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.


Analyzing Polynomial Graphs

Here’s an INB page I created to introduce students to analyzing graphs of polynomials.

Each graph is repeated three times, so we can (literally) highlight different aspects of it. Luckily for me, Sarah is amazing at acquiring classroom supplies, so I have a lot of highlighters for students to use.

The first part was identifying the x-intercepts and the nature of each of the intercepts. I had students highlight the curve around each intercept, to emphasize whether they are simple intercepts, vertices (local minima or maxima) or inflection points.

As students did this, I tried to direct the conversation to figuring out why particular polynomials led to particular types of intercepts. This was actually really easy, as the class were asking and answering these questions without much prompting from me at all.

Next, we found the intervals for which each polynomial is positive, and for which they are negative. Having students visually represent the sections which are positive and negative really helped them in identifying those intervals.

I just (as in, while I’m writing this post) had an additional  idea to help with this part. If I’d given each student a card, they could place the edge of it along the x-axis so that only the positive parts of the graph were showing. They’d highlight those parts of the curve, then flip it over so they could highlight the negative parts of the curve.

Finally, they highlighted the sections which were increasing, and the sections which were decreasing. To find each local maximum and minimum, I just had to quickly teach them some differential calculus…

… just kidding. We used Desmos.

Joking aside, I do like using topics like this to start hinting at the math that students may be seeing in the future. I was able to explain that a big part of calculus is looking at the rate and direction of functions, with a particular focus on where functions are neither increasing or decreasing.

If you’d like these notes, downloads are available here.

I used http://www.graphfree.com/ to make the graphs. I know I’ve made my own graph sketching tool before, but it’s really only capable of parent functions and simple transformations of them, so GraphFree was exactly the tool I needed this time. (To be honest, the main reason I’m mentioning GraphFree here is I’d forgotten what GraphFree is called when trying to find GraphFree the other day, so I want to remember that GraphFree is called GraphFree. GraphFree.)

Following this, we did further practice using section 6.4 of the practice book I’m working on. Follow that link if you’d like to get those practice questions yourself – for free!

 


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.


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.


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.


My most used notebook template this year

Over the last summer I completely rethought my Algebra 2 course. Part of this is my focus on parent functions through the first part of the year, giving students a solid understanding of each function and their transformations.

To help focus on the fundamental properties of each function, we used the following template each time we introduced a new function.

Importantly, I had the students figure out details as a class. After stating the rule for the function, we always filled in the two-sided number line, with inputs on the top and outputs on the bottom. I chose to use a number line instead of a table, as it allows me to point out the continuous nature of the values between each mark on the line.

Then we filled in the domain and range, examining the inputs and outputs to determine these. We also determined if the function is one-to-one or many-to-one. I’m really proud of how my students have become increasingly confident in determining these answers for themselves from their own understanding of the functions and their values.

Next, we plotted the function on the grid. The number line is deliberately aligned with this grid to help students make the connection between the two. I have a SmartBoard template set up with points along the x-axis, with which we move points up or down to plot the function, to emphasize how the graph demonstrates the connection between input and output values. Then knowing the shape of the graph allowed us to easily fill in the rest of the table.

The inverse function section depended on which function we were talking. Sometimes we filled it in immediately, as most students understood x² and √x as inverses. Other times we waited to fill it in, such as with exponential functions which was completed before we talked about logarithms.

The second part of this template are the two “Graphing Example” section inside.

In the past, I’ve found students resistant to showing all the algebra they needed when they sketch a graph (usually because all they wanted to do was copy what their calculator showed.) I wouldn’t say this template has completely changed that, but it has made a big difference. Students complained a bit at the start of the year, but they’ve learned to appreciate the guidance this provides and gained a lot of confidence in their graphing ability. I know students probably don’t need to find the transformations of the parent function for every graph they sketch, but I think having them do this for each question we practice has helped their understanding of why each function produces the graph it does, and has helped serve as a check for the other parts of the template.

We’ve often had to leave the x-intercept section blank, because we’ve started graphing each type of function before we looked at solving equations involving that function. This has actually worked out pretty well. I found students accept my explanation that “We can’t do this yet because I haven’t taught you how.” Then, when we come solving those equations (typically the very next skill), I can use the need to find x-intercepts as a motivation for practicing solving equations. Then we go back to our graphing examples, find the x-intercepts, and add that detail to the graph. My students are sometimes annoyed that we jump backwards in our notes sometimes, but I think they appreciate that I’ve tried to avoid overwhelming them with too many details they don’t need to see all at once.

That’s been my approach all of this year: the idea that students don’t need to see all the detail until they’re ready for it. For instance, while we’ve talked about quadratic functions, we’ve only dealt with the vertex form, as that’s the form that can be explained through transformations, fitting the function pattern we’ve been following. Yes, we still need to talk about factoring, distributing and all that fun stuff. But now I feel we have a structure to build everything else on. I’m actually looking forward to completing the square this year, as I have a really useful motivation for it: it allows us to put quadratic functions into the form my students are already very familiar with.

We’ve introduced our last parent function now, so I’m not going to get any more use out this template this year. To be honest, I felt a little sad when we finished our last one, because it’s worked so well this year. Also, it means I’ll actually have to produce notes for each lesson now, instead of using the same ones over and over again…

You can find PDF and Publisher files here.
Included is a second version that leaves out the parent function template for a third graphing example.

Below you can find all the parent functions from my notebook for this year, as well as a couple of the graphing examples pages.

Linear Function
1

Absolute Value Function
2

Quadratic Function
3

Square Root Function
4

Cubic Function
5

Cube Root Function
6

Rational Functions
7

8

Exponential Function
9

Logarithmic Function
10

Graphing Examples
11