# Geometry and the Holidays

The holidays are upon us, so of course it makes complete sense to look for geometrical connections. Or maybe that’s just me?

As a geometry teacher (just finishing up a Geometry & Spatial Reasoning course), I am seeing geometry connections everywhere. From the wrapped presents, to the origami ornaments, to the snowflake patterns, I am constantly looking for those real-world connections and easy (and cheap), ways to get students working hands-on with math.

We are all familiar with ‘holiday math’ problems that connect to wrapping presents – i.e. how much wrapping paper do you need, how much ribbon, etc. Area, surface area, linear length connections all very obvious. But, as a geometry teacher, I am also curious about the gift boxes themselves. I know it is often difficult to find 3D models for learning, so boxes provide a cheap way to provide students hands-on explorations of nets, area, surface area, volume. So – teachers – get your students to bring in boxes after the holidays – so much you can do with these!!

Another thought – origami. This time of year, teachers often create holiday decorations with their students with paper-folding, which is fun, obviously, but can also be a great way to apply many math concepts. Shapes, fractions, and transformations for example. Take the following two origami designs – a star and a tree. As you are folding, you could be having students think about the individual shapes, but also the dimensions, the fractional parts after making a fold, what types of transformation have occurred – even congruence and corresponding parts.

I

For example, in the star above, after folds #1, what fraction of the square does each smaller square represent? When we fold that triangle in #2, what type of triangle is it? What fraction of the original square is represented in that yellow triangle?  What type of transformation does each fold represent? Are the triangles in #3 and #4 congruent? How do you know?

Again, looking at the tree folding above, what shapes do you see in #1? What fraction of the whole paper is each shape (so squares and triangles)? How about in #2? And which shapes are congruent? How do you know? Lots of great math, that you could really explore with students while they are also doing a fun hands-on activity.

Hopefully you can use some of these ideas with your students. Have a wonderful holiday season!!

# Financial App (Pt 3 in series) – Let’s Talk About Money

With the holiday season upon us, and people often spending beyond their means, it seems appropriate to continue the CG50 (and all Casio Graphing calculators) app exploration with the Financial App.

One thing we do not spend enough time on in K-12 education is financial literacy. I know there are some states that are trying to address this, but it is not enough. This lack of understanding about money, savings, taxes, interest, debt, etc. is a huge contributor to our enormous debt crisis. Take our current political focus on the ‘tax reform’ bill that’s up for a vote soon – most people do not understand the ramifications of this because they don’t really understand anything about finances and how taxes work. We do not in this country teach the basics of financial literacy, which is why we have so many people drowning in debt, losing their homes, barely surviving month-to-month on what they make, and forget about having the ability to save for the future. How many students really understand about saving money? Or how taxes impact their hourly wages (i.e. \$10/hour is not that great when you factor in all the taxes taken out)? Or how not paying of your credit card monthly can make that \$300 dollar purchase become a \$400 or \$500 dollar purchase?

When I taught in Virginia, they started a Personal Finance course ‘elective’ (only for those students technically not on the college prep track – which was silly, as ALL students should take a course on Personal Finance). I was lucky enough to be the pilot teacher in my school, so I could pretty much create the course. My goal was to help students understand the importance of financial planning so they could survive and thrive in the world, no matter where their path took them. We started with learning about different career options they were thinking of, and what a typical annual salary might be (so plumber, electrician, hair dresser, doctor, lawyer, teacher, etc). They learned to fill out job applications, and write resumes, and then we ‘pretended’ they had been hired and were receiving biweekly payments (I actually gave them ‘checks’). We learned about payments, investing, taxes, rent, credit cards, insurance, amortization,balancing a check book (the class had a ‘bank’), etc. They had to determine where they would live, whether they would get a car, how much they could spend on food, entertainment, etc. based on their salary. What they quickly learned is that their wages, after taxes, were often NOT enough to do much else – no fancy apartment and having to make tough choices (i.e. gas or food, no car, no expensive smartphone, taking bus, walking, no movies every week, no fast food, etc.) When a student comes to you all excited about their \$9/hr job and all the things they will buy, and then realize after their first paycheck that it’s going to take months to have enough, it’s eye opening. And scary.

What I learned is that we do not talk to students about real-world, practical mathematics enough –  simple things like saving money, calculating tips, balancing a checkbook, interest, credit card debt, etc. This is math they need in their everyday life. This is math that has purpose. This is math that will help them make smarter decisions about their future. Maybe if we did, we wouldn’t have so many people struggling to survive or believing every unrealistic promise they hear in the news..

My message – let’s get some Financial Literacy into K-12 mathematics programs!

With that said, here is a quick video on the Financial App that is available on the Casio graphing calculators. This video uses the CG50.

# Equation App (Pt 2 in series) – Solving Equations – Why Use a Calculator?

Solving equations is a large part of the mathematics curriculum as students move into those upper-level concepts. If we look at the Common Core Standards, students start solving one-step equations for one variable in grade 6, adding on to the complexity as they move into higher mathematics where they have multiple variables and simultaneous equations and complex functions. It is important to help students understand what solving equations really represents – i.e. determining the values of unknown quantities and to help them solve them in a variety of ways (i.e. graphically, using a table, using symbolic manipulation, and yes….using technology such as a graphing calculator). And connecting those unknown quantities to real-world contexts is a big part of this as well. Students should solve in multiple ways and express their solutions in multiple ways so that they really understand the inter-connectedness of the multiple representations (graphs, tables, symbolic) and what all these quantities mean in context.

