As part of an upcoming “TNIFHS” post, I wanted to include one example of what Christopher Danielson’s approach to a better shapes book might look like in a high school math class. [Update: Christopher’s Which One Doesn’t Belong? is available from Stenhouse; from Pembroke for Canadians.] But then I had some fun with this and created a few more. In each set, a reason can be given for each of the four options being the odd one out. I’ve done this type of thing with numbers and equations before. Worthwhile, but not what I’m going for here. Each set below is pictorial. Also, I went naked; I stripped the graphs of grid lines and ordered pairs. More noticing properties, less determining equations. Aside from graphs, where else in secondary mathematics might this fit? The last two images below are my attempts at answering this question.
I’m picking “TNIFHS” back up. At the end of Part 1, I promised Part 2 would answer “What are the big ideas in elementary school mathematics that students will need for high school?” Instead, I talked times tables.
In this third half, I’ll refocus on these big ideas. Or one of them — one that came up in that initial “they’ll need long division for high school” conversation.
More than the standard algorithm, what students will need is an understanding of division as sharing (finding the number in each group) and measuring (finding the number of groups). More generally, what students will need is an understanding of the fundamental meanings of all four operations.
Here’s part of a task I presented to my secondary math colleagues:
Evaluate, or simplify, each set of expressions. Make as many connections as you can conceptually & procedurally, pictorially & symbolically.
Sticking with division, in this task (−6) ÷ (+3) was chosen to bring to mind sharing (3 groups, −2 in each group) whereas 6/5 ÷ 3/5 was chosen to evoke measuring (3/5 in each group, 2 groups). (This often leads teachers themselves to revisit 6 ÷ 3.)
Flexibility is key. Consider (−6) ÷ (−3), 6/5 ÷ 3, 6 ÷ 0.3, 0.6 ÷ 3, 6x ÷ 3, 6x ÷ 3x, etc. (Note: division of fractions & integers are high school topics in Western Canada.)
(I’m not saying that dividing by a fraction — or decimal fraction — always means measuring. You can think sharing, which can be challenging. Andrew Stadel’s estimation jams are my favourite examples of this. How long is “All Along the Watchtower”?
Did you see 2/3 in the picture? Did you divide by two, then multiply by three? In other words, did you invert and multiply? What’s the meaning of 2:40 ÷ 2/3 in this context?)
The subtraction set above is interesting. Teachers pick up that the expressions are variations on a theme: five “take away” two. Their pictorial representations tend to show subtraction as removal: “if you have five apples/quarters/x‘s/square root of two’s and I take away two…”
Pictorial representations that show subtraction as comparison (the “difference”!) are less frequent, but maybe more helpful.
Consider (+5) − (−2). To “take away” negative two from positive five means introducing zero pairs whereas the “difference” between positive five and negative two means understanding that positive five is seven greater than negative two.
This second meaning is probably more meaningful in high school. For example, subtraction as removal reduces (1.89t + 15) − (1.49t + 12) to an exercise in collecting like terms whereas subtraction as comparison has students contrasting rates of change (e.g., cost per additional pizza topping) and initial amounts (e.g., the price of a plain cheese pizza). Similarly, if F_1(C) = 9/5C + 32 and F_2(C) = 2C + 30, then (F_2 − F_1)(C) compares conversions given by an estimate and the formula. When solving |x − 5| = 2, it’s more helpful to ask “What numbers differ from five by two?” than think missing minuends in take away problems.
Addition and multiplication — as well as other big ideas needed for high school such as proportional reasoning (or “multiplicative thinking”?) — will be addressed in future posts.
Coke is now sold in 20, not 24, packs!
So to determine the best buy, I couldn’t just double. I use that strategy all the time; it’s my Frank’s RedHot. The exclamation point is there because I think that 20 leads to more strategies than 24. (Some of) these strategies are listed in my 5 Practices monitoring tool below. I’m curious if you think that I have anticipated likely student responses correctly. What incorrect strategy could I have anticipated? I wonder how you’d purposefully sequence these responses during the discussion.
More than SWBAT solve problems using unit rates, I want my students to recognize that there are many ways to solve rate problems and understand that we can easily compare rates with one term the same. This big ideas connects the strategies. In the fourth strategy above, we can think of 24 cans as a unit. Call it a “two-four” (Is that just a Canadian convention?) or a “flat” (Are we cool with calling the Pepsi cube a flat?). In fact, Save-On-Foods wants us to think of 24 as one; we’re encouraged to buy two packs of 12, a composed unit. For this task, I’d prefer that they didn’t, so I went back to the store and found this:
Comparing 20 packs with 15 packs is more likely to lead to common multiples than comparing 20 packs with 24 packs as above. Numbers matter. There’s this, but it doesn’t get us a clear winner:
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:
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”:
- 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.
Recommended: “You Can Always Add. You Can’t Subtract.” Ctd. by Dan Meyer
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.
“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?
Some popular solutions:
If you look just right, you can see two arrays:
A creative solution that involves counting what’s not there:
And moving what is:
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).
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.
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.
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.
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.
What say you?