowing students to understand things that are abstract. For instance consider this possible definition for area: “A measure of the two-dimensional space within or occupied by a plane figure or region.” While this seem perfectly serviceable to a math teacher, students may struggle to conceptualize exactly what it means. What exactly does two-dimensional space mean? What does it mean to be “within” a figure or to “occupy” space?

These seem simple, but there are subtleties to understanding what area really is, which I’ve even seen calculus students mess up with. If you ask students what area is, they often respond with something like “area is the inside, and perimeter is the outside,” but then can’t elaborate on that. The common confusion between concepts of length and area is demonstrated by students frequently stating the area of a shape in length units.

This is why I like using the cutting and pasting of paper to represent area. I find students can conceptualize the amount of paper used to make a shape much easier than the abstract idea of “area”, even though they are fundamentally the same thing. Students can understand that one shape has more area than another using the fact that it took more paper to make. Also, if you can demonstrate that two shapes can be constructed from the exact same amount of paper, students can understand that they have equivalent areas.

This activity uses the area of a rectangle to find the areas of other polygons. Students are given a page with templates for the shapes they’re going to create, and a colored half sheet of rectangles and octagons. (The colored half sheet is actually double the number of shapes they need, but it makes things easier to have spares if they make mistakes.)

Everyone also needs a glue stick, a pair of scissors, a ruler and a pencil.

The white sheet can be cut into the six separate sections. On the back of each section, there are instructions on how to construct the given shape from a rectangle.

Students should follow the instructions to construct each shape from one of the small colored rectangles. Reactions will range from “this is easy”, to “I think I’ve got it, can I check what you did”, to “this is impossible!” Make sure you’re cycling around the room a lot, because students can and will fall behind quickly if they’re not paying attention. Frustratingly, I found most of the students who thought the instructions didn’t make sense hadn’t actually taken the time to read them step-by-step.

I suggest doing them in the order of rectangle, parallelogram, triangle, trapezoid, kite and polygon. Once they’re done, they’ll look something like this:

The idea is that each shape has an area equivalent to a rectangle in some way, because they used the same amount of paper. The triangle and the kite are related by a half, because one rectangle was able to make two of these shapes. The regular polygon is a little different, as it starts with the polygon and is deconstructed into half a rectangle; we know it’s half, as we already showed that triangles make the area a half. Add some labels to the shapes, and you get something like this:

In the trapezoid, “m” is referring to the midsegment (or median) of the trapezoid. We’ve already covered the the theorem that says the midsegment length is the mean of the base length, which is why we’re able to substitute (b1 + b2)/2 for m without any further working.

If you’d like to do this, files can be downloaded here. Included are alternate versions for non-Americans who think “trapezoid” is a weird word. Font is Matiz.

 

 
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Quadrilaterals Card Sort

Here’s a card sort I created for the definitions of the special quadrilaterals.

Each group contains four cards: the name of the quadrilateral, a diagram, the definition, and a list of the other shapes quadrilateral is an example of. The diagram shows only the information that’s stated in the definition. Other properties of the quadrilaterals are covered in the following skills (some of which I’ve already blogged about.)

Downloads are here.

 
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Parallelogram and Rhombus Theorems

First day back from Christmas break saw my Geometry classes looking at theorems about parallelograms and rhombuses. We’d already looked at definitions of the different types of special quadrilaterals. I had students divide a page in their notebook in two, and told them to rewrite the definitions of the parallelogram and rhombus in those sections.

While they were doing that, I passed out a set of four Exploragons to each student, with two each of two different colors/lengths. I also made sure that each pair of students received the same colors, which will be important later.

If you haven’t used Exploragons before, they’re plastic sticks with little nubs that allow the sticks to snap together to make different geometric arrangements. Other companies sell them as AngLegs, though I think prefer Exploragons as they have nubs in the middle of the sticks, not just at the ends. When I started teaching at Drumright, I had the opportunity to order hands-on supplies to use. I’ve found that of everything I’ve ordered, these are the most versatile physical tool I have for teaching Geometry.

I gave students the instruction to construct a parallelogram from the pieces I gave them. Thankfully, they (mostly) ended up with something like these:

I then instructed them to write down everything they noticed about their shapes, and to discuss what they notice with the students around them. Answers ranged from what I was hoping they’d notice (opposite angles are the same, opposite sides are the same length) to not as useful (“it’s a shape”), but getting the perfect answer wasn’t really the point. I wanted students to understand that there are things about these quadrilaterals we can know are true aside from just their definitions.

Next, I told students to do the same thing by making a rhombus. Thankfully, they realized I didn’t have the right pieces to do this A few looked at me incredulously, a few demanded I give them more pieces (which I refused), but slowly a few students worked out what they needed to do: trade pieces with the person next to them.

Once students had had time to write down their observations of their rhombus, we started our notes summarizing the theorems for these quadrilaterals. I used the observations as a springboard into this conversation, pointing out that some of the theorems matched what they’d noticed, and some didn’t (particularly the ones involving diagonals.)

After, students started the activity I put inside the notes. For each diagram they needed to identify four things:

  1. What the shape is (admittedly not too difficult, as there’s only two to choose from.)
  2. How they know it’s that shape, based on either one of the theorems or the definition of the shape.
  3. The value of any variables in the diagram.
  4. How they know it’s that value, again by referencing a theorem or definition.

There is a flaw in these questions. All of the parallelograms have a horizontal pair of sides, while the rhombuses are in a “diamond” position. This made distinguishing the two a little too easy. If I get a chance, I’d like to rotate some of the diagrams to different angles.

Downloads for these notes can be found here.