The Guitar Class: MAPs Activity- Cutting out what you don’t need

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.

Guitar Student Handout

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:

Guitar Class

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.”

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Stacking Styrofoam Cups

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.

Stacking Cups

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.

IMAG0482

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.

IMAG0481 IMAG0480 IMAG0478

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:

IMAG0483This 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.

Aside

How many options do I have?

Chris says the following problem has a limited number of solutions:

Directions: Fill in the empty spaces so that you create two distinct parallel lines.  You can use whole numbers 1 through 6, but can only use a number once.

___ x + ___ y = ___

___ x + ___ y = ___

How many solutions does it have?  Provide evidence for your answer.

Is there another way to represent the number of solutions for this problem?

PowerBlocks -3Act Math

ACT 1:

This video was inspired from comments made in Dan Meyer’s My Opening Keynote for CUE 2014.

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.

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

  • Is there a pattern
  • What is changing from one frame to the next
  • Is the pattern constant?
  • Is this an Arithmetic Pattern?
  • Is this a Geometric Pattern?
  • How did you make a cube?

ACT 2:

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

  • Patterns
  • Rules of Exponents

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

  1. Patterns:  Students will notice that I first make a square, each side the same length as the initial picture, and then a cube with each side the length of the number of cubes in the square.  The cubing part is where most classes will struggle, many will just try to create a cube out of the preceding square.
  2. Rules of Exponents: This is what I designed this video for, in an attempt to create a visual representation of (x^a)^b = x^(a*b).  Using the unifix cubes created a quick, easy way for students to quickly see and do the mathematical calculations.  The powers I started with were easily recognizable visually: (x^2)^3.  One thing my students start to see is how the base figures into all of this, we normally pause the video and use the SMARTBoard to draw lines to create the pattern of multiplying the base.  For example on slide 3: we circle the bottom left 2 blocks, one stack represents 2^2, two stack represents 2^3, four stacks represent 2^4, etc.

ACT 3:

  • Frame 9: (4^2)^3 = 4096 => 4^6
  • Frame 12: 5^2 = 25
  • Frame 13: (5^2)^3 = 15625 => 5^6

 

Extensions:

  • Patterns: I ask students to predict what a Step 4 figure would look like
  • Rules of Powers: Have the students determine a third power and sketch what their figure would look like & how many blocks it would take to create it.