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Peter Liljedahl’s Surrey Presentation

On Friday I attended a pro-d presentation in which Peter Liljedahl shared his numeracy tasks. Peter’s tasks get students comfortable with ambiguity, get them writing about math, and get them to stop mimicking the teacher.

Early in the session, I wondered how these tasks could address pre-calculus learning outcomes. Later, Peter answered this for me when the conversation turned to finding time. “Why are we afraid to give up what isn’t working?” he asked.

“Who was your math teacher last year?”

“Uh… you were, Mr. Hunter.”

Despite learning (covering?) things like factoring trinomials or writing equations of lines in Math 10, sometimes my Math 11 students would act like they were seeing these things for the first time. So, why am I holding on to this? Why can’t I make time for numeracy tasks?

Peter works with teachers to design numeracy tasks that require the mathematics that students already have in place. This rules out grade level learning outcomes. He joked about trying to steer students towards a particular method of solving a problem – “Students are very good at smelling a word problem.”

While these tasks may not address grade level learning outcomes, they can be used to address the main goals of mathematics education described in our curriculum. Communication, perseverance, risk taking, motivation, engagement, and problem solving – all of these were listed by teachers as necessary to be able to do these tasks and all of these help define numeracy.

As an added bonus, helping students develop these skills will make teaching and learning grade level outcomes that much easier.

I look forward to trying out Peter’s tasks and developing new tasks with Surrey teachers.

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Bacon And Eggsponents

Recently, a video showing a snappy way to add fractions was shared on the BCAMT listserv. Thankfully, it was panned.

This “butterfly method” also appears in Elizabeth DeCarli’s Ignite presentation, this time to help illustrate that “meaningful representations are greater than cute mnemonics.”

In my previous post, I wrote about one way in which my Math 10 teacher tried to make math memorable for his students. (Yes, I realize that since I still remember this, he was successful.) I also wrote about how this didn’t build any understanding.

As a teacher, sometimes I’d be frustrated/puzzled by what I heard from my students. Negative exponents send numbers “to the basement” (or upstairs if they’re already in the basement). The “Front Door Bomber” has one bomb for each person in the house (the distributive property). Why is a negative times a negative a positive? “When something bad happens to a bad guy, that’s good.”

When something bad happens to a good guy…

But I, too, was guilty.

I’m not talking about the usual suspects, FOIL and SohCahToa. I’m talking about “bacon and eggs”. Secondary math teachers can see slides 3 and 4 below and figure it out. Others probably stopped reading two paragraphs ago.

I imagine my students’ calculus professors being frustrated/puzzled by this. That makes me smile. A little. On the inside.

Aside from being unnecessary, two times out of three it’s incorrect and misleading. For example, in slide 8, is x the exponent or the answer?

I’ll no longer use FOIL in my classroom. Through algebra tiles, I’ll emphasize an area model. I’ll have a tougher time letting go of SohCahToa. It does help students memorize the definitions of the three primary trig ratios. However, whenever I asked my Math 10 students what they knew about trigonometry from Math 9, they would just say “It’s that SohCahToa thing”. No mention of big ideas or similar triangles. Suggestions?

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A Visual Approach to Simplifying Radicals (A Get Out of Jail Free Card)

The radical sign is like a prison. Twelve can be expressed as a product of prime factors so √12 = √(2×2×3). The 2’s pair up and try to break out. Sadly, only one of them survives the escape. √12 becomes 2√3.

That’s how I was taught to simplify radicals. No joke.

I imagined the numbers yelling “All in the name of liberty! Got to be free! JAlLBREAK!” as they scaled the prison walls. To this day, I can’t get this song out of my head when I teach this topic.

Many students are shown this method, albeit without the prison imagery. Write the prime factorization of the number. Circle the pairs. Write/multiply circled numbers outside the radical sign. There is real math behind this procedure. By definition, √2 × √2 = 2. However, I found that students who were taught this method couldn’t tell me why √(2×2×3) = 2√3. Where did the other 2 go?

Instead, I asked students to evaluate √12, then 2√3, using their calculators. Why are they equivalent? Students factored √12 as √4 × √3 (with some scaffolding for some). They understood where the 2 came from. Some began by factoring √12 as √6 × √2. Correct, but not helpful. The importance of finding factors that are perfect squares was discussed.

Marc Garneau shared with me his visual approach to simplifying radicals.

Consider a square with an area of 24. The side has length √24.

This square can be divided into 4 smaller squares, each with an area of 6. The sides of these smaller squares have length √6. Two of these lengths make up the side length of the large square, so √24 = 2√6.

24 can also be divided into 3 rectangles, each with an area of 8. Again, correct, but not helpful. How to simplify √45 as 3√5 and √72 as 6√2 are also shown above. Again, factors that are perfect squares are key.

