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:

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

What do you notice?

What do you wonder?

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: _____ cm

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:

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:

Students were asked:

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

This group worked with a quadratic model. And while it wasn’t the target learning goal for the lesson, it turned out to be a better fit than linear! Made for a great class discussion.

this group was working on a average rate per kilometre after seeing another group try the same

this group started trying to find the average cost per kilometre. they then switched to a linear regression on desmos after seeing others doing that.

love how this group labelled the rate & initial value

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:

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.

Today while Ms. Fahmi, my student teacher, was teaching I went to take a photo of the students at their boards solving in their groups. Then realised that I should try taking some video since there are several of us in the room & I can take the time to do so (I had parents choose at the beginning of the year whether or not they were comfortable with me including photos & videos of their child in class on my professional learning network platforms)

This summer Pear Deck announced the introduction of student-paced mode; the ability for the teacher to allow students to work through the slide deck at their own pace. This is a feature I enjoyed in the Desmos activities I’d been building for graphing (interesting also that Desmos introduced their teacher-paced mode around the same time that Pear Deck introduced student-paced; both platforms now offering both pacing options).

Not sure what Pear Deck is or does? Watch this quick video before reading further:

How to turn on student-paced mode:

Click the 3-dot menu icon on the bottom right of your screen while presenting your Pear Deck, and the option to turn student-paced mode on (or off later) will be there:

Act 1 consists of present my students with a scenario via photo or video & asking them

What do you notice?

What do you wonder?

Then I show them the problem I’ve chosen for the day (usually it’s one that most kids write down for “what do you wonder?” since I’ve carefully selected the scenario to lend itself to asking the question I want based on our learning goal).

Estimate the answer: too high, too low, best guess?

Act 1 happens via Pear Deck in TEACHER-paced mode. Students are at their seats in their visibly random groups for the day assigned by playing cards. They use their own phone or a loaned chromebook (I have 6 that live in my classroom) to answer these questions on Pear Deck. We often have a quick class discussion here too about reasonable estimates and their strategies for that. I, as the teacher, am choosing when to move the slides forward for the entire group.

Act 2 consists of sending each group to their assigned vertical non-permanent surface (ie. chalkboard or whiteboard) to solve the problem. Often groups also need to do some data collection or measurement here in order to solve the problem.

At this point I have a slide with the original picture & the problem to solve written on it projected on the board while the groups are solving. The moment the first group to finish solving heads back to their seats, this is when I turn on STUDENT-paced mode. The rest of the slides will be follow up questions to reflect on their solution or to apply their thinking to extension problems. Students work on these at their own pace at their own desk.

When all groups are done and back at their seats, I lead a class discussion about the solutions from each group using the 5 practices for orchestrating productive mathematics discussions. During or after this discussion, we might also look at some of the responses to specific follow up questions on Pear Deck. If we do, I turn OFF the student-paced mode to bring everybody’s screen back to whichever one we are discussing.

Act 3consists of checking our answer either in real life (as we did for the cup stacking activity) or by showing a video or image answer (as we did for the phone charge activity).

Normally, in Pear Deck, there is a projected screen being shown on the board to the whole class by the teacher. The students see a “response” screen on their own device that is different than the one being projected. When in student-paced mode, the student can see both the content slide AND The student response slide on their own device. On a tablet or laptop the two screens are shown side by side when in student-paced mode:
When using a smaller device such as a phone or iPod, the student will see a blue bar across the bottom of the screen allowing them to toggle back and forth between the “content” & “response” screens:

Have you used student-paced mode in Pear Deck yet? Share in the comments below how you use it with your own students!

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

Not the most exciting problem, but my students were still engaged even if it wasn’t a contextualised scenario.

Act 1:

What do you notice (facts)?

What do you wonder (Qs)?

– The shape is a pyramid that has a square base.
– The area of the triangle is 1 cm square.

– What is the area of the base?
– What is the volume of the shape?
– What is the surface area of the shape?
– What is the height of the shape?

It is a triangle

What is the lenght and height of the triangle

It’s a square based pyramid

how many sticky notes do we need to cover the square based pyramid

It’s a Square pyramid

It’s a triangle and it has 1cm squared

What are the lengths and widths of the pyramid

Its a shape.

what is 10m2?

pyramid

Square based pyramid, with a sticky note that reads “I cm squared)

Why is there a sticky note on one of the sides?

That it is a square base pyramid

What are the other lengths

There is a square based pyramid

What does the 1cm^2 represent?

