Filling it up!

In our grade 9 applied class we are finishing off linear relations and moving into solving equations. I want an activity that is hands-on, engaging, and shows a purpose to solving linear equations.

Here is some thoughts on an activity I want to try. Let me know what you think. Any feedback would be greatly appreciated.

Here it is: Filling it up!!

Show them this picture….

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Let them wonder, let them ask what that thing in the pitcher is.


“How many would be too much?”

“How many would be not enough?”

“How many is just right?”

Have them record the guess. “We’ll compare our answer to our guess”


ME: Let’s find out how many.

Organize them into groups of 3.

ME: What are we going to need?

We’ll need volume of the pitcher, volume of the cup, and volume of the weight.

Have discussion on:

What shape is the cup? … it more like a cylinder or a cone? Which is it closest to? What formula for volume will you use? Will you be right?

What shape is the pitcher? What shape is the weight?

Choose 1 member of you group to find the volume of the cup; choose 1 member for volume of the pitcher; choose 1 member to find volume of the weight.

Have the items around the room like stations:



IMG_2772Each member will find the volume of their object and bring it back to the group.

Allow the students to work

Here are some scaffolding questions I can use (Please feel free to give me some more)

  • What’s changing as you fill up the pitcher?
  • Does it start with zero volume? What volume of the pitcher is already taken up?
  • What volume is left after the weight?

Here is a possible solution….


My idea is this could be great context for introducing solving equations using opposite operations! Use their technique  and show how the volume grows as the cups increase. Use Desmos and relate it to y = mx + b.

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show them how their strategy is the same as solving 5562 = 1511 + 335x. Boom! Context for solving equations!

My ideas for extensions would be to put objects like….


in the pitcher. Count how many cups to fill the pitcher now. Use our equation to solve for the volume of the car. [Corresponding Grade 9 Academic learning goal: Find the y-intercept (initial value) of a linear equation given the slope (rate of change) and a point. ]

What do you think?? Think it would work? I would love some feedback!

The Best Estimates

So Dan sent out this tweet.

Wasn’t sure if he was asking Andrew to make a blog post or anyone, but I decided to share my thoughts!

If you’re not familiar with Estimation180…..become familiar quick!! The challenges/estimates have been great conversation starters, warm ups, and intros to math concepts in my classes for the last couple years!!!

So, to answer Dan’s question…..My favourite Estimates have been the ones that make the students do a double take! They make us say No Way!!! or How is that right?

Here is my favourite…

Day 52

It’s awesome because of the controversy! Very few kids guess that there are actually 12 ounces/355ml in that glass! Most think it must be more than the can! In my class we have had great discussion on reasons. Most say the camera angle in the picture is deceiving. They get angry because they think I tricked them. From this point on they are skeptical about all given information!! Awesome! Love it!

I love this whole line up of estimates. Great discussion come out of why the tall vase has the same volume as the Dessert Dish on day 54….

….and the glass on day 57

I think the kids get a kick out of watching the video answers too!

I’m a huge fan of these types of estimates too …. ”

How many small vases will it take to fill the large vase?

by these types I mean “How many of these fit in there?” These have worked wonders for some of our problem solving skills. After we reveal the the answer we take, for example the total ounces in the large container and try to work backwards and see if we can figure out how many small containers fit. (by dividing). By using these estimates as warm ups it has been an easy transition to solve problems like…

Screen Shot 2014-11-14 at 6.46.49 PMIn the past my grade 9 applied students have struggled with this type of problem. After using the “how many fit” estimates my students’ ability on this type have dramatically improved!

These are a few of my favourite things…

Again ….check this site out now…..Estimation180. Thanks Andrew Stadel!


Amazing Race Review Activity

We should not attach speed as a factor in our math learning but I love the intensity my students show when we do the Amazing Race Review.

I first saw this activity from a friend of mine Brian McBain. He created a review game where students travelled around the school completing challenges.The first to make it back to the room after completing all challenges was the winner…..just like in the show The Amazing Race.

