# Turning Textbook Questions into Problem-Based Learning Activities

Over the last few years I’ve done my best to create a student-centred Math class using a mix of Dan Meyer’s 3 Act Math strategy, Peter Liljedahl’s Thinking Classroom framework and some other routines like Notice & Wonder mixed in, all in a Pear Deck interactive slideshow.

This week I wanted a problem-based activity on volume so I turned to my version of a textbook; Khan Academy practice sets. I picked a problem that my students will see during their independent practice problems on the Khan Academy website and fleshed it out to create a student-centred activity out of it. Thought I’d share the process with you to show that you can take (sometimes boring) problems right out of a textbook & create a student-centred thinking task for your class.

Here’s the original problem from Khan Academy:

So my first task was to find an actual image of a tent and use Google Drawings to add the dimensions as well as the volume to the image:

So this is what I show students to start. I do not tell them yet that I want them to find the height. I have a series of questions we run through every time that I build in a Pear Deck slideshow (where students will be able to answer on their phone & I can display their answers on the board). But you can just ask the questions orally if you like.

Here are the questions/steps:

1. What do you know / notice?
They should tell me facts that they know.
Eg. The tent is the shape of a triangular prism. It has a volume of 70 ft^3.
2. What do you wonder?
What questions come to mind?
Eg. What is the height of the tent? How much canvas is need to make the tent?
3. Now I tell them the question I want them to explore … for this tent the question was “Can you stand up straight in this tent without hitting your head?”
4. Estimate:
– too high
– too low
– best estimate
5. What do you need to
– measure
– calculate
in order to solve this problem? (plan)
Whenever possible I bring a hands-on object in that they can physically measure. This time I gave them the measurements of the tent.
6. Then I send each visibly random group of 3 to their chalkboard or whiteboard section to solve the problem. During this time I’m walking around managing what Peter Liljedahl calls FLOW by giving hints (usually in the form of a question) to those that are stuck and extensions to those that are done the original question (for this tent, how much canvas is needed?). Sometimes this involves calling all groups over to one spot & I do some direct teaching if they need to learn something new or review something to move on.
7. When all the groups have solved the problem, students return to their seats and I debrief / consolidate the activity by “narrating a story” as Liljedahl says of the student work. I found the “5 practices” article really helpful in learning how to do this.
8. At this point I reveal the correct answer (needed more if they are taking their own measurements to see how close their answer is to the real answer; for example how tall the lamppost outside actually is after we solve for its height using shadows & similar triangles).
9. We go back & see who’s best estimate was closest to the actual answer. We celebrate the closest estimate.
10. Which of the overall expectations from our course did we use today? (reflection)
This is where the learning goal of the task comes out – at the END. If I say this up front, then it takes away all the student thinking about what math they can use as a tool to solve the problem.
11. I encourage them to take a photo of any group’s board they wish to save in their notes.
12. Finally, usually the following day, they do some individual practice using some of the problem sets on Khan Academy.

Hopefully that all makes sense and shows a bit about how you can take a typical textbook or worksheet type word problem & turn it into a more student-centred learning task. If you want to see examples of this type of lesson with student work, have a look at my collection of lessons I have blogged about.

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

# Banquet Hall problem #MFM2P #PBL

This is my 2nd or 3rd time doing this activity, but hadn’t yet blogged about it. So here goes … I made lots of changes, even from one period to the next.

Prompt:

Notice-Wonder-Estimate:

Solve:
At your boards (whiteboard / chalkboard) in groups of 2-3 randomly assigned (VNPS & VRG).

Most groups started calculating the cost per person for each teacher. I stressed to them multiple times that all 3 teachers were paying along the same formula or “price plan”. They really struggled with how that could be. In first period there was 1 group whose board had a table on it & they had started using 1st differences to calculate the rate of change. I called all the groups over & led a discussion about the strategy and asked about what sort of deposit (a cost for 0 people) might have to be made by the teachers & sent all the groups back to continue, strongly urging them to explore the table idea. In 3rd period, none of the groups started the table, so I called all the groups over to some spare board & said “here’s something I saw in 1st period” & proceeded to have the same conversation with them.

Here are their boards:

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We got to the point on day 1 where everyone solved for the cost for 150 guests. Time ran out, & bell rung.

Between periods 1 & 3 today I added some extra slides & questions to my slide deck to make it better.

Day 2:

Yesterday I added a slide asking students to graph the 3 points from the original data set in the original prompt. Today we started on that slide in Pear Deck :
I asked them if this was linear or nonlinear. Why? In 1st period this also resulted in a conversation about 1st differences when the x values don’t have a constant increase.
I asked if the line of best fit would pass through the origin?

I then sent groups to their boards with the task of using Desmos to find the equation for the line of best fit . Their boards:

This slideshow requires JavaScript.

Once they had done a linear regression to find the equation, I asked them to use their equation to solve for the number of guests I invited if my party cost \$3545 at the banquet hall. I coached a few groups through the proper format in which to show their work when solving an equation.

