Category Archives: technology

Using Structure to Solve Equations

Last November at the ATMNE 2017 fall conference, I attended a session where Gail Burrill highlighted some TI-Nspire documents that helped kids build concepts about expressions and equations. Sure, they’re targeted at middle level, but you use the tools that your students need, right, not where you wish they were?

Anyway, the one that really caught my eye was called “Using Structure to Solve Equations.” It was the perfect activity at the perfect time because my students were kind of struggling with solving (what I thought) were some fairly simple equations. Their struggle was about trying to remember the steps they knew someone had showed them rather than trying to reason their way through the equation. Just before learning about this activity, I had found myself using these strategies. For example, to help a student solve an equation like 3x + 8 = 44, I found myself covering up the 3x and asking the student, “What plus 8 equals 44?” When they came up with 36, I would follow up with, “So 3x must equal 36. What times 3 equals 36?” This approach helped several students – they stopped trying to remember and began to reason.

There were too many students who were still trying to remember, though. Their solutions to equations involving parentheses like 6(x – 4) – 7 = 5 or 8(x+ 9) + 2(x + 9) = 150 included classic distributive property mistakes. The “Using Structure …” activity was just like my covering up part of the equation and asking the question. But each student could work on their own equation and at their own pace. The equations were randomly generated so each kid had something different to work with. Working through the “Using Structure … ” activity helped kids to stop and think for a moment rather than employing a rote procedure.

In a follow-up to the activity, I asked kids to explain why 8(x+ 9) + 2(x + 9) = 150 could be thought of as 10(x + 9) = 150. My favorite response was because “8 bunches of (x + 9) and 2 bunches of (x + 9) makes 10 bunches of (x + 9).” From there, they saw it as trivial to say that x = 6. These were students who had previously struggled with solving equations like 3x + 8 = 44 because they couldn’t remember the procedure. After “Using Structure …” they didn’t have to remember, they just had to think!

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Why I love my TI-Nspire

As the new school year approaches and I contemplate how I might structure my classes for the coming year, I remember how grateful I am that there are so many resources out there to support me and my students. One resource, though, rises above the rest: TI-Nspire. Why?

Full disclosure: I’ve been using TI-Nspire technology for over a decade. I’m not new to this game. I have also used TI-8* family of graphing calculators as part of my teaching since 1990. I still have a TI-84, but I prefer the teaching power of TI-Nspire. I am also a T3 Regional Instructor. I became an instructor because I am passionate about how TI-Nspire can be used to help students learn math better. Even if I were not an instructor, I would give the same recommendations and say the same great things about TI-Nspire.

Back to the why.

Coding: I’ve created an “hour of code” lesson for 9th grade orientation. Since they won’t have their laptops yet, we’ll be using TI-Nspire handhelds. Sure, some students may already have experience programming, but the materials allow me to easily differentiate. And the Innovator Hub provides an additional challenge for those who need it. Best of all, coding with TI-Basic is pretty straight forward for teaching programming structures.

CAS: Using the CAS capability as a learning tool helps students to see structure in the mathematics. Sometime it causes us to ask interesting questions about whether an unexpected result is equivalent to the result we expected to see.

Modeling: I can import a picture and superimpose the graph of a function. I can drag the graph to conform to the shape I’m trying to model (as long as that function is appropriate for the shape). I can add a point on the graph and identify its coordinates. My students can have some really interesting conversation about what all the numbers mean.

hoop shot

Beyond algebra & graphs: The statistics and geometry applications are unparalleled in a single device. I’ve taught statistics with Fathom and geometry with Geometer’s Sketchpad. The TI-Nspire apps remind me of these two powerful programs. Last year I taught a lot of statistics classes. TI-Nspire was really valuable when it came to representing and analyzing the data. And the geometry app is dynamic, too.

