Table Talk

You don’t teach students the problem-solving strategy of Organize the Information: Make a Table by having them “complete the table.”

The activity “That’s Sum Challenge!” from AIMS asks “What sums from one to 25 can by obtained by adding two, three, four, five, or six consecutive numbers?”

One of the student pages looks like this:

0001EJ

I’ve designed this type of thing before. Fortunately, there’s a quick fix: ask the question, allow students time to work on the problem, ask the groups–or regroup and ask the class–”How can we organize this information?”

Likely, students’ tables won’t match the one above. Some students will probably make a table for two consecutive numbers, then three, and so on. To highlight the impossible sums, the helpful folks at AIMS have done the work of merging these tables into one. In their defence, kinda, the teacher pages has this under “Management”:

  1. If you have a class that functions well with open-ended problems, you can explain the problem to them and have them solve it without using the student pages.

Subtracting the table engages more students at more levels. From “two consecutive numbers are always even and odd (or odd and even) and that gives us all the odd sums” to “the sums made by adding three consecutive numbers are all multiples of three” to “powers of two cannot be obtained because…,” each student can contribute to answering the key question “What sums can be obtained by adding consecutive numbers?” (The ellipsis is there because the reason isn’t immediately obvious to me.)

In the past, I had it back-asswards. Take the “How many different possible meal combinations are there on the kids¹ menu?” problem. I’d give ‘em tables and tree diagrams up front. A problem became practice. Once I “turned the tables” and allowed students time to get started, I could later ask groups to share their tables or I could step in at just the right time with tree diagrams to help make sense of spaghetti nightmares.

¹Kid’s? Kids’? This is why I’m not a prolific blogger.

Related: The more sides you have, the smarter you are.

Recommended: “You Can Always Add. You Can’t Subtract.” Ctd. by Dan Meyer

How many do you see? How do you see them?

This summer, as Gwyneth and I were packing up Othello, I started playing with different arrangements of discs – mostly arrays – and asked her “How many?” I remembered the following arrangement, taken from AIMS’ Cookie Combos activity.

3^2+4*4

3^2 + 4 * 4

“Sixteen plus nine, so nineteen plus six… twenty, twenty-five,” she said. (I don’t think that she actually said “twenty” aloud. That came after my clarifying question: “Wait. Huh?”)

There’s a lot happening in Gwyneth’s bridging through twenty strategy – partitioning of quantities, place value, commutative property, breaking apart to make (a multiple of) ten. All within a three count, standard algorithm be damned.

This invented strategy discussion was a happy accident. The goal of this problem when we pose it to teachers is to see different ways to visualize the group and represent these using expressions. It’s about valuing different methods; the solution – counting 25 cookies – is easy enough.

How many do you see? How do you see them? How many different ways can you find?

25

Some popular solutions:

7+2*5+2*3+2

7 + 2 * 5 + 2 * 3 + 2 * 1

3*5+2*4+2

3 * 5 + 2 * 4 + 2

4*5+5

4 * 5 + 5

If you look just right, you can see two arrays:

4*4+3*3

4 * 4 + 3 * 3

A creative solution that involves counting what’s not there:

7^2-4*6

7^2 – 4 * 6

And moving what is:

output_rf0ouE

5^2

If you plan on using these images with your students, I recommend displaying the photo with just white discs. This leaves the problem open. Two colours were used above to illustrate various visualizations. This can steer student thinking. (See how the use of colour is intended to be helpful here.) If students miss one of the visualizations above, display that photo and ask for the expression (or vice versa).

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. (No “Nearer, My God, to Thee” here.) 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.

Math Picture Book Post #7: Bean Thirteen

A few ago, I was invited to teach a lesson on division (Grade 3). First, I read Bean Thirteen aloud – once just for fun. About Bean Thirteen, from the author:

Ralph warns Flora not to pick that thirteenth bean. Everyone knows it’s unlucky. Now that they’re stuck with it, how can they make it disappear? If they each eat half the beans, there’s still one left over. And if they invite a friend over, they each eat four beans, but there’s still one left over! And four friends could each eat three beans, but there’s still one left over! How will they escape the curse of Bean Thirteen?