Today was a review day on trigonometric expressions. I just grabbed some “Knowledgey” questions from the review section of the text. My goal here was to get them to practice the basics. I wanted to provide them some feedback on the application type questions….so i didn’t include them in the race.

I made clues like this…..



and  like…

Ewhere they are to complete the review question and then use their answer to figure out where to go next.

I also threw in some like….





where they had to come back to my room and complete a challenge.


Each clue was placed around or in the room indicated. With permission from the teachers of the room the students had to actually go in a classroom and look around for the clue.

When found each clue looked like…

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They had to scan the QR code which revealed the clue.

Making the QR codes is pretty easy…..

1. Take a picture of your clues and put them in a Google Drive folder.

2. For each clue, grab the shareable link and paste it into the QR code generator here:

3. Download that QR code and paste it into a sheet like above. And you’ve got a clue!!!

I staggered the start so each group didn’t just follow each other from room to room. I gave them a recording sheet so they could keep track of their clues and work. I set them off and said “Complete all clues in the correct order and you will be eligible to crack the code for the prize.”


Here is the code to crack….

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This years class was pretty intense. The kids were racing each other down the halls and blocking each other from looking at their work. When all groups made it back to the room it was a heated match of “who can crack the code” first!! When finally the group opened the lock …..the class erupted! Some in cheers and some not so much!! You could put anything in that box for the prize and they would be happy!!! Stickers is usually my go to choice!!! Kids have a fun time practicing some skills!!




Stacking Cups!

So we did Dan’s Meyer’s stacking cups lesson in class today!!!  I first saw this activity from Andrew Stadel in his 3-Act math collection. Not sure who first came up with it though. But thanks to both of you!

I started class by stacking the cups up in front of them…..allowed them wonder what was going on. They had questions like

“What are you doing?

“Are we having Hot Chocolate?”

“Are we going to use them to drink something?”

“What are your doing?????”

“How many cups do you have?”

and “How tall are you in cups?”

and bingo there we go!

I told them that is our task for today…To discover how tall I am in cups! I then had them estimate how many cups it would be! They were uncomfortable to start. They wanted to guess perfectly so they wanted to know how tall I was. They tried to put cups next to me as I walked around. They wanted me to lie down! I said just make an estimate to start off! I wanted them to guess so we had something to compare their final answers to. I wanted them to continually checking their work against their initial guess.

After a few minutes of estimating one group asked: “Are we stacking them like this…..



or like this…..

IMG_0965.JPGAwesome!!!! I said “Does it matter?” and they all yelled yes!!! So we then agreed that we had TWO problems to solve. So we put up two sets of estimates!!! We decided to stack them like the second picture first!



I then asked:

“Did you need anything from me?”

they asked for: Rulers, my height, and Desmos!

I gave them all of those things…….everyone wrote frantically when I said I was 183 cm tall!!!

They worked! I saw groups stacking cups, recording values in Desmos, and measuring!

Almost all groups realized that the stack height was only changing by the lip amount and I saw a lot of this…

IMG_2730.JPGwhich had me excited!!! It gave me a chance to say: “Tell me about this, why do you think this is correct?” It was so interesting to hear their responses…..they were convinced they were right so I said let’s plot this in desmos and see if the equation matches the table

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They knew they were wrong…..but what was awesome is that they knew how to fix it!!!! Desmos is awesome for this. It’s like a visual self correction machine! We discussed that the start of the line didn’t seem to match up with our points. Then the ahaa! happened.

“We didn’t use the zero row for our start value.” They fixed it and were visually rewarded with a correct answer.

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After our equations were in desmos, the kids dragged their finger along the line until they reached a height of 183 cm and read off the number of cups! For the kids who seemed ahead of the game this was my chance to introduce solving equations by using opposite operations!

Finally we stacked the cups to verify.




Round 2: Stack the cups end to end.

Most groups divided my height with the height of 1 cup…..21 cups….give or take….So great! It gave us context when we discussed opposite operations when solving equations.

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I found it was great that we had two problems in one! We are discussing how to distinguish between partial variation problems and direct variation problems. And here is one scenario where we got to look at each!!! Such a valuable activity!

