Power bills as sources of math questions.

I’ve been thinking a lot about graphs lately, and how in general, many people are deceived by graphs because they don’t understand numbers, scale, sampling size, etc.  In this very contentious political time, it seems many people are fooled by the statistics they “see” graphically.  In my last post, I quoted Dan Finkel’s line “when we are not comfortable with math, we don’t question the authority of numbers”, specifically referencing people’s willingness to believe statistics they see or hear because they don’t really understand where these numbers came from or what they represent.

We can help our students get a better sense of statistics and numbers by providing them as many opportunities to explore, in context, graphs and statistics and ask questions and make sense of these. That could mean exploring all the statistics and poll results currently happening with the presidential election.  Or looking at weather predictions. As I looked at my power bill yesterday, I realized how easy this type of access to real numbers can be, as I stared at the graphical representation of my gas and electric over the past 13 months. (There is also a numerical table showing daily use of kilowat hours (kWh) and 100 (C) cubic feet volume of gas (Ccf). There alone is a whole bunch of mathematical calculation/conversions/ratios).  What I love about my graphical representations is there is a 13 month trend – so I can see where my usage was last year at the same month, and then see how my usage has changed throughout the year.  Below are my December & January graphical representations for both gas and electric usage.


December Electric

December Electric

January electric

January Electric



December Gas

December Gas

January Gas

January Gas


Just from these graphs, there are a lot of assumptions that can be made, and questions that could be asked, that would then lead to more exploration.  For example, December electric from 2014 and 2015 is about the same, but January 2015 is significantly less than January 2016 – why is that? (hint: my children are home for break, so we use more electricity). Gas use in December of 2015 was much lower – was this because it was warmer in December? Are we having a warmer winter than last year? The gas bills seem to show that – but, we could then go look at the weather temperatures for the same time frames in the area I live and see if there is a correlation between temperature and gas usage (i.e. heat). Why is the electric so much more in the spring/summer months and gas is lower? There are so many questions, and, if we brought in the tables of daily usage, cost of kWh and Ccf (volume) we could be doing math calculations, comparing costs, etc. Maybe compare bills from last year to this year and see if the price in oil/gas has had an impact on the overall monthly charge. I like the idea of bringing in the weather and comparing to the electric/gas usage. You can get average weather for the area you live in pretty easily, but it would be even better for students to collect actual temperatures over time and make their own graphs and comparisons.


Average Climate Chart

The point I am making here is that a simple thing like a power bill can be a powerful tool for visualizing math, doing math, making connections, and asking questions. Or try looking at some statistics from car sales or stocks or polls on the presidential election. It leads kids to ask interesting questions, explore mathematics they care about, and opens them to the real-world aspect of mathematics and how numbers can be used to inform, deceive, and help make decisions.  These types of explorations are interesting and help students become involved in the world around them as well and better prepared for the realities of things like gas bills! Anyway, just another suggestion on how to bring some context into your math instruction in a relatively easy way.











“I Hate Math” – We Need To Stop This Mantra!

If you are a math teacher, you have heard “I hate math” from students, parents, friends….it is often the first thing someone says when you tell them you teach math for a living. Our traditional way of teaching mathematics, through memorization of steps and skills without context or connection, is partly to blame for this.  And, unfortunately, still the prevalent way of teaching today, despite research and standards that encourage and promote thinking, questioning, and multiple approaches. It’s discouraging, it’s depressing, and it’s a disservice to students. No one should “hate math”, and when you hear it from a child as young as 6 (see the video below), it’s even more depressing, because this is someone who hasn’t even really begun to know anything about math and yet they already hate it. Probably because they are being forced to do timed drills, or worksheets (as a friend recently shared with me about their child’s math class).

I just watched this great TEDx talk I found by Dan Finkel, where he talks about bringing joy to mathematics learning.  He begins with discussing how the fear and hatred of math permeates life, and can contribute to poor decisions and immediate trust in deceiving statistics; “When we are not comfortable with math, we don’t question the authority of numbers” (Dan Finkel, TEDxRainier, The Joy of Math).  He points out that the ordinary math class begins with answers – with little opportunity for questioning or creativity. We give students the steps to skills (i.e. steps to multiply, divide, find x, etc.) and our “questions” have set answers, and once skills are grasped, we move on.  “There is no room to doubt, or imagine, or refuse…so there’s no real thinking here”. Sound familiar?  Sound like a topics needed to master for a standardized tests?

