I'm Bon Crowder and the photos above are both of me - in 1989 and today. I'm a Generation X mom of Generation Z kids.

I began peer tutoring in high school in 1984. MathFour.com is the 2015 version of me helping peers be comfortable in math.

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Tag Archives: factoring

Fall Factor Trees!

Turn a dry factor tree into a beautiful fall math craft project! from MathFour.comSeems I’m on a craft roll with this seasonal math stuff.

I found this great fall tree craft and I thought, “Trees? How about factor trees?”

So I got out the supplies and started playing.

Here’s the fun fall factor tree activity I came up with:


  • Scissors
  • Glue
  • 4 Pieces of construction paper in fall leaf colors
  • 1 Piece of brown construction paper
  • 1 Piece of neutral paper
  • Dark crayon or marker

Step 1: Create a composite number.

Choose 4 different prime factors to use. Use a few of each and multiply them together.

I picked 2, 3, 5 and 7. After multiplying, here’s what I got:

2*2*2 * 3*3 * 5*5*5*5 * 7*7= 2,205,000

Keep in mind that each pair of 2*5 makes a zero at the end of your number. In retrospect, I might have created a number with no 2’s or no 5’s so as to avoid this.

Step 2: Factor naturally.

Factor the number like you would naturally. This will be how you construct leaves and branches.

Note: if you have a class do this, each student may do it differently, which’ll make the trees much more fun!

Here’s mine:

Turn a dry factor tree into a beautiful fall math craft project! from MathFour.com

Step 3: Make some factor leaves.

Cut leaf shapes from the leaf colored construction paper.

Use a marker or dark crayon to make each leaf color a different prime factor.

Turn a dry factor tree into a beautiful fall math craft project! from MathFour.com

Step 4: Make the trunk and branches.

Cut a wide piece of brown construction paper for the trunk. Cut many thin strips of brown to make all the branches.

Write all the composite numbers from your factoring on the branches.

Step 5: Arrange your tree.

Arrange your tree. You may have to play with it to get it to fit.

At the end of each branch there should be either

  • Two more branches (two composite numbers),
  • Two leaves (two prime numbers), or
  • A branch and a leaf (a composite and a prime).

Step 6: Glue and discuss.

If you had your class do this, have students compare their trees. How many of them did it exactly the same? What kinds of variations did they have?


Post and link your pictures in the comments, tweet them out or share them on Pinterest. And if you know someone who would enjoy this, pick up the phone and call them!

4 Responses to Fall Factor Trees!

  1. I’ve printed out just a picture of the tree for each student. I am going to ask my students to figure out what is going on. Once they do, they will each make one for the hall BB. I will include a reference to your site with a blurb for teachers to give it a look. Wonderful idea. Thanks.

    • So smart, Beverley!

      Sometimes it’s better to let students figure it out than to give them the step-by-step. In fact I’m often preaching this, so it’s weird I didn’t think of it!

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10 Responses to Factoring Polynomials – FREE Worksheet

  1. Oo, pick me, pick me!

    There’s a quick way to find the missing magic numbers — find the factors of (ac) that add up to (b) — so for 6x^2 + 65x + 50, you’d look at 300; the possibilities are 1 and 300, 2 and 150, 3 and 100, 4 and 75, 5 and 60 (which works), 6 and 50, 10 and 30, 12 and 25, or 15 and 20.

    It’s a little bit more work up front, but it saves you doing several grids.

    • I think that may be the point, Colin. That method totally works for you (I’m guessing) – but it sends me into panic mode.

      My major prof in grad school trained me to “get my hands dirty” (he’d tell me that ALL THE TIME). So now I write every single detail out.

      And for our students – whatever works for them, should be what they use.

      Thanks for stopping by!

      • I had the same issue with many of my students trying to find the factor with a leading coefficient greater than 1. I found the umbrella method works great. Youtube it and it works every time. The kids with the most trouble like this one the best because they can’t get it wrong if combined with the box method.

        • Thanks, Charlie. I found it here: http://youtu.be/FxTiogyhwfc?t=48s I find the details the guy uses a little cumbersome. It took me a while to see that his “find the common factor” on the box method was fancy math words for just figuring out what to put on the left and top.

          However, I see lots of value in making an umbrella on top of the trinomial to help students focus on the important pieces to put in the box!

          Thanks so much for sharing!

  2. Wow! So happy I found this! I’ve been teaching elementary and middle school for about 12 years. Now I’ve added high school math into the mix. Just recently acquired 4 Pre-cal students which I’m enjoying teaching! The challenge is mostly the varied math backgrounds of the students! Factoring polynomials will be our first review/re-teach workshop!

    Thanks so much and wish me luck!

  3. This is a nightmare for algebra 2 teachers when 8th grade and algebra 1 teachers teach kids this method. I’ve taught several student groups multiple years in a row, and the students I taught this or the “split the middle method” did not remember how to factor the next year, but those who I taught the old fashioned “guess and check” method remembered from year to year.

