#3ActMath – What is it?

I learned about a great tool this past weekend at the Ontario Summit; Adobe Spark video. A huge shoutout to Rushton Hurley for the introduction to this tool. It’s a super fast & easy way to combine photos, videos & text and narrate over top of it to create a seamless professional looking video.

I tried my hand and created one about 3ActMath lesson style. Give it a watch & let me know what you think:

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

Chicken & Goat Legs #MFM2P #PBL

Summary (scroll down for more details):2017.01.11 summary.png

Scenario:Capture.JPG

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

captureWe 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: Capture.JPG
They struggled with this so I did some direct teaching about how to build the equation for this and the next slide:
capture

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

Students were asked to develop an algebraic solution using the elimination method:Capture.JPG
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)

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)

Video clip of students at work

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)

Here is a quick (1 minute) video clip of my students working on a visual patterns 3 act math task on vertical non-permanent surfaces in their visibly random groups:

– 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.2016.10.31 summary (1).png

Act 1:

img_20161031_091019

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:2016.10.31 estimate (1).JPG

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:

img_1912

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.img_1915img_1913

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

img_0176

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)

Quadratic Block Pattern #MFM2P

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.2016.10.25 summary.png

Pattern:
img_0074-2

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:2016.10.25 estimate.JPG

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

This is 2nd time blogging about this problem. 1st time-around post is here.

2016.10.24 summary (2).png

Scenario:
LR3

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

2016-10-24-estimate

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:
2016.10.24 desmos.PNG

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:2016.10.17 summary (1).png

Act 1

Scenario:
img_1752

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)

2016-10-17-estimation-1

Act 2

Students were shown this diagram and asked which of these lengths/heights they could physically measure:2016.10.17 diagram.JPG

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)

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)

Not every lesson is awesome :( #MFM2P

I wasn’t going to blog about my lesson on expanding binomials from earlier in October. It did not work out nicely at all. But it wouldn’t be right to only ever blog about the things that go pretty well. So here’s a quickie to share how it all went downhill …

I had students cut out a set of paper algebra tiles for themselves to store in their binder. We started with the 1 tiles and I asked them to show me a rectangle measuring 2 x 3 and then 4 x 5 and so on. Each time I asked them what the area of each rectangle was. So far so good – they got it.

Next we introduced some x tiles & I asked them to represent 3 x (x + 2). And I can’t remember now if it was at this point, or when I asked for a rectangle with width x+2 and length x+4, but somewhere in here my lesson went off the rails because my students were not building the correct rectangles. They were confusing the concepts of area versus width and length. They were getting confused about what each algebra tile represented.

I abandoned my carefully planned out prompts and remembered thinking after this lesson last year that perhaps it would be better to have them factor a trinomial first. That way, you give them a trinomial, have them take out the tiles needed to represent it & then use those tiles to form a rectangle. Once you get the rectangle you simply have to interpret the width & length of it. Boom – factoring.

Well, I’m not sure it went Boom per se. I had to do a lot of direct teaching about where to position the x squared tiles versus the x tiles versus the ones (with no better explanation than because that’s where we put them – help me out on that one!).

By the time the bell rang, some kids had figured out the side lengths, but other groups had not. They were using x^2 in their width and/or length. I did not feel good about the lesson or my students’ learning.

I didn’t assign any homework on the topic as we hadn’t dealt with any negative numbers in there yet, and the first Khan Academy exercise set includes some negatives. I did however put a very simple factoring question on our first monthly test later that week. All positive numbers, nothing tricky. And many students did OK or even well on the question. So all is not lost.

All this to remind those following my blog that not all my activities or lesson work out well. It’s good to reflect and share the times things don’t work out. Next year I will start with factoring first instead of multiplying or expanding. And I’ll hopefully do a bit better. We can always get better.

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