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## Domain and Range Lesson

A few weeks ago, I agreed to teach a lesson on domain and range in a Math 10C class. I told the teacher I’d make it interesting and engaging. Shortly after making that promise, I realized I had no idea at all how to make it interesting and engaging. I did what I always do in those situations. I begged for help on Twitter.

It turns out I wasn’t alone in looking for ways to teach Domain and Range. Marshall Thompson admitted he was also interested in finding something. Don’t worry, Marshall, I’m about to hook you up.

Dan Meyer jumped in and was, as advertised, not all that helpful. And he spelled cheques wrong.

I had given up hope. I was about to plan a typical boring lesson. Then Peter Vandermeulen came through.

Peter’s link was to this file. It’s really all you need. It’s a nice, fun, compelling and engaging way to get at Domain and Range. Peter tells me he got the idea from a workshop in his district. I made some additions and modifications, and I’ll explain the lesson below. I’ll present it how I would do it if I ever did it again. I learned a few things.

Domain and Range Lesson (2 Classes)

Introduction and Hook – Pictionary

Run off the documents below. They contain some blank grids and lots of different types of graphs. Cut them out. The idea is that one student will be given a graph and have to describe it to a partner, who will draw it without looking at it. You make it tougher if you don’t let the describor see what the describee is drawing until the graph is done.

There are two ways you can go from here. Peter’s lesson plan suggests pairing students off and giving them each one graph and one blank. Partner 1 describes his graph, while partner 2 draws. Then they switch. I’d put a time limit of 1 minute on each drawing. When the time is up, they can look at the original and the drawing and see how accurately the drawer was able to replicate the graph based only on the verbal description of the partner.

I tried to make this competitive, like pictionary. I put them in teams of 2 and had them compete against another pair. I photocopied  the completed graphs on card stock and gave each group of 4 the whole set, shuffled and face down. Each student was given the sheet with the blank graphs on it. Then the students took turns pulling the top card, and describing it to their partner. The pair sitting out in a round had to judge and decide if the pair doing the drawing did well enough to earn a point. I tried my best to make sure that the person describing the graph couldn’t see what his partner was drawing. It’s much more challenging that way. We played 12 rounds of 1 minute each, so that each student got to describe 3 times and draw 3 times. The competition was fun, but the noise level got pretty high in the room. Peter’s way might be simpler, quieter,  faster, and every bit as engaging.

Graph Templates

• Word document so you can modify my graphs if you want.
• PDF file in case my graphs look terrible when you open them with your version of Word.

The Lesson

After the game, have a class discussion about what kind of words they were using to describe the graphs to their partners. Students will throw out words like arrow, axis, quadrant, stops, keeps going, points, curves, straight, ends, begins, lowest, highest, farthest right, farthest left and more. Their language leads nicely into domain and range.

Give every student two different coloured pencil crayons for the domain and range lesson. Walk them through several graphs from the game, and show the set notation appropriate to the various types (set of points, between two values, going on forever in one or both directions). All I did was make a quick notebook file with screen shots of some of the graphs from the game. What I tried that was new to me, was using the coloured pencil crayons. I asked students to identify the farthest left and right points, mark them, and then colour the x-axis in that same colour. Then I had them switch colours, find the highest and lowest points, and colour the y-axis in that colour. It really made the domain and range pop out for them.

Formative Assessment

After the brief lesson, give them a short sheet with 3 questions. In a 60 minute class, this will pretty much be an exit slip, which is what I called it. In an 90 minute class, you’ll have time for the next part. Use the exit slip to see who understood the lesson, and who needs more help. As students hand them in, you can sort them pretty quickly. I sort them into three piles – “Got It”, “Mostly Got It”, and “Didn’t Get It”.

Day 2

Practice Time (I use this instead of assigning homework) Group the kids according to how they did on the exit slip. Those in the “Got It” pile are given some higher level questions to practice. Normally, I just pull these right out of the student resource. Those in the “Mostly Got It” pile are given some basic practice questions, as well as some higher level practice questions. Those in the “Didn’t Get It” pile work with me. We will go over some more examples together before I turn them loose.

Closing Activity

Peter’s materials contained a set of cards that I used for a closing activity. Half the cards have domains and ranges on them, and the other half have corresponding graphs on them. I didn’t modify these at all, and used them as-is. Peter’s lesson plan suggests giving each student one card, and having them match up with the person who has the corresponding card.

I went a slightly different way. I copied these cards on coloured card stock and separated the kids into groups of 4-5. I gave them an entire set of cards and had them pair them all off, working as a group.

