# Distance Formula without the Formula

Today in MPM2D our main goal was to discover how to find the distance between two points. But since I’m spiralling the 2D course I wanted to think big picture…I  wanted to tackle this overall expectation: verify geometric properties of triangles using analytic geometry.

We started with this beauty from Would You Rather  — www.wyrmath.wordpress.com

Students argued and discussed which ramp they would rather push that crate up. Most of the class picked A with their reason being it’s less steep and less work. One of the students who picked B said “I want muscles…..so I’m going to push that crate up the steepest slope“. Another student picked B because they wanted less distance and wanted to “get it over with“.

I left the discussion hanging here knowing I was going to come back and revisit this with more ammunition.

I showed them this video

and we completed the Corner to Corner problem (see the lesson plan here) to remind ourselves of the Pythagorean Theorem.

We came back to the Would You Rather problem from above and practiced finding the length of each hypotenuse to see how long each was.

I then presented them with this……and said our goal was to find the length of this line segment.

Find the length of this line segment

I asked…”If I could help you out or provide you with more info what would you want?” Most students said they would want either a ruler or some sort of dimensions or units to look at.

So I  brought up the grid on Desmos and asked if this was enough.

Most students thought it was…..I could see them drawing right triangles on their whiteboards and filling in the lengths of the legs. But one students yelled out “What is the scale?” ….Everyone paused! ….. I brought up the axis!

Students finished drawing their right triangles and said that was easy! We did one more just like this (giving them the grid and axis) to practice.

Here’s the next challenge: I took away the grid but gave them the coordinates of the endpoints. Find the length of this line segment.

I let them struggle a bit here. The majority of the class prevailed and had a similar solution on take up:

Student words: “One leg was the difference between the x-values and the other leg was the difference between the y-values”

We did another in the same format to practice this discovery.

Then I took it up a notch…

The three points shown represent vertices of a triangle. Classify the type of triangle.

And I saw a lot of this…

I’ve been following Mary Bourassa’s Blog and I stole creating my own homework sets from her….so I left the class to complete this. Love how I can ask lagging questions in my homework. Students get multiple opportunities to master skills.

So we’ll take up those questions tomorrow and we’ll summarize the strategy to find the length of a line segment using this formula…

Access: Pre-made Desmos graphs

# Lollipop Lollipop oh la la Lollipop! — & Rates of Change

Last year on twitter I saw that Alex Overwijk and Janice Bernstein with their grade 12 advanced functions classes did this lollipop activity!

I knew that I wanted to give this a try for this semester! What I especially love about this activity other than students experiencing rates of change is that this is an activity that can span multi-grades!

Here is what we did,

## Generating Curiosity

I found this video on YouTube and asked the class to think of great questions we could ask about what we see!

Great questions from the kids and we all agreed to look at

• How does the sucking time affect the radius, circumference, volume, and surface area?
• How long will it take until the lollipop is all gone?

Let’s investigate those relationships starting with the easy to measure (circumference) and also estimate how long it will take until the lollipop is no more!

We had guesses : ranging from 10 minutes through to 35 minutes.

## Gathering Data

I handed out one lollipop per pair of students, along with some dental floss for measuring circumference. We set our timer for 30 seconds and began sucking and capturing data!
We recorded the circumference every 30 seconds up to 7 minutes like Al’s and Janice’s instruct in their lesson Plan.

They also have a great handout for tracking the circumference over the 30 second intervals.

## Analyzing the Data

So we first looked at the Time vs. Circumference and Time vs. Radius relationship

We discussed its linearity and why. Students predicted with more accuracy when their lollipop would run out.
Up to this point this task is great for grades 7, 8, 9, or 10!! (Just edit the file to exclude the average and instantaneous rates of change).

• Grade 7 & 8: Practice plotting points and reading/interpreting graphs.
• Grade 9 & 10: Find lines of best fit and first differences.

We found the average rate of change for each 30 second interval and discussed what this meant. We used the last column to talk about narrowing the interval down to estimate the instantaneous rate of change, and noticed that it’s about the same for all values. Why does this make sense???

We moved on to looking at Time vs. Volume and Time vs. Surface Area

Great talks around how Volume and Surface aren’t deceasing at a constant rate! It changes! Students can see these changes and see in their tables where the volume is changing the fastest.

Overall a great intro activity to get students thinking about narrowing intervals to approximate instantaneous rates of change.