    • I can totally see that. Once students learn to get their hands dirty, they can always go back and dig around to find the answer.

      Thanks for stopping in to join the conversation!

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4 Responses to Polynomial Factoring Practice – with YouTube!

  1. Yes. This lesson is hot! lol And, it will definitely resonate with students and the rhythm that’s playing in their head any way. The use of popular music and combination of math facts/patterns is genius.

    I tell my students K-college that math is all about patterns. Look, compare, and share.

    Keep these great posts coming BON!

    **I learned the Dougie during our Back to School Skating Party… (very similar to the dance they are doing). Kids these days. We must meet them where they are in order to reach them.

    • Thanks for stopping by and commenting, Toni!

      The Dougie – awesome! I remember doing the pony! (Of course that was because it was making a comeback in the 80s.)

      “Look, compare, and share.” <- that’s good!

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4 Responses to How to Use Prime Factoring to Win Friends and Influence People

  1. This is how I habitually multiply and divide, except I worked out how to do this in the fourth grade and call it “cancelling out” in my head, because we didn’t learn about prime numbers or factors until middle school.

    I remember doing it on a test for my IEP and utterly confusing the tester, who outright told me to stop and do it the stupid way – I mean “the way I was taught”. Which was stupid.

    • How wonderful, Uly, that you were able to hold on to your own method. We should encourage kids to develop arithmetic for themselves, as you did, instead of forcing down their throats “the stupid way.”

      Some kids will develop for themselves the way that is taught in the class, or they will want someone to show them so they can follow along. And they should be given the freedom to take what the teacher offers, or create their own method, as you did.

      Just because a child confuses a teacher or tester, does not give that grown-up the right to change the child’s methods.

      Thanks for sharing your story, Uly!

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5 Responses to 2 Ways to Factor Numbers into Primes

  1. I prefer the upside down division way, because you can use it to factor more than one number at once, giving you the GCF and the LCM…the GCF is whatever you can pull out of both, (the stuff on the left side) and the LCM is the GCF multiplied by whatever is leftover at the bottom (the stuff at the left multiplied by the stuff at the bottom–it makes an L shape too).
    You can also use it to simplify fractions. Your simplified fraction is whatever is leftover at the bottom.

      • Maybe this:
        Euclid’s division lemma, states that for any two positive integers ‘a’ and ‘b’ we can find two whole numbers ‘q’ and ‘r’ such that

        Euclid’s division lemma can be used to:
        Find the highest common factor of any two positive integers and to show the common properties of numbers.
        Finding H.C.F using Euclid’s division lemma:
        Suppose, we have two positive integers ‘a’ and ‘b’ such that ‘a’ is greater than ‘b’. Apply Euclid’s division lemma to the given integers ‘a’ and ‘b’ to find two whole numbers ‘q’ and ‘r’ such that, ‘a’ is equal to ‘b’ multiplied by ‘q’ plus ‘r’.

        Check the value of ‘r’. If ‘r’ is equal to zero then ‘b’ is the HCF of the given numbers. If ‘r’ is not equal to zero, apply Euclid’s division lemma to the new divisor ‘b’ and remainder ‘r’. Continue this process till the remainder ‘r’ becomes zero. The value of the divisor ‘b’ in that case is the HCF of the two given numbers.
        Another cool equality is that the product of any 2 numbers is equal to the product of their HCF by their SCM
        I al;so tell my students that you don’t REALLY need the SCM to add fraction. You can do it using the product of the denominators as the common denominator. The result just will not always be the simplest fraction.

  2. I too got a cool trick of “5”
    multiply by 5 means add a zero at end and half it
    Ex- 42×5= 420/2=210
    if can’t half it, just get the previous no. to half and add 5 at end
    ex- 93×5= 930/2=920/2 5= 465

    • Vinayak – nice one!

      You can also do this with multiplying by 5: Add two zeros and divide by 4. (Or do your trick twice!)

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2 Responses to How to Use the Quadratic Formula to Never Factor Polynomials Again

  1. The three examples you use are actually all easily factorisable. What about something like 9x^2 +12x – 1 which is not easily factorised?

    Anyway, why does this formula work
    (I know but does your student)?

    • Thanks, M, for your comment.

      The point of the tip is that you can factor by using the quadratic formula. So that no matter what – you don’t even have to try to factor.

      Using the quadratic formula on 9x^2 + 12x – 1 would be totally fine. Again, if you default to the quad formula, then if they came out as nice numbers, you could show it as factors. If they came out as messy numbers, then you could just finalize your problem by announcing your solution (or zeros, or whatever you were looking for with it).

      Now for the question of why. It depends on the age/maturity level of your kids. And depends on if they are top down learners or bottom up learners.

      You can show them this and then they can investigate the why. Or if they need to be shown the why, you can derive the quad formula and show the comparison with the zeros on the graph.

      This is a tool. It won’t be the best tool for all students, but for some it will be just the thing they need.

      Thanks, M, for your comment and question!

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