(A funny story about beans, that may secretly be about . . . math!)

Bean Thirteen

Next, we revisited several of the pages. I asked students to write an equation to match the picture. I modelled this using magnetic “bean counters.” For the page above, students suggested 2 × 6 + 1 = 13; I introduced 13 ÷ 2 = 6 R 1. We discussed and recorded the meaning of this:

13 divided by 2

In pairs, students then chose their own number of beans (counters) and built different division as sharing stories for this number. They recorded (.doc) their stories using pictures, numbers, and words:

SDEC1-4124-14052811560

SDEC1-4124-14052811561

I called on students to share their stories with the class. They observed that some numbers gave remainders more so than others; Bean Thirteen can also be used to explore even/odd and prime/composite numbers.

This lesson served as the students’ introduction to division. I wrestled with the decision to introduce remainders at this time. An alternative problem – one consistent with both the prescribed learning outcomes and recommended learning resources – might be to start with 18 beans – a “nice” dividend – and share equally among 2, 3, 6, and 9 bugs – “nice” divisors. Note 15 ÷ 3 = 4 R 3 (and 15 ÷ 2 = 6 R 3) above. This mistake would not have happened had I not introduced remainders. I wonder if including remainders makes it more difficult for students to understand division and relate division to multiplication.

Then again, children will have already experienced remainders in everyday contexts.

What say you?

“Selfiest” Cities

Last week, I came across TIME’s ranking of the “selfiest” cities in the world and created the math task below.

Show the first slide–”sharing learning intentions” and all that.

The Selfiest Cities in the World.001

Display and discuss the selfies in Slides 2–8 to provide the context. Also, this will be helpful in the event that some don’t know what a selfie is. Hey, it happens. Each year I’d have to explain rock-paper-scissors to at least one of my Math 12 students (tree diagrams & experimental vs. theoretical probability & law of large numbers).

This slideshow requires JavaScript.

Show the map below and ask students what the data might represent. Given the previous discussion, I’m confident they’ll infer that each yellow dot represents one selfie taken in Anaheim. Yep, that’s Disneyland–Las Vegas for kids!

The Selfiest Cities in the World.009

Provide more details: each dot represents one Instagram photo taken during a 24-hour period tagged selfie that included geographic coordinates.

Put up this picture…

The Selfiest Cities in the World.010…which should lead to this question.

The Selfiest Cities in the World.011

Hand out cards for Anaheim, Milan, and six other cities to pairs of students.

.doc

Present the task.

The Selfiest Cities in the World.018

Call on students to share and justify their rankings. Note that there is (probably) more than one possible ranking.

(Slide 19 is just a placeholder. So is Slide 30, 41, and 43. These signal the end of one part of the task–and stop me from clicking through to reveal too much too soon.)

Provide more information: the number of selfies and the number of selfie-takers. Have students write this information on their cards.

The Selfiest Cities in the World.020

Ask students to revisit their rankings.

The Selfiest Cities in the World.029Call on students to answer the questions above.

Provide still more information: the population of each city. Again, ask students to revisit their rankings.

The Selfiest Cities in the World.031

Fingers crossed, some students will divide the number of selfies or number of selfie-takers by the population. Maybe ask, “Is it fair to compare Anaheim and Milan, knowing what you know about the two cities? What about Manhattan and Miami?”? Again, call on students to share and justify their rankings.

Using the data below, have students compare cities of interest or try on/poke holes in classmates’ measures of “selfie-ness.” Ask, “What’s London’s story?” and “What do the ‘selfiest’ cities have in common?”

The Selfiest Cities in the World.042

Share TIME’s ranking and methodology and have students critique the reasoning of others.

The Selfiest Cities in the World.045

Remember: the required Instagram-hashtag-GPS combination means TIME’s data accounts for a fraction of all selfies taken during a 24-hour period. Also, how many selfies taken in Anaheim (or Manhattan or Miami or San Francisco or Honolulu or Paris or …) were taken by tourists? Is it fair to divide by that city’s population?