I think it would be interesting to try this out. Some students may prefer this method, but most students will likely move towards simplifying radicals without drawing pictures. But by drawing pictures as they are learning this skill, students will be connecting mathematical ideas and building conceptual understanding. New learning (simplifying radicals in Math 10) will be connected to prior learning (concept of a square root introduced in Math 8). Students will have a more solid understanding of why perfect squares are used.

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Revisiting GeoGebra

Four years ago I learned about GeoGebra and made some applets to be used in my classroom. I started by creating applets that demonstrated the effect changing slider values had on the graphs of trigonometric functions. I’d change a value and then ask the class to describe what happened to the graph. These constructions made excellent demonstrations. But that was the problem. The spectator experience was improved, but students remained spectators. (SMART Board fans take note: having one student at a time come to the front of the class does not change this.)

I also posted these applets on my class website. I thought students would try them at home to reinforce learning and check for understanding. They didn’t.

I wanted to move more towards having students themselves do the investigating. I constructed dynamic worksheets to explore slope and circle geometry in Math 10 and 11. Twice, I threw in the towel halfway through the period because of technical difficulties. The 15 laptops had to remain plugged in because their batteries no longer held a charge. The wireless network couldn’t handle having 15 laptops on it. The files were copied from my flash drive to desktops but only worked on some of the computers.

So, we went back to pencil and paper. Each student drew and then measured his or her own angles. Some students immediately observed the relationship. Others observed it after seeing the results of each group member. They asked “What if we move the inscribed angle off to the side more?” and “What if the central angle is larger?” Then, they set off to find the answers. Listening to these conversations, I wondered what this would have looked like had I been able to carry out my lesson plan.

In the four years since then, I’ve seen several GeoGebra/Sketchpad constructions created by other math teachers but very little that really excites me. A new tool to use while I stand and deliver? An e-version of an investigation that my students do using pencil and paper? Okay. I guess. Just don’t try to sell it to me as being more than what it is.

I want to incorporate technology into my teaching in meaningful ways. Here’s something from David Cox that could get me back on the GeoGebra bandwagon. It starts with a great problem that is enhanced because it is posed using GeoGebra. Students continue to interact with the applet as they attempt to solve the problem.

I uploaded some of my dynamic worksheets to GeoGebraTube and was very pleased to see that they worked on my iPad. I’d love some feedback on them. Was my assessment of them correct or are they salvageable? Also, I’m willing to give GeoGebra another try. Can you point me to exemplars?

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Turn it Around

In More Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction, Dr. Marian Small discusses the turn-around strategy to create open questions.

Instead of asking “The legs of a right triangle are 3 cm and 6 cm long. What is the hypotenuse?” the teacher can ask “The answer is √45. What could the question be?”

There are many possible questions. For example,

Determine the length of the hypotenuse.

Determine the length of x.

A square has an area of 45 cm². What is the side length?

What is an example of a square root that has a value between 6 and 7?

Which number is the greatest: √37, 6, 6½, √45?

Students will come up with a variety of questions. However, at first, I imagine the response to open questions such as “The answer is √45. What could the question be?” will be silence. Students are used to being asked questions where there is one correct answer. In math, you either get it or you don’t. It’s not just questions that need turning around. This black and white view of mathematics also needs turning around. With time and practice, class discussions about open questions can help change this attitude.

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Two-Digit Addition – When Do I Show Them the “Real” Way?

Last week, I attended Carole Fullerton‘s parent presentation. She discussed strategies students have for adding two-digit numbers. Carole’s timing was great since I’ve been having similar discussions with teachers in recent weeks.

How many ways can you add 59 + 37?

The most common strategy that I see students use is to add the tens, add the ones, and then combine. Students working with ten frames naturally begin by grouping the 10’s, not the 1’s, together.

Students find other strategies. For example,

  • Add 1 to 59 to make 60. Take 1 away from 37 to make 36. 60 and 36 is 96. (make ten)
  • Add 1 to 59 to make 60. Add 3 to 37 to make 40. 60 and 40 is 100. Take the extra 4 away. (friendly numbers and compensation)
  • 30 more than 59 is 89. 7 more than 89 is 96. (add on)

These are the strategies I use to compute mentally. On paper, I fall back to the traditional right-to-left digit algorithm. It’s the result of performing thousands of such calculations in elementary school.

Students should be encouraged to write their mental math strategies down on paper. Some students will have to.

Teachers and parents appreciate these strategies. They make sense. Teachers and parents want mathematics to make sense to their kids. But at some point they always ask the question: “When should they learn the traditional/regular/real way?” They ask this because they are concerned their kids will not be prepared. “But do these strategies work for three-digit addition?” Yes.

“Relax. This will look familiar,” Carole joked. The same natural left-to-right strategy described above can be written vertically. We start with 50 and 30 is 80. Already, we know the sum is greater than 80.

Compare that with the traditional algorithm. We start with 9 and 7 is 16. We know the sum will have a 6 in the one’s place.