Oh……did you want to know my height in cups (overlapped)??? —–> 128!

Below are the list of Ontario Curriculum Expectations covered in this activity—-> Look at them all!!!!

  • pose problems, identify variables, and formulate hypotheses associated with relationships between two variables
  • carry out an investigation or experiment involving relationships between two variables, including the collection and organization of data, using appropriate methods, equipment, and/or technology (e.g., surveying; using measuring tools, scientific probes, the Internet) and techniques
  • describe trends and relationships observed in data, make inferences from data, com- pare the inferences with hypotheses about the data, and explain any differences between the inferences and the hypotheses
  • compare the properties of direct variation and partial variation in applications, and identify the initial value
  • express a linear relation as an equation in two variables, using the rate of change and the initial value
  • describe the meaning of the rate of change and the initial value for a linear relation arising from a realistic situation
  • determine values of a linear relation by using a table of values, by using the equa- tion of the relation, and by interpolating or extrapolating from the graph of the relation.

Wouldn’t it be awesome…


I can’t stop thinking about the great stuff from Penny Circle, Waterline, Central Park, Desman, and Function Carnival. Specifically the collaboration; the crowd sourcing of data and responses!

In Penny Circle, I love the fact that the student gets to do a few instances of selecting a circle and filling it with pennies. Then the data is grouped with the rest of the class….and voila!! we have a scatterplot!

I would love for this option of crowd sourcing content as a regular option. Wouldn’t it be awesome for when we complete the Vroom Vroom activity or the Barbie Bungee activity that we could ask students to record a few pieces of data in their table….like this,

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but up on the projector the class sees this?

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Wouldn’t it also be awesome if I asked the class to draw me a line with slope -2 ….the student would see theirs….

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but we would all see this?

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I’ve been using PearDeck for some lessons lately and we’ve been able to crowd source some stuff like..

Put the moveable point on A solution to the inequality f(x) > 40

Day 23 - Pear Deck!!

Day 23 – Pear Deck!!

We’ve also been able to crowd source by the old fashion way…….everyone write their points up on the board then we can all graph. This is still great don’t get me wrong……i’m just wishing!

Wouldn’t it also be awesome when we go to make Math Art with our Function Art project…..we all work together to make a picture like…

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I know that Texas Instruments has TI Navigator which tries to link students up ……but in my opinion it’s not as nice or easy as Desmos is to use!!!! Maybe this is already possible in Desmos and I just don’t know it. Or maybe there is something else out there……but I doubt it.

I would love it if my class could all work together…..keep our technology social! These are just some wishes! Love Desmos no matter what!

Changing the Test!

I’ve never been comfortable with how we traditionally evaluate students in math. It has bugged me that I test on a specific date, then move on. We tell the kids to not forget that material, but never really give them credit for doing that!

Yet our curriculum documents say we should do otherwise…..

First one,

From The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007 (revised)

Screen Shot 2014-11-02 at 3.12.02 PMThis document DOESN’T say that students will know __________ skill by Nov. 3 (or any other date). We have traditionally done this! We have set a unit test after we have taught the unit…..the student prepares for the test; writes the test……then they concept and skill is not assessed or evaluated formally again until the exam!!!

We’ve got all year/semester to get them to demonstrate these skills. We should have an assessment/evaluation policy that reflects this.

Second one,

From Growing Success:

“The primary purpose of assessment and evaluation is to improve student learning.”


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Our main goal should be to help students learn math “better”!!! Our assessments should be apart of this!

Traditional teachers are going to argue “We need to prepare them for university”—- My argument is that I want them to know their stuff well! I want to show them that they can always do a little better. I want their minds to grow, their math knowledge isn’t static. (Also, I remember from my university days, professors saying that if you did better on the final then they would take this mark as your whole mark…..rewarding learning!)

My new philosophy:

Everything is upgradeable!!

Here is what I’ve been doing for grade 12 evaluations lately:

1. I still have the test on test day! No different than before.

2. I mark the test and hand it back. I remind the students that all skills/concepts are upgradeable!!

3. The students then sign up for help/upgrade session at lunch, or down time during class.

The student is to bring their test/evaluation with them….we go over it together discussing missed concepts. We pick together curriculum expectations that they can upgrade.

