## Thursday, September 22, 2016

### Analyzing a Bike Rim in 5 Days

At Glebe Collegiate we have tried hard to spiral our MHF 4U course (called advanced functions) over the past 4-5 years. It really has been a collective effort within our department. You can find a whole bunch of information about it here thanks to Janice Bernstein. I posted the whiteboard notes and an early spiraled version of the course here.

We are trying hard BUT I feel like the course is still missing that activity based feeling that I have managed to create in my grade 10 applied course (for example my lesson on T-Shirts). It just feels like we still tell them the things we would like them to discover or experience.

Also I VALUE the mathematical processes in the curriculum document. So.....I started thinking about how I could get my students to discover and experience more of the Trigonometry in the course. One of the activities we have been doing is radian rulers from plates. Here and here is what it looked like.

This is my attempt at creating an exploratory activity 1) to discover and understand radians, 2) to develop the relationship between arclength, radius and the angle measure in radians  3) to understand angular velocity and 4) to develop an understanding of the relationship between the angle rotated and the height of a point on the bicycle rim.

BIKE RIMS

First I went and scavenged some bike rims from a local bike store and cut out all the spokes. Needed 10 but only found 7 so I had to use three tires as well so that each group of 3 students had their own rim or tire.
 Rims I scammed
 Deleted spokes
 Final 10 - one for each group
DAY 1 Creating a radian ruler for their bike rim.
Students were put into random groups of three.( I randomize students by giving them a card from 1-10 which matches the pods of three desks and whiteboards (4 by 8 ft for each group) around my room-these numbers never change all semester just the students at those desks each activity). Verbal instructions ( no paper-throughout all 5 days)

1) Tape the number of their group onto their rim or tire.

2) Create a paper ruler that has the length of the circumference of your rim or tire. Measure that length and figure out the radius of your rim (on whiteboards). Measure the diameter roughly to check.

 Getting the right length ruler
 Measuring ruler
 Finding radius of rim or tire
3) Mark your ruler using a red marker in radius units. HUH? If your radius was 10 cm you would put a mark every 10 cm and mark them 1, 2, 3 etc until you got to the end.

As this was happening I circulated and started having discussions with groups about how many they got. Why? Students started making connections that it was a little more than 6 because the circumference is 2pi*r. Once all were at this point we had a full class discussion that we could put 2pi at the end of the ruler and that was equivalent to 6.28 (about).

4)  Next we talked about it if we folded it in half (pi) and in half again and half again. I asked groups to do this and crease their rulers ( be precise ) and then mark the creases in terms of pi. I actually hesitated with the colour and of course a student hollered " Purple for pi". Me "of course".

5) Next we talked about if we folded in half, then in thirds (like you would fold a letter for an envelope) and then in half-we would get 12 pieces - so 2pi divided into 12 parts - each part would be pi/6. Students labelled these in green. Of course some were all ready marked because they were equivalent to some of the angles from the previous folds.
 Folding and creasing our paper radian rulers

6) I then had groups put a decimal value at every crease (radius units).

7) Lastly they were instructed to put the degree value in orange. Student "Orange for the degree sign". Enough said!

Here is the final ruler with the rim for one group.
 Final ruler with all markings

8) Lastly we wrapped our rulers back around our rims and talked about the angle and the number of radii around the rim.

9) Clean up time-hung the rulers for tomorrow.
 Two class sets

DAY 2 Circular Radian Number Lines

This idea came to me after seeing this tweet from @CHaugneland
So this required some set up. I wanted each group to have a circular number line that they could hang angles in radians on. Here is a pano of the class as they were doing this part of the activity.

 Pano of what the room looked like
Obviously there were many comments about the hanging rims around the room as students entered class.

1) The class discussed that we were going to use these as circular number lines and that they would hook cards of different angles in the right spot relative to each other and then check their placements using their radian ruler from yesterday.

