# Exam Mode on the Prizm CG50 Graphing Calculator

As my last post stated, it’s that time of year for standardized testing. As part of this, certain states require that students use calculators that have been set to exam mode. This means that certain features of the calculator have been ‘turned off’ or are inaccessible to students while the exam is going on.  I remember spending hours setting all my calculators to exam mode for students and then having to spend hours undoing that once exams were over – quite a pain.

The beautiful thing about the CG50 Prizm graphing calculator is that you never have to undo the exam mode – it will automatically turn off exam mode after 12 hours. Which means, you can set it, students can take their test, and then next day, the calculator is ready to go again with full functionality restored.  Another nice feature is that when the calculator is in exam mode, you can actually see it on the screen – there is a green highlighted border when in exam mode. This makes it easy to walk around and visually check that the calculators are indeed still in exam mode (or were set to exam mode to begin with, if you have your students do the process for you).

I made a quick video on how to put a CG50 Prizm into exam mode. I apologize for the lighting – very hard to film the actual calculator (vs. emulator) while holding my computer video camera…and those shadows?!!  But – hopefully you can get the gist of things!!

# Multiple Representations on the Casio Graphing Calculators

One of the key things we try to help students with when studying functions is the idea of multiple representations – i.e. graphical, symbolic (equation) and table.  Ideally, we want students to be able to discern what the function represents or looks at no matter what representation they are given, and to be able to find patterns and important components about that functions from all representations.  Students should never learn about functions just through graphing, or just through symbolic manipulations or just through looking at data points in a table – they should be able to go back and forth and determine which representation is the most useful for the situation.

Unfortunately, too often, the emphasis is on one representation at a time, or at most 2. Let’s look at the graph and find the minimum, maximum, or intersection. Or, let’s find the roots of a quadratic by factoring, or symbolic manipulation. Or, here’s a table of points, where are the x-intercepts or the y-intercepts? Ideally, we want students to be able to look at all of these representations simultaneously so that they see the relationships between the representations and come to understand what the points represent in the table, in the equation, or in the graph.

Technology is one way to show all these representations at the same time, and then quickly manipulate and explore. There are obviously many technology tools out there, but as I have stated in previous posts, the most accessible technology tool for most students and teachers is the graphing calculator, not only because of it’s affordability, but because it is a tool most students have readily available.  It would be nice if all students had computers or tablets for daily classroom use, but that is still NOT the reality.

I have put together a quick video showing Casio’s three graphing calculators – the fx-9750GII, the fx-9860GII, and the CG10/20 or Casio Prizm, and how they can display the equation, graph and table representations of a function on one screen. No matter which model you have, you can achieve the same functionality, allowing students to work with multiple representations and explore relationships quickly and efficiently.

Check it out:

# Permutations & Combinations – Casio vs. TI

I share a twin house with my neighbors (i.e. we are attached) and we like to decorate our front porches for the holidays the same, so that our ‘house’ is coordinated.  Every year we do something different, and this year we decided to hang holiday ornaments along with the lights – so a variety of Christmas balls and various large ornaments hanging from the porch.  As we were trying to decide the most ‘pleasing’ order to hang these, I realized we were basically discussing combinations and permutations, which naturally got me thinking about working with this in math class.

Permutations and combinations are often very confusing for students. Basically you have a group of things (numbers, objects) and you are going to pick a certain amount from that group of things, and depending on whether order matters, you either have a certain number of combinations of things you can make or a certain number of permutations. Combinations are the possibilities of things chosen when order doesn’t matter. Permutations are the possibilities of things chosen when order does matter.

As an example:

We have 3 Christmas Balls – green, red, blue.  If I want to choose two, order doesn’t matter, than it’s a combination, so how many combinations will I get?

3 combinations: Green, Red;  Green, Blue; or Red, Blue

But, if order does matter, then we have a permutation, so how many permutations are there?

6 permutations: Green, Red; Red, Green; Green, Blue; Blue Green; Red, Blue; Blue, Red.

Now, there’s also the whole idea of replacement and no replacement, but I am not going to get into that here. Working with students, you would want to start with small numbers of objects so they can create the combinations and permutations by hand. But then, you’d want to lead into more complicated things such as lottery numbers and chances of winning, where finding all the combinations and/or permutations is hard to do by hand, thus requiring a formula to make it more efficient, and then eventually, if you really want to do comparisons and have interesting discussions about many real-world examples, you’d want to incorporate technology to help be even more efficient. Here’s a nice page I found that discusses the differences between combinations and permutations and the different formulas needed and provides some good examples.

