## Geometry: Which Pasta Does Mueller’s Want You to Buy?

So, I’m walking through Publix about a week or two ago, and I noticed this:

Mueller’s makes both regular length and pot-sized pasta.  Since I’m giving a presentation next week involving problem based instruction in the geometry classroom, I saw this as an opportunity excuse to develop a new geometry lesson (I really miss teaching geometry sometimes).  So, I started talking to my co-presenter and asked her to help me develop something useful for us to present with.

My co-presenter is a first year teacher and I quickly found out that she’s not very familiar or comfortable with problem based instruction.  Since she’s my mentee, I thought it would be a good idea to develop a lesson with her and then team teach in her classroom.

So we started with the question: Which would Mueller’s rather you purchase?

Her student’s sat there and started at me for a few moments.  I encouraged them to talk with each other and figure out how they could answer the question.  Two minutes later, we discussed what they wanted to compare and what questions they had for us:

They were insightful.  The only thing they didn’t ask about (that I was expecting) was a question about shipping.  My mentee and I quickly steered the conversation toward packaging costs.  So, her students wanted some measurements:

I made up the cardboard cost…maybe I can get Mueller’s to give me that info.

Given that this lesson was given during a point in the curriculum that has nothing to do with surface area, they did okay with calculations.  I was surprised to see that 3 or 4 out of the 8 groups thought that volume was the appropriate measurement to use (rather than going straight to surface area).  After some quick conversations, most were right on track.  Ultimately, the most difficult calculation was converting the cost from square inches to square centimeters.

The calculations bear out that the pot sized pasta is cheaper to package.

I’m encouraged that my mentee is interested in continuing to create lessons like these.  I’m taken back a bit that this isn’t the norm in most classrooms.  I have to find a way to keep encouraging my colleagues to keep bettering themselves and their students.

## Summer Lesson Building (Geometry: Distance and Midpoint)

One of the first lessons this coming year in Geometry will be a “review” of distance and midpoint. I’ve spent the last couple of hours trying to figure out the best way to conceptually work with both of these topics and still keep the lesson in the neighborhood of 50 minutes. I’m stuck somewhere between treating this as a review of material I know my students have seen in the past and going full-on inquiry based learning. I’m leaning towards expecting them to need a reminder about from where distance and midpoint formulas come. I am using this lesson to create situations where there is a need for distance and midpoint.

I need to make sure I have my students focus on distance and midpoint on the coordinate plane. To me, the first step is create a need for the coordinate plane. I’ll open with lesson with a subway map of New York City:

Then, I’ll focus my students to a question:

I want them to guess. I want an argument based on what they think they know. I want them to say “it looks like…” and “what’s the scale?” Then, I hope someone finally asks for a way to determine the scaling (I’ll bet they won’t use the term “scale” but whatever works).

Now they can use the length of a block as the unit of measure. But the streets and avenues aren’t equally spaced! I guess they’ll have to come up with a conversion! I expect some students to remember distance, or that there’s a formula for it. That’s not the purpose of the map. I want them to see how much easier it is to estimate with a grid. Now we have a reason for the coordinate plane!

So we’ll start working on a distance formula:

We’ll work with the Pythagorean Theorem, figure out the process, add in the general case, and derive the distance formula. Then we’ll practice in a non-contextualized situation:

Then I’ll give them a “real life” situation that requires them to ask for a coordinate plane:

For part two of the lesson, we’ll focus on midpoint. I’ll give my student the following four graphs:

The only direction I’ll give them is to find the midpoint of each of the line segments. Their goal is to define the midpoint for themselves, then come up with some way to find it:

We’ll come up with a class consensus for a definition and a formula for the midpoint.

Then we’ll practice:

Overall, I’m not thrilled with this lesson. I think it will do its job, but I’ll keep searching for a better approach. Also, I’m expecting this lesson to last more than one class period. I really don’t see how I can expect to finish this in less than a day and half without losing the student centered approach. Thought?

## Summer Lesson Building (Geometry: Surface Area and Volume Ratios)

I’ve been looking for a way to bring up surface area and volume ratios that isn’t a forced “here’s some non-contextual geometric solids to calculate arbitrary values for linear, area and volume measurements.” After looking through some resources and reading some blogs, I found an idea for introducing them: Orbeez!

If you don’t know what Orbeez are, go here: http://www.orbeezone.com/

I’m going to have my students explore their claim…or rather figure out what their claim is:

They’ll have to figure out what’s blacked out. They’ll ask for some measurements, so I had the chance to buy some Orbeez and play with them!

