What pattern do you notice in the picture?

What do you wonder about the picture?

What linear equation does the picture represent?

Give evidence that your equation is correct.

Once again I am using the Shell Center resources for class, this time looking at Modeling Situations with Linear Equations. Once again, I am looking at the student handout and although it is designed as a pre-assessment, I am wondering why they are supplying students with so much information- time to get the scissors out….

The Guitar Class:

To be totally honest here, I’m scrapping pretty much the whole handout and start from scratch, here is what I will show students.

Students should come up with ideas about rent, how many students are in the class, and how much each student should pay for the class. They should also sketch a graph starting at a negative balance (showing the rent for the space) and increasing with each student enrolled. I will then ask them to write down answers to the following questions on their paper:

Teacher Questions:

What values for each factor did you assign?

How did that determine where to start your graph?

How did that effect your Axes? How did it effect your scale?

What do you notice about your graph?

(hopefully they say 2 things, it’s increasing at a steady rate per student, and that is crosses the X axis)

Students will have a varied set of values for each factor, and that is OK, their graphs will all have the same general look, and they should recognize that where it crosses the X axis is where the music teacher breaks even for cost. I don’t want to give away anything with this question, it provides a great platform for students to provide ideas and values, as well as creating their own ideas on how to graph the scenario. Looking at each individual’s work will give me instant information on how much the student comprehends the material and what I need to address.

As students are done with their graphs and our group discussion is when I would put back in the original scenario from their handout:

I do not think I will ask students to write an equation right away, giving students the proposed values instead. I would then ask these questions for students to answer on their paper:

Teacher Questions:

What is the situation the music teacher faces? (using these values)

What is the teacher’s balance at the beginning of the class?

How does each student effects that balance?

Write an equation representing this situation.

I also would not show them questions 3 or 4, I would instead ask verbally these questions, have students record their thoughts and discuss each one as a whole class at the end of the hour:

Teacher Questions:

How can we determine the teacher’s total cost. Write that as an equation.

How can we determine the teacher’s total profit? Write that as an equation.

What factors do we need to consider to determine the cost for each student? Write that as an equation.

Are there values that are unreasonable to charge students for taking his class? Explain.

I will be trying this out with my groups tomorrow. I plan on engaging their creativity and mind- which will produce great ideas to discuss. I realize that this is a pre-assessment for their unit, but students will provide me with enough information on where their skills lie with my student handout and follow up questions. I will write a follow-up to see if I “Cut too Much.”

This is a fun project that I started doing to support my Art teacher and show students that Algebra can make the most amazing things.

*Step 1:*

Have students pick out a picture. This is a great sell to students since they have control over what type of picture, icon, avatar, etc that they use for the project.

This was the example I used, the logo from my college, the Bemidji State University Beaver.

Have student create a free hand drawing of the picture. This allows them to adjust the picture, editing lines and shading that they may not want to show. It allows them to be creative and adapt the picture for the project.

Here was my free-hand drawing of the Beaver.

Students recreate their free hand sketch on graph paper and using only lines. They should try to make shapes end on grid points of the paper. This part can take a while, if students created a large enough free hand sketch, you could allow window tracing to help transfer the image to graph paper. I have students make 2 copies of the graph image, one they can decorate and one for the next step.

Here is my colored graph paper image.

**Step 4:**

Have students create an X&Y Axis on their picture. I allow them to create them wherever they want, and usually they have them right in the center of their drawing. Depending on the patience level of the students, I have them label points that create their drawing. There are times I only have them do a set amount (20 or so) and times I have them label each line segment ending. This is totally up to you.

My XY Axis image:

**Step 5:**

DESMOS! Need I say anything more. I used to have students write the equations of the lines that create their image, but with the totally awesome program DESMOS, I now have students create their image with it. DESMOS has helped students understand how changing the slope or intercept effects the line, and with the instant drawing of the line when they enter the equation, it allows them to visually see where their line is. This is a great error check for students, and they accept mistakes more readily than if they are writing equations on paper.

My DESMOS image.

To complete this project, I have students create a collage of their sketches, and a printout of the equations from DESMOS. I then hang these posters out in the hallway for everyone to enjoy. This attracts students from all over the building to come check out what kind of cool activities we do in 8th grade. I am even getting new 8th grade students asking me when we will start this project!

So students were a bit overwhelmed with Desmos at first, so I had them do a quick activity with horizontal and perpendicular lines: create a logo from our school initials, CL.

Here are some examples of work I received.

Thanks again Desmos, the Students are really loving playing around creating things. It’s been a great end-of-year activity. I hope this encourages them to use Desmos for their high school classes.