Closing Activity Materials

My Thoughts

Thanks to Twitter, I think I delivered a way better lesson than I used to do on this topic way back when I was in the classroom. I’d like to try this lesson once more, exactly as described here. It is ready to go. If you try it, let me know how it goes for you.

## Surface Area vs. Volume

At a recent session in Wainwright, one of the participants, Mary Frank, showed me this demo. It’s similar to Dan Meyer’s Popcorn Picker, but I really like the payoff in Act III. Presented in Dan’s three act format, here’s the materials.

Act I – Video

Act II – Information

Have the students make predictions about whether the wider cylinder will overflow, fill right up, or have space left in it. If they want to run calculations, the tubes are simply 8.5 x 11 pieces of overhead paper. One is rolled vertically, and the other is rolled horizontally.

Sequels

• How tall would the skinnier cylinder have to be to completely fill the wider one?
• By what factor are the volumes different? Why?

## RAFT in Math

Several years ago, I attended a session on differentiated instruction with a wonderful woman named Dr. Vera Blake. One of the suggestions she made was that we use the RAFT model as a check for understanding. For those of us more mathematical than Englishal,  RAFT is a writing tool typically used in English classes. RAFT stands for: Role, Audience, Format, and Topic. It helps students focus their writing by clarifying what their role is, who their audience is, what format is appropriate, and what topic needs to be covered. I may have just demonstrated a rudimentary understanding of the process, but I’m a math teacher…

I asked Dr. Blake to tell me how I might use it in a Math class. She showed me how to use it as a review, and a check for understanding. I was wrapping up a quadratics unit with an11th grade class, and she helped me write a set of RAFTs to use as a review with them.  I created as many as I could think of, and had pairs of students randomly select one RAFT.  For example, one pair was given the role of the discriminant. Their audience was a quadratic function, and the format was a letter from a stalker.  The topic was “I know all about you!”  A pair of quiet and shy girls wrote a really creepy letter from the discriminant to the quadratic function. Their letter clearly demonstrated understanding of what the discriminant indicated about the graph of the corresponding function.

The class had a lot of fun with it. We had songs, raps, poems, letters, posters, radio ads and many other things performed in class after one day of preparation. Some other examples included a quadratic formula writing a cover letter to a quadratic equation to apply for a job, a dating ad written by a quadratic formula who was looking for love and understanding, and a workout plan devised by a personal trainer aimed at making a specific quadratic function skinnier. What all of them had in common, was that they showed an understanding of the class material.

There were some bumps. The group that had to write the dating ad had no idea what a dating ad looked like, so they searched personal ads on my computer. That probably wasn’t a good career move. Another group had to design a twelve step process in the manner of AA to solve a problem. They also searched on my computer for addiction programs. Despite it all, I managed to keep my job.

I had also forgotten to consider assessment, so I forced a rubric on their presentations in the end. I should have left it as a formative assessment.

If you are interested in trying one of the two I created, feel free. I’d love to hear from you about how it went.

Quadratic Equations and Functions RAFT Topics – Math 20-1 and Math 20-2

Relations and Functions RAFT Topics – Math 10C

Here are some samples of student work from the 11th grade class on quadratics. They were a little better when seen performed live, in front of the class, but you will get the idea. The girls who wrote the first letter are clearly better students of English than I am.

## China Wedge – Three Acts

While in the Pennsylvania Convention Center for ISTE 2011, I came across this intriguing piece of art wedged under an escalator.

I have since learned that the piece of art is by Mei-Ling Hom, and is called “China Wedge”. It  is composed of many Chinese cups, bowls and spoons wedged into a space under an escalator. It appears that the space is not entirely filled with cups, bowls and spoons, but they go about 2.5 deep all around. The sculpture, which was commissioned for the Center’s Arch Street Concourse, pays homage to Philadelphia Chinatown, which is adjacent to the Convention Center. (Source (Spoiler Alert):  http://philadelphia.about.com/cs/artmuseums/a/paconvention.htm)

I took some photos, because I wanted to turn this into a math story. I didn’t get all the shots I needed, so I have to give a big thank-you to Max Ray, who went back and took care of the Act II photos with his girlfriend, who happens to be a photographer and got great shots. The Act II photos containing referents and shots of Max are all the work of Kaytee.

Here is my attempt at a three act math story as described by Dan Meyer here.