Which piece of information is more important? Carole made the point that accuracy is important. Always was, always will be. But it’s not where we should start. Strategies should be built on conceptual understanding. The emphasis of a left-to-right strategy is on number whereas the emphasis of a right-to-left strategy is on digits.

In her new blog, Amy Newman also writes about this. As well, she shares Carole’s key messages for parents helping children at home.

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Teachers Make Excellent Pirates – Two Treasures from Blogs I Follow

If I had asked my Principles of Math 11 students “What can you tell me about linear inequalities?” I bet most would have said something about graphing a boundary line and shading one side of it. If I had asked “What does the shaded region represent?” I bet many would not have been able to answer correctly.

Similarly, if I had asked “What can you tell me about absolute value?” most would have said something about changing negatives to positives. Few would have been able to give a satisfactory explanation if I had asked “What does |2 – 7| = 5 mean?”

After reading John Scammell’s recent post on linear inequalities, I realized that I had it backwards. I’d begin by graphing the line. I’d explain that a line cuts the plane into two regions. Together, we’d determine which region to shade. I’d tell students that each point in this region is a solution.

Instead, John’s colleague begins by having students find x– and y-coordinates that satisfy the inequality. Then, each student plots these ordered pairs on a grid at the front. It becomes clear to students that each solution is a point in a half-plane and that a boundary line exists. I robbed my students of this discovery.

John’s scatterplot reminded me of something from Kate Nowak’s back catalogue. Kate’s activity involves having students guess the number of M&Ms in a container. Plotting the points (guess, distance from correct value) results in something like this:

Kate has students write equations and inequalities that model weather forecasting. “Today’s temperature will be more than 10 degrees off from the usual temperature” can be modelled using |T – 68| > 10. In a later post, she has students write an inequality that models this scenario:

She scaffolds this by asking students to explain why |t – 12| ≤ 1993 and |t – 12| ≥ 1993 are not good models and by having students write a sentence that begins with “The distance from…”.

Context is important. Not because it must answer “When am I ever going to use this?” but because it helps build conceptual understanding. I’m guessing Kate’s students can tell her more about absolute value than “it changes negatives to positives”.

By the way, the inequality for music that I find tolerable would be |t – 1985| ≥ 6. I graduated from high school in ’91 not ’85. I’m just not a fan of 80s music.

(The title of this post was “borrowed” from Nat Banting.)

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“Under the M… the square root of 12”

On this blog, sometimes I share my thoughts about transforming math education. This is not one of those times.

Here, I’m using my blog as a digital filing cabinet.

One activity that my students enjoyed was MATHO (and its variations FACTO and TRIGO).

Have students select and place answers from the bottom of each column to fill up their MATHO cards. In some versions, I pulled prepared questions from a hat. In other versions, I translated answers to questions on the fly. For example, if I grabbed 2√3, I called out “Under the M… the square root of 12”. After a student shouts “MATHO!” ask potential winners to read aloud their numbers. (Remember to keep track of answers you have called.)

Nothing revolutionary here – just a fun way to review content.

Squares & Square Roots
Exponent Laws
Simplifying Radicals
Rational Exponents
Factoring Trinomials x^2+bx+c
Factoring Special Products
Trig Functions

By the way, if you are looking to read about changing things, please check out Sam Shah’s recent post, The Messiness of Trying Something New.

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BCAMT Conference 2011

Here’s the list of puzzles & games from my session at the BCAMT Fall Conference.

Together, we brainstormed some ideas about connecting games to mathematics. I listed some of these ideas in an earlier post.

Also, I attended a couple of sessions in the morning…

“It’s the most interesting thing a graphing calculator can do and we don’t even have kids do it.” Dan Kamin was talking about creating a scatterplot to determine the linear, quadratic, or exponential regression equation. While lines/curves of best fit are no longer in the curriculum (regression functions were in Applications of Math 10 & 11), it is expected that students will solve problems by analyzing linear, quadratic, and exponential functions. This provides opportunities to have students use technology to answer questions such as:

  • Is the data best modeled with a line or with a curve?
  • What is the equation of the function that best models the data?
  • How does your best fit line/curve compare with the those used by experts?

As a person inhales and exhales, the volume of air in the lungs can be modeled with a periodic function. Dan asked his students to write the equation of the sinusoidal function. They couldn’t. It wasn’t a Ferris wheel. “No one had asked them to create something before.”

I’ve had difficulty making domain and range relevant to 15-year-olds. David Wees shared an interesting activity at his presentation. Have students draw a picture of an object using line segments. Then, for each line segment, have them write the equation and determine the domain and range. Finally, have them enter this information using GeoGebra and compare the result with the drawing.

A reminder… the 2012 BCAMT New Teachers’ Conference will be held on February 11 at Queen Elizabeth Secondary School in Surrey.