Buying Calculators Problem #MFM1P/#MPM1D #PrBL

As an introduction to linear direct variation, I put together a quick problem-based learning task that was proportional for my combined academic & applied class:summary-2017-02-15-m9-1

Scenario:

Buying calculators.jpg

What do you notice?

Capture.JPGI had to use the Pear Deck dashboard to hide some responses that involved calculating the price per calculator as this was part of solving the later problem. I suppose I could have left them up, but I wanted to leave the calculating part until later when students were in their groups.

What do you wonder?Capture.JPG

How much would it cost to buy a class set of 25 calculators?
Best estimate: ________$
capture

Solve:

You can find the Pear Deck slideshow in this folder. Also in the folder is a follow up slideshow exploring the concept of Direct Variation.

– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)

Height VS Foot length #MFM1P/#MPM1D #3ActMath

As an introduction to Linear Relations with my combined 1D/1P grade 9 Math class we investigated height VS foot length and the guinness record holder for the tallest woman:Summary 2017.02.08 (1).jpg

I asked students to measure their height and foot length and record it on a Google Spreadsheet we had up on the projector:

Capture.JPG

What do you notice?Capture.JPG

What do you wonder?Capture.JPG

I posed this question:

Zeng Jinlian was born in 1964 in Yujiang village in the Bright Moon Commune, Hunan Province, China. She holds the record as the tallest woman. She measured 2.48 m (8 ft 1.75 in) when she died on 13 February 1982. How long were her feet?

Estimate: _____ cmCapture.JPG

Students were sent in their VRG groups to their VNPS boards to solve. Here are their boards:

Since it is still early in the semester I scaffolded the activity a bit by instructing them to create a scatter plot of the data on their board to help them solve the problem. I did not however instruct them to use a line of best fit, although many groups used that strategy to help them come up with an answer. Some groups had graphs with a Height axis that went high enough to lookup Zeng’s height and find the corresponding foot length from the line of best fit. Other groups made an educated guess based on the trend the points were showing.

Each student was asked to determine her foot length based on their graph:Capture.JPG

Her real foot length was 35.5cm!

The following day students were assigned some individual practice with scatterplots on Khan Academy.

Activity materials (include a slide deck for Pear Deck) available here.

– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)

Flight Costs #MFM2P

I’ve done this activity once previously. I changed how I did it for this second go. I will change it again for next semester.

Here’s how it went this time …

Students were presented with this data:copy-of-lr-flight-distance-vs-cost

Students were asked:

2016.12.21 notice.JPG

2016-12-21-wonder

The task for day 1: Determine the initial value & rate, on average, for flights with Air Canada.

Some groups went to Desmos straight away. Others needed some reminding that Desmos can be very helpful with data like this.

On day 2, groups were asked to determine the distance they could fly for $500 using their equations from the previous day. I only took a photo of one group’s board that day:2016.12.22 summary.png

I think next semester I will change this up. I think I will present the name of a city & ask students to estimate the cost of flying there. Then I’ll give them the set of data for cost & distance for multiple cities, but with the first city blanked out; perhaps allowing them to adjust their estimate if they like. We’ll do notice & wonder, and then proceed to solve for the price. I won’t specify modelling algebraically but will perhaps create a Desmos activity builder they can do to practice that in the case where they don’t use an algebraic model to solve.

Activity available here.

– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)

Cup Stacking #MFM2P #3ActMath

This activity is another one that stems from one by Al Overwijk, but a bit more directed than his. In Al’s activity, he opens it up to any model of cup stacking which results in relations that are linear, quadratic & beyond. Maybe my class can come back to this later in the course & do it that way. For now, we stuck to a nested cup model that I knew would result in a linear pattern since that was my learning goal for this activity.2016.10.11 summary.jpgAlso different from Al is we used a stool simply because we don’t have enough solo cups to get to the height of a person when they’re nested inside each other 😦

The Scenario:

lr2-cup-stacking-prompt

Act 1: Notice, Wonder & Estimate

What do you notice (facts)? What do you wonder (Q’s)?
What is the height of the chair
The cup is way smaller then the stool How many cups do you need to get the same height
Each picture has more cups How many cups does each picture have
there more than 1 cup. why do you have only 3 chairs?
Each picture has different number of cup How many cups that can be same hight with the chair
there’s more cups in each picture how much cups does it
Stool and cups How many cups we need to put together so that they have the same height with the stool?
What’s the height of the chair
The number of cups increase Why are there cups
– There are a different number of cups in each picture.
– The stool is much taller than the stacked cups.
– The only thing that changed is the number of cups
– How many cups will it take to stack it as high as the stool?
More cups in each picture How many cups do we need to cover the hight of the chair
cups what is the volume of each cups?
The number of cups is increasing How many cups would it take to be the same height as the stool
3 stools, cups is increasing How many cups it will take to reach the height of the stool
In each picture there is a certain amount of cups

The chair is always the same

How many cups it increases at a time

2016-10-11-estimates

Act 2: Collect data & solve

Each group received 8 cups and were instructed not to share their cups with any other group. They quickly grabbed rulers & began measuring. They made tables of values on their boards to record their measurements. But quickly most groups began to look for patterns after taking 2 measurements & then continued their tables based on their assumptions of how it will continue to grow. But I let them continue with that thinking for now. Here were there boards:

So every group had an answer but the strategy was lower level (tables and making assumptions about rate of change). Next I had them log in to a Desmos activity I had built to get them to perform a regression on their data. I expected them to input heights for 1, 2, 3, …, 7, 8 cups. But of course they input the tables they’d made on their boards that were full of inaccurate assumptions about how the stack would grow. So we had a discussion about whether their tables were data or assumptions. I asked which one would we rather base our regression on? They decided data. So I handed 8 cups back out to each group (this was now onto day 2 & so their groups had changed since we do “visibly random grouping”) and asked them to measure the height for 1, 2, 3, …, 7, 8 cups and adjust their tables in Desmos. The activity asked them to decide if the relation would be linear or not (they agreed yes) and then asked them to perform a linear regression. I love the recent Desmos feature where it carries their current graph & work forward to the next slide/question! Here are some samples of their Desmos work:

2016-10-11-regression2016-10-13-desmos

I then asked them to take one of the equations for line of best fit that they’d found in their group and go to their boards to solve for the number of cups needed to match the stool’s height:

Act 3

We stacked the cups as a group next to the school and the actual answer was 73! We discussed possible reasons for differences between our first answers, our answers after using Desmos & the real answer.

A third day was spent practicing two skills that had been assigned via Khan Academy over the previous days of work:
Graph from slope-intercept form
Slope-intercept equation from graph

The entire activity is available here.

– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)

Problems We Solved in #MFM2P

My plan was to blog about every problem-based learning activity I did this year. I did not succeed; I think I blogged about two from the my MFM2P course? So as a runner up to a full blog post reflection on each, you’re getting one post with a summary image of each activity or problem & a link to my materials for it.
I’ll group them by strand here, but they are not listed in the order that we did the activities. If you’d like to see the progression of activities I used, you can see that here.