Orbeez’s claim is about how much the volume grows. I wanted to leave that little tidbit of information missing. My hope is that the students start with the diameter grows approximately 4.8 times larger. Next, I hope they go to where the surface area grows approximately 23 times larger. Finally, I hope they figure out that the volume grows by a factor of around 109. We’ll come to some conclusion of a class as to which one is the best to help sell the product and they’ll see:

This isn’t meant to be stand alone as a surface area and volume ratio lesson. It’s going to give us a starting point to look at the relationship between the values (the 4.8, 23 and 109). Hopefully they’ll infer what the relationship is and we’ll finish the lesson.

## Summer Lesson Building (Geometry: Naming Basic Geometric Figures)

So, I put in for a presentation at the Florida Lab School Drive-in Conference and managed to get accepted. My session title is “Creating Need in the Mathematics Classroom” and is centered on an approach I started to focus on last year in my curriculum. I’ll give more details as I really dig into making my presentation. But in order to help my presentation, I need to start building lessons that lend themselves to helping create my presentation. So here we go:

Although it’s technically not a standard, the first topic we’ll discuss in geometry is how to name basic geometric figures. My students need to be proficient in naming points, lines, rays, line segments, angles, etc.; otherwise, they won’t succeed in Geometry.

So I figure there’s two options in how to present this to students.
Option 1: I can tell my students how to name figures. If they ask my why, I’ll tell them because I said so…or someone said so some time ago. I can hear my students falling asleep already. I can hear them saying “who cares?” I can see them struggling to remember how to name geometric figures. I need a better option.

Option 2: I can create a need for naming geometric figures. Here’s how:

I’ll ask one student to pick a point and not tell anyone which one they chose. I’ll then ask them to describe which point they chose to someone else on the other side of the room without using any physical cues. My hope is that this will be a giant pain in the rear for my students.

Then I’ll show them this slide:

I’ll ask them to repeat the process. I’m betting they’ll have an easier time.

To really emphasize my point (and to practice the whole writing and communicating thing), I’m going to give them a partner project:

One person in the partnership will get this:

They’ll get three minutes to write a set of instructions to have their partner recreate this picture. Once the instructions are made, their partner will have to use this:

and recreate the picture.

My hope is that they’ll fail miserably, and this activity will create a need for them to find a better way to quickly and efficiently name geometric figures.

Then, they’ll have to describe which one they picked to a fellow student.

They’ll decide this is easier. I’ve purposely set this up to where they’ll need two points to describe which line they chose. Once they’ve made this decision (and only when they’ve decided how to name the line), I’ll help them create that definition:

By now, I think my students will have gotten the point. We’ll discuss how to extend this idea to line segments and rays. To finish, we’ll go back to this activity:

I’ll have one student write a set of instructions and their partner will recreate the picture.

My hope is that this lesson design will create a need for naming geometric figures. The way we name figures is not arbitrary, and hopefully my students will discover that.

## How Far to the Horizon?

In geometry, the focus is on circles. We have finished a unit consisting of parts of circles, arc length, angles in a circle and segments in a circle. I decided to give them the following problem as a review:

I’ve been to Chicago a few times, and have gone up to the observation deck in the John Hancock building each time.

I originally played with this question in Algebra II and used this to set up a need for solving square root equations (which google was able to give me for the distance to the horizon…and a formula which one of my students found quickly on their phone). This time, in Geometry, I wanted to calculate the distance without any formulas. My students’ first question was how far off of the ground was that picture taken:

We discussed which of those heights was most appropriate and decided to use the height of the observation deck.

This is where I had to help them out a bit. I had to give my students a nudge in the right direction.

They figured they needed to use circles, so they asked for the radius of the Earth.

As we drew on the Earth, my students noticed that my line of sight from the observation deck created a tangent line. We were able to create a right triangle, and use what we know about right triangle trigonometry to calculate the central angle created between the building and my line of sight. Once we had the central angle, we used arc length and calculated the distance to be approximately 39.3 miles.

And here was the answer based on what I was able to find online:

## Mr. Cloud Likes to Run

In geometry, we started a unit on angles and segments in circles. Since we are at the end of the softball season, I decided to use a situation that I’ve encountered to help them figure out how to calculate arc length.

I like to work out and run during the hour I have after school and before needing to be at softball to start pre-game warm ups. In order to save some time, I usually run around the school or on the softball field. So the question arises:

I define a pole as from the left field foul pole to the right field foul pole and back. My students immediately asked for some measurements and some sort of “proof” that the field was circular.

My students quickly figured out that the 90 degree angle made by the foul lines was a quarter of the circle and tied that to the idea that they were using a percentage of the circumference. As a group, we came up with a general formula and then worked some practice problems dealing with arc length.

