Kaleb Allinson, with a great end-of-class factoring exercise:
I tell my students that my dice add to 19 and multiply to 88 and ask them to guess my dice. I try to play this at the end of class for a week or two as I have time leading up to factoring. Then when they discover how to factor, this dice guessing skill is very helpful. They always realize what I’ve done and think I’m really tricky.
There are other ways to do this, of course, but the dice randomize the factors and I think that’s important. It says to the student, “Whatever algorithm you’re rolling around in your head right now –Â it’ll work for any whole numbers. The teacher isn’t putting her thumb on the scale. She’s giving you numbers she can’t control.”
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Another extension is to ‘fake the dice results’ and have students determine whether or not there IS a solution to Kaleb’s problem; can they prove they have exhausted all possibilities?
21 Comments
Jared Cosulich
September 7, 2012 - 6:35 am -Why are techniques like this on the periphery of pedagogy? This process of allowing students to figure out the solution to a challenge any way they can is fun for them. Why don’t we use techniques like this through out the curriculum? Have the students “figure out” challenges that get harder and harder and, when necessary or when they ask, make information available that can help them solve the challenges. If you just point out that there is some information that exists that can help them with the challenges without force feeding them the information then they’re very likely to seek out that information on their own. That has to have a positive effect on how effectively they learn and understand the material.
Chris Robinson
September 7, 2012 - 6:58 am -Jared:
“Have the students “figure out” challenges that get harder and
harder and, when necessary or when they ask, make
information available that can help them solve the
challenges.”
I think this goes along with what Dan and others are doing with 3 Act Math. Pose a challenge, and only give students what they need when they ask for it. From my experience, with students that have not had lessons like this, it can be painful at first, but once they get used to the process, they enjoy the challenge and their engagement and learning increase exponentially.
Marshall Thompson
September 7, 2012 - 7:26 am -Hmmm. Might use this. Probably should. Good stuff.
Ian Kerr
September 7, 2012 - 7:52 am -Embedding Algebraic thinking with simple concepts…love it!
I wonder when kids could start playing this? 3rd grade with single digits? “Add to 4 and multiply to 3.” The number sense it provides is awesome.
Mike C
September 7, 2012 - 10:14 am -Love the dice idea.
This same kind of thinking is what is required for Diamond Problems
(here is sample of them
http://www.rocklin.k12.ca.us/staff/cmcnabbnelson/prealgebra/ch05/Diamond%20Problems.pdf).
The factors or addends are at 9 and 3 o’clock. The product is at 12 o’clock and the sum is at 6 o’clock.
Here’s is a simple one all filled out:
45
9 5
14
But you would present them with two of the diamond spaces blank, like
45
14 and ask students to find the left and right numbers.
These are great for differentiating, reviewing fractions, working with positive and negative integers.
Julia Tsygan
September 7, 2012 - 11:24 am -If you also flip a coin, you can make the second factor positive or negative. But maybe that will complicate things too much?
Kaleb
September 7, 2012 - 12:02 pm -Actually, I forgot to mention the negatives. Once they are kicking butt on the regulars I up the dice to more sides and or go back to smaller dice and make one of them negative. I have two different colored dice, so I just make one color negative. I don’t tell them this, but they figure it out pretty quick when my numbers multiply to a negative number. Then I do both as a negative.
timstudiesmath
September 7, 2012 - 12:57 pm -I hope it’s not unnecessary to point out a direct application: factoring trinomials over the integers without blindly guessing, which teachers I’ve met are afraid their students can’t do. Given the polynomial ax^2 + bx + c, call a*c ‘the product of two integers’ and b ‘the sum of two integers. This is the only lifting required for what I’ve seen called the Key Number Method or the Airplane method.
Hopefully, exercises like this allow us to completely remove the dreaded slide method from use. It makes no intuitive sense, does not produce congruent polynomials with which students can check their work or graph at any step, fails if a student can’t remember every step, and does not reinforce traditional algebra skills.
I love any teacher putting “I’m thinking of a number” problems in Algebra; hopefully they are incredibly valuable and intrinsically appealing.
Michael Paul Goldenberg
September 7, 2012 - 2:43 pm -Ian: Robert B. Davis created materials in the late ’50s through at least the mid 1960s as part of the Madison Project (yes, one of the various projects that collectively were called “New Math” and hence have been buried together as “failed,” even though there are certainly a lot of worthwhile things in that mass grave) that were designed originally for middle school students but fortunately wound up being used in lower grades as well. You can download his 1964 book, DISCOVERY IN MATHEMATICS: A TEXT FOR TEACHERS as a free pdf here http://bit.ly/RaJlpw The teacher materials include all the student materials. In chapter 3 (starting on p. 38 of the teacher book, which represent pp. 5 – 6 of the Student Discussion Guide, students are asked to guess a number and substitute the number they choose to see if it will work, for problems like
( __ x __ ) – ( 5 x __ ) + 6 = 0
where those underlines represent squares in the actual text. As the investigation proceeds into this first problem, the second question raised is “Have you discovered the secret? Remember – don’t tell!”