2) I had prepared 4 sets of cards.
• First set had positive angles only with multiples of pi/4 between 0 and 2pi. So pi/4, pi/2, 3pi/4, pi, 5pi/4, 3pi/2, 7pi/4 and 2pi. I also included some that were equivalent with a higher denominator, for example 6pi/8.
•  Second set had positive and negative angles with multiples of pi/4 between -2pi and +2pi. This time I included equivalent ones that were positive and negative, for example +pi/4 and -7pi/4. And again some that were equivalent with a higher denominator.
• Third set had positive and negative angles with multiples of pi/3 between -2pi and +2pi with equivalent angles.
• Fourth set had positive and negative angles with multiples of pi/6 between -2pi and +2pi with equivalent angles.
• For the fifth set I had groups make a set of cards and place them based on instructions that I gave them. For example
1. decimal values between -6.28 and +6.28.
2. between -4pi and -2pi or 2pi and 4pi with multiples of pi/4
3. between 0 and 2pi with multiples of pi/8
4. etc
3) The card sets were placed around the room on the ledges of my white boards. A group would pick up a set, place them on their circular number line and then check placements with their radian ruler. Groups then checked in with me-I typically drilled them and asked lots of questions. Groups then returned the card set and picked up the next one. Repeat.

Here are some pictures.

 View from above tire
 View from below rim
 Equivalent angles
 Checking placements of angles with radian ruler from day 1
 Big smile
Some thoughts:

There were many groups that wanted to know where to place zero and what direction was positive? negative?

Checking with the radian ruler was tricky but doable.

Mathematical discussions and productive struggle were through the roof.

High level of engagement.

A colleague thought I was crazy because of the work required for just this one day. It was well worth the experience as students participated whole-heartily in rich mathematical discussions. Would I do this again-ABSOLUTELY.

DAY 3

Believe it or not it took some thought for me to clue into this. Plot Arclength(y) versus Angle in radians(x). The slope of the line will be the radius of their tire, and the equation will fall out. Here is how it went down.

1) Same groups as the last two days. Get your bike rim, your radian ruler, whiteboard marker and a measuring tape.

2) Measure the arclength for a particular angle in radians and make a table of values.
As I was demonstrating this by wrapping the radian ruler around the tire and the tape measure around the tire (which was difficult-see picture) this conversation ensued.

Student asks and states "Do we really have to wrap both around each time? Couldn't we just lay it flat and measure - it is the same thing".

Me "What do you think?"

Other student " As long as we realize that when we measure the distance on the (radian) ruler we are really measuring the arclength around the rim"

Beautiful!
 Measuring arc length for a particular angle in radians by wrapping around bike rim
 Measuring arc length for a particular angle in radians using the radian ruler laid out flat
3) Make a table of values and graph the data. Find the slope of the line. Find the equation of the line using technology (or algebraically) (or both)

5) At this point I asked if there was any connections between the slope of the line and the radius of the bike rim. We went around the room and each group yelled out their slope followed by the radius of their tire. AHA the slope of the line was the radius. And there you have it

6) We briefly worked on an application.

Some samples of  groups whiteboards at the end of the period.
 Sample 1
 Sample 2
 Sample 3
 Sample 4
 Sample 5
 Sample 6
DAY 4 PART ONE
Distance A Point On The Wheel Travels

1) We grabbed a bike rim and the entire class went out into the hall. I asked some students to time how long the rim rolled for while I counted the number of revolutions. We picked a starting point. One of the students marked where we stopped the wheel. I rolled the wheel and I counted revolutions while some students timed the roll until I yelled stop.

5 revolutions of a wheel with a radius of 25.9 cm in 8.68 seconds

2) Groups were sent back into the classroom to work on whiteboards and find the total distance travelled by the point on the rim. I also asked them for the angular velocity in radians per second. I also asked them to figure out the total distance travelled by the point on the rim using the angular velocity. Here is the work of one group for the data given above.
 Group work: Total distance a point travels 811 cm
For a different rim
7 revolutions of a wheel with a radius of 26.1 cm in 10.11 seconds

3) Once all groups had done the calculations we debriefed. All groups went out in the hall and measured the distance we had marked earlier. The calculations were right on.

DAY 4 PART 2
Relationship between the ANGLE ROTATED and the HEIGHT OF A POINT ON THE BIKE RIM

1) I did a whole class discussion on how we could collect data. Place the point at the wheel on the ground at zero on our radian ruler. Roll the rim on the ruler and stop at particular angles and then measure the height of the point. Looks like this.

2) Groups made a table of values, graph and generated the equation using technology. If they finished this I asked groups what the theoretical equation should have been. If they finished that I gave them an angle and asked for the height of the point.

DAY 5
ANGLE ROTATED and the HEIGHT OF A POINT ON THE BIKE RIM (AROC and IROC)

Here were the instructions I had written on my whiteboard for groups to do this day. We discussed average rate of change and I said I would talk to any groups about instantaneous rate of change if they got there.

