linear functions – Chris Hunter

September 4, 2012September 30, 2016

Hey, I just met you and I wanna rock your gypsy soul

Carly Rae Jepsen’s “Call Me Maybe” passed Van Morrison’s “Into the Mystic”.

I’m referring to my iTunes library, of course.

It wasn’t me. Meet the culprits:

First, “Van the Man”. On October 13, 2008, I added “Into the Mystic” to my library (‘Date Modified’ in iTunes). I’m calling this t = 0. I’ve played it 62 times. I last played this “song of such elemental beauty and grace” 1284 days later on April 19, 2012.

Jepsen’s up next. “Call Me Maybe” was added (not by me) on February 28, 2012. This is 1233 days after I added “Into the Mystic”. Seventy-five days later, on May 13, 2012, I listened to this sugary pop tune for the 63rd time. This is 1308 days after adding “Into the Mystic”.

NB: Screenshots of the iTunes Summaries for both songs would make a better first act. Here’s the summary for “Call Me Maybe”:

My initial questions were:

When did this happen?
Could I have predicted this?
How will the number of plays compare in the future?

I modelled this situation using a system of linear equations. For the Irish singer-songwriter, we get p = 0.05d, where p is the number of plays and d is the number of days. For the Canadian Idol, we get p = 0.84d − 1035.72.

Comparing slopes is an obvious discussion topic. The line for “Call Me Maybe” is much steeper than the line for “Into the Mystic”; the rate of change is 0.84 plays/day versus 0.05 plays/day.

This problem can also be used to explore unit rates. Unit rates can be expressed in more than one way. It’s about what one is one.

I wanted to express the equation p = 0.84d − 1035.72 in the form p − 63 = 0.84(d − 1233). Slope-point form tells a better story than slope-intercept form in this situation but my GeoGebra skills are rusty.

Having students look at their own iTunes libraries might make a better investigation than practicing solving catch-up problems like this:

I assumed that this situation could be modelled using linear relations. For “Into the Mystic”, fair enough. I think this reasonably approximates the real data. Outside of perhaps when I was commenting on Michael Pershan’s blog, I didn’t go through a Van Morrison phase. Van Morrison is in my wheelhouse and “Into the Mystic” is just in the rotation. The number of plays per day is (almost) constant.

For “Call Me Maybe”, this assumption is likely incorrect. The song’s got legs but the instantaneous rate of change has to be decreasing, right? For my mental health, I hope it is. That many plays would surely take its toll.

And what if Carly has competition?

What if I modelled this using a logarithmic function? Check this out:

Note that ≈ 5½ years after first being added to my library, “Into the Mystic” can be expected to pass “Call Me Maybe”. The natural state of the universe is restored.

Update: I learned how to animate my GeoGebra construction. Also, I corrected a math mistake. (What was my misconception?)

April 4, 2012September 30, 2016

A Linear Functions Lesson Across the Grades

How many people can sit at 100 (or n) triangular tables? Square tables? Hexagonal tables? What if you join the tables so that one side of the next table touches one side of the previous table?

I appreciate this problem for a few reasons:

I can present it in grades 4 through 10. In grade 4, students write a recursive relationship (e.g., for joined hexagonal tables, start at 2 and add 4 each time). In grade 6, students write a functional relationship (e.g., 4n + 2). In grade 8, students graph a linear relation (e.g., y = 4x + 2). In grade 10, students interpret the slope and y-intercept (e.g., each added table provides 4 additional seats, there are 2 additional seats at the ends of the table). When I teach and discuss this lesson at different grade levels within a school, I think a common activity helps teachers connect the big ideas across the grades.
I can easily adapt and extend the task. When I have taught this lesson in grade 6 (see three-part lesson plan), most students can write an expression for joined square or hexagonal tables. Some students may choose to solve a simpler problem and write an expression for joined triangular tables. Other students can be challenged to write an expression for tables with any number of sides. All students can participate in the class discussion.
The use of pattern blocks can help students gain a deeper understanding. Most students were able to make sense of the 4 in 4n + 2. Each time a table is added to an end, 4 seats are added. (Two seats are lost when tables are joined.) When one student showed how he added tables to the middle rather than an end, this helped his classmates make sense of the 2 in 4n + 2. There are two more tables at the ends. Pattern blocks allow students to make sense of the expression beyond “add 2 to make the numbers in the table of values work”.

This problem appears in several resources including The Super Source.

September 12, 2011November 11, 2011

The more sides you have, the smarter you are.

“How does shape affect your place in society?”
“The more sides you have, the greater your angles. So, the smarter you are.”

