# Category Archives: geometry

## “Real-Life” Annulus Problem!

I just can’t type “real-life” without quotes because I’m yet to resolve what “real-life” means in a math class.  But for this post it means – “Math someone needed to do at their job”.  And in this post that someone is my wife – who much to her distain and my joy – does a lot of geometry as a project manager at a construction firm.

So here’s what we need to know:  How much cement is needed to make that border?  We need the answer in cubic yards because you buy cement in cubic yards.

The image below provides some of the context for the problem – The cement border is being used to circle an existing tree.

I was actually surprised how open the middle was on this problem.  Yet students used two main strategies.  The first was the standard subtract the areas and multiply by the height, or they subtracted the volumes of both cylinders directly.  The second was the find the perimeter, think of the wall as a rectangle (dimensions of 2*pi*r X 1′), find the area of that rectangle and then multiply by the height.

One student using the second strategy used a radius of 14ft and got a solution of 16.2 cubic yards.  He told me he knew the answer would be a little too small because of only using the inner radius.  Another students used 14.5 and got the same solution as the area subtracting syndicate.

Converting from cubic feet to cubic yards is a great time to practice your perplexed I wonder why you divide by 27? face.

By the way the wall ended up costing around \$35000.  I can’t believe how long I would have to work to put a wall around a tree 🙁

Cheers!

– B

## Quick, Fun, Artistic Geometry Review

Here you go.  It’s a review of the fundamental vocabulary of geometry.  Basically students need to use all of them to draw a picture.  It is a simple idea and it works very well.  When I described it as “quick” in the title I was alluding to the fact that students are quick to understand what they are suppose to do, and they just go for it.  You don’t have to do much from a teacher standpoint – just get out of the way and see what their creative energies do.   It was made by my department chair – who along with everyone else in my department is an amazing, blogless (is that even a word?) teacher.  I suppose that’s where I come in.

The Goods:

Review Drawing Assignment

## This Is Always On My Whiteboard…

…  in geometry, for the first month.

because I can do things like this quickly

And then students get a tone of mileage out of a worksheet like this (thanks Walt!):

And it’s not “always” on my whiteboard, but it’s on there a lot  :0

Right at the beginning of geometry I want to focus on helping students draw conclusions from diagrams and given information.  Plus I have done a lot of programming and have a natural love for the IF THEN statement.

IF this THEN what?  So we got a piece of information – what does it do for us?  What does it tell us?  And since the IF THEN is not on my main whiteboard (my main one is the big one on the right of the picture above), I have no problem leaving up my IF THEN all the time.  And probably most importantly – Having it already on the whiteboard reminds me to use it.

“If angles a and b are complementary THEN a + b = 90”  I am reinforcing the point that all we know is that the two angles add up to 90.  We don’t know where the angles are located, we don’t know if they are adjacent, we don’t even know how big they are, or what color they are…  we just know that they add up to 90 degrees.

Here’s the worksheet above as a pdf:

DrawingConclusions

## Monomial Partners

This is a great activity that was inspired by Matt Vaudrey’s Equation Speed Dating.  In this lesson each student gets to create their own monomial – which I constrained to having to be even and with a variable.  Then they break up their paper into three columns:  Partner / Our Binomial / Our Rectangle.  The students pick a partner and join each others monomials together to create “Our Binomial”.  Then they factor their binomial and represent it as a rectangle by labeling it’s dimensions and indicating the area.  I circulate the room and once it appears every group is finished, I have everyone get up and find a new partner.  I’m demanding here that all students get up out of their seats and move somewhere new.

After a couple rounds I started having them draw their monomial and their partners monomial as separate rectangles, and then draw them together.

I have been focusing on a geometric approach to factoring, so the rectangle column was a great addition to previous times when I have done this activity but only asked for the solution.

The column “Our Binomial” does a nice job reinforcing that a binomial is the combination of two monomials.

Don’t require them to say “what’s your monomial?”, “do you agree that our binomials is….  “, but inspire them to say it by modeling it.  A lot of my students were saying it because I was giving them messages that anytime they get the chance to say “monomial” or “binomial” they need to take it.

Tell the students not to move onto a new partner when they are finished.  They need to wait until you tell them to switch partners.

Remind them that you are really counting on the partners to catch any errors! Because you can’t do the problems on the board since every pair is working a different problem.  “And yes, you are the partner I am counting on for someone else.”

“What’s your name, what’s your monomial?”   No that’s not a pickup line for Speed Dating…  or is it?

## Tangent Line and Circle Problems

I will categorize this post as “sometimes you just need a worksheet”.  #SYJNAW for my twitter peeps.

I have always kind of disliked teaching the circles unit in geometry because of all the different rules – tangent / secant angles, chord-chord sides, chord-chord angles, blah blah.  This year I put together a learning segment on circles that involved satellites in geostationary orbit.  It was based on my experiences working at Lockheed Martin and my engineering background.  I will write about it when I have time.  But for now I will just attach a couple worksheets I made of problems that I put on a homework, or threw in a test. I figured I would just share these, because you know… some times you just need a worksheet.