For my first attempt, students ranked the data once, armed with all of the data–map, number of selfies/selfie-takers, and population. I wasn’t confident that students would use the population. Number of selfies per capita is a bigger leap than, say, party over business. Marc suggested the three successive rankings. I think this will work better.

The full slideshow:

.key .ppt

“They’ll Need It for High School” (Part 2)

So Part 2 was supposed to be about the big ideas in K-7 mathematics that students will need for high school. But that’ll have to wait for Part 3. Instead, more on times tables.

Three oft-used arguments for the importance of memorizing times tables:

  1. When learning higher levels of math, there just isn’t time to use calculators or strategies to determine basic facts.
  2. Besides, thinking taxes working memory which means by the time you’ve worked out the first part of the question, you will have forgotten the… Where am I?
  3. Because factoring.

1 & 2 are gospel. Well, so is 3; nevertheless, it’s the focus of this post. I have a couple of thoughts on times tables and factoring trinomials.

The reason some students struggle with factoring trinomials is not because they haven’t memorized products to 10 × 10. I can get away with this if we’re talkin’ Pythagoras. But factoring?! I mean, that’s all it is, right? To factor x² + 7x + 10, you just have to ask yourself, “What two numbers multiply to 10 and add to 7?”

HS math teachers, try this: give your students a quiz on factoring. Include both x² + 10x + 24 and x² + 25x + 24. Get back to me. For extra credit (yours, not theirs), throw x² + 6x + 5 in there. If your students are anything like mine, I bet x² + 25x + 24 gives them at least as much difficulty as x² + 10x + 24. What does this mean for these students? More practice multiplying by one?!

Of course, 1 × 24 falls outside most times tables. Recall of products to 10 × 10 gets us the factors of x² + bx + 60 – if b = 16. But x² + 17x + 60, x² + 19x + 60, x² + 23x + 60, and x² + 32x + 60 are fair game, right? Try c = 48. Or 72. Or 96. Or 100. What role does memorizing times tables play? What role does being flexible with numbers play?

My point, I think, is that these are different, albeit related, skills. In other words, the “it” they’ll need for factoring (trinomials) is factoring (numbers). And number sense. This has some implications for K-7: not necessarily more “What’s 4 × 6?” but more “A rectangle has an area of about 24 square units. What could its length and width be?” or even “The answer is 24. What’s the question?”; not thinking digits/standard algorithm but thinking – and talking! – factors/mental math strategies, e.g. 16 × 25 = (4 × 4) × 25 = 4 × (4 × 25) = 4 × 100 = 400 (via Sherry Parrish).

origami by @Mythagon nothing to do with post @k8nowak says put pictures in posts

origami by @Mythagon
nothing to do with post but @k8nowak says put pictures in posts

Say you’re still asking, “How am I supposed to teach them factoring when they don’t even know their multiplication facts?” When I introduced polynomial division in Math 10, some of my high school students didn’t even know long division. So I taught division of numbers and polynomials side-by-side, highlighting connections. Can the same miiindset (channeling my inner Leinwand) be applied to factoring trinomials and times tables?

And what about something like x² − 2x − 24? If that – asking yourself, “What two numbers multiply to -24 and add to -2?” – is all it is, why not factoring trinomials to teach multiplication (and addition) of integers?

Part One

[TMWYKS] Rainbow Loom

Christopher Danielson brought you #tmwyk, or talking math with your kids. I bring you #tmwyks, or talking math with your kid sister.

It happens to every parent, I think: the kid says something and nobody has to ask “Where’d she hear that?” Maybe it’s the kid’s choice of words. Or maybe it’s the tone, pitch, or rhythm that gives you away. Rare is it for me that these are proud parenting moments.

A recent exception:

Gwyneth (9 years old): What patterns do you see?

Rainbow Loom

Keira (6 years old): Red, white, yellow, red, white, yellow, red, white, yellow.

Gwyneth: Great! Can you find another pattern?