Logarithmic Warm Up

Our goal in Advanced Functions today was to graph y = log(x) and transformations of y = log(x). Here was what we did as a warm up/minds-on. Everyone started with a whiteboard and a device of some sort (SmartScreen Shot 2014-12-08 at 2.16.38 PMphones, iPads, tablets—I have access to a set of iPads for anyone without a device. This class being a grade 12 class….just about everyone has their own device).

I had them log into a PearDeck file I created.

I gave everyone in my class a number. Some got integers 1-20 and some got fractions 1/4 through 1.

My instructions:

1. Take your number, n, and find log(n). Write your number and log(n) as an ordered pair! (n, log(n))

I started the PearDeck presentation which showed them this slide……and gave them all a movable point.

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2. Move your point to the location of the ordered pair you have! So my students started moving their points around and on the projector screen we can see everyone’s points all at once in real time! So we are basically watching the creation of y = log(x).

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Sorry for the poor quality of pic…it was on the fly

You could see the looks on their faces as the graph was being created….pretty cool

From here we took a note on the properties of the function….then kept using PearDeck to analyze transformations of y = log(x). We saw Graphs then –> wrote equations and then saw equations –> drew graphs. We could do everything right in PearDeck so we could see all of our answers all of the time! PearDeck works through your Google account…..give it a try!

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I thought it was pr

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Projects from 2013-2014, iTunesU, iBooks

I feel like since I started using Twitter for Pd and following the #MTBos my learning curve has been steep! As a result, when I look back at resources I have created I find myself wishing that I had done it “this way” or “that way” or used a different task here or there.
This is no problem for creating lessons, activities, tasks to use with MY students because I can always modify, change, manipulate!
My regrets show up when I’m involved in a project that gets published or shared out beyond my reach and afterwards my thinking has changed by seeing a great blog post, or a new activity, or by just having a discussion with a colleague. I wish I had all the time in the world to keep everything “up to date”

Our blogs, Google Drive Folders, Dropboxes are easily update-able and where are current lessons are!

Here are a few of those projects that I hit publish and find it hard to keep up to date: Each of these has lots of stuff I’m proud of, and some stuff I wish I could find the time to “update”

ITunesU Course – MPM1D (created April 2014)
Last spring I created an iTunesU course for my school board (Lambton-Kent).
Purpose: To share some digital resources I have used, or, are using in my grade 9 Principles of mathematics course.

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https://itunes.apple.com/ca/course/principles-mathematics-public/id946920145

iBook – Measurement (created July 2014)

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Whatcha Thinking – independent use of 3 Act Math Tasks for MFM1P

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iBook – Linear Relations (created July 2013)

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Popcorn Pandemonium

My afternoon grade 9 applied class (as a group) is very outspoken, loud, and restless (maybe it’s because it’s the afternoon and they have been sitting at desks all day). They have been a challenge to keep on task. So….I  am trying to find opportunities for them to be outspoken, loud, and restless.

A few weeks ago I came across this post by John Berray. Using/eating marshmallows to compare rates of change. I loved his idea of “experiencing rate of change” I decided to re-purpose his lesson to meet our goal of—> “I can solve a linear systems of equations by graphing.” I also took his recommendation of using popcorn instead of marshmallows…..and it paid off!!

Here is the low down…. we start the “Math Dial” off low.

ME: OK you are going to have a good o’ fashion popcorn eating contest!

Start with this video:



Ask for questions:

Here are a few from math tweeps

here are a few questions we can address with this problem.

  • When will Tim and Don eat the same amount as Jon?
  • Who will eat the most when the minute is up?
  • Will there be a time when Tim and Don eat the exact same amount?
  • When would Don eat more than Tim?

ME: Ok lets figure out who will eat the most in the 1 minute. But I want to recreate the video with you guys.

So I made a giant bowl of popcorn. (Don’t have time to make enough popcorn? — have kids give high fives to a timer instead)

Arrange groups of 2 or 3 and everybody grabbed some popcorn to start!

Round 1:

In each group kids are to choose who to mimic, Jon, Tim, or Don. They are to eat just like them! Allow them to ask about how fast each person is eating….or how much did each start with, etc.

Show Act 2 to answer those questions:


Tell them to get their timers ready….because they will eat just like one of those guys. Ready…..all you Tims and Jons eat your starting amount … Set….Go!

Start the timers and eat!

Question 1:
After they are finished, have them work out on their whiteboards who would eat the most in a minute.

Question 2:
When would Tim & Don eat the same as Jon if ever? (Great potential here for integer solutions talk).

Question 3:
During the minute, at anytime did Tim and Don eat the same?

If there was no time limit find when Tim & Don would eat the same?

Used this handout so they could create tables of values. Had them graph in Desmos!

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The awesome thing was that my students were desperately trying to find the equations to match their graphs….they didn’t want to plot all the points. I visited each group helping them find the equations if needed. Once the equations were in desmos they knew where to look.

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Act 3 – The reveal of who ate the most in a minute


Round 2: Do it all over again with new eating patterns!
Here are two possible eating pattern cards to give out:

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Students who finished early worked on our Crazy Taxi  vs. a new Insane Cab

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(@mathletepearce has a nice write up on using the Crazy taxi problem in class.)

