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Ask an Expert (Teacher Edition)

It can be challenging to plan activities for workshops with secondary math teachers. I like to have teachers first experience learning mathematics as my students do. There’s the rub – if I share an activity from my classroom, teachers already know the math. They can opt out of explorations designed to construct understanding – they know how the story ends.

Marc Garneau (@314Piman) and I have two strategies to deal with this. First, we can have teachers look at a familiar topic in a new light. For example, have teachers:

A second strategy is to have teachers solve a problem that is similar, but not too similar, to something they teach. For example, I wanted to model how I use expert groups to have students develop the exponent laws in Math 9. Having teachers do this would be iffy. Instead, Marc and I came up with this:

Each expert group of teachers was responsible for learning and teaching one set of ‘pop’ rules. For example,

(a + c) ‘pop’ (b + d)
= 2(a + c) + (b + d)
= 2a + 2c + b + d
= 2a + b + 2c + d
= (a ‘pop’ b) + (c ‘pop’ d)

0 ‘pop’ a
= 2(0) + a
= a

Later, we asked “Okay, so ‘pop-ifying’ is not in the WNCP curriculum, but where could you use this teaching strategy?” Teachers answered “It would be great for teaching exponent rules or log laws.”

Mission accomplished.

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My 7-year-old daughter keeps beating me at Spot it!

I have an excuse. While playing, I start thinking about the mathematics behind the game rather than the cards in front of me.

The goal of Spot it! is to be the fastest player to spot and call out the matching symbol between two cards. There are 55 cards, each with 8 symbols. Between any two cards there is one, and only one, matching symbol. How did the designers accomplish this? Sue VanHattum explores this question on her blog, Math Mama Writes.

In addition to thinking “How did they do that?” I started thinking about creating a smaller math version of Spot it! What if, rather than symbols, students matched equivalent expressions? A game might consist of 21 cards, each with 5 expressions (e.g., \sqrt {64}, 2^{3}, \dfrac {4} {3}\div \dfrac {1} {6}, \left( -2\right)\left( -4\right), and 8).

I began by creating 7 cards, each with 3 letters. While I was trying to create 13 cards, each with 4 letters, I finally asked “Why am I doing this?” Okay, so the game might be fun for some students, but would it increase their conceptual understanding? Of course not. We’re talkin’ about practice.

I have decided to walk away from creating these types of activities. It won’t be easy. The card stock! The laminator! The paper cutter! I love these things more than a grown man should. I’m quitting. Cold turkey.

But first, check out my latest Tarsia jigsaws…


factoring trinomials tarsia (normal)
factoring trinomials tarsia (larger)
factoring trinomials tarsia (solution)


rational exponents tarsia (normal)
rational exponents tarsia (larger)
rational exponents tarsia (solution)

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A Linear Functions Lesson Across the Grades

How many people can sit at 100 (or n) triangular tables? Square tables? Hexagonal tables? What if you join the tables so that one side of the next table touches one side of the previous table?

I appreciate this problem for a few reasons:

  1. I can present it in grades 4 through 10. In grade 4, students write a recursive relationship (e.g., for joined hexagonal tables, start at 2 and add 4 each time). In grade 6, students write a functional relationship (e.g., 4n + 2). In grade 8, students graph a linear relation (e.g., y = 4x + 2). In grade 10, students interpret the slope and y-intercept (e.g., each added table provides 4 additional seats, there are 2 additional seats at the ends of the table). When I teach and discuss this lesson at different grade levels within a school, I think a common activity helps teachers connect the big ideas across the grades.
  2. I can easily adapt and extend the task. When I have taught this lesson in grade 6 (see three-part lesson plan), most students can write an expression for joined square or hexagonal tables. Some students may choose to solve a simpler problem and write an expression for joined triangular tables. Other students can be challenged to write an expression for tables with any number of sides. All students can participate in the class discussion.
  3. The use of pattern blocks can help students gain a deeper understanding. Most students were able to make sense of the 4 in 4n + 2. Each time a table is added to an end, 4 seats are added. (Two seats are lost when tables are joined.) When one student showed how he added tables to the middle rather than an end, this helped his classmates make sense of the 2 in 4n + 2. There are two more tables at the ends. Pattern blocks allow students to make sense of the expression beyond “add 2 to make the numbers in the table of values work”.