Group work for 5 of the 20 groups (two classes):

 Group 1
 Group 2
 Group 3
 Group 4
 Group 5
Final Thoughts

My original intent for this activity was to redo the radian plate activity and the radian war activity from this site. This is where I have grown. I am thinking what else can I do with this (thank you #MTBOS for #WCYDWT) This post reflects my creative juices in squeezing curriculum out of an activity. Hope you enjoyed. Honestly - this activity feels like what I envisioned for a spiraled course and wrote about back in 2013. #makeitstick #spiraling #activitybasedlearning #interleaving

Also I feel like I am getting closer to a Thinking Classroom that Peter Liljedahl has been researching about.

I teach so differently now then I used to. I can't even start to describe how different my role in the class is compared to years ago when I lectured. While doing this activity I actually sent out this tweet.

Any and all feedback welcome. What else could we do with this?

## LESSON STUDY

With all the great stuff happening with number lines around the #MTBoS, I thought it might be an interesting idea to bring to our cross curricular lesson study group in first semester of this year (2015-16). The original lesson was designed for a 75 minute period. I just repeated this lesson again this semester with two classes of grade 10 applied's but carried it over two periods. This was much more manageable than the original lesson which was done in one period and that felt very rushed.

Kudos goes to Robin McAteer (@robintg) for the original idea which I experienced at a summer math camp for teachers in Barrie Ontario a few years back. I specifically asked Robin to join in this lesson study (she is an instructional coach at our board - and I am a huge fan) as I thought she would have some great insights (as she always does). Also, as always, Robin documented our debrief and summarized the exit cards. You really can thank her for the depth of this post.

This particular lesson study was a bit different. Let me tell you why. Normally we would plan a lesson one week and then the following week deliver and debrief. This particular lesson was planned on the afternoon of January 7th, 2016 to be delivered on the morning of January 8th, 2016. The reason being that we had a guest coming from LA to join us for the two days. Judith Keeney (@JudithKeeney) whom I met at a session she presented on at #TMC14 in Oklahoma on lesson study (and hit it off with by the way) was interested in joining us to observe and participate in a lesson study at our school.

The two of us had discussed this at NCTM in Boston in April 2015 and it became a reality. I am so glad she was able to experience our group and visit my classroom and experience all that we try to do at Glebe Collegiate. Plus her insights were awesome.

This presented a slight challenge as the lesson had to be done by the end of the first day, including any classroom set up. That being said I had an idea of what the lesson would look like before the meeting ( so..... unfortunately our group did not really feel like they had ownership of this lesson). None the less something cool did happen. Normally we would talk for a whole afternoon and then the teacher delivering the lesson the following week would iron out the details and set up the classroom for the lesson on their own. This time though we planned for about half the afternoon and then we all went to my classroom to set up the physical learning environment. OK something cool happened - not sure how to describe it- because we were just chipping in on the set up we started bantering a little. Lot's of jokes, pokes and other things. There was something about setting up the lesson together not just planning it together. A real sense of team - hard to describe. Lot's of talk the next day as to how to incorporate the set up as a group as part of our lesson study model. Of course here we are 6 months later and we have not managed to do this (would cost more \$\$\$), but it is on the back burner.

Anyhow can't even start to thank both Robin and Judy enough as well as all our team members from first semester - it was a unique experience.

## THE LESSON

Here is the planning document for the lesson.

Number Lines: A Lesson for Connecting Representations and Developing Number Sense
·        Provide all observers with:
Photocopy of names with pictures
Observation sheet
Lesson plan
Random Groupings

 Timing Teacher Moves Observations / Improvements Before students arrive ·         Desks arranged in pods ·         Random Groups of 2/3 by cards ·         White Boards, colored markers, number line paper, Number lines pre hung for groups, Clothespins As students arrive 8:50-9:05 Inroduction to number line activity (day 1) ·         Place 10 and -10 and 0 together ·         Give each group (random) of two/three students 3 numbers and have them place it on the number line (from -10 to 10)  Decimals, Fractions, and Whole numbers ·         Clothes line all the way across the room. ·         Discussion about placements - scale 9:10-9:45 Group number lines (day 1) ·         Give groups expressions and colors associated with each expression. ·         Quadratics and linear expressions. ·         Give groups Table Markers (8 colours) and one non permanent marker. ·         Give groups a number from -10 to +10. “I am going to give you a y number line- it has the x value you will be working with” ·         For each colored equation find/calculate the y value on your whiteboard for your given x value-indicate the equation on the whiteboard so that I can verify your work. ·          Once you are finished and Mr. Overwijk has verified. Then place the values with the appropriate color on your number line. ·         If beyond range of values it is not included on the number line. ·         Need to do minimum three ·         Once the group has the number line done for that x value they repeat for a different x value until time is up. (7 groups - hopefully 3 per group) 9:50-10:00 Generating graphs of expressions using the 21 number lines (day 2 - second time through) ·         Students bring their number lines and place them vertically on that x value- this should create a graph of the y=the expressions ·         Look at characteristics of the expressions from the equations and the graph - connections 10:00 Home Base ·         Exit Card

