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)

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 – Area #MPM2D #MFM2P

CaptureFor my MFM2P group this followed the Perimeter activity I did with the 26 Squares manipulatives (partially pictured at right). For my MPM2D group, this was their first introduction to working with the 26 squares manipulatives. For both groups this was their first introduction to Quadratic relations and parabolas.

 

 

 

 

Predict: What is the relationship between side length and area of a square?

Capture.JPG

Create a table of values:

This was done in their groups at their boards.
ToV
I had to encourage groups to count the grid on their squares. Many were calculating the side length times 4, while others were trying to square the side length but doubling instead. For each of those groups, I redirected them to our physical squares cut out w/ grids [pictured at top of post] & asked them to count the area of a 2×2 square, then a 3×3 square, and so on.

Graph: Back in their seats students were given this handout & asked to graph by hand the data from their table.

Linear VS Quadratic: Students were asked to choose which type of relation they thought this was.
Capture
And why:Capture.JPGWe then discussed the shape of the graph being a curved line & the first differences being not equal (which only some students had pointed out).

First & second differences: Groups were sent back to their boards & their table of values with this prompt:
Capture
IMG_1123
We discussed that second differences being equal means this is a Quadratic relation; a new key term for us. The black writing on the whiteboard above is my own addition during the class discussion.

Desmos & Quadratic regression: Back at their seats, individually students used Desmos to perform a quadratic regression on their table of values. They had this prompt on their handout from earlier:Capture.JPGThe 2P students had practiced performing a linear regression with Desmos the day before during the Perimeter investigation. The 2D students had mostly never seen Desmos before. I walked around helping students that got stuck or couldn’t find where they’d mistyped something & gotten an error. The result was:Capture.JPGat which point I did some direct teaching about how to use the a, b, and c value determined by Desmos to write out an equation for the relationship between side length and area. I also introduced the word parabola to them while we looked at the graph from Desmos, zooming in & out.

In their groups at their desks they had 4 application questions to work on:Captureand this became the homework for the MFM2P class as we ran out of time in class.

Key features of a quadratic graph:

With the 2D students I had time left to do some direct teaching about y-intercept, x-intercept / zeros, vertex, max/min, & axis of symmetry. Their homework was on Khan Academy to identify these key features given an equation that they could graph using Desmos.
For my 2P students this lesson came a few days later with class time to work on the Khan Academy exercise set.

My reflection: I wish I had asked at the end of the activity for students to restate in words the relationship between side length & area.

Folder w/ handout & Pear Deck interactive slideshow here.

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

Toothpick Triangles #MFM2P #3ActMath

Back at the beginning of this month we explored Dan Meyer’s toothpick triangles task:

Summary (9)

Act 1

What do you notice?
Capture

What Mathematical questions can we ask & solve?Capture

The question I chose for us to investigate (decided ahead of time to meet the curriculum learning goal for that day):Capture.JPG

Act 2

Groups were given toothpicks to play with & this image with the number of toothpicks (250) in the jar displayed:act2 (1).jpg

It’s important to give students enough toothpicks to extend the pattern seen in the video, but not as many as 250. My goal is for them to model mathematically, which they won’t do if they can complete the physical model w/ the manipulatives given.

I have also found lately that my students prefer to draw their model on their boards, rather than use the physical manipulatives. Perhaps I should have them work for 2 minutes with the physical manipulatives first before sending them to the boards? Thoughts?

The solutions from each group:IMG_0491.JPGCapture.JPGCapture.JPGYou can see that they all used a table of values with second differences to continue the pattern & find an answer. CaptureCapture.JPGThese 2 groups were trying to use some calculations but the top group could not explain to me their reason (within the context of the toothpick triangles) for dividing the total number of toothpicks by 2.
The bottom group was trying some linear proportional reasoning and using the idea of 3 toothpicks per triangle in their calculations. They didn’t make the connection between their constant second difference meaning that linear reasoning wouldn’t work here.

Act 3

The answer:Untitled drawing.jpgThe video version (which we watched) is here.