The rest of the period was dedicated to individual practice on a Khan Academy problem set called “Slope intercept equation from graph“.

Find the whole lesson here (the unassociated file is Pear Deck).

As a final note, this whole problem-based teaching can be hard for the student to grasp sometimes. Today this happened:

How do you handle it when students question your teaching skills or pedagogy? Let me know in the comments below!

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

# Push-Back to Student-Centred Learning. #sketchnote

I’ve often said that I would hate to be a learner in my own classroom. I was a very strong student in high school. I didn’t need to be in class; if I missed class I would read the section in the book & do the homework problems & learn it myself. I made beautiful pages of copied notes from the teacher’s board and was able to understand the content as I copied. I did not enjoy group work; hated relying on partners to do their bit. I am still the first person to roll my eyes at ice breakers in a staff meeting or workshop.

And yet, my classroom is the opposite of this. I ask my students to work in groups, beginning with a getting to know you question every day since we change groups daily. I don’t give many notes, rather I give students time to summarize their new learning in their course packs. We do problem-based learning with hands-on components whenever possible. This is a far cry from the teacher notes followed by homework problems routine from my day.

But many to most of my students are not able to learn that way (although a small number of them are & would prefer a more traditional teaching style). Most can’t understand the notes they’re copying down because they’re too busy copying. (Have you ever asked your students if they’re able to listen to the teacher while they copy notes? My students tell me straight up that they are not able).

So over the years I have searched for strategies & pedagogical methods that would transform my classroom to be a better learning environment for my students. But my students haven’t always been eager about my methods; group work, problem solving, critical thinking, feedback separated from marks, etc. The workings of our Math classroom are so different from their experience so far that they sometimes push back. And for many teachers, this push back stops them from continuing to pursue different teaching methods. For example, I’ve had students say “you don’t teach us!”. But upon drilling down further as to what they mean, it becomes clear what they really mean, is you don’t write long, detailed notes on the board to copy down. They think that is teaching and don’t view the careful orchestration of a student-centred classroom as teaching also.

My advice to teachers: keep trying! Don’t let that student (or parent) push-back stop you from pursuing new & innovative teaching methods. It’s normal – it happens to all of us! But eventually students (most anyway) get past it. Alice Keeler shared this great article entitled “NAVIGATING THE BUMPY ROAD TO STUDENT-CENTERED INSTRUCTION” by Felder & Brent that likens the student push-back during student-centred teaching to the 8 stages of grief. I love sharing the article with teachers that are frustrated by students that are reacting negatively when they try to transform their classroom to a student-centred learning environment. So to make the ideas even more shareable, I put together a sketchnote version:

But I really do encourage you to read the whole article as the authors go on to explain some suggestions as to how to mitigate the push-back, such as sharing with students the reasoning behind the methods, and modelling & establishing criteria for the successful use of the critical thinking skills expected of students.

I’ll finish by including a few of the tweets from other teachers on the topic:

What push-back have you experienced in your classroom and how have you dealt with it?

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

# 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:

Scenario:

What do you notice?

I 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?

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

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)

# Yard Space #MPM1D/#MFM1P #PrBL

I took the typical “find the largest area given a specific perimeter” problem and created a hands-on, problem-based learning task for my combined grade 9 Math class (academic & applied combined):

Scenario:

Ms. Wheeler wants to build a fenced in yard for Sally to run around in.
She buys 16 1-metre long sections of fence.

What do you wonder?

Physical & Visual Representations:

The yard must be fully enclosed. Use toothpicks to create show different ways of placing the 16 pieces of fencing (I forgot to take photos of this part but they made stuff like this):

Draw your shape & label its dimensions:

How should the pieces be set up to create the largest enclosed area possible?

What shape offers the largest area?

We discussed that while a square was the largest rectangle possible, there were other shapes possible with greater areas.

How should the pieces be set up to create the largest enclosed area possible if Ms. Wheeler uses a wall of the house as one side of the enclosure?

We have some more exploration to do here. I left this pretty open and they explored various shapes. But I’m not sure they’ve drawn any solid conclusions just yet for the case where we have 1 side of the shape already accounted for.

Get the Pear Deck slideshow here.

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

# Chicken & Goat Legs #MFM2P #PBL

Summary (scroll down for more details):

Scenario:

I asked some questions on Pear Deck to get students thinking about the parameters of the problem:

We discussed some of the above responses that did not meet the criteria of a total of 70 legs and why.

Students went to their boards in their small groups to solve this problem:

#### She has 26 animals all together.There are 70 chicken & goat legs all together.How many chickens? Goats?

Most groups were very unsure as to how to proceed in their solving. Most were simply guessing & checking various pairs of numbers. After a few minutes of allowing that productive struggle, when I noticed frustration setting in for some, I asked if anyone had considered drawing animal bodies & assigning legs to them? Here are the student boards:

We returned to our seats and our Pear Deck session & I put it into student-paced mode. I asked them to create the equations for the various parameters of the problem:
They struggled with this so I did some direct teaching about how to build the equation for this and the next slide:

Students were asked to use Desmos to graph their 2 equations & then sketch the graph and point of intersection:

Students were asked to develop an algebraic solution using the elimination method:
Not all of my students are comfortable with the algebra still (even though we’re at semester’s end now).