Instead, we need to give students a question and make it authentic.  (His example with the numbers 1-20 and the colors is great, so be sure to watch that). I’ve written about this previously – making math relevant, authentic, and focus on questioning (Real-world Math Applications vs. “Naked” Math; Math Questioning to Support Mathematical Practices). Finkel’s point is that math and the beauty of math can be found by asking questions. “Thinking happens only when we have time to struggle”. Time is so important – it’s the only way to teach students to be ‘tenacious’ and to persevere.  So those of you out there still stuck in this obsession with ‘timed’ skills and rote memorization, pay attention to this video and what Finkel is saying. There are many others saying the same thing (i.e. Jo Boaler from Stanford), but Dan’s message is well worth listening to.  He has so many great quotes, I could go on and on writing them in this post, but probably better if you listen yourself and take away from it that which speaks to you. For me, I am just more committed to the message that math should be about thinking, connections, questioning and providing students the opportunity and time to really explore, question and pursue authentic problems to spark their creativity. Let’s please stop the “I hate math” mantra and instead try to create joy and wonder about math so that instead we hear “Wow, look what I learned about math today!”

Graphing Piecewise Functions – Casio Prizm vs. TI-84+ CE

2016-02-04_16-50-11In my explorations of hand-held calculators and how they can support mathematics learning, I want to continually share when I learn new things. Why calculators? Well, the obvious answer is because I am working with Casio. But the real answer is, if you actually go around the country and go into math classrooms, calculators are still the most-used and available technology to students.  I know, I know -we hear about iPads, tablets, laptops, etc. in use in classrooms, but the reality is these are NOT readily available to most students.  I think I did a post already about this (Calculators, A thing of the past?), but from my own personal experiences, teaching and working with teachers (some of these in the last couple of months), most math classrooms are still working with the following technology: one computer with projector/screen (sometimes a whiteboard, most often NOT), and then hand-held calculators.  And, unfortunately, not even enough of those for each student.

So – yes, despite the ‘edtech revolution’ we hear about in the news, in the real, every-day classroom, students are most often using calculators, and this will be the case for quite a while unless there is some funding-miracle, which, as we know, is very unlikely.  It’s a sad reality – as an edtech supporter, I would love more than anything all 2016-02-04_16-22-37students to have access to technology on a regular basis that allows them to quickly research, explore, practice and visualize mathematics, whether that be via tablets or computers or calculators. But as most of us who work in/with schools know, that is NOT what’s actually happening in most math classrooms.  That said, let’s focus on the great technology that is accessible to a majority of students – and if not, should be, since it’s affordable, portable and can do much of the visualization and exploration that students should be doing in mathematics – graphing calculators.

Now another reality is that TI seems to be the go-to calculator found more often in schools, a lot of this due to brainwashing and really good marketing and the old “change is hard” mentality in education. I myself was a TI graphing calculator user the whole time I was teaching in public schools because that’s what we had. What I am now finding more and more, as I learn the Casio and compare it to the TI, is that I can remember what to do on the Casio way more so than I can on the TI.  That’s just one thing, though admittedly a pretty major thing.  And – while many of the steps for using the TI and Casio are often similar, the Casio is often quicker and more efficient than the TI, and can usually provide a visualization on one screen that helps make a connection which might otherwise be impossible to see when having to look at separate graphs (i.e. graphing  y= and r= on one graph).

My goal here is to point out places where Casio has an advantage over TI (and I am comparing the Casio Prizm and TI-84 CE, which are the graphing calculators most similar and also both are accepted on standardized tests). Obviously, my opinion is probably considered biased – though I am speaking as someone with over 26 years experience, one who has used many different technologies and only ever taught with the TI (Navigator included). I honestly find the Casio more fun and easier.  More intuitive. I just can’t remember where things are with the TI – it’s frustrating! As they say with many things – once you go Casio, you’ll never go back! But – I don’t think I would feel this way if I wasn’t constantly comparing the two side by side, something most teachers never get the chance to do.  With that said, here is another side-to-side comparison of the Casio Prizm and the TI-84 CE showing how to graph a piecewise function, something I believe Algebra II teachers are probably getting into about now, that helps illustrate my preference for the Casio over the TI.

Mathematics and History – Cross-curricular Learning

Obviously with it being February and Black-history month and Presidents Day, a lot of teachers are trying to find ways to bring some of that history into their classrooms, no matter the subject. The sad thing here is that we should be doing these types of cross-curricular learning regularly, not just when there is a designated day, or week, or month. In math and science in particular, there are so many historical events and people that have impacted the study of these subjects, therefore bringing in history, writing, and art really shouldn’t be that much of a stretch. And, vice versa – if you are a history teacher or an English teacher or an art teacher, there are mathematical and science connections that you can be using regularly.  Heck – the Common Core ELA standards actually have a huge focus on students reading in social studies, science and technical subject areas. Cross-curricular learning helps students make connections to not only where these subjects they are learning came from and who helped develop them, but how they work together and apply to life and future skills.