Operates like a computer: The operating system is file and menu driven, so it’s easy to think about TI-Nspire documents like computer files. All of the same keyboard shortcuts apply, too, which kids really love. There’s even a touch pad that operates like a mouse. When I first introduce the handheld to my students, I point out the important buttons: menu, esc, tab, ctrl. These can get you out of anything you’ve messed up by mistake. You can keep undoing until you get back to what you want. Just keep hitting ctrl-z.

Beyond the handheld: We only have a few classroom sets of the handhelds, but we have enough software licenses for all of our students’ laptops. That’s where we use TI-Nspire most often. It’s great because the handheld and computer versions are functionally identical and the computer version offers a lot more screen real estate. Sometimes that comes in handy when you’re comparing a lot of variables.


Advanced options: Coding with Lua can extend document design/creation options for more experienced programming students. Using science probes for data collection helps to integrate the two disciplines. The TI-Nspire Navigator is a powerful tool for formative assessment and student feedback.

TI Support: There is a vast library of activities available for free on the TI website. These are curated, organized by topic, and searchable. Each activity includes a student directions sheet in Word format so that any teacher can modify it for their students or context. If I ever have a problem, TI Cares is right there to help. I’ve never had an issue renewing a software license (because my school laptop was reimaged over the summer) or getting help working through a network issue getting the Navigator up and running. People, right there in Dallas, ready to answer my questions and help me out. I really appreciate that they listen and aren’t just running through a script.

So, why do I love my TI-Nspire? Because it’s powerful, flexible, and backed by a company with over 30 years in education: one that listens to teachers and continues to improve.

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“I can’t wait to find out!”

As stated in the last post, Learning from Failures, I decided to adjust my approach to having students analyze and discuss data. We’d put a lot of time into working out many of the kinks, but it was really time to move on to scatterplot representations of data. My students already knew a lot about scatterplots and best-fit lines, so this allowed me to dive right in with some data.

Rather than stating a claim, I started with a statement and four questions:

I have the following measures (in cm) about 54 students: height, arm span, kneeling height, hand span, forearm length, and wrist circumference.

  1. Which pair(s) of variables do you think might show the strongest correlation? (And what would a strong correlation look like in a scatterplot?)
  2. Which pair(s) of variables do you think might show the weakest correlation? (And what would a weak correlation look like in a scatterplot?)
  3. Which variable (from the list above) do you think would be the best predictor of a person’s height (in cm)?
  4. Write one claim statement about the class data variables.

These questions forced them to think about the data and make some predictions about what they might see once they were able to access it. We hadn’t really talked much about correlation, so I was really interested in their responses to what strong and weak correlations look like on a scatterplot.

Generally speaking, they said that strong correlations

  • look like a line
  • can almost see a line
  • looks like a more defined line
  • looks pretty linear

and weak correlations

  • look like randomly placed dots
  • have points that are far from the line
  • looks more spread out and scattered
  • has dots all over the place

As for question 3, there was quite a debate between whether arm span or kneeling height would be the best predictor of a student’s height. One side (6 students) argued that arm span would be the best predictor because “everyone knows that your arm span is about the same as your height.” The other two students claimed that kneeling height would be a better predictor because “it’s part of your height.” Both sides stuck to their convictions – neither could be swayed, not even by what I thought was the astute observation that kneeling height is probably about 3/4 of height. This prediction was made by a student in the arm span camp!

Students each received their own copy of the data and investigated their claims. During the next class, we took a look at a couple of those claims, together. The plot on the left is height vs arm span, with the line y = x (height = arm span). The plot on the right is height vs kneeling height, with the line y = (4/3)x (kneeling height = 3/4 height).

More debate ensued, though most admitted that kneeling height had a stronger correlation to height than arm span did (for this data, at least). And maybe the 3/4 wasn’t the best estimate, but it was pretty close. They also talked about those outliers, which led to a conversation about outliers and influential points.

Moving from Class Data to Cars

I took a similar approach with the next data set.

I have some data about cars, including highway mpg (quantitative), curb weight (quantitative), and fuel type (categorical: gas, hybrid, electric). Think about how these variables might be related and make some predictions.