This is asked again as the 7th question (after other examples are looked at.
Q8 is “How many secrets are there?”
In Q13 & 14, Jerry and Marie contend that there are one and two secrets, respectively.
I don’t think this is quite as gripping as using dice, but if you look at the material leading up to this activity, particularly the first 20 pages where Davis gives background on the project and his view of math and education, you might feel as I do that this might have been exciting for kids compared with the “regular” math they were taught (these materials were intended for enrichment, not replacement of the regular curriculum, according to Davis’ introduction).
So, yes, people have explored algebraic thinking regarding quadratic equations with elementary students going back at least as far as when *I* was in elementary school. I personally find it humbling and somewhat sad that interesting work like Davis was doing is mostly unknown or ignored today.
Of course, a lot of the fabulous ideas Dan and others are writing about these days couldn’t have been explored easily or at all 60 years ago due to limitations of technology. But the underlying mathematical ideas? The math in the New Math wasn’t new. Realizing that traditional math instruction was not engaging for a lot of students and that it failed to stimulate the thinking capacity of students about mathematics is not new, either. This is in no way to depreciate the work of today’s forward-thinking math educators, but only to remind us of how easy it has been for the deeply-embedded culture of American math teaching and overall culture of schooling to resist innovative and exciting ideas about math teaching and learning.
The good news is that there are young and not so young teachers using the Internet to share great ideas. If they can keep spreading and implementing and fighting for those ideas, MAYBE we’ll finally get the kinds of makeovers that as Dan has said, math class really needs.
Belinda Thompson
September 8, 2012 - 7:25 am -I really like this idea and agree that it could be used in lower grades. Flexibility in decomposing numbers is quite useful. The opportunity to extend to integers, and to all rational numbers, marks this as a winner in my book.
I’m thinking about how the conversation goes when the ta-dah of “we can use this when we factor trinomials” is revealed. Is it just “remember that game we played?” or is there an algebraic structure conversation? If it’s the latter, what does that look like?
Tom
September 8, 2012 - 8:47 am -This is a great idea that is so easy to implement. It really makes me think of what other simple 3-minute games can I use for other concepts.
What are some games for we can use to “warm-up” for linear functions?
Maybe an “if then” game to get ready for written proofs? Is there a way to pull up random, funny “if” statements and let them finish? Prior to conditionals I read the “If you give a mouse a cookie” series. Maybe read one of those a day in the week leading up.
Lots of options if you are willing/able to sacrifice that few minutes at the end of class.
timstudiesmath
September 8, 2012 - 9:29 am -Hi Belinda,
There is a demonstration here and another one here. It may look intimidating (I worked with teachers who didn’t follow me on it), but the only tools used are Kaleb’s problem and the distributive property.
The method works for all integer polynomials with rational roots. Students practice the distributive property several times and finding greatest common factors after grouping, which are all good skills. It also introduces students to factoring by grouping, which is sometimes taught in Algebra or Precalculus and can occasionally be used to factor third-degree polynomials.
Another extension is to ‘fake the dice results’ and have students determine whether or not there IS a solution to Kaleb’s problem; can they prove they have exhausted all possibilities? Is it faster to satisfy the product first then search for the sum, or vice versa? As a teacher, thinking about Kaleb’s problem allows me to create examples of polynomials with rational roots on the fly.
http://misscalculate.blogspot.com/2011/12/factoring-ax2-bx-c.html
http://patternsinpractice.wordpress.com/2011/04/25/factoring/
Dan Meyer
September 8, 2012 - 9:32 am -timstudiesmath:
Ugh. That’s awesome.
Belinda Thompson
September 8, 2012 - 10:41 am -Thanks for those links, Tim.
I was thinking more along the lines of this:
If I multiply (x+m)(x+n) I get x^2+(m+n)x+mn. (I chose 1 as the coefficient for x for simplicity). When I have to work the other way, do I just notice that the game I’ve been playing shows up and is useful here or is there a way to connect factoring to the puzzle “m+n=b and mn=c, what are m and n?”. What I’m looking for is a connection of the generalization of the game structure to the factoring process. Maybe there isn’t one, and it’s just something that shows up as a result of applying the distributive property.
What would be great is to somehow connect this further back to multiplying multi-digit whole numbers.