Let me provide a little more detail.

The minds on activity with the physical number line was to get students thinking about placements of different numbers on a number line.

Once we had the numbers placed students were allowed to come move numbers they thought were in the wrong spot and explain why they were moving the number. Lots of great discussion about numbers being spaced properly.

Next students were given the 8 equations that they would be working with, each associated with a colour, and a number line from 10 to -10 and finally a number for x. They would calculate the value for y for all 8 equations and then get it verified by Mr. O and then make any corrections. This was all done vertically on white boards.

Here are the eight equations and the colors associated with them.

Here is some groups work calculating the values for all 8 equations using their given x value. We spent a period getting all the number lines together this most recent semester. (21 in total from -10 to 10)

Once students had the y values correct for their given x value they would place the appropriately coloured dot on the number line if it landed between -10 and +10. By the end of the period the class had produced the 21 number lines.

At the start of the next day I asked if anyone could tell me what we did yesterday.
S "We placed dots on number lines"
Me "Where did the numbers come from?"
S "We had 8 equations and an x value and we placed the y values on the line based on the colour of the equation."

Me "Ok great. So if we were going to place these from -10 to 10 would they go vertical or horizontal?"

Long period of silence.

S "They would go vertical because the dots represent the y values for the x value you gave us. The y values are up down."

And there you have it. So with some student help we slowly placed (taped) the 21 number lines. Here is what it looked like.

So some dots were in the wrong spot despite me checking their values on the whiteboards. It appeared as though most of the mistakes were groups putting a positive value as a negative value and vice versa. I will say as we put the lines together there were lots of comments like, "Oh we are getting the graphs of those equations!" I think I even heard "Holy S&it."

Here is what it looked like once we connected the coloured dots for the two different classes.

I then asked students to do some characteristic finding for the 8 equations. Once they were done they could go look at the graph and verify their answers.

Here is some sample work from groups for one of the linear examples.

And sample work from a group on a quadratic one.

## THE EXIT CARD

At the end of the activity we asked students to fill in an exit card that looked like this.

Exit Card Name:__________________

 About your Learning Red Yellow Green I understand how to place values on a number line (positive, negative, decimals, fractions) I chose an effective tool or strategy to calculate the y values given an x value I understand the connection between the equation and the x and y values I understood how to  place my y values on the table of values for the different equations I understand the math vocabulary used today. I worked well with my team (asking questions, explaining, on task, encouraging etc.)

 About the Lesson Liked Didn’t Like No Opinion Using the number line strips Learning with my group / working together Building the tables of values as a class Other?                __________________________________

Something I learned today was _______________________

Something I’m wondering about is _______________________

We got these results:

## THE LESSON STUDY DEBRIEF

Notes from the lesson study as they happened. (student names deleted - names included are teachers)