  1. How might the highway mpg and curb weight be related?
  2. how might the curb weight and fuel type be related?
  3. how might highway mpg and fuel type be related?
  4. Do you think there might be any outliers or influential points? If so, what might they be?

Through some class discussion, we came up with the following claims and predictions.


Students still had not seen the data and one of them said, “I really can’t wait to see what this looks like!” Another said, “Yeah, I’m not usually all that interested in cars, but I really want to know.”


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The Power of Interesting Questions

Today I led two groups of students through an introduction to spreadsheets as part of our Baxter Foundations workshops. Our framing question was, “How much is that Starbucks habit costing you?” Many students, of course, said $0, but we widened the question to include other vices, like Monster drinks, Red Bull, going across the street to Portland Pie every day, or down the street to Five Guys for lunch. And we broadened the question to, “What if you put your money into a retirement fund instead?” To make this real for my students, my friend Tracy admitted to her Starbucks habit and offered to be our real case study.

Before we started creating anything, I asked the students to complete this quick survey to figure out what they knew and what they didn’t. Then we looked at the results as a group. Here’s what we found:

Group 1: Mostly sophomores

Group 1: Mostly sophomores


Group 2: All freshmen


Group 1: Mostly sophomores


Group 2: All freshmen


Group 1: Mostly sophomores


Group 2: all freshmen

Clearly, the sophomores were bringing more to the table than the freshmen. After all, they had been instructed in spreadsheets in their engineering class last year, but they were still a bit unsure of what they knew. They thought they probably knew more than they had indicated, but didn’t know what I meant by “cell reference,” for example. And remember, I teach in Maine where 7th graders are given their own digital device. It used to be a laptop, but last year many districts changed to iPads. I would have expected the 9th graders to have had much more experience with spreadsheets, but I’m seeing that the switch to iPads is having an impact on that. Very sad.

I began by explaining the situation: Tracy spends $x each day on her Grande Soy Chai at Starbucks. If we want to figure out how much she spending, and what she could be earning instead, what information do we need? And then I had them brainstorm for a couple of minutes.

Information needed: cost of the drink, how much spent each month, and interest rate for the investment.

We made a few assumptions:

  • Tracy could find a mutual fund, or other investment, that earns an average of 7% annually
  • that she is 25 years from retiring (I don’t actually know this)
  • that the price of coffee would not change over the life of the investments (we knew this was unreasonable)
  • that Tracy would invest the same monthly amount for the life of the investment (also unlikely)

But this is also part of problem solving. Take a few minutes to watch Randall Monroe’s TED Talk and you’ll understand what I mean.

So here’s the spreadsheet that we came up with.

So what did we learn?

  • Tracy spends a lot of money on her Grande Soy Chai. But, it’s possible that the drink adds some value to her life and is worth the price.
  • Investing early and for a long time really can pay off, even if the amount invested isn’t all that much each month.
  • Learning about spreadsheets can be fun if you have an interesting question to answer.

Do I think the students in this 90-minute workshop will remember everything that we discussed? Of course not – I’ve been doing this job way too long to think that. But here’s the beauty of it all – they have their own model to reference, be it Google or Excel, they all created one and can take another look at any time. I heard from another teacher that a couple of his advisory kids started talking about making their own coffee instead. A couple of my advisory students commented on the experience at the end of the day. One said, “It was interesting to see how the numbers involved in the Starbucks added up if invested in a retirement fund. The actual application was nice.” Another said, “The spreadsheet exercise this morning was fun. I think it was the funnest way to learn how to do a spreadsheet I have ever done. So thank you.”

You’re welcome.

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Function Carnival

I haven’t been doing a good job posting this year. Something awesome happens in class and I think, “I have to write that up.” Then I get home, and start planning the next few lessons, and I forget all about the awesomeness. It’s been a busy year.