Say, for example A student received 1 out of 4 on a question testing the skill  “Solving polynomial Equations”. We discuss the mistakes and the students can re-do that question. Once that question is correct I give the student another, new question that tests the same concept. If that student can prove they know the concept two times in a row….then I give the original test marks to the student….they now get 4/4 on that question. (I keep helping and giving questions to the student until they can complete it on their own. )

5. I then go into my markbook and change the marks!

Here is a recording sheet (nothing revolutionary) I use so that I can track their upgrades. Most upgrades span multiple days.

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They can do this for any concept/skill from the test….they can do this for every test.

So can a student get 100% on a test? My answer: Sure! Does every student in my class get 100%  Not even close…. I wish they would! They have that option.

If my students know and can demonstrate the skills and concepts of my course consistently why wouldn’t I give them full marks!!! Their mark is supposed to represent their learning and knowledge…..not their work ethic!

The students’ attitude toward test days have changed. Some are working harder before the test so they don’t have to spend their lunch time upgrading…..some have told me their anxiety towards their tests have been lifted!

Other teachers who helped my with my thinking:

Dan Meyer:  The Comprehensive Math Assessment Resource

Evan Weinburg: Standards Based Grading: Bridging the Gap

Most recently: Mary Bourassa: Rethinking Tests – who inspired me to write this post.

This process is most likely debatable and definitely can be improved upon. I am still learning this process myself and would love to discuss these ideas, so please leave some feedback!


Describing Relationships – Active Learning

We have started discussing graphing relationships in my grade 9 class.

Our lesson goal was to

I can…

describe a situation that would explain the events illustrated by a given graph of a relationship between two variables

I have loved and wanted to incorporate them…and some of Dan Meyer’s graphing stories …into my lessons! This is the perfect place…’s how we did it.

Day 1: We start with an estimation…..

I play the video (Poor quality) about 6 different glasses filling up with coloured water.

I pause it after a few seconds and ask: “Which glass will fill up the fastest? Draw the shape of your winning glass on your whiteboard” We take some guesses from the class…..then watch some more video. About 3/4 of the way through I pause again. “Anyone want to switch glasses?” Some of them will switch and some stick to their original guesses. After watching the final result…..I get some boos and “You tricked us”. We discuss why they filled up at the same time….which leads us into saying Volume of all  = Same and comparing the volume of the glass vs. time……which leads us to discuss the rate at which the glasses are filling = Also Same!

From there we jump right into and the WaterLine Activity.

This is one of the most engaging tasks on this topic I’ve seen. The predictions of height of water vs. time and the ability to self correct is amazing! As the kids work through the activities I always take time to discuss their answers to:


Screen Shot 2014-10-27 at 7.59.55 PM


We have a great discussion on how this new graph looks and I get them to draw one one on their whiteboards. (wish Desmos could add that prediction/check into the activity!).

WaterLine and its discussion takes the remaining of the period to complete.

Day 2: Graphing Stories & Distance Time!

We begin by watching and drawing different graphing stories from Dan Meyer (Elevation, Distance vs. Time). Check them out and download them! (He doesn’t even charge $4.99 for them!).

We then use Vernier’s LabQuest2 and the motion detector (borrowed from Science). Students walk in from of the detector and the rest of the class watches in real time how their distance changes vs. time. We get the kids to walk slow….then fast, walk away/towards….so they see how the graphs relate to real motion!




Lastly….. The kids are to create their own video graphing story and have peers draw the graphs to match.IMG_2664-0.JPG

  1. They pick a card that has some sort of motion described: “student walks fast away….then backs up slowly”
  2. They create a video that shows that motion. They use their phones or our iPads to capture that motion.
  3. They trade that video with another group and then they have to draw the graph to match!
  4. They keep trading and graphing different relationships!

The first time I ran through this series of activities I freaked out after because we just used two days and the kids didn’t solve/answer any problems from the textbook/worksheet! They don’t have anything written in their notes!!!!


