CAS with Proportions

This lesson deals with teaching not only proportions but proportional reasoning that will have connections from 6th grade through 12th grade.  This is a very important foundational skill for not only future math courses but science course and elective courses as well.  Let’s start with connecting to fractions.  Students will have seen fractions and their equivalents on a basic level so let’s connect with that prior learning.  Place the slide below on the board and ask the students:

  • Why is each one true?
  • What math is being done to the first fraction to get the second fraction?


Now ask if they can share a false example.  Let them use a CAS device and they can investigate before answering which gives them more confidence to answer.

Now let’s look at these fractions in context.  Context is always an important part of a lesson so students can relate to the numbers.  For my example I am using boxes and crates.  This can be modeled easily with blocks and cups, M&M’s on a certain size paper, even real boxes if you can get the different sizes.  The key is nothing fits perfectly.

Discussion Question:

If 6 boxes fit into 4 ½ crates, how many crates will it take for 1 box? for 100 boxes?


Using the CAS students can easily use the words “boxes” and “crate” as variables to do the investigating and keep it contextual.  Students can look at the different representations and make connections between the numbers and the application.  For example the graph below is created from the spreadsheet above without retyping anything.  Create a graph page and SELECT your variables created by the spreadsheet.  It reinforces the relationship between the data, coordinates, and graphing.  All graphs come from data not randomly generated numbers.


After this introduction some good activities to use throughout the unit are TI’s Recipe: Unit Rate, Proportionality in Tables, Graphs, and Equations, and/or Proportions in Stories activities.  I really like the Recipe:  Unit Rate because it ties the previous unit rate knowledge into proportional reasoning and it is contextual.  The other two activities move into the more advanced graphing and equations part of proportionality but still shows the students that slope is just using proportional reasoning for more complex problems.

I feel that we as educators sometimes tend to treat graphing as a separate entity, so the students see it as something completely new.  Actually it really is an extension of proportions and proportional reasoning.  “Slope” is not a new concept it is just a different use for proportions.  If we build on student’s knowledge of equivalent fractions and unit rates to progress into proportional reasoning then proportion calculations and graphing, students will see a natural flow instead of a new beginning.

Blended Learning with CAS

I have been reading about more classrooms using blended learning as technology becomes a natural part of education.  One article stressed that it doesn’t always have to be students 1:1 on computers.  It could be a YouTube video, a teacher made video, Twitter chats using personal devices, etc.  Of course I immediately thought, “What about CAS?”  Although there are CAS applications on computers, there are CAS apps on tablets and of course the TI calculators.  I have yet to come across a teacher in middle school or high school that did not have access to a calculator.  The key is asking for the correct device.

Blended learning according to Google is “a formal education program in which a student learns at least in part through delivery of content and instruction via digital and online media with some element of student control over time, place, path, or pace.”  This definition varies with different sources but all have the same basics:  The student is not sitting and listening to a teacher talk for the whole class and some part of the learning is digital.  Most sources state there is a student-controlled path and pace also.  Using CAS covers all of these things.  TI has activities for their devices that allows students to learn by just using the device and following instructions.  Teachers can create lessons ahead of time and send to the device for students to use.  CAS apps allow for downloading pictures from the internet to be used with investigations of skills.  It is not meant to be the ONLY tool in a blended learning environment, but it definitely should be in your toolbox.

This part of the blog post is going on a little tangent, but I get this excuse all the time.  “My school does not have the money for technology.”  Money is always an issue with technology but there is state and federal money allotted to schools for instructional technology.  There are also grants that are fairly easy to get.

  • The first step is to let your administrator know that you want new devices.  Be prepared to negotiate.  The word “calculator” is usually easier to sell than “device” or “iPad”.  If you feel your classroom will benefit more from the tablets, agree to 5 or 6 and use in groups but don’t forget the money needed to purchase apps.  If needed suggest stages, i.e. get 3 each year until we have a classroom set.
  • The next step is be willing to follow through by finding out when the district orders technology and keep gently reminding your administrator.  Do the leg work and find out the costs of different solutions and have it ready at all times.
  • Look for grants! is designed for schools and has companies just waiting to give you money without you having to find them yourself.  A science teacher in our building received TI-Nspires and probes this year by using this website.
  • Finally be patient.  It may not happen over night, but eventually your school will get you what you need.  Assuming your school does not have the money will get you no where.  Not asking will get you nothing.  I have only worked for two different districts but both were not “rich” schools.  Both were rural and one has even been in financial distress.  I have had to wait 6 months, but have always been able to get some technology that I have asked for.  One year I had been asking for a TI-Navigator system which then was going to run around $6000 for the handhelds and systems.  My administrator said they would try but didn’t know when.  I was in my classroom when the office said they were sending a sub because the superintendent wanted to see me.  I tell you I was scared to death!  What had I done that I had to go straight to the top?  It turns out the school had to spend some technology money by the end of the month or they were going to lose it and the amount was very close to the amount I needed for my system.  He asked for a summary of what it was, what it did, and how it would help our students.  He liked my answer and I had the system before the end of the school year.

Blended learning is the future and has been proven to motivate students.  What we once thought would be impossible is now not only possible but inevitable.  Are educators ready?  Ready or not it is here.