There is a triangle

What is the value of this pyramid

what’s the area of the square based pyramid

Estimate:

Act 2:

Each group of students was given a plastic pyramid like the one in the picture. They began measuring dimensions of the pyramid and using the formula from their formula sheets in their binder. They solved the problem on their boards:

I asked the group why they thought we got different answers in different groups and they commented that some of our plastic pyramids were slightly smaller than others. I did a little direct teaching about the net of a square based pyramid and how that translates into the formula on their formula sheet:

Act 3:

I then handed out grid paper and asked the students to draw a 1 cm by 1 cm square at the top left of the page. They told me that the area was 1 cm^2 and determined that every 4 squares of our grid paper made a 1 cm^2 area.

I asked them to trace all of the faces of their pyramid onto the grid paper to create a net. Then to colour in alternative 4-square blocks to allow us to count the area in cm^2.

We counted up the area and found the answer to be 114 cm^2; right on with our calculations!

Students were assigned a “surface area” practice set of questions on Khan Academy; different ones depending on whether or not they had completed the previous set I assigned earlier in the semester.

The materials for this activity are available here.

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

Last week we used similar triangles to find the height of lamp post out front of the school:

Act 1

Scenario:

What do you notice about the lamppost? (FACTS)

What do you wonder about the lamppost? (QUESTIONS)

The pole is taller then the person

What is the height difference between to man and the lamppost

It’s a lot taller then the person

how much taller is the lamppost compared to the person?

its a tall lamppost

How tall is the lamppost?

The iamppost tall than the boy

What height the lamppost and what the height of the boy

there’s a person beside the lamp post

how much of that person does it takes to get the height of the lamp post

The lamppost is tall

What is the height of the lamppost?

What’s the height of the lampost

What’s the height of the lamppost

– The post is taller than the person
– The structure of the lamp post is sturdy

– How much taller is the lamppost than the person?

– How tall is the lamppost?

– How many persons will it take to reach the height of the lamppost?

The lamppost is taller than the person

What is the hieght of the lamppost/person

A person is next to the lamp

What’s the height of the person and lamppost?

the lamppost is tall
The lamppost is black

How tall is the lamppost
(who is that person)

Act 2

Students were shown this diagram and asked which of these lengths/heights they could physically measure:

Then we headed outside to measure whatever we could with metre sticks & record on a handout of the above diagram in our small groups.

We returned to class & students solved at their boards (red/orange annotations on boards are mine during the whole class discussion afterwards):

We discussed the different boards & their strategies. We grouped the boards by strategy; proportion solving vs scale factor.

Act 3

The next day I poked a hole through a foam stress ball & fed some string through it – leaving the roll of string trailing behind. We went outside & took turns trying to throw the ball over the top of the lamppost. It took a good 20+ minutes, but we got it (“we” is a strong word since my throws did not work & my student Ahmed got it over!) and the students then measured the length of string that hung down to the ground; 10.16 m was the actual height (which was fairly close to their solutions on the boards).

The rest of the day 2 class was dedicated to individual practice. Some students never completed the first practice from earlier in the semester on similar triangles, so they were assigned the basic exercise set on Khan Academy. Those that had completed that skill were assigned a more advanced exercise set involving similar triangles nested inside of one another.

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.Also 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:

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

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:

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.

From 3 weeks ago, here is the filing cabinet post-it activity. It was originally created by Andrew Stadel and available on the 101 Qs website here. I’ve made my own photo prompt for act 1 so that students can do the measuring on our classroom filing cabinet in act 2.

Act 1

Act 2

Each group was given 1 sheet of paper. Students got busy measuring the filing cabinet and their sheets of paper. They worked at their boards:

There was some confusion to start about how to “read” and thus use the formulas for surface area on their formula sheet. A few groups worked through the areas of each face instead of the formula. I did a little direct teaching about nets and they can be more intuitive to use than the formulas.

Most groups got answers around 60 sheets.

Act 3

After all my years of using this activity, I have yet to get a group interested enough to take the time to cover my filing cabinet with paper to get the actual real life answer to see how close their work is. They always seem content that their Math has found the answer. Perhaps I just need to do it myself one of these days.

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.

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)

I’m catching up on blogging about a couple of activities I did before my student teacher took over my classes. Here’s a brief overview & reflection about our Gummy Bear problem for linear systems.

I gave the groups access to some fake coins and some blue & red blocks to represent the candies. I didn’t get shots of everybody’s work, but here is an example from one group:

Act 3

The solution:

Consolidation:

Using some direct teaching, I asked them to come up with an equation for each purchase if x represents the cost of 1 red candy and y represents the cost of 1 blue candy. Then I asked them to graph the two equations in Desmos & we looked at & talked about the point of intersection.

A colleague suggested showing students one purchase at a time and asking them what some possible prices for each colour could be.

I wondered whether or not this is a good context, because in reality, the blue & red gummy bears would not have different costs. Thoughts? Might this be a problem for students trying to understand the problem & context?