Sorry if I sucked the fun out of the activity for everyone :(
Michael Paul Goldenberg
September 8, 2012 - 4:12 pm -From DISCOVERY IN MATHEMATICS (remember this note is for elementary school teachers):
“This chapter. . . returns to the topic of quadratic equations. Every child is not expected to have found the secrets by now. Some people never learn about quadratic equations, and quadratic equations are not among the minimum essentials for productive adulthood — or for promotion into the sixth grade.
We would suggest this attitude toward a student who has not yet found the secret: Quadratic equations are a game. If the students have fun at them (most students do) and if they are good at them (most students are), why, that is very nice. But if the students do not enjoy them and are not good at them, don’t worry. There are lots of other things — linear equations, signed numbers, graphs, identities, derivations, and so on — that they will find exciting and amusing.
Pedagogically, it is desired that the students get *experience* with mathematical material, learn from this experience, and enjoy as much of it as possible.
*We are convinced students do much better work when they are kept away from too much pressure.* They pull us along after them: we don’t push them. . . .
It is a mistake to expect that we will ever *know* exactly what should be taught in grade 4, or in grade 5, and so on. The teacher who is not surprised by his students’ ability is probably not observing them carefully (or sympathetically) enough.” (p. 67)
Those last two paragraphs make an interesting contrast to the current pressure-cooker philosophy that serves as educational policy in this country.
mr bombastic
September 8, 2012 - 5:04 pm -I like Kaleb’s activity as … an activity.
I find a lot of the examples of factoring methods really distrurbing. Factoring by grouping, airplane method, slide method are all “magic” as far as most kids are concerned. The reason the methods work is almost never explained, and it is very unlikely the kids would be able to follow an explanation anyway.
Being able to factor a messy polynomial isn’t really that important, but it is important to understand that polynomials can be written in different forms. The guess and check method is easy to understand and reinforces that polynomials can be written in different forms.
Michael Paul Goldenberg
September 8, 2012 - 5:41 pm -I’ve never heard of the airplane or slide methods, unless under other names.
I was always dissatisfied with doing trial-and-error for quadratic equations where the leading coefficient wasn’t equal to one, and enjoyed learning how to attack those when I taught an Algebra I class for the first time (the book called it “the Master Product Method,” which sounds pretty heady, but I’ve not bumped into that name since first seeing it in the late ’80s).
That said, I agree with mr bombastic, but if students are exposed to an approach that demystifies the alleged magic, and if they have been able to see that one of the things that goes on a lot in algebra is “doing-undoing,” then maybe they’ll learn some things that are useful. Certainly a “black box” approach isn’t something I’d advocate.
But also, it shouldn’t be surprising that it’s easier to multiply polynomials together (if you’re systematic and careful) than to factor. After all, we know from cryptography that it’s a heck of a lot easier to multiply two large primes together than to factor their product without knowing either of the original prime factors.
Michael P
September 9, 2012 - 6:37 am -I’m not crazy enamored with building factoring trinomials on the foundation of sum/product problems. I worry that it allows students to skip the hardest, but most important part, of factoring, which is seeing it as the inverse process of multiplying binomials. True, if you understand the relationship between multiplication and factoring at a deep level then you’ll have access to the sum/product intuition, but the converse isn’t true.
What I did last year (and I liked) was I just gave kids a set of factoring problems and told them to keep their eyes out for patterns. The changes from one problem to the next were subtle (e.g. change the value of c; change the sign) and they picked up the sum/product patterns on their own.
At the same time, I do love the activity in this post. I see myself using it in more or less the way that you describe, except that I don’t see myself explicitly making the connection between sum/product and factoring.
Karl Mason
September 9, 2012 - 7:43 am -Obviously the decentralised extension is a pair of dice for each pair of students, starting at the 6-sided and going up to the beast of 20 sided. You could even make a negative dice if you’d like, in fact you could make fractions dice, decimal dice, anyways, yes, allow the pupils to take control of their own ‘game’ then ask them if each set of numbers is fair.
Make it 3 dice, 4 dice, so much extension actually.
With 3 dice are there any problems which have more than one solution?
Jerrod B.
November 1, 2012 - 7:08 pm -Mike C,
I used the diamond problems that you spoke of when I taught Algebra 1 several years ago. I remember not mentioning anything about factoring and merely telling the students that we were going to spend time solving some math puzzles. When the students though of it as a game, all of the students were engaged and it turned out to be a “race” of sorts to see who could solve the puzzles the quickest. In fact, kids would take great pride in being able to figure out the difficult ones. So whether it’s the idea of dice, diamonds, or whatever else, it’s good to show students any form of application of mathematics in various contexts.
Thanks for sharing!