This morning, on CBS Sunday Morning, I learned about a truly extraordinary man, Jim O’Connor, a high school math teacher who volunteers his time at the local Children’s Hospital. What made me sad, though, was his comment, “It drives me crazy when people say that school should be fun. I mean it’s nice if it could be, but you can’t make school fun.” Watch the video. Mr O’Connor really is an amazing man. I just think that it might be time for him to retire from teaching.

I mean, if learning math can’t be fun, then why should anyone consider doing it? Kids and their parents already think that learning math is a drag, so shouldn’t we math teachers be working hard to change that thinking, not perpetuate it?

I’d like to think that my students have had fun learning this year. From dissecting chocolate chip cookies to writing graphing stories to rolling balls down ramps, they’ve collected and analyzed data and created function models. They’ve studied some statistics and some functions (linear and non-linear) and now we’re working on right triangle trigonometry. With 9th graders. I’ve worked hard to make learning fun and challenging.

Thankfully, others are also working hard to make school mathematics not only interesting and fun, but helpful for us teachers to diagnose student difficulties. Take the Function Carnival currently under development by Christopher Danielson, Dan Meyer, and Desmos. Honestly, I don’t know how they do it over there at Desmos, but these little animations will tell me more about what my students understand about functions than anything I could have come up with. And the beautiful thing is that they’re engaging for physics, too. That’s awesome for me and my students because at Baxter Academy, my 9th graders are also learning physics. Imagine my glee at learning about this interesting new tool. I will definitely have them exploring (in a few weeks) and sharing the results with my physics teacher colleagues.

In response to suggestions from the many commenters to Dan’s post, the Desmos team got busy creating more scenarios, including graphing velocity vs. time along with height vs. time. I’m looking forward to these new situations being included in the current Function Carnival site. Maybe they’ll be ready when I need them in a few weeks. It will also be fun to have my students attempt these graphs before we go off to Physics Fun Day in May.

Here are a few more challenges in development:

Try them out. Give feedback. Encourage your students to have fun while they learn.


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What Time Will the Sun Rise?

This week I begin Exploring the MathTwitterBlogosphere. I’m looking forward to these missions and challenges because I need someone pushing me to find the time to write in this blog. It’s good for me. Like spinach.

This week’s mission: What is one of your favorite open-ended/rich problems? How do you use it in your classroom?

One of my favorite open-ended/rich problems comes at the end of a unit on trigonometric functions. After exploring, transforming, and applying trig functions to Ferris wheels, tides, pendulums, sound waves, … I assess my students’ understanding by giving them some almanac data of  sunrise and sunset times for a specific location on Earth. Their job is to analyze the data and create a trig function to model either sunrise times, sunset times, or hours of daylight – their choice.

The data looks like this

and that makes it somewhat challenging for students to even begin. They are reminded that they should have “enough” data to know if the model they develop fits well. I point out that the times are given to them in hours and minutes, but that they probably want a single unit (hours or minutes after midnight). From there, they are on their own to solve the problem. Usually, they work with a partner.

In the classes that I’ve used this task with, we’ve modified the amplitude, period, and midline of the sine and cosine functions. We haven’t introduced phase shift, yet. So, there is also a reminder about selecting a convenient “Day 0” for the function they choose to model with.

What I love about this task:

  • Students are talking math, asking each other about the number of data points they should use: “Should we just pick the same day every month? Are 12 data points enough?” or “Do we just go every 20th day?” or “What should we use for the first day?”
  • Students are problem solving. They have to convert the times into a single unit. They have to make decisions about which variable to model, when to start, which type of model to use. Then, they can collect the relevant information to modify their chosen function.
  • Students are using technology. Although they don’t have to, it’s really easiest to have the kids making scatterplots on calculators or computers and then graphing their model on top of that. Then they have a built in way to check their work – they don’t have to ask me (the teacher) if they are correct. It shows up in the picture that they create.
  • Students think that working with trig models is really hard, so they feel very proud when they are able to complete this task without any help from the teacher.
  • It’s really easy to grade. Either the model fits or it doesn’t. Kids turn in their data tables and work showing how they calculated the necessary values for their model. This precludes anyone from using the old SinReg command.
  • Even though I’ve used this task for about ten years, it’s a perfect fit with the Common Core math standards (trigonometric functions) and practices. And since I live in a SBG world, this is a very good thing.

