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.

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.

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!

TI-84 Guides for Univariate and Bivariate Statistics

Here are some guides I made for my students to help them remember how to use the TI-84 Plus CE (though I think it’s pretty much the same for all TI-84s) to calculate statistics and create graphs for univariate and bivariate data. They’re designed for interactive notebooks, but I’m sure they’d still be useful for teachers who don’t use INBs.

1-Var Page 1: Entering Data and Calculating Statistics

1-Var Page 2: Histograms

1-Var Page 3: Boxplots

2-Var Page 1: Entering Data and Calculating

2-Var Page 2: Scatterplots

2-Var Page 3: Regression

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.

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.

Linear Regression Intro Activity

Today in Statistics we started discussing linear regression. Before getting into the details of how it works, I wanted to help my students understand what we are trying to use it to achieve. That is, creating a linear equation that models bi-variate data.

To start with, I posted this data in Desmos on my smart board:

There’s nothing special about the data, they were just the numbers I happened to type into Desmos at the time. If I was going to do this again in the future, I think I’d want to source some real-world data to use. But as this activity didn’t have a whole lot of preparation go into it, there wasn’t a whole lot of opportunity to find that data. (That said, if you’d like to use my artificially created data, go for it.)

I had students create their own scatter plots for the data. Once they had done this, I told them to rule a line through the data that they thought summarized and modeled the data as well as possible. I informed them that there is an objective way to determine which equation does the best job of this, so it was now a competition.

Then the moment that revealed to us just how rusty some of their algebra skills are: I had them each find the equation of their line. After they complained that it’s been too long since Algebra 1 (despite most of them seeing this in Algebra 2 or Geometry with me last year), and a quick recap tutorial on slope-intercept form, they were able to find their equations.

Then to compare them, I typed their equations into Desmos so that we could visually compare them. I’m happy to say that most of the equations fit the data reasonably well, at least to the naked eye (which is, of course, all my students had to work with for the activity.) Then, I added one more line: the regression equation created by Desmos itself (in orange.)

One of the lines (the blue dashed one) is actually very close! I also changed all of my students’ equations into regression equations, so we could compare their R2 values. For now, I just told them that this is a measure of how well the model fits the data. In future lessons, I will explain the more formal meaning to them.

To finish our discussion, I had Desmos plot the residuals for the linear regression equation, as well as for some of the students’ equations. I explained that what Desmos was doing was trying to make these residuals as close to zero as possible. Over the next few days, we’ll start to get into the details of how the mechanisms of regression actually work. But for this lesson, I wanted to give students a sense of what regression is about, rather than how it works.

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.)

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.

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

Absolute Value Function

Square Root Function

Cubic Function

Cube Root Function

Rational Functions

Exponential Function

Logarithmic Function

Graphing Examples

Functions INB Page

I’ve decided to make functions the key focus of the first few units of Algebra 2 this year. I mentioned this in my post about my SBG skill lists:

This year, the start of the [Algebra 2] course is going to
focus on functions, transformations and inverses. For quadratics that means only dealing with vertex form, as well as showing the relationship to square roots. We’ll cover all the functions that way, before coming back to all the other algebra we missed along the way. So we’ll cover quadratics again, including all the factoring and solving stuff.

With that in mind, I thought I’d better make sure my students understand what a function is.

This is the front, basically just explaining what a function is. The examples on the bottom section were actually done after the notes inside. You could really use any examples you wanted. I chose these ones because this first unit focuses on absolute value functions, and I figured they should at least be able to evaluate quadratics, even if they don’t know anything else about them yet.

This is the part I’m most proud of. I think it lets students see clearly what the difference relation types are. I know it’s traditional to talk about the vertical and horizontal tests in this situation (and I did mention them briefly), but I decided to go back to the basic definitions of one-to-one, etc. and let students develop an understanding from those definitions. I was glad that by the time we did the last few diagrams, students were suggesting what the diagrams should be without any prompting.

Though they sometimes have to be prompted with “does each output have exactly one input,” most students have been good at recognizing the difference between one-to-one and many-to-one, which is critical as inverses are coming up soon. They often get many-to-one and one-to-many mixed up. They know which is which, they just get the names confused. I had a student say at one point, “So, it’s the number it goes from to the number it goes to, right?” Then after realizing what she’d just said, added “Oh, I guess that makes sense.” So, at least one of them has made that connection!

I know that it’s typical to make a mistake as a teacher and just claim it was to “see if you’re paying attention,” but in this case, having to cross out the bit that said “function” for one-to-many and many-to-many really was deliberate. (I promise I’m not just saying that!) Making students have to edit the page like this was an attempt to drive the point home that these are not functions. Thankfully, some of my students did notice that these sections contradicted the notes on the front of the foldable, and suggested it needed changing before I did.

Files are here.