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!)

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:

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:

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.

“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.

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!

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…

…which should lead to this question.

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

.doc

Present the task.

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.

Ask students to revisit their rankings.

Call on students to answer the questions above.

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

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?”

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

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 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?

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

Gwyneth: Great! Can you find another pattern?