IMG_2667.JPGEvery year I realize that there is great thinking and reasoning going on here! We have great discussions on what the graphs should look like. We need less notes and more active learning!! The kids are completing problems because they created them! They lived them! I feel they have understood the main concept at a deeper level!

anyways……thats all for now!









Pumpkin Time Bomb – Data Collection

I was watching Jimmy Fallon on the Tonight Show last week and saw this…

After wanting to try it myself (I did on the weekend with my kids – messy) I had a “wonder question.” I wondered would it be possible to predict accurately the number of bands to destroy any pumpkin?

After a few tweets:


I decided to gather some data…..maybe other math teachers out there would blow up a pumpkin and send me the data. I put together a form/activity for people to fill out. See the form below.

So….when the data files in I will send out a spreadsheet so teachers can analyze the data with their class. Hopefully we will see some relationships….and be able to predict how many rubber bands to blow up pumpkins next Halloween!

Here are some videos so far….


Please contribute to the data

Fill out the form


Kyle Pearce (@mathletepearce) has written on his blog about this activity. Check it out!

Here is the open spreadsheet with all entries so far

Here is a desmos file showing some relationships (or non-relationships)

This is my class and our result —- 407 bands


Pattern Talks

I was blown away by my grade 9 class today. They have been so insightful and brave when sharing their strategies when solving problems!!!

On Thursdays we have been starting class with Fawn’s site Visual Patterns. Today we were working on this one….

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How many boxes will be in the 100th frame?

Here is one solution from a student:


ME: “Tell me how you’re right”

KID A: “I saw it was growing by 4 and knew to multiply the frame number by 4…..but it didn’t give me the right number… i adjusted it by adding 1. So in the 100 frame I multiply 100 by 4 and then add 1….gives me 401″

Honestly,  this is how I expected everyone to do it…… here is the next one I saw (2 students had this solution).




ME: “Tell me about this”

KID B: “I noticed that one the first one there is 1 box on each corner of the middle box. And on frame 2 there are two boxes on each corner…and on frame 3 there were 3 boxes on each corner. So I knew that on frame 100 there would be 100 boxes on each corner…..thats 400 and add the one in the middle”


I was so surprised……I loved that these students looked at the pattern that way! It was refreshing that they used a strategy that worked for them and not waited for the teacher to show them how to do it!

Here is another one from the same class….




ME: “Tell me about this”

KID C:”Well, you are adding 4 every time…..if you forget about the first frame and imagine that block of 5 removed from the second frame….then you have 4 then you are really just adding 4 99 times……4 X 99 = 396. Then just add the original 5 back on to get 401.”

I’ll admit I was blown away with not only the solutions but the courage the students had to get up and share their methods. (oh…..We haven’t “formally” discussed Linear Relations YET—-just warm ups for now)

If you haven’t already checked out Visual Patterns… better…..they are great warm ups or warm ups that turn into full periods!

Subtract Before You Add

I’ve been reading Dan’s posts for a while now and this is one series that has stuck with me…..  “You Can Always Add. You Can’t Subtract.” Ctd . Here is one small way I used this concept today…

Here is the original problem from EQAO

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My applied students in previous years have struggled on problems like this.  I simply “whited out” some….

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That pic is what I start with. It’s amazing what happens! With the original problem students immediately are turned off….it looks tough, and confusing. It’s got a few things going on!  With this new one… are voicing their thoughts on a bike priced at $175. They’ve got lots of ideas on this statement. They start sharing stories and are comfortable! It’s pretty awesome that this “subtraction” strategy has changed the tone in my classroom. It’s more inclusive……more safe!

I then reveal a little more….

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We had another discussion on what this means…..but most students jumped right to finding that percentage (we calculated percentages last day). Kids were engaged! just because we started with that small simple statement.

I then revealed everything once they had calculated that percentage.

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Again, they were off and working…..I had a student put their solution on the board and we discussed it. It was awesome to see that there were not intimidated on this problem! It was also awesome to see that they could apply their knowledge when we saw another problem like this!

thanks Dan!