Even though our curriculum doesn’t require students to determine a quadratic equation from a table of values, it does expect them to graph the data (and I like them to use Desmos to perform a regression & find the equation since Desmos makes it so easy!). Instead of the typical handout I’ve used in the past to consolidate these sorts of activities, I created a Desmos Activity version which you can find (& use) here:CaptureThis was their homework after the toothpick task.

On day 2, we used this handout to work on describing the steps in some of our solutions:
Student solution:Student Sol'n.jpgDesmos solution (which I created):Desmos Sol'n.jpg

As always, all materials for this activity can be found here.

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

Quadratic Visual Pattern #3ActMath Activity #MFM2P

On Monday we worked through a quadratic visual pattern with blocks (blog post here), and so today we tried a harder one, again with linking cubes. Here’s a summary of their work:Summary Quadratic visual pattern linking cubes MFM2P

My students used Desmos to find the equation since doing so algebraically is not in the 2P curriculum. In fact, determining the equation for a quadratic relation isn’t required in 2P at all, but since Desmos makes it so easy, we do it! I recreated their work in Desmos for the image in the summary above, but the blackboard work is theirs.
I think it’s time to clean my boards 😉

After taking up each group’s board as a class, we looked at this image to try and make connections between the geometric pattern and the equation we had found:
y=x^2+2x+4visualising the pattern

My files can be found here.

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

Quadratic Visual Patterns #3actmath #MFM2P with Linking Cubes

Act 1
Today’s 3 act math activity was in the style of Fawn Nguyen‘s Visual Patterns. I presented the class with this:Captureband asked what Mathematical questions could we ask & solve? Students responded via Pear Deck:CaptureaI want my students to be thinking about good questions to match a given scenario as this is what our MFM2P board-wide summative task will require of them.

So I then asked them to estimate the number of cubes in step #62 (being careful to pick a high enough number that they won’t want to count up like they did last time):Capturec

Act 2

I asked them to solve at their boards with their groups. All of them had a table of values created and had started to determine the first differences for the most part. One group also had the 2nd differences listed. Most were stuck as to how to continue. I overheard the following:

“We can’t [make a table] all the way to 62 …”

“There’s got to be a better way to do this”

One student realized they could multiply the term number by itself & multiply by 2 (effectively 2x^2). But they were only counting blue blocks (when they should have been counting red + blue). I told her to keep that in mind but that they needed the red blocks too.

At this point I asked them to take a seat at their desks & I did a quick “teacher-centred moment”. We reviewed the linear & quadratic regressions with Desmos that we learned & practiced last week. We also talked about 1st & 2nd differences and how they allow us to determine whether or not a relation is linear, quadratic or neither. It looked like this:IMG_0084

Then I sent them back to their boards. A few groups took their chromebook with them & got to work with Desmos. Here’s a group that worked their way through the quadratic regression w/ Desmos w/ some prompting from me along the way:IMG_0086

Another group had a member that recognized a pattern:IMG_0085

I asked them to explain how they got 124 at the bottom of their table. They said they counted up knowing it was increasing by 2 each time. I asked them if they could figure out that number just based on the step number without counting up. I pointed out the pattern 1 x 2, 3 x 6, 4 x 8 … and they realized that it was twice the step number (which I scribed on their board for them as they explained it to me).

That group then decided to go around and explain their pattern to the other groups so you can bits & pieces of their calculations (without the pattern or thinking behind it) on these groups’ boards:IMG_0082IMG_0083

This last group didn’t quite get to the end point that we wanted:IMG_0081

Act 3

The actual answer was 7691 blocks, which the groups had figured out. I called students back to their seats even though the last group didn’t yet have an answer. I reviewed the different solutions on the first and second boards pointing to things as I went, clarifying their written work. I showed how the pattern the 2nd group came up with could be written as n x (n x 2) + 3. I also wrote on the original image to show them how it could be n^2+n^2+3 or 2n^2+3.

They reflected on their original guesses with a little help from my prompts about expectations of linear VS quadratic: Captured

And then I asked them to consolidate their learning with some practice using this handout for homework tonight.

The whole set of materials is here.

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