Act I – The Video

• Word document containing all the information I could find about the China Wedge.
Sequels
I struggle with this one.  Any feedback would be greatly appreciated. I’ve only one idea so far.
• If the entire area under the escalator was filled with cups, bowls and spoons, how many more would have been needed?
In his last one of these, Dan asked whether this broad outline is enough for teachers to go by. I know it is enough for lots of us to run with. If you need more details about how to make this work in a classroom, contact me and I will spell it out a bit more. I would present it in much the same way I discuss in my learning through problem solving explanation.

## Roller Coaster

I got this idea from Frank Sobierajski at the ISTE 2011 conference.  It fits nicely in our Math 10C and Math 20-3 curricula. His take on the subject is available on this teacher tube video.

I took his idea and found a Canadian roller coaster.  Let’s take our students on a virtual roller coaster ride.  Most roller coasters have Point of View (POV) videos available.  I suggest finding one near you.

Bring up some facts about the roller coaster.  This one claims to have a 75 degree drop.

Find a picture of it, like this one.

Source: http://en.wikipedia.org/wiki/File:900behe.jpg

Have students import that image into GeoGebra, and check the math behind the 75 degree claim.

This 75 degree claim is either false, or the angle on the picture I found isn’t right. It’s possible that I haven’t measured from the same places they did. No matter what, it’s a good conversation in class.

I would follow up by having students find their own and check the claims made about them.

The comments on my previous posts where people suggested spirals reminded me of this activity. When the department head group I support was looking at ways to help implement our revised curriculum, we decided to do a lesson study. Two of the lessons they planned are rolled into this one idea. The majority of this activity comes from Deanna Matthews at J. Percy Page, who tells me she got the idea from one of my professors, Florence Glanfield, from the University of Alberta. This post should probably have preceded the last one, but I never think that far ahead.

Creating a Pythagorean Spiral

Have the students create a Pythagorean spiral on card stock, by following the steps below.

1. Use one piece of card stock to draw the spiral on, and use a different colored piece of card stock as a “ruler”.
2. Using the “ruler”, draw a horizontal line one unit long towards the middle, right hand side of the other piece of card stock.  The length is arbitrary.  I usually tell students to make this longer than an inch, but shorter than 3 inches.  If they start too big, it won’t fit, and if they start too small, it’s hard to work with later.  Make sure to mark this length of 1 unit on the “ruler”.
3. Arrange the “ruler” so that you can create an isosceles right triangle. We are using the corner of the “ruler” paper as a guide to make that right angle.
4. Use any straight edge to draw the hypotenuse of this triangle, which will be $\sqrt{2}$ units long.  Notice that I labeled the sides of this triangle on the inside, because we are going to cut it out later.
5. Place the “ruler” on the hypotenuse of the 1,1, $\sqrt{2}$ triangle as shown, to draw another leg that is at a right angle to this hypotenuse. Make sure students position the “ruler” correctly. I’ve seen them trying to guess at a right angle, rather than using the corner of the paper.
6. Use a straightedge to draw the hypotenuse of this new triangle. It will be $\sqrt{3}$ units long.
7. Repeat the process. This new hypotenuse is $\sqrt{4}$ units long. I usually leave it as $\sqrt{4}$ on the label, because one of the things I’ll use this activity for is to talk about the difference between rational and irrational numbers.
8. Extend this pattern as far as you can until you run out of paper.  I went to $\sqrt{12}$.

