Wanted Parabola

As much as I love mathematical modelling, so much of Math 10 to 12 is contextless stuff like this:

Determine an equation of a quadratic function with vertex at (-5, 3), passing through the point (-7, 15).

Lately I’ve been looking for activities that address this sort of naked math yet engage learners in processes similar to those in a mathematical modelling cycle.

Consider the exercise above. What questions could you ask? If I were to ask a student about their equation, I’m likely to hear play-by-play, not colour commentary: “… and then I plugged -7 and 15 in y = a(x + 5)² + 3. Negative seven plus five is two…”

Instead, I could have students try to figure out a quadratic function that satisfies a set of criteria, gradually revealed to them as “clues.” Throughout, students would check their quadratic functions and make changes when necessary. This is the gist of Wanted Parabola, my adaptation of Cathy Marks Krpan’s Wanted Number:

I started with a very general clue: the direction of opening. I anticipated a variety of parabolas, which I got when I tried this activity out with math teachers in my district. When I tried this activity out in the MathTwitterBlogoSphere (#MTBoS), I got a bunch of y = x²s. The biggest difference was that my colleagues were invited to draw a parabola (on whiteboards) whereas my tweeps were asked to write an equation (in a Desmos activity). It’s interesting to think about this activity in terms of freedom and constraints. When I revealed the next clue, it pushed my colleagues’ thinking together. However, from my tweeps, it triggered new and diverse ideas, simulated here:

I like this as a blank-page (or whiteboard) activity but a Desmos activity (1, 2, 3) does provide the opportunity to talk about some interesting overlays. If using vertical non-permanent surfaces (#VNPS), I’d stop partway through to hold a “board meeting” where students would share possible parabolas.

In general, I progressed from providing more general to more specific clues. For example, “vertex in QII” divulges p < 0 before “axis of symmetry x = -5″ gives away p = -5. Most clues add new information and move students closer to the Wanted Parabola. Some confirm earlier decisions. For example, “vertex (-5, 3)” before “axis of symmetry x = -5″ and “minimum value of 3.” This last clue is anticlimactic. An earlier clue, “never enters QIII,” is much more interesting. It might feel like new information. But it must be true given preceding clues; a parabola that opens up and has no x-intercepts cannot contain points in QIII (or QIV).

You can play with the order of the clues. A second Wanted Parabola:

Here, the direction of opening clue is revealed midway through the set. It doesn’t add new information but is reasoned to through “two x-intercepts” and “vertex in QI.” I meant to delay students determining the direction of opening a bit, hoping to surprise them after a few clues. In a third Wanted Parabola, “passes through” is the first clue; I anticipate that some students will place the vertex at this point.

Instead of “How did you find a?” you could ask “Which clues were helpful? Which clues were necessary?” In my mind, helpful ≠ necessary. A clue might be helpful if it pushes students in the direction of the Wanted Parabola despite not providing the values of a, p, or q. Or a clue might be helpful if it tells students that they’re on the right track. In the way that the first Wanted Parabola plays out, three pieces of information are necessary (to determine three unknowns): “minimum value of 3,” “axis of symmetry x = -5,” and “passes through (-7, 15).” If some students don’t argue that only two clues are necessary — “vertex (-5, 3)” and “passes through (-7, 15)” — you could ask “What is the fewest number of clues you need?”

This activity helps students develop an understanding of the different attributes parabolas can have. It provides an opportunity for students to solve problems, reason, explain, justify, and connect mathematical ideas in ways that “Determine an equation…” does not.

Wanted Parabola (.pptx) (.pdf)

References

Marks Krpan, Cathy (2013). Math expressions: developing student thinking and problem solving through communication. Toronto, ON: Pearson Canada.

Update

Fair Share Pair

A couple weeks ago, I was discussing ratio tasks, including Sharing Costs: Travelling to School from MARS, with a colleague who reminded me of a numeracy task from Peter Liljedahl. Here’s my take on Peter’s Payless problem:

Three friends, Chris, Jeff, and Marc, go shopping for shoes. The store is having a buy two pairs, get one pair free sale.

 Chris opts for a pair of high tops for $75, Jeff picks out a pair of low tops for $60, and Marc settles on a pair of slip-ons for $45.

The cashier rings them up; the bill is $135.