These problems themselves involve tangents, central angles, and trig functions.  The actual learning unit is very similar, but requires the students to contextualize and decontextualize.  So without further comment – here’s some of the practice problems I used:

The Goods: (sorry I only have pdf’s, I create things with Adobe Illustrator)

SatelliteGenQ1

SatelliteGenQ2

SatelliteGenQ3

## The Centauri Challenge

I’m posting this because my students enjoy it and I can’t find it anywhere online.  I got it from a colleague a few years ago.  I have no idea where it originally came from.  It’s is a great intro to logic and proofs.

CentauriChallenge

## My Day 1 Lessons For 2013

This year I am teaching algebra and geometry again – new school, same subjects.    I have decided to do no introduction or ice breaker activities.  No syllabus on day 1 (which I never have done) either.

Last year I did the straw bridge challenge and I loved it and definitely recommend it.  But I am scrapping it this year due to the time constraints of  focusing on CCSS in a district that is still giving the STAR test.  So I choose these two activities for their more direct relationship to standards I must teach, as well as their low entry point and interesting hooks.

Algebra

Day 1 in algebra is going to be my Getting to Vegas problem, which is simply a personalization of a problem Dan Meyer describes here.  When I was living in Forestville some friends and I decided to go to Vegas.  There are two airports that we could have used – the smaller local airport in Santa Rosa, or the larger airport in San Francisco.  Which airport should I have took, or will I take next time?  I have screen grabs of all relevant information in the slides.

A couple extensions:  How long would the Vegas trip need to be in order for the Santa Rosa airport to be cheaper?  (Eventually the more expensive parking at SFO takes over).  Or Dan’s scenario of taking a shuttle from Santa Rosa to the San Francisco airport vs. driving directly to San Francisco.

The Goods:

GettingToVegas

Geometry:

In geometry I’m starting with Dan’s Taco Cart problem.  I am just going to go to keep it as Dan vs. Ben, because I am a bit intimated on day1 to follow Fawn’s more interactive implementation allowing students to choose their own paths.

This is an exercise with the Pythagorean Theorem, which is great for day 1 because they have all seen it before.

The Goods:

TacoCartWS

TacoCartWS

## Math Council

This is a group activity I created and have a lot of fun using in my classes.  It can be used as a review activity at anytime because all you need to do is create 10 or 12 problems.  It came from a desire to have a collaborative activity where each person in the group had their own distinct role to play.

The Overview

Students form a Math Council in groups of 4.  Each student then gets or chooses a role to play.  Each role has unique things that they are graded on.  The end goal is to create a poster that highlights the problems that the group worked on.  But in the creation of that end product, each member of the council has certain responsibilities.

Each student must do every problem on their own paper.  Then as a group they decide what the right method was and then that gets transferred onto a poster.

There are four roles:  Leader, Scribe, Sage, Runner.  Here are descriptions of each:

The Leader is in charge of making sure the poster finishes, as well as selecting the problems that group works on.  Each group is initially given an envelope of problems and the Leader chooses which ones the group works on.  Typically the envelope will have about 10 problems of which the leader selects 4 or 5 for the group to work on.  Interesting for me to see which ones they choose.

The Scribe is in charge of the poster.  Other members can help work on it, but their grade is most directly tied to the quality of the poster.

The Sage is responsible for coming up with key points for each problem.

The Runner is the only person who can ask me a question.  The Runner must report my answer back to the group.

– I do not answer any of the Runner’s questions near the group.  I make the Runner come to me and report my answer back to the group because I want to make sure that the group must rely on the Runner to explain to them what I’ve said, rather than them simply hearing me say it.

– I give each group an envelope with lots of problems in it, and then have the leader choose 4 or 5 for the group to work on.

– Each student must complete every problem on their own paper.  That is the “work shown” grade.

The Executive Council

This is a fun thing I added to mess with the students help students work on their collaborative ability to adapt to changing circumstances.  I have what I call the “Executive Council” and they call me periodically during the activity and make new demands upon the students.  I pretend to be receiving a phone call and then announce that the Execute Council just contacted me and they now want this or that.  Here are some common things the Execute Council calls for:

– They demand that every council has a name and that name goes on the poster.

– They require a certain problem out of the bunch to be on every poster.

– They require the Sage’s key points to be on every poster.

– They want a picture of a penguin drawn on every poster.

– They give extra credit for their favorite posters.

It’s pretty fun – after awhile when I act like I’m getting a phone call the students will call out “If that’s the Executive Council don’t answer it!”.  They will continuously ask who the Executive Council is – but of course I am not allowed to reveal that.

The Goods:

Runner

Sage

Scribe

## So I Guess I’m Making Videos Now

I had to be absent a couple days last week, and I wanted to the class to learn some new material.  We had just spent two days simplifiying radicals, and I wanted them to move onto adding and subtracting.  So I decided to make a quick video of myself explaining how to do it.