This problem appears in several resources including The Super Source.

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Ask an Expert

This was my go-to review activity. I picked it up at an un-unconference as a student teacher.

First, have students get in groups of four. This is their home group. Have students number themselves from one to four.

Home Groups

Have students move and form groups so that each student in the group has the same number. This is their expert group. Each expert group is responsible for one part of a review assignment, such as this. For example, the 1’s (Adele, Ellen, Lea, and Oprah) may be responsible for becoming experts on solving quadratic equations by factoring, the 2’s on solving using the square root method, the 3’s on solving using the quadratic formula, and the 4’s on the nature of the roots. Emphasize that each member of the group must understand and be able to explain the solution to each question in this part of the assignment. I play up that I will only help students while they are in their expert groups.

(Classroom Management Tips: Ask just the first four home groups to move and form their expert groups. Have the remaining home groups remain seated until this is complete. You will be able to see if each student is moving to the correct group. I’ve used this activity in classes of 24 to 32 students. Plan for this. For example, 12 students will form three home groups of three and will move to form four expert groups of three.)

Expert Groups

Have students return to their home groups to complete the assignment. If any student needs help with any question, he or she is sitting with an expert. For example,

  • Ashton needs help factoring when the leading coefficient is not equal to one?
    Adele’s an expert.
  • Barack has difficulty using the square root method when there are brackets?
    Ask Ashton.
  • Beyonce struggles with simplifying expressions when using the quadratic formula?
    Barack knows.
  • Adele can’t remember which condition results in two equal real roots?
    Beyonce can help.

I may have gone to the well one too many times with this as a review activity. Time to try Kate Nowak’s speed dating activity. Also, I’d like to use expert groups to have students learn, rather than review, concepts. I’ve used this activity, with some success, to teach exponent laws in Math 9.

A Leibniz-Newton Moment

As I was about to hit publish on this post, Mathy McMatherson published his own post on expert groups. He even mentioned @k8nowak’s speed dating. Please read his post for a more in-depth reflection on expert groups and jigsaw activities in general.

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Tarsia Jigsaws

Last year, one of my former student teachers told me about Tarsia, a software program that allows teachers to create jigsaws (and more). He remembered that I created similar jigsaws using MS Word (no small feat) and experienced this joy himself as a new teacher. I wish I knew about this tool several years ago.

Tarsia includes an equation editor for entering matching expressions. Teachers may also enter distractors so that corner and edge pieces are not easily determined. The activity cards are scrambled when outputted, ready to be cut out by students.

Here’s one that I quickly created:
logarithms jigsaw (normal)
logarithms jigsaw (larger)
logarithms solution

In my classroom, I often used jigsaws to review a topic. In addition to providing students with opportunities to practice, these activities get students talking mathematically. As a teacher, I am able to listen to students making mathematical arguments about whether or not pieces fit together and observe them checking and revising their work. Also, eavesdropping on these mathematical conversations will tell me if there are topics that need to be discussed further (e.g., rational exponents).

Formulator Tarsia (for Windows only) can be downloaded here.

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One of these things is not like the others

When you read the title of this post, did you think Sesame Street? Foo Fighters? Or, like me, both?

Recently, Geoff shared seven (sneaky) activities to get students talking mathematically. One activity, ‘odd one out’, involves having students pick the one mathematical thing that doesn’t belong. This reminds me of one strategy used by Dr. Marian Small to create open questions – asking for similarities and differences.