Below is how you can calculate permutations and combinations when you know your sample size (n = number of things you have) and how many you are choosing (r) from that group of n things. This video shows how to do this on both the Casio Prizm and the TI-84+CE.

Clearly in my front-porch, neighbor decision making, order actually mattered. We wanted a pleasing arrangement. We therefore were looking for permutations – how to choose six balls from a possible 10, so 10P6. There are a staggering number of permutations – 151,200.  Who knew holiday decorating had so many choices!!!  Needless to say we did not try to look at all of them – but good to know we have so many options for the years to come!

# Using Connections to Build Understanding

I am teaching a Geometry & Spatial Reasoning course for Drexel this semester for their math masters program for teachers. Absolutely love it because I am learning so much from my students/peers, but because it really is bringing home the importance of prior knowledge to help build connections and real-world connections in helping students learn versus memorize, and construct and reconstruct based on their ability to make connections.

My students, who are a mix of very new math teachers, experienced teachers, and even some career-switchers still in the early stages of teaching, are having this great discussions on the importance of using prior knowledge to help student make their own connections. Some have been doing this all along, but others, as they themselves struggle with some of the geometric concepts we are ‘learning’ (relearning in some cases), are coming to understand the value in helping students use what they know to build on and connect to new information. Makes it easier to recall, and builds a confidence in students that when faced with an unknown situation/problem, they have the skills and confidence to look at it, break it down or add in things to make the unknown familiar and then look for and make use of structure (see what I did there….Common Core Math Practice #7!) to help reach a solution or develop a new conjecture/conclusion.

As an example, we’ve been doing a lot of work with inscribed angles in circles and how do you help students use prior knowledge to build the idea that an inscribed angle is half the measure of it’s intercepted arc if you don’t want students just memorizing formulas? Basically, the conversations revolve around constantly using prior knowledge to make connections, which might mean you need to add in an auxiliary line to a given shape to ‘see’ something familiar (i.e. a linear pair or a triangle, as examples). A strategy that really helps students look for and make use of the structures they are familiar with to help them make sense of a problem.  Here’s an example of just one way to explore inscribed angles, using previously knowledge about triangles:

• In Fig 1, we have an inscribed angle and its intercepted arc a. How could you show that angle 1 (the inscribed angle) is half the measure of it’s intercepted arc? Here’s where students need to make sense of this structure – what prior knowledge can they use to help them?
• In Fig 2, they add in a radius (auxiliary line), because they know all radii in this circle (any circle are equal – doesn’t change the original inscribed angle….but now – we have a triangle and a central angle (angle 2).  What do they already know? Well, they know the central angle 2 is the same as the measure of the intercepted arc, which is the same intercepted arc as angle 1 (inscribed angle).
• In Fig 3, students are looking at the triangle created and using prior knowledge – we can mark the two radii equal, making this triangle an isosceles triangle, which they already know from prior knowledge has two base angles that measure the same (angle 1 & 3). Angle 2 is an exterior angle to the triangle, and angles 1 & 3 are remote interior, which they know from prior knowledge sum to the measure of angle 2. Since angle 1 & 2 are equal (isosceles triangle), that makes them each half of angle 2 (Sum divided by 2). Angle 2 is equal in measure to the intercepted arc, so angles 1 & 2 are each half of that, so the inscribed angle 1 is half the intercepted arc.
• Fig 4 shows that the relationship holds true even if you change the size of the inscribed angle.

This is of course just one example for an inscribed angle, but they can then use this to show that inscribed angles that are not going through the center of the circle have the same relationships – ie add in auxiliary lines, use linear pairs, or triangles or other known things to help make sense and show new things. Prior knowledge, connections – they really matter.

As teachers, it is our duty to make sure we are modeling and helping students use what they know to build these connections and see the relationships. It takes deliberateness on our part, it requires modeling, it requires setting expectations for students till it becomes a habit (habits of mind) to look for and make sense by pulling in previous knowledge.