How much should each friend pay? Try to find the fairest way possible. Justify your reasoning.

Sharing Pairs.pdf

I had a chance to test drive this task in a Math 9 class. I asked students to solve the problem in small groups and record their possible solutions on large whiteboards. Later, each student recorded his or her fairest share of them all on a piece of paper. If you’re more interested in sample student responses than my reflections, scroll down.

The most common initial approach was to divide the bill by three; each person pays $45. What’s more fair than same? I poked holes in their reasoning: “Is it fair for Marc to pay the same as Chris? Why? Why not?” Students notice that Chris is getting more shoe for his buck. Also, Marc is being cheated of any discount, as described by Student A. (This wasn’t a happy accident; it’s the reason why I chose the ratio 5:4:3.)

Next, most groups landed on $60-$45-$30. Some, like Student A, shifted from equal shares of the cost to equal shares of the discount; from ($180 − $45)/3 to $45/3. Others, like Students B, C, and D, arrived there via a common difference; in both $75, $60, $45 and $60, $45, $30, the amounts differ by $15. This approach surprised me. Additive, rather than multiplicative, thinking.

Student C noticed that this discount of $15 represented different fractions of the original prices; $15/$75 = 1/5, $15/$60 = 1/4, $15/$45 = 1/3. He applied a discount of 1/4 to all three because “it’s the middle fraction.” Likely, this is a misconception that didn’t get in the way of a reasonable solution.

Student D presented similar amounts. Note the interplay of additive and multiplicative thinking. She wants to keep a common difference, but changes it to $10 to better match the friends’ discounts as percents.

Student E applies each friend’s percent of the original price to the sale price. This approach came closest to my intended learning outcome: “Solve problems that involve rates, ratios and proportional reasoning.”

In spite of not reaching my learning goal, I think that this lesson was a success. The task was accessible yet challenging, allowed students to make and justify decisions, and promoted mathematical discourse.

Still, to increase the future likelihood that students solve this problem using ratios, I’m wondering about changes I could make. Multiples of 20 ($100-$80-$60) rather than 15 ($75-$60-$45)? Different ratios, like 4:3:2 or 5:3:2, might help; the doubles/halves could kickstart multiplicative thinking. (Also, 5:3:2 breaks that arithmetic sequence.)

Or, I could make changes to my questioning.

Sharing PairsWhen I asked “What do you notice?” students said:

  • the prices of the shoes are different
  • Chris’ shoes are the most expensive
  • Marc’s shoes are the cheapest
  • Chris’ shoes are $15 more than Jeff’s, which are $15 more than Marc’s
  • Jeff’s shoes are the fugliest

Maybe I could ask “What else could you say about the prices of Chris’ shoes compared to Marc’s?” etc. to prompt comparisons involving ratios. If that fails, I’m more comfortable connecting ratios to the approaches taken by students themselves than I am forcing it.

BTW, “buy one, get one 50% off” vs. “buy two, get one free” would make a decent “Would you rather?” math task.

h/t Cam Joyce, Carley Brockway

Sharing Pairs - Sample Student Response A
Sharing Pairs – Sample Student Response A

Sharing Pairs - Sample Student Response B
Sharing Pairs – Sample Student Response B

Sharing Pairs - Sample Student Response C
Sharing Pairs – Sample Student Response C

Sharing Pairs - Sample Student Response D
Sharing Pairs – Sample Student Response D

Sharing Pairs - Sample Student Response E
Sharing Pairs – Sample Student Response E

Always, Sometimes, Never

This week should have been my first official – third unofficial – week back. Instead, I’m starting this school year as I ended the last – walking the picket line. I haven’t been up to blogging since this started.  Below is a draft from June. I never got around to finishing it. The ending has a “pack up your personal belongings” feel. I left it as-is; seems fitting that this post should come up short… I mean, 10% of my pay – and my colleague’s – was being deducted at the time. 

Recently, I invited myself to a colleague’s Math 8 class to try out Always, Sometimes, Never. In this formative assessment lesson – originally by Swan & Ridgway, I think – students classify statements as always, sometimes, or never true and explain their reasoning.