Using the Spiral to Create a Radical Number Line
Once students complete their spirals, I have groups of three decide which one of them has made the best spiral.  A good check for the accuracy of a spiral is to look at the $\sqrt{9}$ side and compare it to the “ruler”. It should be 3 units long. Whichever student’s is closest to 3 is the one they should use. This is the one we will cut out to create a set of triangles.
We are now going to create a Radical Number Line from these triangles. I like to use a little something I discovered in consulting services called “Sentence Strips”. These strips are foreign to high school teachers like me, but apparently they are used in elementary schools extensively. They are 2 foot long pieces of card stock that come pre-lined on both sides.  Students can create their number lines, and then fold them in half and keep them in their binders.
We will start by putting whole numbers on this number line. We’ll use one of the triangles we cut out and mark the multiples of 1 across the number line. There was some discussion among the group that planned this lesson as to whether we should include negatives or not. My preference is to start on the left side at 0.
Next we will add the multiples of $\frac{1}{2}$ to the number line. Ask students how they could do this without estimating. I folded one of the one unit sides in half, as shown.
At this point, draw their attention to the fact that every other multiple of $\frac{1}{2}$ lands on a whole number. Ask them to try multiples of any fraction they want. They will discover that eventually they hit a whole number. For example, a student who starts with $\frac{2}{9}$ will get $\frac{2}{9}$$\frac{4}{9}$$\frac{6}{9}$$\frac{8}{9}$$\frac{10}{9}$$\frac{12}{9}$$\frac{14}{9}$$\frac{16}{9}$, and finally stop at  $\frac{18}{9}$, which is equivalent to 2. We are steering them towards a definition of rational vs. irrational numbers.
Now we will add radicals, and their multiples to our number line. Starting with $\sqrt{2}$ on the left, we will add $\sqrt{2}$$2 \sqrt{2}$$3 \sqrt{2}$ and so on. Students should notice that these multiples of $\sqrt{2}$ never hit whole numbers (although, this is somewhat dependent on the accuracy of the spiral they created).
Next, add $\sqrt{3}$ and its multiples to the number line.
Next look at the $\sqrt{4}$.  It should hit at 2.  Ask the students why $\sqrt{4}$ is different than the other two we sets we have plotted so far.
Add $\sqrt{5}$ and its multiples to the number line.  Continue to add as many as you think fit without cluttering it up too badly.
Things you can do with this Radical Number Line:
• Notice equivalent radicals.  Above, we can see that $\sqrt{8}$ is equivalent to $2 \sqrt{2}$.
• Discuss rational vs. irrational.
• Order radicals without converting or using a calculator.  Below, we can see that $3 \sqrt{6}$>$4 \sqrt{3}$

## Can I Make it to Calgary?

This one didn’t get much love when I threw it out on Twitter under the #anyqs hashtag.  I still think it has nice use in a Math 10C or 10-3 class on measurement.

It’s a bit of a shaky video, so I paused it at the crucial parts.  If it’s still hard to tell, I’m sitting on the side of the road.  I have about a half tank of gas.  My truck tells me I can go 163 miles until empty.  The sign I’m stopped at tells me it’s 16 km to Leduc, 137 km to Red Deer, and 275 km to Calgary.

Let’s fit this into the 7 steps of LTPS.

1. Play the video.
2. Ask the students what they wonder.  They will likely wonder where I can get to before running out of gas.  I was wondering if I could make Calgary.
3. Ask them to guess how far I can get.  Near Calgary?  Past Calgary?
4. Ask them what other information they require to solve the problem.  They will likely need some conversion factors.
5. Students solve.  Teacher circulates and offers support and/or extensions.  Extensions could involve litres per 100 km or miles per gallon based on the fact that I have a 100 gallon tank in that truck.
6. Students share answers. The teacher can’t play an answer video, because I didn’t actually drive until I ran out of gas.  Sorry I didn’t take that one for the team for the sake of math education.
7. Teacher summarizes learning with a brief wrap up on metric to imperial conversions.

## The Sandbox

Since I was filling my daughter’s sandbox anyhow, I decided to film it and turn it into a math problem.  I’ll fit this into the 7 steps of the Learning Through Problem Solving approach discussed previously on this blog.  Here’s how to make this work in your classroom.

• Play the question video.
• Ask students what question they want to explore.  They will likely come up with “Does he have enough bags of sand?” or “How many bags of sand is he going to need?”
• Elicit student guesses.  Students may assume the answer is 20, because that’s how many bags are stacked up.  You should tell them that the guy in the video is a notoriously bad measurer, and he could have way too many or way too few. As a class, agree on a range of reasonable answers.
•  Ask the students what further information they need to answer their question.  Provide them the measurements of the sandbox, and the information from the bag of Play Sand as shown on this handout.
• Allow students to work on the problem.  Students who finish could be given an extension like this Google image of a local playground.  Tell them that the sand was put in at a uniform depth of 15 inches.  Ask them how many bags that would take. I would use a park near their school that they might remember playing in as a child.
• Share student solutions.  Have students share solutions with other students, or with the whole class using a document camera or chart paper.
• Play the answer video.  Discuss sources of error.
• Summarize what was learned about volume.

@dandersod posted the following video on his blog last week.

He asked, of course, What Can You Do With This?

I decided that there was a lot I could do with it, so I worked on some editing of the video, and then I rolled it out with a group of teachers this morning to see where it would go.

