Factoring Trinomials is so easy it's child's play...
Factoring trinomials the easy way. As you read this page, bear in mind that all the pictures depict work done by a 7 year old. The child who did this work was very much more impressed that he could make 8's with a single stroke instead of two circles and draw "some epic twos" than he was with factoring trinomials. He also wanted to make sure I got the crabs in the shot. "Those are some nice crabs." Factoring polynomials: taken for granted. How cool is that?
Algebra really is child's play. When you have the proper tools abstract concepts become concrete and we can cross teach math concepts via what were formerly abstract algebraic expressions.
For the 7 year old this lesson was about addition, and multiplication as much (or more) than it was about factoring trinomials. Also check out this blog post and click the link for videos showing very young students being taught math via algebra.
This was just part of a one hour session that included lessons on place value, number identification, fractions, building addends, and practice with writing numbers neatly.
These four problems are in order of increasing degree of difficulty, and are used to teach addition, addends and multiplication. By allowing for trial and error a lot of math gets done even though there are only four problems here.
The last one for example,
x2 + 11x + 28
took quite a while to solve as he tried many combinations to try and get 28. On his way to solving this he figured out addends for 11 and various products for the addends he was making (2)(9) = 18 and 2 + 9 = 11 but the problem asks for 11x and 28.
He also had to figure out how many more x he needed in the first place; since the problem before it had 9x...he had to get out 2 more x. Simple for you and me but required just a bit of thinking for a 7 year old. So there is one addend of 11 right off the bat: 9 + 2. Then he tried 5 and 6...which was easy because he can count by 5's...but harder when I asked him (just for fun) to count by sixes. It is also still some what amazing to the mind of a 7 year old that 6 times five really is the same as five times six. After several more tries he got to 7's. Then he got out four 7's but was quite unsure how many four sevens was.
So he lined them up and measured with tens...three tens minus two...28!!
"Got it!" he exclaims excitedly.
I also got him to add by using the "wants to be a ten" method. 7 wants to be a ten, takes the three out of the next 7 which makes one 10 and 4, 14. Add another seven. The 4 wants to be a ten so it takes the six out of the seven and we get two tens and 1 which is 21, then 21 and another 7 is pretty easy 1 plus 7 is 8 for 28...
"But it's just as fast to measure," he says. Great, we also discover that 7 is contained in 30, 4 times with 2 left over. "Measuring" is very much the same as that operation we call "division".
Lots of math being discovered here without tiresome worksheets or drills. We will do these problems several times over, until they are easy and don't require much trial and error at all. Trial and error being an expression because you may note the method makes it so that the child is never WRONG just getting more information.
Here we see the first attempt at the first problem
x2 + 7x + 12
He has gotten the right number of x, but find he can't make it work with two sixes...he does note that it would work if he had
x2 + 8x + 12
but, alas, he only has 7x to work with. "Problems" like these allow for a lot of discovery.
x2 + 7x + 12
Next he trades the 2 sixes for 4 threes...and finds that three is contained in 12, 4 times and that 2 threes are indeed 6 (and that 2 sixes are 12)...this will give him more flexibility when making his rectangle which is how he is factoring trinomials so easily.
Can you see how we get all learning styles involved and therefore more of his brain.
x2 + 7x + 12 = (x + 3)(x + 4)
Counting the sides is actually the easy part when you are 7. Factoring trinomials can be a way to increase confidence build self esteem and remove fear in students who have had poor math experiences in the past. This particular student knows no other way and will never know or even believe that students find factoring trinomials hard.
Now all he has to do is make the symbols match the blocks and make a quick drawing.
This part of the process engages more understanding, the same way drawing a picture of the story helps with understanding in English.
It doesn't matter which comes first ordinarily, although some teachers like descending order when factoring trinomials, what is important is the across and up mirror whatever is built in physical reality. In this case it's (x + 3) across (even though that symbol doesn't look like a 3, it is upon closer inspection) and (x + 4) up.