Next up: We’ll relate what we did here with the tables to the graphical interpretation of rates of change (secant and tangent lines) and then on to the algebraic!

# Desmos Challenges in iTunesU, Multi-Touch Book, and Web Version

For many years now my classes have been completing a course wide project on Picture Modelling. Before Desmos (B.D.) I use to send home copies of Geometer’s Sketchpad for students to generate a picture using only functions.
After Desmos showed up it was now super easy for students to generate art and access graphing software from any device.

The project has been so successful at engaging students to learn about various functions and their transformations I extended it to all grades! For the last few years the project spans grades 9 through 12. Each year learning new functions and creating art.

This summer while at the Apple Distinguished Educator Institute in Miami I started a project that would create a digital resource that would link the Modelling Functions with Art Project with function challenges created by Michael Fenton, Dylan Kane, and myself.

If you are in an one-to-one iPad room or have access to iPads the resources are in an iTunesU course and multi-touch book for iPad  otherwise they are linked on this site for any device (see below).

Each chapter starts with linking patterns, tables, graphs and equations in pre-made Desmos graphs or in pre-made Desmos activities made using Activity Builder.
Following that, activities ask students to match functions to specific criteria like Michael Fenton’s Match My Line or in my Match My Trig Function. Again the teacher can choose to use the activities in the Multi-touch book or from the pre-made Desmos activity.

Every so often in the challenges students are asked to show their thinking by uploading a picture of their work on a Padlet page. Students can crowd source different ways to solve the same problem.

Finally, at the end of each chapter students are to create a working piece of art and share it on a Padlet gallery page! Students can see each others work and comment.

Each chapter covers different functions but many chapters can be done in the same course:

### Ontario curriculum suggested chapters:

• Chapter 1 – Linear Functions  – grade 9 & 10
• Chapter 2 – Quadratic Functions – Grades 10 & 11 & 12
• Chapter 3 – Various Functions (function notation, cubic, square root, reciprocal, non-functions).  – Grades 11 & 12
• Chapter 4 – Trigonometric Functions  – Grades 11 & 12
• Chapter 5 – Exponential & Logarithmic Functions – Grades 11 & 12 (Coming soon!).

The project page has more details on how to access the course, book, and web resources.

[aio_button align=”center” animation=”none” color=”blue” size=”medium” icon=”none” text=”Go to the Project page” relationship=”dofollow” url=”http://wp.me/P3az6g-14y”]

# R2D2 – Pear Deck/Desmos Mash Up!

School is just right around the corner for us up here in Ontario and I can’t stop thinking about that first day. As for my grade 9 applied class’ first day I have ran the R2D2 problem in the past with great success.
Now, over the summer I’ve seen great improvements in Pear Deck and wanted to get into it! Also Desmos has been busy and released Activity Builder!! So let’s mash these two apps up with some R2D2!!

So here is the R2D2 problem presented with Pear Deck and an extensions with Desmos….

Act 1: The video

and this is what Pear Deck will show after you insert the video…..love how the video will be displayed on the projector and not on each individual device!!!

I like using Pear Deck here for asking for wonderings and notices because it allows students who normally won’t shout out answers to have a voice in the room. Students get to input their responses and the teacher can show them on the projector.

For generating estimates I absolutely love how they put our Too high and Too low guess on a number line…..it gives us the visual of where our actual estimates will lie.

Act 2: Gathering the Info

In the new version here I get students to draw their estimates of the dimensions of both the board and the post it note…..this pushes them into drawing diagrams.

Revealing the dimensions….

Students are ready to solve….

Act 3: Revealing The answer

The Extension: How many rectangles can we make that have an area of 609 post it notes?
To extend I want students draw out different rectangles and label their dimensions! They can use Pear Deck’s white board!

But then they can enter them into Desmos through a pre-made activity I created in Activity Builder. (the Pear Deck file links to the Desmos activity).

For each rectangle the student can come up with they find the perimeter and plot the length vs. perimeter in the Desmos graph. The teacher on the projector can use the Overlay function and show all the different rectangles students are coming up with…essentially showing the pattern that emerges! Using the pattern students can read off the minimum perimeter!

If you have a Pear Deck account Grab and download the file below!

[aio_button align=”center” animation=”none” color=”blue” size=”medium” icon=”star” text=”Pear Deck File” relationship=”dofollow” url=”https://drive.google.com/file/d/0B9g0jeaVwshveDVhWktzdTRudE0/view?usp=sharing”]

Link to the Desmos Activity