That said, many teachers are reluctant to use the equation solver that is often part of a graphing calculator because, as I have heard multiple times, it does the work for the students and just gives them the answer. True. But – there are ways to utilize the equation solver so that it supports the learning, not just ‘gives the solution’. The obvious way, and probably the most frequent way, is to have students solve the equation (s) by hand, showing all their inverse operations/work, maybe even sketching a graph of the solutions, and then using the graphing calculator to check their solution. Very valid way for students to both do the work, show their steps, and verify their solutions. But – the reverse is also a great way to try to help students learn HOW to solve equations. Working backwards, so to speak.

By this, I mean, use the equation solver to give students the answer first, and then see if they can figure out how to use symbolic manipulation and inverse operations to reach that outcome. As an example, start with a simple linear equation, such as 2x – 5 = 31. Have students plug this into the equation solver and get the solution of 18. Then, in pairs or small groups, have students look at the original problem and try to figure out how they can manipulate the coefficients and constants using inverse operations to get to that solution of 18. So maybe, plug the 18 in for the x.  What would they have to do to the other numbers in order to isolate that 18?  This forces students to use inverse operations to try to ‘undo’ the problem and end up with 18. In doing so, they are discovering the idea that to isolate a variable, you have to undo all the things that happened to it.  Give them a harder problem. Same process….and let them get to a point where they try to solve using their ‘understanding’ of inverse, and then they use the calculator to ‘check’.  The idea here is students are figuring it out by starting with the solution and working backwards to understand the process for solving equations. And they develop the process themselves versus memorizing it.

Rather than thinking of the calculator as a solution tool, think of it as another way to help students discover where those solutions come from.

Here’s a quick video on using the Equation App (solver) on the CG50. The process is the same on Casio’s other graphing calculators. This is another installment in the app exploration series, started last week with the Physium App.

# 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!

# 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:

# Creating a Classroom Culture That Encourages Student Discourse

I just spent two days earlier this week working with middle school math teachers. Our focus was on the 6 – 8 Common Core Geometry standards, and how they build on elementary geometric concepts and continue to build that understanding that students need when they get into high school geometry. As part of our work, we also focused on the Math Practices, because it’s the intentional alignment of practices and content to create engaging mathematics activities that really help students develop the deeper understandings. By that I mean you shouldn’t be teaching the content standards in isolation – they should be combined with helping students make sense of the problems, choosing appropriate tools to explore and apply the standards, and really explaining and justifying the conclusions they make.  Practices and content go hand-in-hand.

In our many collaborative discussions these past two days, as we really dived into both practices and content, what was very apparent was how important it is to create a collaborative, safe, classroom. Mathematics classrooms should constantly focus on vocabulary use (by both teacher and students), modeling, discussing your thought process (in many ways – spoken, written, pictorally), explaining and clarifying your thinking, asking questions, and really focusing on all types of communication. Mistakes or misconceptions that students have should be expressed freely, without fear of embarrassment, and students should be free to try multiple pathways to solutions and multiples ways to express their understanding. Students are not going to talk about mathematics if they feel they will be laughed at or considered ‘stupid’ – and that requires a classroom culture that fosters real communication between students that involves listening, ‘arguing’ against someones responses in a constructive, polite way, and a sense that it’s okay to make mistakes because we are all in this together, learning.

What the teachers expressed as their “ah-ha” from our days together was that in order to create these types of classrooms and math learning, you have to start the process right at the beginning of the year, during those first weeks of school, when students are new and class is unknown, and math concepts are relatively familiar since we are starting, in theory, where we left off at the summer. Those first couple weeks of school are the perfect time to create that collaborative classroom culture.

My suggestion was to start, day one, creating the idea that in this class we will be talking with each other, sharing ideas, and learning together, but we need some guidelines. So day one, don’t go over your rules – students know the rules. Instead, start having conversations – putting kids in small groups, and practice how to work and talk together using something non-math related to help foster the idea that in your class, communicating and listening are encouraged. For example, small groups of 2-3, and each group must decide what the best and worst movie they saw this summer was and give reasons for both.  Then have groups share out, using some simple group share out routines like the person in the group with the longest hair tell us about your groups best movie.  Be very explicit in how they communicate – ie., one person talks, everyone listens; write down your ideas, so one person records, etc.  Day two, do the same thing, but this time maybe use a simple math concept – i.e., in your groups, explain why a square might be a rectangle, or why is a circle different from a square?  Something appropriate for your grade level, but something that you know most of the students are familiar with.  Again – the idea is to learn how to communicate in the classroom – how to create that culture of collaboration, listening, and justification.  Day 3 – maybe introduce a math tool students will be using this year – like a calculator, protractor, compass, ruler, software, etc.  Have students explore and write down things they notice (give them simple things to do), things they have questions about.  Have other students explain what they found or answer each others questions.  Model the use of the tool (s) yourself. Lot’s of options, but the point being to get kids thinking, talking, listening, and understanding that if they have questions or concerns it’s okay to voice them.

Do a little bit each day – change the groups up, do it whole class, do it with partners. The idea is that you are helping students learn to talk to each other constructively so that when you get into the real learning of new math concepts, they are already comfortable with each other, with some of the learning tools they will be using, and understand that in this math class, we work together and listen to each other and support each other.

Learning is not an isolated activity – we, as teachers, are there to facilitate learning and help students become active, productive, problem-solvers. This happens in classrooms where it is okay to communicate, it is okay to make mistakes, it is okay to have your own approaches to problems but that requires justification of those approaches so others can learn from them.  The more you create this type of learning environment, the more your students will persevere in tackling those tough learning moments.