Drawing them is a crucial part of the process and method, it bridges the gap between the concrete, that is, the base 10 blocks with the semi-abstract (drawing) on the way to the completely abstract where we only use symbols.
Quite quickly he goes through the problems. (Factoring trinomials is no problem with this method though...tee hee.)
They are self correcting, that is he knows he has them right when he shapes the blocks into a rectangle, he is learning multiplication and because he does not know his facts yet he is slower to build the problem. He is motivated because when he is done he will get free time to draw.
Kids of all ages love to draw on white boards. This simple motivation will help him to remember and learn his multiplication (and addition) facts. Because next time he will be able to go faster and therefore get more free time to draw.
All done...on the white board. Now he has to record them n his notebook. He didn't put them in as he went along so now he has the symbols but no pictures. We erased the drawings as we were finished with each problem. Factoring trinomials with the blocks is not as difficult as drawing and writing. He used the symbols to help him remember what the drawing was supposed to look like. This is hugely important: the symbols represent physical reality.
My favorite part about this portion of the lesson was when he said, "hey, look what I can do!" And began drawing 8's. To the child this was much more impressive than the fact that he could build some silly rectangles and count the sides. He will never know that algebra is supposed to be hard, nor will he know the fear it can invoke in some of my older students. It's child's play.
And then he got free time and as you can see at the very top he drew a pirate ship and then he drew two turtles, three fish and four crabs. "Hmmm, 2 plus 3 plus 4...9! I drew 9 sea creatures."
On the one hand I am amused with the arguments the teachers in California are putting forth against introducing algebra into the earlier grades. Believe me, I understand: most of them don't remember it and if they do remember it they didn't learn it the easy way, and trying to impart this knowledge on unruly second graders would be no easy task...on the other hand I am bemused because this method isn't new. Factoring trinomials to teach concepts like addition and multiplication is wholly foreign to their way of thinking. Just like it is yours.
Above is a page out of the students notebook from a different day. It shows some of the same problems. The red checks in this case mean CORRECT. The first one isn't checked because that's the example that I did. The student was very excited to get them all correct and wanted additional confirmation...note how the pictures begin to incorporate the symbols, because the symbols really are faster to make than the drawings. The blending of the concrete and abstract leads to a deeper understanding of the concepts and demystifies algebra for the students. This generalized patterning helps them with factoring trinomials of any kind positive or negative. We will eventually cover what to do with negative coefficients and how to use the blocks to represent these expressions.
Also looking forward to the calculus, when we are playing with "complex" concepts like derivatives going from something like
(x + h)2
x2 + 2xh + h2
and back again is duck soup because they've done so much, and because they know what it looks like they don't forget it as easily as if they only had symbols to work with. Imagine trying to speak a language where all you get is the symbols CASA but never get to see the house.
(x + h)2 = x2 + 2xh + h2
Some teachers of the "higher" mathematics may recognize a piece of the the first derivative when f(x) = x2.
Which brings us to another point that I will touch on only briefly: how does a teacher prepare a student for calculus or the "higher" mathematics when they have no idea what the "higher" mathematics will entail? Home schoolers often get this objection too: how does a parent teach a kid calc when they don't know calc. With great difficulty. My first calc teacher was a nice guy but hadn't a clue, because like most people who took calc in college he couldn't remember much of it, much less answer my pointed questions that began with WHY. Soon this site (soon being a relative term) will have instruction on this part of the mathematics too: it's all one language. Hence the quotation marks around the word "higher"
This factoring trinomials page is under construction more links need to be added, more descriptions...etc but for now...good enough. Updated Oct 2010.
Go to BLOG post showing factoring binomials with 4 and 5 year olds.
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“The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.” ~Aristotle
We used to think that if we knew one, we knew two, because one and one are two. We are finding that we must learn a great deal more about 'and'.Sir Arthur Eddington (1882 - 1944), The Harvest of a Quiet Eye (A. L. Mackay), 1977