Reflection: Representations of Relations

The one thing I am really nervous about when starting this new school year is the bell schedule, 35 minutes for one period. What can I get done in 35 minutes? Can I get a Desmos activity done in 35 minutes? Could we do a class project in 35 minutes? I feel like I will be cutting out important discussions or big "ah-ha" moments with less time.

So I went back to thinking about when I start the second day of class with mathematics, I want there to be context in what the students see and do in math. In my last post about Representations of Relations I received this comment:

They changed the focus of that first lesson to make the rest of the year one cohesive goal.

Image result for gif that's what i want

As part of the curriculum group for Advanced Algebra, we set the pacing guide and decided that representations of relations along with the distributive property should be taught first. One thing I want to get across to students that first week is everyone having their voices heard and problem solving.

So each wall of the classroom will have the same layout from the previous post,

  • one wall will have ordered pair along with a piece of butcher paper with notice/wonder at the top. 
  • second wall will have a mapping with a different piece of butcher paper with notice/wonder.
  • third wall will have a table with butcher paper labeled the same way.
  • fourth wall will have the New York Times graph and butcher paper.
I want to have students stand (and gather by the board) and take 1 minute to look and 2 minutes to discuss with a partner what they see. I will ask what students notice first, then wonder. At the end I want them to discuss what was similar or different with the four different relations.

I still have to cover distributive property at the end, but as an exit ticket I want them to reflect on the experience and answer the following question:
Why did the New York Times select a graph to represent this relation?  
I want a connection more to relations, what question should I ask that encompasses what they learned and that representing functions is useful?

Representations of Relations

This is going to be my fifth year teaching Algebra 2, this year I am changing schools so it will be called Advanced Algebra. I was doing some curriculum writing with my new team of teachers and the first section we are going to cover as a class is Representing Relations my first reaction was...

Image result for why gif

At this point with just coming back to school students may not remember what relations are, what functions are, or domain and range.

I thought back to Dan Meyer's talk of headaches and aspirin and why do students need to know there are different ways to represent relations: ordered pairs, tables, graphs, and mapping.

I took the second graph from New York Times: What's Going on in this Graph?  and re-organized the information differently.

Give the students the following information:
This data is organized from by: country (guns per 100 people , mass shooters per 100 million people).
United States (85, 28)
Canada (26, 9)
Afghanistan (2, 20)
Iraq (37, 4)
France (35, 15)
Yemen (55, 40)

Ask the students what they notice? what do they wonder? As the teacher write down everything they say. One question I have is what is the data saying? Is there a different way to represent the data?

Give the students the following information:

What do they notice and wonder now? What has changed? You can show them mapping as well, but eventually you will need to introduce other things, but the last one is the graph from The New York Times.

How do the three representations differ, do they all tell the same story? Do some tell the story better? I'm not sure if this is the way to start the year out, but nothing is perfect. I want students to feel that there is some context to mathematics other than its day 1 therefore we do lesson 1.

"The Grasshopper and the Ant" teach Exponents

What do you notice, what do you wonder when you see this image?

It looks cold, how do you know which one is the ant? Which one is the grasshopper? Which one looks warmer? How do you think the discussion is going?

Next have the students read the fable that is associated with it:

A Grasshopper gay Sang the summer away,
And found herself poor By the winter's first roar.
Of meat or of bread, Not a morsel she had!
So a begging she went, To her neighbour the ant,
For the loan of some wheat, Which would serve her to eat,
Till the season came round. "I will pay you," she saith,
"On an animal's faith, Double weight in the pound
Ere the harvest be bound." The ant is a friend
(And here she might mend) Little given to lend.
"How spent you the summer?" Quoth she, looking shame
At the borrowing dame. "Night and day to each comer
I sang, if you please." "You sang! I'm at ease;
For 'tis plain at a glance, Now, ma'am, you must dance."

What do you notice? What do you wonder now?

Where math comes in to play is the idea of where the grasshopper says, "I will pay you, she saith,
On an animal's faith, Double weight in the ." When you can't afford something say, the full price of a car, how do you afford it?  Most students will talk about saving money or a loan. If the grasshopper wants to survive, she wants a cut of the ants food savings and next season the grasshopper will give double back. Ask your students is this fair? Is this how a bank works?

There is a way to calculate it mathematically, but right now I want students getting use to the idea of exponents. The equation I would have them use is the compound interest equation.


Where A is the amount
P is the principle
r is the interest rate
n is the number of times it is compounded per year
t is the time in years.