My favorite kind of assessment is one where students have to apply what they’ve learned to a different situation. Even though we create lots of different trig models in class, sunrise, sunset, and daylight hours represent a new application. And a new challenge.


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Spreadsheets – Part 2

As a reminder, here’s the original project descriptor.

Here’s a report from one pair of students.

Mistakes made: Originally we had started out with a more expensive, brand new car. Our budget was about $100 a month to put towards a car, so it would have taken us 20 years to pay off the car, which is unreasonable. We had to downsize and settle for a less luxurious car, but one that still met almost all of our standards.


      • 4 weeks is a month.
      • We have enough money saved for the down payment.
      • We take 10 years to pay off the loan
      • annual rate of 2.29% for loan

Mid-size Sedan

2. We have $100 left for a monthly car payment. Income- expenses. 3400-3300=100

3. Typically for a down-payment you would need about 11% of what the car is worth. So for our car that is worth $11,000 we have about $1,210 saved for the downpayment.

4. The requirements that our car has to meet are 4 door, 30 mpg, mid-size, seats 5.

5. We would want our car to have a working air conditioner, heat, sunroof and radio.

6. We decided to get a used Mazda6i Touring for $11,000 but with 10% off it would cost $9,900.

7. We would need to borrow $9,900 and we would get this money by taking out a loan with Bank of America.

8. (On Spreadsheet)


9. Our dream car was originally a new fully equipped midsize sedan with a sunroof, but after we found that with our budget of $100 dollars a month, it would take 20 years to pay off. So we decided to get a used car. Mazda6 i Touring which still had air conditioning and heat but did not have a sunroof sadly. This car still meets most of our requirements and was much more affordable so we would be able to pay it off in 10 years.

10. Like we said in number 9, one of our problems before was a 20 year long loan. We fixed this problem by selecting a cheaper used car to buy. If the payment stays the same ($100) and the down payment is the same (11%) then the more expensive the car, the longer the duration of a loan.

11. Our dream car is a stretch hummer limo. The cost of this car brand new is $300,000 after the down payment of 11% costs $267,000. This car would be impossible for us to pay off because the interest that we would have to pay is more than we make monthly so the payment would keep increasing and we would never be able to pay it off.

12. To pay off our dream car in six years we would have to earn about $7320 a month. So subtracting expenses that leaves about $4025 a month to put into the dream car, which will pay it off in 72 months. Assuming that the interest rate and bargaining rate are the same.

What I like about this solution:

  • They stated their assumptions.
  • They made decisions.
  • They made adjustments.
  • They analyzed their results.
  • They dreamed big.
  • They used absolute addressing as part of their spreadsheet formulas.
  • They knew the difference between an annual interest rate and a monthly interest rate.
  • They understood that if their payment doesn’t even cover the monthly interest, they’ll never pay off the loan.

There were several solutions like this one. Not enough, though. Something to think about next year.

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Authentic Assessment?

My junior level math classes have begun working on a project called Buying a Car. For the past few classes we’ve been problem solving using spreadsheets. They’ve been working in teams, using Google spreadsheets to solve problems like this and this (which I adapted from our Core-Plus Mathematics text). My teaching colleague and I decided to jump into this spreadsheet mini-unit before our students had to turn in their laptops for the year. (We are a one-to-one school.)

Here are some pictures of my students hard at work.

photo7  photo6



Some things I heard as the students were working:

  • Oh, so the bank pays for the car and then you pay the bank. I get it now!
  • How much does gas cost right now?
  • Where’s the best place to get the loan from? What’s the lowest interest rate we can get?
  • Are we going to buy that truck? What’s the gas mileage on it?
  • How do we figure out the payment? What did we do before?
  • So we have to add the interest and then subtract the payment.
  • We can cut back on the money for entertainment. We can be cheap. There’s only two of us, we don’t need that much food. It’s not like we’re feeding any children.
  • How do we determine how much for a downpayment?
  • Can we afford a monthly payment of $875?