Another thing we need to do is make connections to real-world. My students are sometimes struggling with this idea of relating prior knowledge and new ideas to real-world applications, but if you get in the habit, its not so hard to do. Since I am focused on circles and the lines that intersect them now with my class, I pulled up a ready-to-use lesson from Casio’s lesson library that is a great example of a real-world connection to circle concepts that would force the use of previous knowledge.  The lesson is briefly described below:

The Perfect Glass Dome: (here’s a link to the complete, downloadable activity)

• – Use coordinate geometry to represent and examine the properties of geometric shapes.
• – Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

This activity uses the Prizm Graphing calculator and picture capability to help build understanding.

The kinds of questions and connections to prior knowledge that can be asked of students just by looking at the image are pretty endless. What relationships do you see (i.e. lots of diameters, or straight angles, lots of central angles, all the angles are 360, are their auxiliary lines we could add to find the areas or relationships or angle measures, etc.).

If you look around, you can probably find a real-world example of most math concepts your are working on with your students. Show them pictures, show them real objects they can get their hands on. Start asking questions. Ask them what they recognize or think they already know. Ask them if they could add something or take away something to see a familiar object/concept. How does that help them? What relationships and connections help them get to something new or interesting?

My Drexel course and student are reemphasizing for me (and them) the importance of prior knowledge to help build connections on a continuous basis, all the time, every day. It helps students think mathematically and consistently use vocabulary and math concepts to deepen and create new understanding and relationships. It also promotes logical reasoning and problem solving – win-win!

# Conics – Casio Prizm vs. TI-84+CE

I am currently teaching a course at Drexel University and we are starting a unit on circles. I loved using Sketchpad when teaching because it allowed for dynamic manipulation of objects (shapes, functions) so that students could visually see the impact of variables to the shape, size, position of the object. Unfortunately, my students (math teachers in a Masters Math Teaching Program) do not have access to Sketchpad, though one does use Geogebra, and as this is a course focused on teaching, they need to use what they have access to in their own classrooms with their students. For many of them this does not involve any technology at all, which is sad, but for some, they do have access to graphing calculators.

Naturally, this got me exploring what the graphing calculators could do, and surprise, surprise, I noticed quite a difference between the Casio Prizm and the TI-84+CE graphing calculators, which are the ones my class seems to have. I was investigating conics, and in particular circles, and what options the graphing calculators gave me, especially when thinking about dynamically modifying the variables to see how each impacts the graph of the circle. Here’s is a quick summary of what I found:

1. Both TI-84 & Casio Prizm can graph conics (circles, ellipses, hyperbola, and parabola, though how to access these conic graphs is different on both.
• It is more apparent/easy to find on the Casio (there is a Conic Graph menu).
• TI requires knowing that there is a Conic app in the app menu, which is a button on the calculator. It is not seen from the main screen, and if you don’t know it exists, you won’t know it’s available.
2. Both provide more than one equation form for each conic.
3. Both show the graph of the conic, but how is very different.
• Casio shows the graph on the coordinate grid, where you can see the whole grid, see values on each axis, and identify quadrant and key points on the graph
• TI shows the graph in the entire window with a weird yellow frame around it. It is difficult to determine where on the coordinate grid the graph appears – there are axis marks, but no values, not origin, making it difficult for students to understand where on the coordinate grid the graph is. Very difficult to identify quadrant and key points on the graph.
4. Both allow you to enter different values for each coefficient variable,
• Casio has a modify feature that allows you to see the equation, graph, and coefficient variables on one screen. You can then modify one coefficient at a time and see it dynamically change on the graph, allowing students to visually see how each impacts the graph and see the conic change shape, location, and/or size.
• TI84+CE only shows the graph or the equation/coefficients – never together.  You have to go back and forth between them when changing values. The TI does not clearly show where on the grid the graph is, does not show a size change (all conics look the same size, but the grid scale is changing). It’s actually very confusing and would be difficult to help student visually see the impact of changing coefficient variables on the size, location and shape.

Below is a video I made showing how to graph a circle and modify the coefficients on both calculators so you can see the differences I am talking about.

Hopefully you will come to the same conclusion I did – Casio Prizm is far superior when graphing conics than the TI84+CE.

# Casio Graphing Calculators – Which One’s For You?

It being the start of the school year where everyone is getting their school supplies, one question that gets asked by parents and students seeking to get a graphing calculator is which one should I buy? I’ve already done several posts comparing Casio graphing calculators to TI graphing calculators, so there’s no question when comparing these – buy Casio!  So….now that you’ve made the smart choice to go with Casio, which of the models is the right one for you? What’s the difference, aside from the cost? If you go with the most affordable version, the fx-9750GII, will you be able to do all the things you need to do in your math and/or science courses? What’s the advantage of the fx-CasioPrizm model, that costs a bit more, over the other two?