Linear Relations

26 Squares: This one I did manage to blog about.Summary (11).jpg

Banquet Hall2016.04.22 2P summary.png

Phone Charging2016.05.19 2p.JPG

Phone Plans2016.04.27 2p.png

Gummy Bears: I did blog about this one here.Summary 2016.02.29 2P.jpg

Measurement & Trigonometry

Lamppost: w/ shadows 2016.04.15 2P Summary.jpg

School Height: w/ mirrors 2016.05.25 (1).JPG

Tree Height: w/ clinometer 2016.05.11 2p summary.jpg

Wheelchair ramp2016.06.08 2p (1).JPG

Filing cabinet post-itsSummary Filing Cabinet 3-Act.jpg

Pyramid Post-its2016.05.13 2p (1).JPG

Quadratic Relations

26 Squares: I did blog about this one2d 2016.02.08 (1).JPG

Visual Pattern2016.04.12 2p summary (1).JPG

Not every lesson we did was problem-based. Sometimes I need to do some direct teaching right from the get go, like with expanding & factoring. Other times we explore & investigate by drawing & cutting out shapes, like with similar triangles & trigonometry. But in any case, maybe someone new to the MFM2P course (or not so new to it) will find these activities useful!

– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)

26 Squares – Perimeter #MFM2P

26 squares is an introductory investigation I use in MFM2P. It comes from Al Overwijk & Bruce McLaurin. The idea is that you can use the same set of manipulatives – 26 squares of varying sizes with an overlaid grid – to run investigations/activities to introduce each of the 3 strands in the course; linear relations, quadratic relations, and measurement & trigonometry (similar triangles, Pythagorean Theorem, trig). Today’s first investigation introduced linear relations.
Summary (4)   Capture

Investigation Question

What is the relationship between the side length & the perimeter of a square?

Students were asked to predict the relationship. A sample of responses:Capture

Table of Values: Groups were sent to their VNPS station to create a table of values of side length & perimeter using their squares to collect data.

Some groups correctly counted the perimeter using the grid. At least one group was squaring side length, so I went over and we talked about counting perimeter using the grid & they changed their table of values. One group (red marker) decided to measure the lengths with a ruler instead of counting w/ the grid.

Graph: Back at their desk students graphed their data by hand on this handout (forgot to take photos of student work here). I then had them all decide whether or not this was a linear relation & why. This led to a class discussion of the graph being a straight line as well as the pattern in the perimeters. At this point, groups were sent back up to their boards to determine the first differences for their table & we discussed their findings (again, I forgot photos here).

Equation: Back at their desks once again students worked their way through this short Desmos activity I created asking them to create a graph & perform a linear regression to find the line of best fit. A summary of the student work from Desmos:Summary (4)Students then completed 4 practice problems on the earlier handout to solve for either perimeter or side length given the other. This all took 2 days and they had time at the end to start the homework which was a Khan Academy exercise set titled “Slope Intuition”.

Update: I added a 3rd day to wrap-up this activity and talk about representations. Students completed this handout:RepresentationsThey had to name the 3 different representations & explain how they are all related to each other. After 5 minutes of working on it themselves, I had them get up & walk around the room to read each others’ sheets in a gallery walk type style. Then they returned to their seats & could add, change or erase anything from their own notes. I then led a class discussion about the connections of slope & y-intercept between the 3 different representations.

Reflection: I wish I’d included a “word” representation such as “Perimeter is equal to 4 times the side length”.

Handouts & Pear Deck interactive slideshow here
Desmos activity here
26 squares here

– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)

Phone Charging activity #MFM2P #3ActMath

Summary (3).pngThis activity comes from the awesome Michael Fenton. He also has a Desmos version of it which I posted on my website so that any absent students could do that version of the activity at home.

Act 1

Charge.002-1024x768 (2).jpgI asked “What do you notice?”:CaptureThen I asked “What Mathematical questions do you have?”Capture

I then presented them with the question I had chosen ahead of time to address the curriculum expectation “Create graphs and equations of linear relations” & asked for an estimate that was too high, too low & best estimate:Capture

Act 2

I asked what information they would need in order to solve this problem & got answers such as . . . Capture . . . so I gave them:Charge.007-1024x768 (3).jpg

They went to their boards & solved. Some groups counted up the entire way using a table of values:IMG_0387.JPGIMG_0385.JPG Other groups determined a rate of charging that they used to multiply by the % charge left to go:IMG_0386.JPGIMG_0388.JPGThese two groups above had some guidance along the way from me in terms of questioning the values they picked for the change in charge and the change in time; both groups reworked their solution at least once.
One group, with help from a discussion with me, created a proportion:IMG_0384.JPGThis group had the proportion idea on their own but needed some guidance in how to set it up properly.