Here’s what I really like about this assessment (having never done it before):

  • There is a high degree of choice.
  • There isn’t a definitive solution.
  • Students have to make (and state) some assumptions in order to solve the problem.
  • They have to think about lots of things that go into a household budget and buying a car.
  • Students working together and helping each other to succeed.

What I’m not so sure about:

  • The quality of their results.
  • If they’ll really apply what they’ve learned during the past 4 classes learning about spreadsheets.
  • How much understanding they’ll walk away with.

It will be interesting to see what they produce as a result.

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Ask different questions

Rather than assess whether my students can do matrix multiplication by having them multiply matrices without a calculator, I decided to ask a different question. After all, the point is about understanding and not computation, right? So, instead of giving them a non-calculator section on their quiz, I changed the question:

Given that
\left[  \begin{array} {cc}  5 & 2\\  7 & 3\end{array}  \right]  \left[  \begin{array} {cc}  1 & -4\\  -6 & 9\end{array}  \right]=  \left[  \begin{array} {cc}  -7 & -2\\  -11 & -1\end{array}  \right]

explain the calculation that gives the entry in the first row, second column of the product matrix.

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The Evolution of a Teacher

When I began teaching high school mathematics in 1988, there were no such things as affordable graphing calculators. A mere four years later, I had a classroom set of TI-81 graphing calculators.

Actually, in my second year, one of my Honors Calculus students showed me his HP Graphing Calculator. It looked nothing like what we would all soon know as graphing calculators. It had this tiny screen that handled about four lines of text – amazing by 1989 standards – and it had two keypads that were connected across a folding spine. Amazing!

So, five years into my teaching career, I have this classroom set of graphers – TI-81’s. What was I supposed to do with them? I mean, I had taught Algebra 1, I knew what the kids were supposed to learn. They had to learn how to draw graphs of lines. They had to learn how to manipulate symbols. How was this new device supposed to help without undermining me? I didn’t have a clue. Sure, it was cool, but the kids were supposed to be able to manipulate a pencil and a ruler – not this new, cool device. It’s not that I was anti-calculator; I was new. And I didn’t want to lose my job. But this was too interesting a tool not to use. So I learned. I read – journal articles. It was 1993, after all. Web? What’s that?

I bought my own graphing calculator: A TI-85. I know a lot of people didn’t like that model, but I did. I liked the menus. I liked what it could do that the TI-81 couldn’t. But it was more expensive, so schools went with the TI-81, which evolved into the TI-82, TI-83, TI-83 Plus, and TI-84 Plus Silver and TI-89 Titanium. That line has been pretty much developed out. It’s where we are right now. We’re comfortable. We know how to use them – as teachers, as students, as test developers. We use them to analyze graphs, to solve systems of equations, to crunch data, and manipulate algebraic symbols, if we have a CAS.

In 2006, I received an invitation to participate in a field test of a new piece of TI classroom technology: The TI-Nspire. Never heard of it; jumped at the chance. It was both computer software and handheld device. Literally. The first exposure my students had to the TI-Nspire was at a computer in the lab. At the time, I couldn’t imagine how this new tool would revolutionize my classroom. Frankly, the first models were so clunky, that I just wanted my TI-84 Plus Silver. But they (at TI) listened to us – we teachers in the field test and my students, too. There were some things were really didn’t like – those green alpha buttons – and things we really liked – being able to grab and move function graphs around, for example. Think about that for a second. We could grab a graph and move it around the screen, changing the slope or changing the y-intercept. As we did that, the function rule would change. That means that I could graph a line, grab it, move it, and see the effect on the rule. Holy cow! That’s a game changer. There was so much that this new device could do, my head was spinning. After all, it’s just a tool. If I can’t use it to teach something, then what’s the point? How was I to make the best, most effective use of this new tool? Still working on that. Every day.


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