During my middle-school teaching years, it was easy to focus on cross-curricular learning because I was part of a cross-curricular team where we made a concerted effort to focus the learning on themes.  So maybe if we were learning about astronomy in science, then in math we were talking about planetary distances and gravitational forces, and in history they may have been learning the history of space travel and writing about it in English.  As I moved into high school and teaching, where there was more isolation of subjects, I still created opportunities for my students to connect history, science, art, and writing to mathematics. Some of my favorite resources were the AIMS Historical Connections in Mathematics books, which had summaries of various historical mathematicians, some sample problems related to their discoveries or work, and provided connections and timelines.  These were great starting points, often with hands-on activities, that helped support student projects or investigations.

What is probably of most importance, is to try to find historical figures or artistic applications of the math/science your students will be learning so that they see relevance to either a) what they are learning and where it came from; or b) why they are learning it and what they can use it for in the future. Learning is so much more interesting if there is a reason why or and understanding of how the math/science contributed to some point in time, some event, etc.

So – in light of the fact that it is February and Black-history month, I thought I would share a few influential mathematicians/scientists, with some links to what they did to maybe get you or your students thinking and connecting. Let’s write.  Let’s research.  Let’s connect math/science to other subjects so that learning is not an isolated topic but an interconnected experience.

  1. Benjamin Banneker (1731-1806) – mathematician, astronomer, writer of Almanacs
  2. David Harold Blackwell (1919-2010) – mathematician, first African-American inducted into the National Academy of Sciences
  3. Kelly Miller (1863-1939) – mathematician, first African-american to attend Johns Hopkins University
  4. J. Ernest Wilkins, Jr. (1923-2011) – nuclear scientist, chemical engineer, mathematician; contributed to The Manhattan Project; the youngest ever student to enter The University of Chicago (at age 13)
  5. Elbert Frank Cox (1895-1969) – first African-American to earn a Ph.D. in Mathematics
  6. Marjorie Lee Browne (1914-1979) – noted mathematics educator
  7. Charles L. Reason (1818-1893) – mathematician, linguist, educator; first African-American professor to teach at a predominantly white U.S. college
  8. Katherine Johnson (1918 – ) – physicist, space scientist, mathematician; early application of digital electronics as part of NASA space program

This is just my first posting trying to connect mathematics/science to other subject areas. Each month I will share some historical mathematics/scientists to help provide some resources for those of you who wish to incorporate cross-curricular learning.



Linear Programming – Great Real-world Applications

I’ve been playing around with graphing inequalities on the Casio Prizm (and fx-9860GII & fx-9750GII), realizing 2016-02-11_11-54-52how easy they are to graph and how easy it is to find the area of intersection. In doing so, trying to find some relevant problems that would be applications of graphing inequalities, I was reminded how relevant linear programming problems are to both graphing inequalities and finding intersections, but more importantly, to real-world situations.

I spent the majority of my teaching in K-12 with Algebra and Geometry, so linear programming was never on the required curriculum back when I was teaching.  It’s usually in an Algebra II course other ‘higher level’ mathematics courses.  And, if I remember, it is often a content that is ‘skipped’ because it isn’t on standardized tests. In looking at some of the activities in Fostering Algebraic Thinking with Casio Prizm (Goerdt, Horton), trying to find some inequality problems, I found some activities under Linear Programming that made me stop and think – these problems are much more relevant and connect to actual situations than those “naked math” inequality problems we tend to give our students. Find the intersection of these two inequalities is not very engaging, but find out number of servings of potato chips and peanuts you can have while staying within your recommended daily allowance (I know I am asking this question!!). Or, if a company wants to advertise in the SuperBowl, based on their budget and cost of an ad during the first half or the second half, when should they advertise? Still looking for intersections of inequalities but now making a connection in context where there is a purpose for using the math.

These type of activities require a lot more math than just graphing inequalities and finding intersections.  You have to translate the situation into inequalities. You have to convert these inequalities to a form that  will allow you to graph them (i.e. y=). You have to consider ‘implied’ constraints (if you are dealing with money, you can’t have x or y less than zero, for example). There’s a lot of sense-making about what is being asked and the constraints given, modeling a situation with the correct mathematical equation, seeing if your answer makes sense, understanding what the graphical representation of the intersection means, and what do the intersection points mean in the context of the situation.  Linear Programming problems are really robust applications. There’s a lot of Common Core content and more importantly, Mathematical Practices going on here.  Students can then apply this type of thinking to situations that impact them – for example, selling girl scout cookies….if we have to make a certain amount, the cookies cost this amount, the troop gets this percentage, how many do we need to sell to make our profit?  I think these are a lot more interesting than ‘find the intersections of these inequalities’.

I have attached an activity  from Fostering Algebraic Thinking called “The Snack Problem” (Download: Linear Program Snack Prob).  You can look at it in depth by downloading it.  I am just going to state the problem here, identify the inequalities, and then do a quick video on how to graph and find the intersection.  There is a lot more to the activity then whats below so I encourage you to take a look and try it with your students.