Today I wanted to revisit Algebraic equations with my students. I decided to use Andrew Stadel’s Styrofoam Cups 3 Act lesson. The thing I like most about this lesson, is that it has a lot of “hidden math” conversations available- more on that later.

I showed my students the video, and asked them to write down 3 questions they had about it. I really stress doing this with 3 Act lessons, student always hit upon what you want them to learn and they have the credit for it! I also like it when I throw up questions and students start to evaluate them and can tell me which ones are similar or even which questions actually answer other questions submitted.

One of the hidden math conversations that I use during this time is questioning: what makes a good question? Since my students have not seen Mr. Stadel before, I can’t even begin to tell you how many questions I got on his appearance; where did he buy that shirt, why is he wearing khakis, who is he, where is he at, do you know him? I am a believer that all questions are good as long as they are asked with a desire to really obtain knowledge. Many of my students were asking these questions because they wanted to gain attention from the class, and we talked about if these were good math questions. *Students defined good math questions as ones that focus on what we are trying to learn and how we can solve problems.* I was OK with that- I may even make that a mini-lesson next year and create yearly posters on questioning. After we got through the silliness, we got down to business.

This is what the board looked like today. Even though there is teacher writing, this was all student driven ideas. I have found that I like students showing WORK on my whiteboard, but for idea expression I need to regulate it to be efficient and on task. The ?’s were ones they finally decided on to be good math questions for the video. They then wanted diagrams showing the measurements of the cups and door.

The first great conversation that came from this video what what measurements we needed to solve this. Students needed to know the height of the door and height of the cup, nothing more. When I asked them to take 2 minutes to come up with ideas on how to solve this, a great discussion ensued. I asked my students to relate what their concerns were, and they were confused about the stacking process and how to properly address it. One student showed me this model to represent the difference in stacking cups and the data they were using.

This represented the height of 2 stacks of 5 cups, and students decided they needed the measure of the lip of the cup to continue with the problem.

After this discussion another great discussion came about when we looked at units. The cups were in centimeters, the door inches. Students knew they had to do a conversion, but many could not remember what the conversion rate was. One student remembered that it was 2.5cm = 1in so we rolled with that. I knew this would only serve for great discussion for Act 3.

The next task was to figure out how many cups would fit in the threshold. As you can see from the whiteboard, there were a couple of variations that students wanted to try. We worked on the first idea for a while, but students realized that this was a guess and check method. They quickly decided that was too much work and I gave them another 2 minutes of partner thinking to come up with a different solution path.

Idea 2 is one that I have struggled with all year. Students take formal algebraic equations and convey them in simple terms using basic operations. None of my students considered writing an equation to model the situation, most just knew what steps they needed to solve the problem. We have struggled with this concept all year, they are great with “fill in the blank” or “empty box” expressions, but those darn * X’s*. Cognitively some students are not ready for that type of representation. After they solve this problem, as a class we go back and formalize the notation- and I will also show them how their solution method connects to the properties of equality in generating a solution.

One side-note here, when we start in the solution process many students start complaining about the long decimal in the height. This generates another good conversation about precision and what is appropriate to use. Even though students may not believe it, they have a great feel for what precision is.

Here are some student samples of work.

A couple of things to note, students forgot to add the last cup on the stack back into their answer. When they did the math, they envisioned the cups at the pattern, or 1.3. Out of all the errors to find this one proved to be the most elusive.

One thing I feel needs to be brought to 3Acts is reflection. While we can reflect on why our answer was different, it is a totally different thing to reflect on what we did for the project, what math was involved and why we did the activity. Right now I am reserving 15 minutes of class for good reflection time, and for this activity I put up an reflection outline for students to model and help organize their thoughts. I hope that after a couple of Lessons with 3Act, I will be able to just hand them a blank sheet and they can write away. Here are a few examples:

This really gives me an idea of what students took away from the lesson. Some were very superficial, claiming the height of the door and cup height were key ideas. Others connected the rim as a slope or pattern of the activity. Other students focused on the rounding and unit conversion as important things. Even though the main focus of this lesson was linear models, each student gave me a different perspective of what they noticed during it. I will go back tomorrow and have a review of what we did and the mathematics involved, and as a class we will fill out a reflection form. We will practice linear problems, and I will hit them up the next day to see what things truly stuck in their learning.

So today we are reviewing Linear Functions; and when asked what those are, I get a textbook-like explanation.

“It is where each x only corresponds to one y and graphs a straight line.” Another student asks, “What does that really mean? How does it even relate to what I would do when I’m done with school?”

