# 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)

# 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:

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:

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.

Update: Find the Desmos activity builder follow-up here

Activity available here.

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

# 26 Squares – Area #MPM2D #MFM2P

For 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?

Create a table of values:

This was done in their groups at their boards.

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.

And why:We 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:

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:The 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:at 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:and 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)

# 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.

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:

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: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:They 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)

# My Favourite: Desmos! #MTBoS

While I’m not technically participating in the Explore MTBoS blogging initiative, I have been reading others’ posts & liked this idea of writing about “my favourite” something. I also realised that I haven’t ever posted about the ways I incorporate Desmos into my class except in passing when describing the activities we’ve done in class. So here is a list of the ways I use & love Desmos w/ my students:

New sketchnote created 2016.11.03

Basic Graphing & Solving Problems

Here’s an expectation from my grade 10 applied math class:
The old way: Graph by hand by making a table of values & interpret the graph. Or learn the more difficult algebra for solving for zeros, etc.

The Desmos way: Use Desmos to graph the equation given & identify key points in order to solve problems such as maximum height, landing distance, etc.

Investigating with Desmos:

Here is an example of a curriculum expectation for my grade 10 academic math class:
The old way:
– Draw graphs of many different equations by hand to compare & conclude
– Use school-loaned TI-83 to graph multiple equations, compare & conclude

The Desmos way:
Have students use sliders to see in real time the effects of the various parameters.And the best part is you don’t even have to make one yourself. Just search “vertex form” in the handy dandy search bar to get a pre-made graph to use right away:

Regression Models

The old way:
In past years I was reluctant move to far towards Demos from our TI-83 graphing calculators because the TI’s regression models were useful to my students. Especially in the applied class where the focus is less on algebraic manipulation and more on understanding & problem solving, being able to use the TI-83 to find the equation given a table of values was great. However, it did require some serious steps to follow on the graphing calculator.

The Desmos way:
Desmos makes linear & quadratic regressions easy for my students. Input your table of values, and one line of “code” . . . et voilà! Click here for a tutorial from Desmos.

Activity Builder

Often when I do a 3 act math problem with my classes, for example a linear pattern problem, they solve using a table of values. I’m happy that they’ve selected & implemented a valid strategy. Perhaps not the most efficient, but the one that makes the most sense to them at the time. But the curriculum asks them to use a graph & equation to solve these problems.

The old way: So I used to create worksheets that asked students to draw a graph the data points, determine an equation, etc. the day after the 3 act math to ensure that we explored all the possible strategies for solving the problem. I would then do some direct teaching pointing out how the graph & equation relates to their table of value solutions so they could see the parallels between them.

The Desmos way: For years Desmos has been creating pre-built activities you can run with your classes; Polygraph, Marbleslides, etc. But new this year is that you can build your own activities with the activity builder. So I’ve been making online “worksheets” where each student can work through the activity using Desmos to create a graph and perform a regression to find the equation for the relation. Click here for a Desmos activity I created to consolidate our learning after the Toothpick Triangles activity. The dashboard allows you to see the work of each student:

Testing with Desmos

So far this year, I’ve allowed my students their phones, a loaned iPad from me, or a loaned Chromebook from me on tests (in replacement of the TI-83 they used to be allowed during evaluations). They have to cover all camera lenses with paper & masking tape. They are supposed to only use their calculator app & the Desmos app. But I know of at least one student that has used an app that will expand/factor quadratics for her. I would be fine with that if I could make the questions harder (more critical thinking than calculation) but the course has a board-wide exam so my hands are tied.

The exam for that class will be this Friday (in 2 days). And this morning I’ve been experimenting with iOS guided access mode in which I can lock them into 1 app only (Desmos) as well as Android’s Surelock app in which I can lock them into multiple apps (calculator & Desmos) with my own password. I am thinking I will try this for the exam on Friday; no other apps, no internet access.

Have I missed any ways that you use Desmos in your classes?
Share them in the comments below.

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