Sometimes-Always-Never-3

Because it’s June, we created a set of statements that spanned topics students encountered throughout the course. Mostly, this involved rephrasing questions from a textbook, Math Makes Sense 8, as well as from Marian Small’s More Good Questions, as statements. That, and stealing from Fawn Nguyen.

.doc .pdf

To introduce this activity, I displayed the following statement: When you add three consecutive numbers, your answer is a multiple of three.

Pairs of students began crunching numbers. “It works!”

“You’ve shown me it’s true for a few values. Is there a counterexample? What about negative numbers? Does it always work? How do you know? Convince me.”

Some students noticed that their calculators kept spitting out the middle number, e.g, (17 + 18 + 19)/3 = 18. This observation lead to a proof: take one away from the largest number, which is one more than the middle number, and give it to the smallest number, which is one less than the middle number; each number is now the same as the middle number; there are three of them. For example, 17 + 18 + 19 = (17 + 1) + 18 + (19 – 1) = 18 + 18 + 18 = 3(18).

I avoided explaining my proof: x + (x + 1) + (x + 2) = 3x + 3 = 3(x + 1). This may have been a missed opportunity to connect the two methods, but I didn’t want to send the message that my algebraic reasoning trumped their approach. “Convince me,” I said. And they did.

To encourage students to consider different types of examples, I displayed a ‘sometimes’ statement: When you divide a whole number by a fraction, the quotient is greater than the whole number. Students were quick to pick up on proper vs. improper fractions.

Next, students were given eight mathematical statements. We discussed some of the statements as a whole-class. Some highlights:

A whole number has an odd number of factors. It is a perfect square.

I called on a student who categorized the statement as always true because “not all of the factors are doubled.” We challenged doubled before she landed on square roots being their own factor pair. For example, 1 & 36, 2 & 18, 3 & 12, 4 & 9 are each counted as factors of 36, but 6 in 6 × 6 is counted only once.

The price of an item is decreased by 25%. After a couple of weeks, it is increased by 25%. The final price is the same as the original price.

Like the three consecutive numbers statement above, students began playing with numbers – an original price of $100 being the most popular choice. I anticipated this as well as the conceptual explanation that followed: “The percent of the increase is the same, but it’s of a smaller amount.” I love having students futz around with numbers; it’s so much more empowering than having them “complete the table.”

The number 25 was chosen carefully in hopes that some students might think fractions: (1 + 1/4)(1 − 1/4) = (5/4)(3/4) = 15/16. None did. There’s a connection to algebra here, too: (1 − x)(1 + x) = 1 − x². Again, I didn’t bring these up. Same reason as above.

One side of a right triangle is 5 cm and another side is 12 cm. The third side is 13 cm.

All but one pair of students classified this as always true. That somewhat surprised us. More surprising was how this one pair of students came to realize

BTW, the blog-less Tracy Zager has a crowd-sourced a set of elementary Always, Sometimes, Never statements.

Update: I stand corrected.

Sinusoidal Sort

On Monday, I was invited to Sandra Crawford’s Pre-Calculus 12 classes to try out an activity we created together. Thanks, Sandra!

Sandra’s students were familiar with how transformations of functions affect graphs and their related equations. They’ve stretched & shrunk (vertically & horizontally), flipped (in the x-axis & in the y-axis), & slid (up, down, left, & right) linear (& piecewise linear), quadratic, absolute value, reciprocal, & radical functions. These were topics in prior units. In this unit, students were previously introduced to radian measure, the unit circle, the six trig ratios, & the functions y = sin x, y = cos x, & y = tan x. Next up: determining how varying the values of a, b, c, & d affect the graphs of y = a sin b(x – c) + d & y = a cos b(x – c) + d.

Such was the case when I last taught trig functions (in Principles of Math 12). Back then, my approach was to provide clear and concise explanations, connecting these transformations to those transformations (or, better, transformations of these to transformations of those). But was this necessary? Shouldn’t students be able to make this connection? On. Their. Own.