Before I describe the lesson, I should point out that my first edit of the video involved crossing out some of the measurements the original video contained, as well as beeping out the commentary that mentioned the measurements.  I was limited by my video editing skill, and the software I was using. It ended up being awkward and kind of annoying, so I re-did it simply by deleting the parts I had originally wanted obscured.  The new video was much cleaner, and it had the extra benefit of not being so obvious about what I wanted students to explore.  With the beeps and blocked out numbers visible in the first edit, it was painfully evident what I wanted people to find.  The second edit allows for many more directions to be taken in the exploration, and it’s not even obvious that I have deleted anything.

Here was my lesson plan.

1. Provide each group a ruler and a 60 g bag of gummy bears.
2. Play the question video.
3. Ask the students what they wonder about after seeing the video.  In this edit, they will wonder about a whole bunch of things.  I’m hoping they get to, “How many small gummy bears are equivalent to the giant bear, and what are the  dimensions of the giant gummy bear?”
4. Elicit guesses, lower bounds, and upper bounds of reasonable answers.
5. Ask them if they need any clarification or information that might help
6. Turn them loose.
7. Work the room.  Help those who are struggling.  Provide extensions for those who hammered through it.
8. Have students share their answers.  They will be all over the place here.  Those that work with the calorie count will be closest to the “right answer”.  Those that worked with the mass will be a bit farther out.
10. Discuss discrepancies.  Is it measurement error?  Problems with the calculations?  False advertising?
11. Eat the gummy bears.

The lesson is fun, engaging, and has great curricular fit to Alberta’s Math 20-2 course in the measurement and proportional reasoning units.

At step #3 above, some members of the group I was working with really wanted to go a different direction.  Their suggestions became cool extensions for those that got done quickly.

1. How tall is the man in the video, who is the same height as the gummy bear from 30 feet away?
2. How many times would you have to walk around the school to burn off the giant gummy bear?

Enjoy

Thanks to this post on Dan Meyer’s blog, and an ensuing conversation between Dan and Curmudgeon, I was pointed to an article that I think would make a pretty compelling problem in Math 10C or Math 10-3 measurement.

The article describes a 17 year old driver who was given a $190 ticket for going 62 miles an hour in a 45 mile an hour zone. His parents, however, had installed a GPS system in his car to track his speed and driving habits, and they claim the GPS proves their son was only going 45 miles an hour at the time the ticket was issued. It appears to have taken two years of legal wrangling, before the ticket was finally upheld, and he had to pay the fine. I wouldn’t tell the students that yet, though. Here’s a link to the article: Speeding Teenager Lesson Plan 1. Present the problem. Give the students the following excerpt from the article: Shaun Malone was 17 when a Petaluma police officer pulled him over on Lakeville Highway the morning of July 4, 2007, and wrote him a ticket for going 62 mph in a 45-mph zone. Malone, now 19, was ordered to pay a$190 fine, but his parents appealed the decision, saying data from a GPS system they installed in his car to monitor his driving proved he was not speeding.

What ensued was the longest court battle over a speeding ticket in county history.

In her five-page ruling, Commissioner Carla Bonilla noted the accuracy of the GPS system was not challenged by either side in the dispute, but rather they had different interpretations of the data.

All GPS systems in vehicles calculate speed and location, but the tracking device Malone’s parents installed in his 2000 Toyota Celica GTS downloaded the information to their computer. The system sent out a data signal every 30 seconds that reported the car’s speed, location and direction. If Malone ever hit 70 mph, his parents received an e-mail alert.

Malone was on his way to Infineon Raceway when Officer Steve Johnson said he clocked Malone’s car going 62 mph about 400 feet west of South McDowell Boulevard.

The teen’s GPS, however, pegged the car at 45 mph in virtually the same location.

At issue was the distance from the stoplight at Freitas Road — site of the first GPS “ping” that showed Malone stopped — to the second ping 30 seconds later, when he was going 45 mph. Bonilla said the distance between those two points was 1,980 feet.

2.  Ask the students to discuss the article.  In the end they will come to the question we want explored.  Was young Shaun guilty of speeding?

3.  Let them answer the question.  Have them prepare a defense for Shaun, or an argument for the prosecution.

4.  Show them the Commissioner’s conclusion, based on mathematics.

Bonilla said the distance between those two points was 1,980 feet, and the GPS data confirmed the prosecution’s contention that Malone had to have exceeded the speed limit.

“The mathematics confirm this,” she wrote.

Teacher Resource

A possible solution

An extension, eventually.

I have been attempting to contact the person mentioned in this local article, but so far he hasn’t responded to me.  Similar mathematics could prove he wasn’t driving as excessively fast as the red light camera claimed, but I would need to get a copy of his ticket to show that.

Red Light Camera