I like that we used 3 different methods of solving this problem; diagramming, graphing & algebraic. I want my 2P students to know they can always fall back on “less sophisticated” methods to solve these problems at evaluation time (as opposed to the algebraic solution).

The resources can be found here (including the Pear Deck interactive slideshow).

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

# Pyramid SA #MFM2P #3ActMath

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)

This past week I did an activity inspired by Fawn Nguyen’s Visual Patterns work. The last time I did this activity, I blogged about it here.

Pattern:

Notice & Wonder:

 What do you notice (facts)? What do you wonder (Qs)? each step the cubes increase how come the reds arent increasing? How many blue blocks will they add on the 4th step Each step, more cubes are added Why is there always 3 red but the blue always increases? Why Do we only have 3 reds There are 3 steps in the picture there are red and blue cubes how much the sides go up each time The number of blue blocks increase as the number of steps increase. why isnt red increasing? Cubes, there is steps How many cubes added every step? I notice there a step 1 , step 2, step 3 Why does the blues always increase and the red stays the same Always 3 red in the middle the outside length increases by 1 each time Why is the red not increasing – The number of blue cubes increase each step. (2, 8, 18, etc) – The number of red cubes stay the same each time. – How many blue cubes will there be at step 10? – What is the formula? 3 red squares on each step , There’s always 3 red in the middle How much blue cubes will it be in step 4 the red blocks stay they same but the blue blocks increase every time how much the blue blocks are going up by

Estimate:

Solve:

Groups used tables to start. Then, most could see the pattern of the two squares on each end with a side length equal to the step number and they used this pattern to calculate the number of blocks for step 57.

A follow up question in the Pear Deck slides asked them if the pattern was linear, quadratic or neither. We discussed how we can determine this, and I sent students back to their boards to find the first & second differences.

The next question in the slides asked them to use Desmos to find the curve of best fit and its equation. I reviewed how to do both linear & quadratic regression on Desmos on the board for them. After students found the equation with Desmos, they were asked to go to their boards one last time and use their equation to verify how many cubes would be needed in step 57.

We then had a whole class discussion on how the terms in the equation represented the visual pattern.

Individual practice on quadratic relations was assigned from Khan Academy; different exercises depending on whether or not they had finished their homework from the last time we worked on quadratics.

Activity available here.

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

# Coffees & Muffins #MFM2P #PBL

Scenario:

 What do you notice (facts)? What do you wonder (questions)? different prices why are they buying these Adding 3 extra coffees cost more by a little How much with it cost for 4 coffees and 4 muffins ? why is the kid buying coffee? whys the kid buying coffee? The totals are different on each side. One side has less drinks. How much is each item? nothing 2 different cost How much the cupcake cost each one How much the coffee cost each one there’s money, drinks, cupcakes how much each coffee and cupcake is For the first indivdual, it costs \$8.85 for three cupcakes and three coffee cups. For the second person, it costs \$5.35 for three cupcakes and one cup of coffee. Im curious about what brand of coffee that they are buying. It seems potentially no name or even something like a corner store kinda coffee. ew. oh yeah also how much do they each cost? The total cost are different How much it cost in each item I notice cupcakes , coffee , a boy and a girl how much is each The person on the left has more coffee and is going to spend more What is the individual price of the coffee and the muffins – They both ordered 3 muffins, but one had 3 cups of a drink and the other ordered one. – The one that ordered 3 cups, have to pay more. – How much does one drink cost? – How much does one muffin cost? – Does the person on the right have a better deal than the person on the left? Different prices and different subjects How much does it cost for each item Adding 3 more cups of of coffee is a little bit more than getting one cup of coffee how much is one cup of coffee Different objects in both pictures Why did the person on the left buy more diffent how much money does it cost to for one cup of coffee and one muffin

Solve for the cost of 1 muffin as well for the cost of 1 coffee (red/orange annotations are mine during whole class discussion):

They all solved it by subtracting what was common to both orders & splitting the remaining cost amongst the remaining coffees. The follow up questions on Pear Deck asked them to create an equation for each order. I then did some direct teaching on the side showing them how to do elimination using 2 different equations. Then I asked them to go to their board and use elimination to solve this problem. They started this on day 1 above but we class ended & we hadn’t finished. So on day 2, with a new group of partners & fresh boards, I sent them up to use elimination to solve fully:

We compared the solutions of the different groups and picked out the one board that had the most correct formatting of an algebraic solution. I drew parallels between their work during elimination and their earlier logic, pointing out how they are both eliminating something (I explained this more in depth & more eloquently).

We then did a quick check with Desmos:

The individual practice to wrap up was a Khan Academy exercise set on elimination not involving any multiplication of equations.

Activity available here.

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

# Lamp post height #MFM2P #3ActMath

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.

Lesson materials available here.

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