In small groups, students were handed a set of equation cards to sort and were asked to explain their sorting rule. We designed the equations so that there were plenty of similarities and differences in terms of whether or not there were leading coefficients, coefficients of x, brackets, etc., as well as in terms of the values of a, b, c, & d themselves. After all that, most groups just sorted the equations into sine and cosine functions — to be expected, I guess, given the focus of the prior lesson.

sort1

Next, students were handed graph cards and were asked to match each to the corresponding equation card. We encouraged students to make predictions, then test these predictions using technology. Interestingly, few reached for their graphing calculators or phones. We asked students if, having seen the equations and their graphs together, they wanted to re-sort.

sort2

This process was repeated with characteristic cards. Note: The terms amplitude and period were introduced the lesson before; phase shift and vertical displacement were not. Hence, horizontal translational and vertical translation at this stage of the lesson.

For the most part, students were communicating and reasoning mathematically, making connections, and problem solving. They were engaged with mathematics. A minority probably would have preferred to be engaged with taking notes.

Groups shared their sorts the following day. In the end, the functions were sorted in a variety of ways, which allowed Sandra to highlight each transformation.

sort3

A few groups struggled with matching all of the cards. Therefore, I reduced the number of functions. If finished, some students could be given two additional functions. Each of these is actually a phase shift of one of the initial eight (e.g., y = cos x + 2 ↔ y = sin (x + 90°) + 2). I wonder what they’d do with that.

Sinusoidal Sort (doc)
Sinusoidal Sort (pdf)

(Note: I’ve triple-checked these. Still, no guarantees.)

Less Play-by-Play, More Colour Commentary

To many, Explain your thinking = Tell me your steps.

Which got me thinking about hockey.

In sports broadcasting, the play-by-play announcer gives a detailed account of the action. The colour commentator provides expert analysis and insight. The sideline reporter does this.

Listen for the difference (play-by-play vs. colour commentary) here:

From ‘Doc’ Emrick, play-by-play announcer, we learn:

  • Sidney Crosby tries to split the defence
  • Ryan Miller steers the puck into the corner
  • Crosby “crunches” the puck along to Jarome Iginla
  • Crosby scores
  • the game is over
  • Canada wins the gold medal

Emrick’s enthusiastic call certainly added to my enjoyment of the broadcast, but it did little to add to my understanding of the events. It’s the stuff of who, what, where, & when. I didn’t really need ‘Doc’ for this; I saw it for myself.

From colour commentator Ed Olczyk, who comes in at 0:50, we learn:

  • a two-on-two turns into a one-on-nothing
  • Sidney Crosby beats Ryan Miller under the pads
  • Jarome Iginla, as he’s falling down, makes a beautiful pass to Sidney Crosby
  • it’s man-on-man coverage in overtime
  • Crosby gets offensive position on Brian Rafalski

Olczyk answers how & why Crosby scores.

Back to the math classroom…

Explain your thinking.

Two fictional responses at two extremes:

Doc: First, I minused 5 from both sides. Then, I divided by 2 and got x equals 3.

Ed: We modelled open & closed using red & yellow counters. We looked for a pattern and noticed that the first three open lockers–1, 4, & 9–are perfect squares. We tested 24 & 25. Switching has to do with factors. Only the perfect squares have an odd number of factors: you only count the 5 for 25 once.

In many math classrooms (mine included), student explanations can sound more like the former than the latter; more detailed account of the calculations on the page than insight into mathematical thinking.

Math teachers can work backwards and determine that Doc completed a practice exercise; he solved 2x + 5 = 11 for x. They’ll also recognize that Ed solved a problem–the well-known locker problem. Students are more likely to explain their thinking if they are being asked to think.

But practice or problem, creating a culture of why–consistently asking “Why?”/”How do you know?”–can also insert colour.

At first, I thought this analogy might be helpful to students–a small part of conversations that also involve post-game analysis of shared student responses (formative feedback, exemplars, etc.).

Whiteboard apps, such as Explain Everything or Show Me, can be used to capture and share student thinking. Student-created videos shared with me (so far) are more play-by-play than colour commentary. There is a place for a description of events as they happen. In fact, I just used a step-by-step video tutorial to help me repair my dishwasher. But we’re talking about mathematics, not home appliance repair. Behind the bench of each student-created tutorial that gets a “meh” from me, there’s a teacher passionate about mathematics and/or technology. I think we have different gameplans. Maybe the sports broadcaster analogy would be helpful to teachers, too?

Got a student-created video that’s more colour commentary than play-by-play? See you in the comments.

And just for fun, the finer points of hockey:

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.