Next day! Solving Multi-step equations…..will solve this systems of equations algebraically.

Let’s Start with the Easy Ones

Here’s how I taught students how to solve trigonometric equations in our grade 12 advanced functions class.

Started with this Ferris wheel problem

From 101qs.com find it here

What has been working well is starting our “math” at a very low level…..like on a dial…..then we slowly turn the dial up….adding more “math” in. Read more about the Math Dial from a comment on Dan Meyer’s blog here.

Starting with this video the math on the “math dial” is very low.

I asked: What questions do you have after seeing this….

Answers:

How fast is it spinning?

What’s the radius?

What’s the period?

Where will the red dot be after 3 min?

And that last one is the question we studied.

Act 2:

From 101qs.com

Almost all kids solved this problem using proportions! They kept the dial in the low position still!  They realized that it takes 5 seconds to travel from dot to dot. Therefore it takes 40 seconds to go all the way around. They divide 3 minutes up into 40 second sections and get 4.5 rotations. The dot will end at the top of the Wheel!!  But the Trigonometry in me was screaming to get out……I asked, “Did anyone create a trig equation to model the height?” — cue crickets!

So we cranked the math dial up a tad!

I said:
When I go on a ferris wheel I always look for my house.” We talked about how high that might be in relation to Dan’s problem….we settled on about maybe 40 feet.
My question: How long will it take to get to that height?

Guesses? Will it be a nice number? No? Why not?
Crank it up a bit more …
Let’s create an equation for the height in terms of time (we had already learned how to do this and it was no problem for the class) .daum_equation_1417134251422

Now, to solve our question we have to solve this equation!

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Student: That looks super hard!
Me: It does doesn’t it!

Let’s make that our goal!
We don’t want the math dial going up too quickly!

Let’s start with the easy ones, like this:

Screen Shot 2014-11-27 at 7.27.23 PMGotta keep the math dial low for a bit more…

Screen Shot 2014-11-27 at 7.29.22 PMWe solve this as a class, then another, and another, slowly building up our skills; slowly bringing the dial up. We stop at the end of the class. I assign a few more like the ones above. “Let’s get good at these so we can do the super hard one… Practice these for homework….”

Next day:
We take up the assigned questions then get back on track! We then solve these:

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We have a discussion on how many solutions there are here… and plop down a graphical solution in Desmos

Screen Shot 2014-11-27 at 7.30.00 PMThe math dial is getting up there…

Me: “Are you ready to try the big one?”

We do it! And everyone is into it….they have been waiting two days to see the answer! And the dial is pretty far up there!
After:
One student says: “That was pretty awesome! ”
That was my highlight of the day! Best compliment for a teacher!

We then show the graphical solution in Desmos. IMG_2795.JPG

Oh…..and we started class playing Pictionary (It’s our Wednesday thing) there was a tie and we have a good o’l match of Rock, Paper, Scissors to declare the winner. It was Intense!!!

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Many Many Volumes

In our senior math classes (advanced functions & calculus) we come across a problem like this….Screen Shot 2014-11-26 at 7.00.47 AM

I really like these problems, they have great potential but not really in this form. Let’s jazz it up and spend an entire class with this

Start with this video:

Ask What questions do you have about this?

Them:
What size is that rectangle?
Why are the corners cut?
Is volume always the same?

Etc,

My question:

What size of that square do we cut out so the box has the biggest volume?

Play the video again and have them yell out when they think the box has the largest volume.

Have them guess
What is too small?
What is too large?

Have them take their guess for the size of the corner and find the volume of the box

Draw a picture of the “card board” label the dimensions.

Draw the squares to cut out. Optional (Cut them out) make the boxes.

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What’s the new length?
What’s the new width?
What’s the height?

What’s the volume?

Is this the max?
How can we check?
Have them do another? And another.

Have them come up to your computer and enter their height and volume in the Desmos page for each box.

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Now, let’s generalize!
This time let your guess be x and find an expression for the volume.

What’s the new width? Take 8 and subtract twice your guess. (8-2x) Now the length? What is the height now??

Put that expression into Desmos and let them see the function, let them point to the maximum.

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For calculus: have them find the maximum using derivatives!

Show them this video to check their guesses.

From here we can solve problems like our original textbook question. The kids are invested now and they are ready to use the equation to find the value of x where the volume is say 24 cubic units.

Further reading: Jonathan Newman’s volume of a box Activity 

Credits: Algebra in motion for the Geometers Sketchpad file. Dan Meyer – this lesson mimics his Circle Square lesson.

UPDATE [Nov. 27, 2014]

Luke Walsh created a Desmos Sketch that seems super useful!


 

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.

ME:

“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”

Next,

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? …..is 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:

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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….

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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….

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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…..

A

 

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….

D

 

 

F

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: http://www.qrstuff.com/

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!!

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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…..

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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!

Estimates

Estimates

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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Oh!!

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.

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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…

Desmos,

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…

Screen Shot 2014-05-13 at 9.53.09 PM

 

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!