Great questions – questions we get frequently, especially when we are out at workshops and conferences. The short answer is they will all do what you need in all K-12 courses and on standardized tests (ACT, SAT to name a couple), so you wouldn’t go wrong purchasing any of the three. And, they all follow the same keystrokes, so knowing one means you know the others. But, there are some differences, which might matter to you, depending on your preferences. You can see a complete comparison of all our graphing calculators to each other and to the TI graphing calculators in our program book, pg 16-17.

What I have done in this post is compile a short list of the major differences between the three Casio calculators (Casio Prizm, fx-9860GII, fx-9750GII) and made a quick video so you can see both their similarities and their differences.

Short-List Comparison  (for all the features, refer to our program book, pg 16-17):

 Feature Casio Prizm fx – 9860GII Fx – 9750GII Display 384×216 128×64 128×64 LCD Color High Color Monochrome Monochrome Storage Memory (Flash Memory) 16MB 1.5MB – Rechargeable Battery Available Yes No No Exam Mode Yes Yes No Natural Textbook Display – input/output Yes Yes No Simultaneous/Polygon Results Yes Yes No Irrational Number Natural Display Yes Yes No Modify Yes No No

This is just a few of the features that differ. The obvious one being color in the Prizm, the size of the display, the Flash Memory capabilities. But for the most part, if you check out the complete list of features, you will see that all three they have comparable functionality and many features/functionality that the TI calculators do not. So – if you like color, want more flash memory (for pictures, movies) and the ability to modify one variable at a time, then the Prizm is your choice. If color is not important, but you like the natural display, then go with the fx-9860GII. If your school requires exam mode capabilities for standardized testing, then the Prizm or the fx-9860GII would be your choice. But – the fx-9750GII, for its lower cost, is going to meet most of your functionality needs, so if the extra features aren’t necessary for you, go with that calculator. You won’t go wrong with any of them.

Here’s a quick video showing some of the differences:

# Visuals to Start Interesting Conversations & Problem-Solving

I realize most teachers and students in the U.S. are just beginning their summer vacations, so thinking about math and problem-solving is most likely the last thing on their minds. But – if any of you are like me, the summer was always a time to regroup, rejuvenate, and come up with new and brilliant ideas to utilize in math class starting in the fall.  I often spent my summers taking a class or finding projects to use/create, so always looking for ways to enliven my math instruction.

This morning, with all the news about UK voting to leave the EU, shocking news to be sure, I couldn’t help notice the many different visuals being bandied about to visually show how the votes were laid out.  It’s fascinating to look at these different representations, and then to just consider all the possible questions that arise.  Here are some examples of the visuals I have seen:

The interesting thing with these visuals is they are all showing the same “results”, but from different perspectives or different ‘groupings’.  I love the map one – it clearly shows how the countries played out in the vote.  Now – this is NOT a post about the referendum – you will have to go to your news sources for information there.  But – from the math teacher side, all these visuals about the same results just got me thinking about how really great questions and problem solving could arise from the simple act of putting up a graph of some results and asking students “what do you think or wonder?” and letting them then investigate. For example, if we look at just the map, and don’t give them any numbers, they might wonder is it half blue/half yellow? How could they then determine the actual area of each colored portion of the graph?

Here’s a couple more pulled from the Prizm Resource Page:

If you were to just throw these up on the screen at the beginning of class and ask the students to come up with some things they wanted to know about these visuals, it would lead to some student-generated questions that then would require the use of mathematics and possibly some background/related research, to find the answers.  If we are thinking about the mathematical practices, or habits of mind we are trying to instill in our students – such as analyzing, communicating, persevering, applying, arguing, critically-thinking, problem-solving, rather than giving them all the information and then asking them to ‘calculate’ the solution, why not let them find answers to questions that interest them? They would be applying mathematics in several ways, perhaps incorporating skills they have not yet learned but need – and in the process realizing that mathematics is useful and interesting.

Try it – find an interesting visual – graph, picture, etc. that spark in you some interesting questions that need math to solve. Put them in your “things to add to my class for the fall” and then get back to summer!