Here’s my ‘odd one out’ question:

Which of the following quadratic functions doesn’t belong? (Dr. Small might ask “Which of these four functions are most alike?”)
y=2\left( x-1\right) ^{2}+3
y=\dfrac {1} {2}\left( x-3\right) ^{2}-5
y=3\left( x+2\right) ^{2}-4
y=-\dfrac {3} {2}\left( x-4\right) ^{2}+6

Students might say,
y=2\left( x-1\right) ^{2}+3 because it does not cross the x-axis
y=\dfrac {1} {2}\left( x-3\right) ^{2}-5 because it is a vertical compression of y = x²
y=3\left( x+2\right) ^{2}-4 because it is a horizontal translation to the left
y=-\dfrac {3} {2}\left( x-4\right) ^{2}+6 because it opens down

Do the graphs of these functions strengthen your choice or make you change your mind?

I carefully chose the values of a, p, and q in y = a(x – p)² + q so that students could reasonably argue that any one of the functions could be picked as the odd one out. Because I am not looking for one particular answer, each student should be able to confidently answer the question and contribute to a mathematical discussion. Planning disagreement is key; it means students will have to justify their mathematical thinking.

Sneaky.

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Communicating Effectively?

From my textbook I had as a student teacher taking my teaching mathematics course:

“Ms. Spencer’s body language suggests that she doesn’t take what she’s saying seriously enough to face her listeners. Ms. Castillo’s body language clearly tells her students, ‘I’m talking to you and I expect you to be listening to this important message!'”

“Managing to focus your eyes on each student regularly during the course of classroom activities, and occasionally making positive expressions and gestures (a smile, a wink, a thumbs up) when you’ve caught the student’s eye, helps establish an atmosphere of mutual respect.”

I don’t know. I think Ms. Spencer’s body language suggests she is asking “You think you’re better than me?!” I think Ms. Castillo is saying “It’s go time!” Am I the only one who wants to shout “Mandelbaum! Mandelbaum! Mandelbaum!”? Ms. Castillo may have an important message. Her students may also have something important to say. Maybe this wasn’t an important point to make back in ’96.

Not sure I’d recommend winking at students who’ve caught your eye.

Maybe Sal Khan is on to something when he says “the face is hugely distracting”. I know mine is.

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Mystery Transformation

A Principles of Math 12 learning outcome states “It is expected that students will describe and sketch 1/f(x) using the graph and/or the equation of f(x)”. In my classes, students were more likely to ask ‘the question’ at the beginning of this lesson than during any other lesson.

Over the years, I simplified my explanation. Three steps:

  1. asymptotes
  2. signs (i.e., if f(x) is greater than zero, then 1/f(x) will be greater than zero)
  3. invariant points (and other important points)

Most of my students were able to successfully sketch reciprocal functions. I had successfully prepared them for the provincial exam. Still, I wasn’t satisfied with this lesson. My students weren’t learning mathematics, they were just following directions – following my steps.

Together, Marc Garneau and I created the activity below, probably inspired by this book.

Warm-up:

  1. How is the blue graph related to the red graph?
  2. Write an expression to represent this transformation.
  3. The point (5, 3) is on the graph of y = f(x).
    What point must be on the graph of y = -f(x)?

Activity:

  1. Have students work in groups of 3-4. Give each student in the group one of six cards.
  2. Have students record any observations they make about the graphs that are on their cards.
  3. Have students take turns sharing their observations. Encourage them to look for similarities and differences.
  4. Ask students to describe, in words, how the blue graph is a transformation of the red graph. Ask students to write an expression that represents this transformation. (The extra 2-3 cards can be used to test and confirm ideas.)
  5. Have students share their strategies with the class.

 

Discussion:

Sunita Punj invited us into her class to try out this activity. (Thanks again, Sunita!) Her students made some key observations, such as:

  • “There’s an asymptote at the x-intercepts”
  • “When the y is 1 or -1, it stays the same.”
  • “Points on the red graph that are 2 spaces up become points on the blue graph that are 0.5 spaces up.”