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 ap, 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.

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.

iPad as Mathematical Communication Tool – Part Deux

I have been learning about educational uses of the iPad. My daughter has been learning about repeating patterns at school (Grade 1). Also, she has been asking me to show her how to use iMotion. A win-win situation.

She built four patterns, taking a photo each time she added a piece. Then, she created a video which I dragged into iMovie. Finally, I recorded her as she talked about her patterns. The movie would be better if the audio were synced to the video, but I wanted to see what we could create in ten minutes. Here it is:

In primary classrooms, students could share their videos and have classmates describe or translate the patterns. Similarly, in high school mathematics classrooms, students could build functions and have classmates determine equations. See an interview of UC Berkeley Math Education Professor Dor Abrahamson for the inspiration behind this idea.

These student-created movies could be used by classroom teachers to assess what students are able to do. There are nine mathematics learning outcomes in the BC Kindergarten IRP. One addresses Patterns:

B1 demonstrate an understanding of repeating patterns (two or three elements) by
– identifying
– reproducing
– extending
– creating
patterns, using manipulatives, sounds, and actions [C, CN, PS, V]

In Grade 1, two small changes are made to B1 and a second PLO is added:

B2 translate repeating patterns from one representation to another [C, R, V]

What judgements could you make about my daughter’s performance in relation to the prescribed learning outcomes? A rhetorical question – I’m not expecting or even wanting a reply.

My daughter also told me that sometimes shape and size can be used to describe patterns (e.g., “circle, circle, square, circle, circle, square” or “small, big, small, big”). Our movie doesn’t demonstrate this knowledge. This speaks to the importance of having conversations with our students – from Kintergarten to Calculus.

iPad as Mathematical Communication Tool

When children think, respond, discuss, elaborate, write, read, listen, and inquire about mathematical concepts, they reap dual benefits: they communicate to learn mathematics and they learn to communicate mathematically(NCTM)

In general, I’ve been disappointed with many of the iPad apps categorized under Education. With new apps being added (270/day in June 2011), I’ve got to admit it’s getting better. A little better all the time.

As Orwell Kowalyshyn and/or Kevin Amboe mentioned last spring, apps from other categories such as Games or Photography may provide richer educational opportunities for students.

My daughter (6) is currently enjoying the game Slice It!. The goal is to slice shapes as evenly as possible. The number of slices you are allowed and the number of pieces the shape is to be sliced into is given. The challenges get increasing difficult. I can imagine using this app to explore mathematical concepts such as area, fractions, percents, and line symmetry. Perhaps students could take screenshots and explain their strategies to their classmates. Maybe they could explain how they know the pieces have approximately the same area. (The FAILED text that appears when not sliced into the correct number of pieces may turn off some educators. No noticeable signs of this affecting my daughter, at least so far.)

Students will benefit from iPads in the classroom not because there’s an app for practicing number operations, but because there’s an app for communicating their thinking. ShowMe, ScreenChomp, and Explain Everything have been listed/discussed in many math ed blogs. Students, not their teachers or Sal Khan, can create video explanations using these interactive whiteboard apps.

Meeting with Surrey & Vancouver secondary math teachers this summer, one teacher showed us a picture of two containers each filled with chocolate eggs. The number of eggs in the smaller container was given and we were asked to guess the number of eggs in the larger container (see Dan Meyer’s blog). Using her iPad 2, the teacher filmed us giving and justifying our estimates. In a classroom, teachers or, better yet, students could interview peers, administrators, parents, members of the community, etc. and then share and discuss these guesses and strategies.

One app that I had fun with this summer is iMotion HD. This app allows you to create and share stop motion movies from pictures you have taken. In the video below, I show why 1/2 + 1/3 = 5/6 using pattern blocks.

Using iMovie, I could have added narration but I chose not to. Why? Because I have no plans to share this with students*. I chose not to narrate my movie because students, not the teacher, should be doing the math. In this way, students communicate to learn and learn to communicate.

Khaaan!
photo by pong0814

*Also, I have only one nephew. He is 18 months old and so far has been able to complete his algebra homework without asking his uncle to tutor him. The Khan Academy has already been widely and deservedly criticized by others. Please check out Karim Ani’s An Open Letter to Sal Khan on his Mathalicious blog.