Al’s Class - Debrief

Complex detailed lesson
- couldn’t have done it all with one person - would have taken 2 classes
- needed 2 or 3 people walking around checking
- lots of details had to be worked through in the morning
ex. doing up answers for quick checking - checking was time consuming!
Al
• Reflecting on planning a lesson and leaving an individual to finish it up vs. the team effort of putting it together like we did this time
• Maybe we should be doing this more often instead of leaving it to one person
• it was cool for us all to do it together
France
• How - maybe staying after school - it worked in this case because it wasn’t all of our lesson
• Can’t sacrifice the time spent making up the lesson
Robin
• Some groups have a rule that they don’t leave someone with a pile of work - they do what needs to be done
Dana
• France’s idea - stay after school to help that person - get it done - it’s fresh
Paula
• The tweaking is sometimes more than the whole design - liked that we started with the lesson somewhat designed and had time for tweaking and perfecting
• Sometimes when we build from the ground with so much input you don’t get to the point of tweaking
France
• Maybe we’re at the point where we can build on an existing lesson because we understand that the kids are doing the work… we are starting with a fairly solid idea - we’ve been doing this for a while.
Robin
• agreeing that the details / intricacies are important
Al
• Reflecting on Ball Roll lesson - people were taken aback when I did a teacher move that I had thought of by myself - they weren’t expecting it
France
• Have seen some lessons where the plan get’s abandoned because the person left with the planning loses confidence / doesn’t fully understand / gets scared
• Do you find that there are times when you don’t get to all the details so you aren’t pushing the envelope as you hope?
Judy
• Yes - people work in much more isolation
• it took three years to get to a point similar to where you are now
• it was really bumpy at the beginning - because we were learning how to collaborate
• we got to the point where someone comes with a lesson and we focus on the details
• we do a lot of electronic collaboration
Discussing pros and cons of various timing options
• need to finish up the planning fairly quickly while it’s still in your mind, but leaving a bit of time for it to settle and ideas to come is good too
• Al:  still likes it being a week between the planning and the delivery, but sometime during that week do the in between work
• Paula:  Sometimes I start to forget when it’s later in the week
• Anneke:  Plan on monday, revisit on Thursday after School, implement next Monday - to give it a bit of thought but not too long

Reflecting on the planning
• turning point was when non-Math people said that they weren’t getting it
• we all slowed down - realized orientation was a problem - worked it through a bit - Paula’s idea to pull the strips off the wall was generated

Flexibility
• sometimes you need to start from scratch, sometime a partially planned lesson can work
• depends where the participants are at - for newer people the lessons take more time to plan and everyone benefits from the whole team being involved

How important is it for everyone to understand the lesson?
• observation is improved with deeper understanding, but it isn’t reasonable to expect non-subject teachers to be experts in the nuances of the curriculum
• important thing is for everyone to understand the big idea / learning goal
• ex. in this lesson, making connections to the characteristics
• if the whole purpose is to watch kids learn and move, then the more you know the better

Al’s reflections
• intricate lesson, super busy for me, needed to make sure all of the pieces were right
• breakthrough in the last 15 minutes - they were getting quicker
• second time was way faster
• bailed on the table of values - that wasn’t going to happen
• it was hectic - I didn’t have time to notice who was learning what

Robin
Student A and Student B
• two weak students together randomly
• took a while go get going, but they were engaged for almost the whole class
• payed attention to the number line - quite engaged / focused
• Student A has a knack for knowing when things don’t make sense
• Grabbed the graphing calculator, but they didn’t know how to use it
• They were excited and meticulous after that
• both learned about the calculator tool

France
Student C, Student D, Student E
• Student C had absolutely no idea from the beginning to the end
• she did nothing - she filled out the exit card
• Dana:  she looked busy - from afar
• Al:  She can play the game
• Student D did the thinking
• None of them understood how to start
• they saw the boys
• Student D started doing the calculation
• Student E started copying from the boys board
• was on her phone - facetiming at points
• Might be interesting to have her followed around to collect evidence about the time she spends on the phone
• Student D asked Student E a question at one point - she said “we’ll do both”  she was involved to some extent
Paula - observing same kids
• it never ceases to amaze me how Student C can hide
• Al:  has warned her that she has to show me something - she’s a worry for me
• She bluffs it
Al / Dana - pros and cons of battles with kids over the phones
Al:  There are 5 or 6 in the class with this problem - decision to battle or preserve the relationship
France:  Would be interesting to follow her around and document phone use as evidence for a phone addiction
Dana:  I don’t think you destroy relationships … they resent it…
Al:  It’s a battle - I know where this ends up - those kids stop coming and you lose them - it’s tough - the energy it takes -
Anneke:  I take ____ phone all the time - I provide a free service to kids by taking their phones from them

Student F  / Student G  / Student H
• Student F was really involved
• Al:  He’s a smart kid - has missed so much - when he’s there, he’s there
• Anneke / France - it’s anxiety
• Student H - why is she in P?  Al talked to her about it  early on
• one of her first questions - can I just make x^2 + x  x^3… maybe that’s why she’s in applied...