Two years ago, I created a lesson on Angles in a Polygon. The ‘hook’ was the opening minutes of the animated film Flatland: The Movie. In the story, Arthur Square asks his curious granddaughter if she has memorized her ‘laws of inheritance’.

Hex replies “Isosceles triangles have baby equilateral triangles. Equilateral triangles have baby squares. Squares have pentagons. Pentagons have hexagons, like me! And each new generation gets one new side until they get so many sides they look like a circle and become a priest.”

This film interestingly addresses many mathematical concepts, such as points, lines, and shapes in zero, one, and two dimensions as well as larger themes such as critical thinking.

Here it is:

I think it’s a pretty good lesson, but I decided to tinker with it. Here’s the new and improved version:

Yep. That’s it. Blank space.

I learned that from Sandra Ball when planning together for elementary school demonstration or team-teaching lessons. Just one of the many things I have learned from Sandra since joining the team a year ago.

The first activity is overly scaffolded. In the second version of the activity, the scaffolding is removed. Students will ask “How can I solve the problem?” versus “How does Mr. Hunter want me to solve the problem?”. Some students may need scaffolding, but I can better support these students by listening to and observing them. In the first assignment, I assumed all students would need scaffolding. And, really, if my students can’t think of using a table to organize information, what does that say about how numeracy is taught in my classroom?

Here are the documents as well as the three-part lesson plan:

Flatland Assignment
Flatland Assignment 2.0
Flatland Three-Part Lesson Plan

September 6, 2011September 7, 2011

Revisiting Pictorial Representations of Functions

The K-7 word walls were developed by my Numeracy Helping Teacher colleagues to help students and teachers communicate mathematically. They were not meant to ‘teach’ concepts but to help make visual and conceptual connections. The cards have been very popular with Surrey teachers. The Math 8 cards have been created and we will be sharing them with Surrey secondary teachers starting in September.

See the sample cards to the right. In an earlier post, I mentioned how concrete and pictorial representations of linear functions can enhance understanding. For example, in the expression 2n + 1, the coefficient of 2 can be interpreted as adding 2 tiles as the pattern continues.

The coefficient can also be visualized in another way. It may be easier to describe by looking at the card for constant. In the first figure, we can see two groups of one (one white square above the red square and one white square to the right of the red square). In the second figure, we can see two groups of two (one group of two white squares above the red square and one group of two white squares to the right of the red square). Similarly, in the third figure, we can see two groups of three. Finally, in the nth figure, there will be n groups of 2, or 2n, white tiles.

This can also be an interesting investigation when teaching quadratic functions (or a challenging extension when teaching linear functions). In the pattern to the right, the red squares in the first figure make a 2-by-3 rectangle. The red squares in the second figure make a 3-by-4 rectangle. We can see a 4-by-5 rectangle in the third figure. In the nth figure, there will be a rectangle with width n and length n + 1 . In each figure, there are also two white squares. Therefore, the expression is n(n + 1) + 2.

This pattern, too, can be be visualized in another way. For example, in each figure, the red tiles can be seen as being made up of a square and a rectangle. In the first figure, we can see 2 squares on top of a 2-by-2 square. In the second figure, we can see 3 squares on top of a 3-by-3 square, and so on. In the nth figure, there will be n squares on top of an n-by-n square. Remembering the 2 white squares, the expression is n^2 + n + 2.

The two expressions are equivalent but reflect different ideas.

How do you know that a relationship is linear? quadratic?
How are the pictorial representations of linear and quadratic functions the same? different?

To see more on this approach, visit I Hope This Old Train Breaks Down.

One more thing… I purposely did not circle the groups and shapes discussed above… I didn’t want to take away the fun of visualizing them for yourself.

September 15, 2010September 5, 2011

Linear Functions – Concretely, Pictorially, Symbolically

Welcome to my blog!

I really enjoyed Marc’s Patterning the Blues activity (taken from Marian Small’s Big Ideas book that department heads received).

Teachers often talk about how manipulatives can help the struggling learner. I’m suggesting that having students solve problems concretely can assist all learners.

When I experienced this problem using the blue and yellow tiles, I gained a deeper understanding of the problem. The equation y = 3x + 2 now had meaning. I was able to find the pattern in the table to determine the number 3. By modelling the problem using tiles, I was able to see this as adding an extra 3 blue tiles every time the figure grew.

In the past, I had difficulty explaining to students where the 2 came from. I could convince them that it had to be there. For example, take the point (2, 8). Multiplying the 2 by 3 gives 6, so we need to add 2 more. Looking at this concretely & pictorially, the 2 now has meaning. For me, it is how many blue tiles there were before we start adding yellow & blue tiles. (See the photo below.)

Your students who used to get it symbolically will still get it if they approach it concretely. However, what it means to “get it” in your classroom will start to change.