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 courses 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?

Pg1

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?

Pg2

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.

Pg3

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.

Advertisements

Characteristics of Digital Curriculum

This blog post has some key technologies and methods that every school needs to consider.  Students today just don’t learn the same way we learned and it is time to take that into consideration.

Check out this blog: Characteristics of Digital Curriculum

 

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!  Donorschoose.org 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.

Can I do CAS?

I promise I am working on another CAS lesson to share (Hint: think proportionality) but I also love to share stories of students’ first impressions of a CAS device.  As most of you know by now for me this is the TI-Nspire CX CAS handheld or the TI-Nspire CAS iPad app.  In this case it was the handheld.

This week my students took a categorical self-assessment on categories such as utilization of  class time, level of effort, learning new skills, staying focused on assignments, etc.  Next the students averaged their rating for an overall percentage.  They had their choice of a TI-84 Plus, a TI-84 Plus CE, their phone, the computer calculator, or the TI-Nspire CX CAS handheld to do the calculations.  Fifty percent of the students chose the CAS handheld.  Now our CAS handhelds are new to all of these students at this time.  I did not give any instruction I promise.  I just handed them out.  Only one student had a question about how to operate and that was because it opened on the setup screen.  The only other question I received was, “Why does it give me a fraction instead of a decimal when I divide?”  Now talk about a teaching moment!  This happened in six different classes, grades 7th-12th, gifted students to students with 504’s and IEP’s.  Merry Christmas to me!

Moral of the story

One of the biggest obstacles I have to getting teachers to use CAS is they claim that students won’t know how to use them and they do not have the time or knowledge to teach them.  I just proved no EXTRA teaching is necessary and the teaching you do need will be math skills and practices that should be in every class.  I also want to add that Texas Instruments has getting started docs and tutorials for all of their devices for anyone willing to look at them.

So can I do CAS?  The answer is YES!  Everyone can do CAS.

#ACC2015

I attended the Arkansas Curriculum Conference last week, and it was awesome as usual!  Thanks to Lynne and all the people that put it together.  There were several sessions I wanted to attend and had to choose.  From all the sessions I attended and presented, I gathered a common theme: Students learn more when they create the “rules” themselves.

By “rules” I mean properties, algorithms, theorems, postulates, etc.  This is why CAS should be a part of every math and science classroom.  CAS can enhance any lesson and any activity to draw those conjectures from the students and motivate them to discover the “rules” for themselves. In my sessions I met the same old stereotype that the TI-Nspire CX CAS is just another calculator.  I hope I enlightened some educators to the fact that CAS devices are a teaching and learning tool not just a calculator.  If you are assessing the students on if they know their multiplication facts, do not give them a device. (Just to be clear I have strong opinions on this topic and am in no way advocating requiring students to memorize facts.)  If you are assessing the students on how well they understand the math, having a device should not matter if you ask the questions correctly no matter what it is called.  CAS devices such as the TI-Nspire series are TOOLS and should be used appropriately and strategically…Did I just quote a best practice? (MP5 for CCSS and computer algebra system is stated specifically under this practice.)  Feel free to check.

In closing I enjoyed all of the sessions and appreciate the educators who attended mine.  A special thanks to Linda Griffith for leading the ACTM group and Tony Timms for being my sidekick in my sessions.  See everyone in Orlando!

CAS with Probability

Using CAS with probability lessons can make learning fun and allow students to do experiments on a device instead of gathering data, flipping coins and rolling number cubes the old-fashioned way.  Sorry teachers but the students like the devices, and it will save you money and storage space if you keep it digital.

Let’s start with random samples involving population.  TI has a FREE activity called Interrogating the Data that works great.  The data is a random sampling of 7th grade math scores.  It starts with the question:  “Why might one use a sample instead of collecting data from an entire population?”  The handheld generates a random sampling for the students.  You do not have to come up with the numbers.  Let the technology do the work for you.  Here is a screenshot of the samples:

10-05-2015 Image001You can use the TI-Navigator CX CAS system to capture student handheld screens to display on your screens for discussion and self-checks.  A CAS handheld will allow a student to compare not only the table but different data displays to compare which is the best display to analyze the data.  A dot plot and stem plot is shown below, but you can also look at a histogram.

10-05-2015 Image002       10-05-2015 Image003
You can even compare all of the samples on one page.

10-05-2015 Image004

With technology students can customize the color and axis to investigate to their individual style.  Next the CAS will allow students to calculate the mean of the sample and absolute deviation using formulas in the spreadsheet which leads into major discussions on the data.

10-05-2015 Image008

Instead of focusing on how to do the calculation at this time you can focus on the following:

1.    What does the formula mean?

2.    Why is it necessary to calculate the absolute value?

3.    What is learned by calculating the Mean Absolute Deviation?

This is applying the mathematics instead of just doing the mathematics.  Students can then make predictions using their analysis of the data instead of a textbook created problem with no meaning.  The CAS handheld will also allow the students to interact with the graphs by hovering over different parts of the graph on the screen to see other important values as shown in the screenshot below: 10-05-2015 Image006

This particular activity gives students other types of population data to study also.

Now let’s look at experimental and theoretical probability.  Using the One and One Equals Win activity from the TI website, the CAS handheld has a basketball player simulation to analyze probability of making a shot and a basic spinner simulation.  Screenshots are shown below.

10-05-2015 Image005     10-05-2015 Image007

The CAS handheld allows students to calculate the theoretical probability then do the experimental probability and compare the two.  This will save time in the classroom and still allow the most important discussion to take place:  Are the experimental outcomes different from the theoretical and why?  Then finally give the students a basketball scenario and let them use the probability experiment to predict the outcome of the game based on the data.  The TI activity provides a scenario but you can create your own if you choose.  A sample table is shown below:Probability TableIt is easy to see that technology is the way to improve motivation for learning probability.  The TI-Nspire CAS handheld comes with pre-made activities and simulations for this purpose and has the calculation and graphing capabilities working together.  Whatever data is put in the table is automatically connected to a graph for manipulation and analysis.  I will never use physical number cubes again.

Why I Use TI Technology

Although this goes for CAS and nonCAS, I thought it worth sharing and she says it so much better than me.

Easing the Hurry Syndrome

I was asked a few weeks ago why I use TI Technology in my classroom. I have never felt articulate when it comes to extemporaneous speaking, but I agreed to talk with the reporter because my experience has been that it’s good for me to have to justify what I’m doing in the classroom. Why do I continue to use TI Technology in my classroom when other free technologies are available for teachers and students to use?

A Google search of “technology speeds up life” results in about 142 million results in less than half of a second. What I find in my classroom, however, is that using technology actually slows down the pace.

In the midst of my usual rush to cover all of the required standards, when we use TI-Nspire Technology to explore a difficult concept, the questions that students ask slow us down. The platform that we…

View original post 732 more words