Anneke
Student I, Student J, Student K
• Student K had the marker and did most of the work - was the boss / leader
• Student J stood on the side with the graphing calculator
• Student J was very engaged - trying to learn - the whole time she was on
• Lots of conversation in Spanish
• Exit cards showed positive reflections
Student I
• was trying to figure it out using tiles
• Al:  That’s a skill that they have in my class - she was turning it into length/width form
• it would help them to sub into a factored (more familiar?) form
• Robin:  Was she understanding why she was doing it?
• Al:  They have to sub x values in… so they are used to doing it when they are doing the vertex… so she wanted to rewrite it so she could sub in.
• Exit card for Student I - is this going to be on the test?

Student L and Student M Student N
• Student L was engaged
• Student N came in late, hung out with student O
• Making comments about everyone else including Mr. O with his harem
• Student M was engaged right from the beginning - started measuring right a way to figure out where the zero was - knew that he had to put zero in first

Dana
Student O
• wants to wander around the room the whole period and socialize
• he’s a huge distraction
• other teachers have inquired via e-mail about him
• was quite engaged until the end when Student N came in

Exit Card
• they liked it, they took time to do it
• some kids just checked it off  (ex Student O ) - have to question the validity

Graphing Calculator
• some were using it as a tool by putting in the equations and going to the table
• others were using it just to calculate
• most weren’t using it to get a table

Dana:
Student N, Student P, Student Q
• Student P was putting values on the white board while Student Q told him  the numbers
Is the process of putting in the numbers helping him?  He’s really just dictating.
• Logistically - might be an idea to change who uses the calculator
• Student N wrote explicitly on the exit card that he liked the explanation
• hard to say what he meant exactly
• Student Q was focused, asking questions, very much engaged
• exit card -  yellow on math vocabulary
Judy
Student R and Student O
• Took a long time setting up the equations
• Student O was driving the calculator - putting in every single operation
- missed negatives and operation signs
- During pauses (ex. waiting for checks) Student R took the calculator and tried to figure it out
- diligence - she was watching what he was doing, then practicing
- The second time they worked through it together in about 5 minutes
- Took a long time with the number line - getting the dots coloured in
- Student O was engaged the second time around
- Student R said she liked working with the group - had a positive impact
- Student O didn’t like working with the group - put yellow for worked well with his team

Thach Thao
Student N, Student Q , Student P
• Student N was the first at the board, but Student Q corrected some of his answers, so he stepped back and Student Q took over
• Student P just stood there watching
• Wanted to go to Student P and just say “does that answer look reasonable”  would be useful to get the EAs to do that prompt
• Thach Thao asked them to reflect on the impact of repeated removal (negatives)
• Student P said “that doesn’t make sense” - he does have some understanding
• Student P - language barriers / processing
• when he understands what your asking, and when you sit and listen, he can tell you
• he can do a lot of the stuff - he needs time for processing and articulating
• Paula : he’s under investigation for an IEP - he’s ESL so it’s hard to assess
• Al:  interesting about the language - sometimes if I wonder if he is just being a goof and playing people
• Paula - this question has been raised before
• Lot’s of wondering about him - He’s a conundrum
• France - it would be nice to have a bilingual tutor in there
• Al:  I’d like to know what Student O saying about me - he’d be badmouthing me non stop

Student R
• was very late
• with a group who was struggling
• jumped right in and tried stuff

Student S  wasn’t there today
• maybe didn’t want to be exposed

Student T
she’s anxious - she refuses support of a scribe
• during tests - she needs me to ask her the questions individually or she’s not going to write anything down
• Paula: if she could connect really nicely with an EA - who could force themselves on her - get in there and support - she has a lot of potential

Lesson Reflections
• great to have the repeat built in to the lesson
• immediate feedback / positive reinforcement

Robin:  Wondering about the learning of people in groups where there is a large gap in readiness
• ex. Student H, Student F, Student G
• Student H was chatting with others
• Student F was holding his own - learned how to convert fractions to decimals
• Student G - not sure - said that something he learned today was nothing

## MY THOUGHTS ON THIS LESSON

1. Loved this creation of the graph. It would be a cool way to introduce the graphs of functions that you will be exploring in a course. For example in MHF4U do a rational, polynomial, exponential, logarithmic, trigonometric. Tons of potential with vertical y value number lines.

2. What about later in courses where students then knew how to solve for x given y. We could give them the y value, they would solve each equation for the x value, put on the coloured dot, and then place the number lines horizontally at the given y value. Again we would be generating the graphs.

3. HMMM      Desmos activity Builder anyone?

Huge thanks to all involved. Never would have generated all this on my own. So much learning for me.

Also nice to blog about my classroom again!