Problem:  Assume you like snacks and insist on having at least one serving of dry roasted peanuts and one serving of potato chips each day.  Each serving of the peanuts contains 15% of the recommended daily allowance of saturated fat; each serving of potato chips contains 10%.  Each serving of the peanuts contains 12% of the recommended amount of dietary fiber; each serving of potato chips contains 5%. You determine you want to consume no more than 60% of the recommended allowance of dietary fat from these two snacks, but you want to get at least 30% of the recommended allowance of fiber from them.  Sketch the feasible region.

(please refer to the handout for the other questions).2016-02-11_12-39-57

Inequalities: x represents dry roasted peanuts, y represents potato chips

  • 0.15x + 0.10y ≤ 0.60 (saturated fats) –> y≤(0.60-0.15x)/0.10
  • 0.21x + 0.05y ≥ 0.30 (fiber)  –>                        y ≥ (0.30-0.21x)/0.05
  • x ≥ 1 (at least 1 serving peanuts a day)
  • y ≥ 1 (at least 1 serving potato chips a day)

The video below shows how to graph all four of these inequalities on one graph (yes, even x ≥ 1)! And how to construct just the intersection (vs. the union of these four inequalities) and find specific intersection points.  The attached activity includes all these steps as well.


The Election Process – Confusing process but applies very real-world math!

I will admit that I find the U.S. election process very confusing – primaries, caucus’, delegates, voting. With the Iowa Caucus’ last night I actually learned something I didn’t know before – that Democrats and Republicans hold their caucus’ in very different ways. Democrats actually physically move around the room until there are viable candidates (those with over 15% of the vote) where as Republicans do secret-balloting and the percentage of votes for each candidate determines how many delegates they receive. I found this great article that helped me understand things a little better – “Caucus math: An NBC Primer” by Carrie Dan and Mark Murray.

Basically, in a nutshell, here’s what happened last night at the Iowa Caucus’.  At least I think….I could still have some misunderstanding, I admit.

Democrats: At each caucus site, Democratic attendees physically went to their candidates ‘corner’ to show their support – i.e. 2016-02-02_11-50-14O’Malley corner, Clinton corner, Sanders corner and an ‘uncommitted’ corner. Attendees can speak out and try to convince others to come to their side, and after all the debate and moving about, a final count is taken for each candidate. Based on the total number of attendees, a percentage for each corner was calculated, and if any candidate did not have at least 15% (making them a viable candidate), then that candidate was out, and their supporters could move to either of the other candidates or go to the uncommitted corner, and new percentages calculated. Based on all the caucus sites, the number of delegates awarded to the candidates is their percentage times the number of available delegates.  So, according to the results last night, Clinton and Sanders had a virtual tie, with Clinton getting 49.9% and Sanders 49.6%, and O’Malley only .6%.  This means Clinton gets 22 of Iowa’s 44 delegates, Sanders gets 21, and apparently the last delegate is “uncommitted”.

2016-02-02_11-49-29Republicans: At each caucus site, there is a set number of delegates being fought over that adds up to the total Republican delegates for the state (there are 30 Iowan Republican delegates). A secret ballot is cast, and a simple formula is used to determine how the delegates are divided up for that site/district.  Basically, the formula is: #of votes for each candidate at the site x ratio of delegates for site/30.  There is rounding done and each candidate ends up with a portion of the available delegates, which then, when all sites report, give a grand total for each candidate.  Last nights results for the Republicans are as follows: Cruz, 8 delegates; Rubio, 7 delegates; Trump, 7 delegates; Carson, 3 delegates; Bush, 1 delegate; Paul, 1 delegate; and everyone else, 0.


Here are some graphs that are more visual represented:

2016-02-02_11-50-48                   2016-02-02_11-51-12


But this is just a small blip in the overall race to get delegates, which then determines who eventually becomes the official candidates running for President for each party. It is  a long, confusing process but interesting! And then we have the race for presidential votes and the electoral college after that….. My point here is look at all the math!?!  And it’s real-world, and it’s happening right now and has an impact on students lives, so it’s relevant and interesting and can be a great source of real-world application and learning in your math class. Why not today in your classes (or sometime this week), hold a mock class-caucus? One the Republican way, and one the Democratic way.  See who the winners are based on your class results.  Discuss the process – is it fair, unfair? What’s the math? And as each primary/caucus comes along, involve your students and do the math.

I just want to plant a seed. Explore with your students some of the math involved in the election process, no matter what age/grade you teach. Getting students involved in our country’s electoral process in a way that lets them apply what they are learning and feel a part of the world around them can only be a good thing. Hopefully, down the road, it will help them become interested, informed and voting citizens.