Sunita’s students were then able to use this information to discover that the blue graph could be obtained from the red graph by taking the reciprocal of the y value. I enjoyed listening to, and participating in, the mathematical conversations that were happening at each table.

  • “Is that always true?”
  • “I have a theory…”
  • “But why don’t the blue graphs touch the dotted lines?”

Students were not simply following directions. Nobody asked ‘the question’. (Still, if there is a real-world application here, I’d love to learn about it.)

There may be limited opportunities in Math 8-12 to have students identify a ‘mystery transformation’. However, I think it’s worth exploring the bigger idea – giving students questions and answers and then asking them to talk about how the answers may have been determined.

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The first step in adding fractions is to find a common numerator.

“Okay, listen up! Today’s lesson will be on adding fractions. Let’s start with an easy one like 1/3 + 1/6. The first step is to find a common numerator, which, in this example, we already have. This becomes the numerator of the sum so let’s write a 1 up there. The denominator is, of course, itself a fraction whose numerator is the product of the denominators and whose denominator is the sum of the denominators. This gives us 1/(18/9), or 1/2.

Let’s kick it up a notch and try 2/3 + 1/4. Remember, the first step is to find the lowest common numerator, or LCN. You guys look a little puzzled. You remember learning this in grade 7, right? Since the LCN is 2, we have 2/3 + 2/8. Write a 2 up top. To determine the denominator, simply multiply and add to get 24/11. We have 2/(24/11). This is a tricky one since 24/11 doesn’t reduce nicely. Multiplying the common numerator by the denominator of the denominator gives us 22/24. One more thing… if you don’t reduce to lowest terms, I’ll have to deduct half a mark. 22/24 should be written as 11/12. I’ve typed up some notes. Take one sheet and pass the rest back.”

Christopher Danielson over at OMT shared the method above with me earlier this year. Recently, I presented it to a group of secondary math teachers. Christopher’s algorithm brilliantly initiates conversation about what is important in teaching and learning mathematics. For example, one teacher said “It works. I can prove that it works. But, it doesn’t make sense.” Another asked “It’s quick and easy, but does that matter?”

I think Christopher (@Trianglemancsd) plays it straight when he shows his algorithm to pre-service teachers. I couldn’t pull this off – more of a tongue-in-cheek thing for me. This elicited some (nervous?) laughter as teachers put themselves in the role of their students learning about LCD’s.

This segued to activities that do build conceptual understanding of fraction operations. We looked at:

  • using an area model to represent multiplication,
  • using pattern blocks to explore quotative division, and
  • using a common denominator to divide fractions.

These last two are connected… more on this later.

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A New Bumper Sticker – My child is in the same classroom as your honour student

“Your rockstars aren’t the ones who put forward the best ideas. They put forward the safest ideas.”

Peter Liljedahl said this while discussing ‘A’ students and his numeracy tasks at last week’s pro-d presentation. I thought this was a great argument against ability grouping, in a bumper sticker kind of way. If we teachers were able to create problems and tasks with low floors and high ceilings, would that spell the end of honours classes? Probably not. Parents like the stickers. And they should.

I’m no rockstar, but I was an honour student. I know, I know, what happened? As either a student teacher or a new teacher, I attended a workshop (Kanwal Neel maybe?) and we were given the challenge of designing the largest popcorn box (or open rectangular prism) from a single sheet of paper. My partner (a secondary science teacher) and I (a secondary math teacher) were confident we had the winning design. We had a function, V(x) = x(8.5 – 2x)(11 – 2x). We had the quadratic equation. We had calculus! We had everything but the winning design. Beaten by a pair of primary teachers who had tape and the ability to ‘think outside the box’. (Sorry.)

Update: Kanwal sent me his largest box problem. The context has been changed to create the largest bentwood box.