I would have students think about what each of them means and how the rate effects how much the grasshopper would way in the long run.

Ruth and the Taco Cart

I love Dan Meyer's Taco Cart if you have never heard of it, here is the task it is great it is wonderful.

We were recently doing this activity in Algebra 2, because I love how it uses the Pythagorean Theorem and one of my students asks, "what if you split the angle in half and walked that, because that is what I would do." I thought that would be a wonderful geometry question.

Ruth and the Taco Cart

I posed this question to my geometry students, we weren't really learning about angle bisectors, but we had already learned it as well as trigonometry. It took them about 15 minutes to use pythagorean theorem and find the time it would take. However, Ruth was much more difficult.

After students completed the time section I asked what they needed to find the angle and the distance Ruth would have to go.

There were lots of good questions. I gave the rest of it as homework and only a few were able to do it completely.

For students that did the work, I had them work in a group and work on a couple of extension problems:

  • If Ruth didn't want to talk alone, how long would she have to wait for the other to catch up?
  • What if she took a different angle?
  • Does changing the angle effect the time it takes for her to get to her destination?
The other students continued to work on the problem and were given some of these extension problems as they got the answer.

I loved the use of geometry, I will definitely use this when we are learning trigonometry.

3 ACT Task: Beat The Freeze: Circumference

The Situation: 
During Atlanta Braves games, one fan has a chance to race "The Freeze." Who is the first to reach the end?

Act 1: First 10 Second of the Video:

The video shows the contestant running the warning track in the outfield. It shows the lead the contestant has over "The Freeze."

Have the students discuss who they think will win.

How can we prove who will win or lose? What would we have to know in order to solve this problem? Are there properties of a baseball field that we need to know before beginning?

Act 2:
At the beginning of the video it shows the distance to the left field wall. The distance to the right field wall is 325. You can use 335 or round to 330. 

The Freeze runs at 22.5 ft/sec
The contestant runs at 19.1 ft/sec

The contestant is given an 70 foot head start.

Act 3: 
Show the full video. 

Extension Questions:
How long can The Freeze wait and still win?
What if the rates changed at the midway point?
What is the biggest lead you could give the fan and win?
If you were the contestant what strategy might you use to win?

Peer Teaching with ELL Students

I have been teaching one student who is an English language learner who came to me at semester how to add and subtract proper fractions for the past three days. Everyday they come in I feel like I am starting from square one each day. I tried teaching by one example at a time, didn't work. I tried teaching using visuals like fraction circles and bars, that failed. The student was getting more and more frustrated, because they weren't moving forward.

I tried a different way. The other students in my class are to graphing linear inequalities. A majority of students in this class speak limited English and/or struggle with mathematics. One of the students finished early and I asked them to help this student.

This was their discussion back in forth in Spanish. It was a great way for both students to move forward mathematically and feel confident going forward.

Link to conversation in Spanish: . The sound byte is a minute and a half of the whole conversation which took about 5 minutes. 

The girl in the audio does an excellent job of breaking down the problem and used fraction bars to represent the fractions in the problem. You can hear her counting out the fraction bar in the first part of the audio, eventually she moves towards release of instruction where they did a problem together, then she watched as the student did one guiding through the entire process.

I need to find ways of incorporating more peer teaching for my other students, I wonder how I can help guide them through the steps of asking questions and dialogue between each other better?

Orthographic Projection with Merge Cube

Merge Cube has been a hit with stores like Walmart offering the simple flexible cubes for a dollar a piece. Merge cube is a simple way to get students using augmented reality in a QR code way. Students scan the Merge Cube with an app and a magical world appears.

One of my favorite apps using the Merge Cube is Dig!

Using the app changes a simple cube with a bunch of symbols looking like hieroglyphics into another world. You can build and deconstruct the cube that looks almost exactly like Minecraft. The reason I like this app the most is that students can build using the app.

My students at the beginning of the year struggle with this concept of orthographic projection and being able to correctly sketch the block layout. Having students use the Merge Cube students can grasp that conceptual understanding that they don't get from a sketch.  Last year I borrowed some of the wooden blocks our construction teacher uses and it was a great way for some of the students to see the finished product. The app allows students to see around each object looking at it from the sides and from the top.

What I would like to see is have students create their own and have stations at each group where students correctly draw the orthographic projection of their groups creation. 


TIP: If you can't get a Merge Cube for each student, there is a shortcut. I printed a picture of one side of the Merge Cube and you can't rotate it like a Merge Cube showing each side, but students can use that one side to create especially using the Dig! app.