## Mr. Cloud Makes Dinner Part 2

So yesterday’s and today’s lesson in geometry started with this picture:

The question asked of them was “Will the pot of sauce overflow when the meatballs are added?” My students were quick to break that question down to “how many meatballs will make the pot overflow?” They’ve gotten to the point that they ask for measurements very quickly:

They knew that we’ve been working with volume, but we had not discussed cylinders yet (they were at least calling the pot a cylinder). They asked about how to do the volume of a cylinder and that turned into a 30 minute lesson involing some basic calculus ideas. The bell rang and they didn’t get their answer.

Today’s lesson started with a review of the homework from the previous night. Once they felt grounded with cylinders, we brought back up the measurements. They calculated their volume of the pot. They quickly realized that the meatballs are spheres. On went a lesson about the volume of a sphere. Once they were okay with that, we got some measurments:

I didn’t give them every meatball’s diameter, but they decided that 4.5 cm was a “good enough” estimate (they actually found a use for an average). Then came their work:

The general consensus was that 11 would be okay but meatball #12 would be too many. Tomorrow we get to see if they were right!

## How Long Until Mr. Cloud’s Out of Water?

Today in geometry, I had an empty 2.5 gallon container of water. I raised the question “How long did it take for me to drain the container?”

The first step was for the students to decide what was a reasonable guess:

Once it was time to get to mathematical business, they quickly asked for some measurements. We’ve just started with calculating volume, so they decided for length, width, and height:

There was a quick conversation about the water level at the start of the draining, and a quick conversation about the accuracy of our calculations since the container isn’t a perfect rectangular prism.

Finally, someone asked how fast the water drains out of the container. I couldn’t answer that question for them, but I could give them this:

It took me 13 seconds to fill up the smaller container. My students were content at this time and went about their calculations. Most of the calculations were along these lines:

Seemed reasonable. Most people were in this ballpark. So we got our answer:

This confounded them. They weren’t right. They looked to me for answers. No calculation errors were found. They concluded that the flow rate wasn’t constant and that the amount of water in the container helped determine how fast the water comes out. Huge insight!! Now if only we offered physics!

## Mr. Cloud Cooks Dinner

My wife was going to be late getting home from class and I was on my own for dinner. I was going through the fridge and noticed that it was time for a cleaning. We had some veggies and such that weren’t quite in a condition to eat…so I thought that it was a perfect opportunity to make a geometry lesson on cross sections!

The idea is for students to have to give their best interpretation of what they think the cross section of each cut would be. This lesson isn’t anything too special or ground breaking. Just a nice twist to give my students a real-world look at what cross sections are and how they’re already familiar with them.

## Rumors and The Pentagon

Two short problems from class today:

First, in Algebra II, we started discussing logarithms. Rather than just jumping straight into the definition, I gave my students an example (with a lot of pitfalls) that made a logarithm a necessary tool to use. Here is the situation:

Josh has decided that it would be great not to have to attend school next Wednesday (April 9th). Therefore, he starts a rumor that schools will be closed that day. So, today, he told two of his friends that school would be closed. On the next day, each of these students tells 2 students and on consecutive days, each of the new students tells 2 more students and so on. If there are 1,730 students at Florida High, was started early enough for everyone to have heard it?

A lot of interesting conversation ensued. After a few minutes working with the problem, my students ended up at the following consensus:

They decided that, at most (which took them a while to realize that some students may tell the same friends), 1023 students would hear the rumor, which is not enough. One student pointed out that if Josh had started the rumor one day sooner, everyone would have found at (at least by the numbers). There was great conversation about the pattern for the number of new people that heard the rumor and the pattern for the total number of people that heard the rumor were exponential. Some formulas were derived. Then this happened:

The first question turned into a simple plug and chug in the formula issue. The second question led to some good discussion. My students decided that they needed something new…something they didn’t have…something that would allow them to find a missing exponent…a logarithm!!

Secondly, in geometry, we derived the formula for the area of a regular polygon. The interesting conversation came from an application problem.

I gave them some interesting information about The Pentagon. I gave them the area of the building and the apothem (without telling them that’s what they had) and asked them for the perimeter.

I know, it’s a really straight forward textbook style of question. I was excited, however, to see how interested some of my typically unengaged students were with this question. The really exciting part was when I asked them how long it would take Bethany to walk the perimeter of the E-ring. There wasn’t guessing, there wasn’t assuming…they went straight to: get her up in the front of the room and let’s do some measuring. They decided to use proportions and they decided that they needed to convert their answer to minutes. It’s really wonderful to see how they’re taking ownership in the problems and are driving the conversations. I did very little probing. I was truly facilitating and only offering suggestions. I can’t wait until this model can be fully implemented!

**Notes: I didn’t think up either situation. I found the rumor problem in a textbook and the pentagon idea in another blog. I just took them and made them fit my classroom.**