I will post later on how my conversation went, but I’d like to hear your ideas before I give away any secrets. (For reference, I teach 8th grade students in a public school on the reservation with high poverty rates.)

**ACT 1:**

Recently I had to go to a conference and when I walked in, this is what I saw

*What questions do you have when you look at this picture?*

Ask students to write down their questions, I normally ask students to find at least 3. When I observe that most students have questions written, I ask them to share those questions with their neighbor. I then throw up a Microsoft Word document and start typing down questions students supply. Students from my classroom came up with all sorts of different questions, some we can easily answer and others that we can’t. I am looking for a key question or questions to start this lesson. If students do not ask one of these questions, I tell them that I hope I can answer most of the questions provided, but that I need them to consider one of these questions first.

*How many tables are there?*

*How many people are there?**How many people are in a row?**How many people can sit at a table?*

Any of these type of questions will lead students down the inquiry I hope to explore with them.

There are 512 attendees at this conference, how many tables are needed to seat them all?

How would this change if I needed to seat 900 attendees?

**ACT 2:**

Linear Equations is the topic I wish to address with my students. Students will need to think of counting strategies needed to determine the number of people at the conference. There are many different combinations of ways to figure this number out, but here are the stats:

*There are 8 people seated at the head table, but they needed to use 6 tables (extra space was used for the podiums). Otherwise there are 4 sections of tables, 9 rows of tables to a section, 14 people to a row, and the table size is 6′. *

These are hard to identify by the picture so the students really work on the following pieces of information: 6′ tables, 9 rows and 512 attendees.

I also ask students to create an equation for the number of tables that are needed.

**ACT 3:**

For comfort reasons, 2 people are seated per table.

*You will need 7 tables per row.*

*4 x 9 x 7+ 6 = 258 tables.*

Students will typically write this type of equation based on attendees:

512 = 2x + 8

504 = 2x

252 = x

Then I typically ask them how many tables per row, and what type of configurations makes sense for the conference (does one LONG row really work?).

**Extensions:**

After we decide the general equation for the conference, I then ask the students how many tables are needed for varying amounts of attendees: 200, 900, 1200, 2500. What type of configurations would work for these new conventions?

**Order of Operations:**

Using mathematical operations, grouping symbols and equality symbols, create as many mathematical equations as you can using the numbers:

6 5 1 4 1 3 6

**ACT 1:**

This video was first brought to my attention from Dan Meyer’s My Opening Keynote for CUE 2014. Turn the volume on mute when you show this to your students. I also do not show the individual run, I start the video 15 seconds in. I show the clips of Jacoby Ford and Terrence Cody ending at a minute in.

*What questions do you have when you watch this video?*

Ask students to write down their questions, I normally ask students to find at least 3. When I observe that most students have questions written, I ask them to share those questions with their neighbor. I then throw up a Microsoft Word document and start typing down questions students supply. Students from my classroom came up with all sorts of different questions, some we can easily answer and others that we can’t. I am looking for a key question or questions to start this lesson. If students do not ask one of these questions, I tell them that I hope I can answer most of the questions provided, but that I need them to consider one of these questions first.

*How fast are they running?**How much of a lead does Rich get on run 2? run 3?**How much does Rich lose by each time?**How big of a lead does Rich need to tie? to win?*

Any of these type of questions will lead students down the inquiry I hope to explore with them.

**ACT 2:**

Once again, this video can create a few different paths of exploration. We can explore:

- The rate of the runners
- Graphs of the runners
- Equivalent equations

These are all excellent topics and students generate a lot of classroom discourse discussing each one.

- Rates: This is one that creates a lot of arguments about precision. Students normally start trying to time Rich by using the clock on the wall, or their wrist watch. Some will break out the stopwatch feature. I have other students use the watch feature on their phones. Timing issues like accuracy starting or stopping the time, cause quit a disturbance with the students.
- Graphs: I love this part. I normally show a clip twice and have students graph the race. Independent and Dependent variables, scale factor on axis, and the solution of two lines are great topics to discuss. Students really enjoy graphing the races and are really good at evaluating work and refining the process and answer.
- Equations: This normally involves at least one of the first two processes and builds upon that. Finding the rate of each runner (their slope) and setting their expressions equal to each other leads to when Rich is overtaken. Having these expressions will also allow us to find the exact time Rich is passed and how much of a head start he would need to either tie or finish first. I can’t think of a better introduction into solving systems of equations.

**ACT 3:**

I normally show the opening video to answer how fast Rich runs, and turn up the volume to allow students to know how much of a head start he is given in the other races.

**Extensions:**

Show them the 3 man race (1:10 into the video) and let them loose, it’s fun to watch.