Act 3

Answer reveal:Charge.010-1024x768 (1).jpgMichael does a great job of breaking down why the actual answer is so different from the models my students made here on his original blog post.

The following day we worked through this handout which forced them to create a graph & determine an equation for the relation.

And, as always, here is the folder with all the materials for this activity.

– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)

Tile Pattern Activity

If you haven’t checked out Fawn Nguyen‘s Visual Patterns website before, you should do that right away! It’s pretty awesome.
Today’s MFM2P activity was built off of a pattern from her site:Tile Pattern MFM2P Activity

Most groups came to the correct answer. We have to work more at showing all our steps & explaining all of our thinking. We consolidated with a handout that asked for a table, a graph, and the y = mx + b equation. We did not build the model to check if we were right (act 3) … students don’t seem to miss act 3 when it involves a long building process?!

All the materials for this activity are here.

– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON

Toothpick Houses

Here is the 3 act math task we did in my MFM2P class today:Toothpick Houses

2 of the 5 groups found a rule (albeit two different rules) to calculate higher values like 500 houses. One group (on the chalkboard) found the rule I was expecting using slope & y-intercept of 4 x step number + 1. The group on the whiteboard above used 5 x step number – (step number – 1) even though they didn’t write it out in that fashion. I had them explain that to me orally (which required some prodding & questioning from me to get it out of them). I love that the group came up with this 2nd rule as it wasn’t one I predicted; so awesome when they surprise me & see patterns in a new or different way from me!

Mostly all of the groups counted up in pictures or tables to solve for 30 houses. Note to self: pick a higher number next time, maybe 99?

Here they are making a row of 30 houses to check their answer (act 3):Toothpick Houses Act 3

– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON

Equation Headbands

Last year, I read a post about Quadratic Headbanz by Mary Bourassa and thought it sounded like a great game!

I’ve been teaching mostly all MFM2P classes (applied grade 10 Mathematics) for the last two semesters. In the 2P course we only get into quadratic equations superficially. So I’ve been making mostly linear headbands for my groups. I’ve used the game in both my grade 9 and grade 10 applied classes so far. Here’s how it works:

I bought wide ribbon from the Dollar Store & cut lengths long enough to tie around their heads in a bow at the back; about 1 meter long I think? Then I wrote out a variety of linear equations on strips of paper that I taped to the ribbons:IMG_7869

Playing the game:

  • Each student is given an equation headband.
  • They are instructed to put the headband they were given on someone else who is not seated at their group ensuring that the person can’t see the equation you are putting on them.
  • Students walk around the room asking yes/no questions of their classmates. Questions such as “Is my slope positive?”. Classmates may answer yes, no or I don’t know. They are not allowed to ask the same classmate two questions in a row.
  • When they think they know their equation, they come to me and tell me their answer. If wrong, I send them back out to their classmates to keep trying. If they are correct, I remove their headband for them and send them back out to answer the questions of those students still working to determine their equations.

In the past few weeks I added a new step to this game: graphing. Not only did you need to determine your equation, but you had to create a correct graph on a handheld whiteboard with the Cartesian plane.
IMG_7824
My students found it tough but they did it! A good number of my kids knew their equation but were struggling to graph it. It was awesome to watch the stronger students that finished first go back and help teach their peers how to use the slope and y-intercept to make their graph (I had to remind them often not to graph it for them, help them by explaining & asking questions … “don’t touch their marker!”).IMG_7823

I use this game as a bellwork (although it takes longer than the usual bellwork task) on days when we might be doing more individual practice and thus fairly sedentary for the rest of class. This is a great way to have everybody up and moving around the room, talking to different classmates before settling in to the main seat work on a given day.

– Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)