I forgot to post about the last lesson - I gave students a quiz on radicals and complex numbers, and I also taught them how to factor trinomials in the form ax^2 + bx + c when the gcf = 1. I graded the quizzes over the weekend, and they weren't that great, so I decided to spend today reviewing with them.

So after reviewing the homework carefully, we spent the rest of the class doing an activity called "Showdown". I got this from another teacher, and I'm not sure where he got it from. When managed properly, I find that this activity works extremely well for review. Students always want to know if it's a competition, and it's not - it's a collaborative activity. Though competitions are fun and have their place, they often lead to one student in the group doing all the work so that the team can get the highest score. This activity helps students communicate with each other and lets them go at their own pace. I made a set of 16 problem cards (with answers on the back) for each group, which I'll post on ILoveMath. Here is the overhead of the rules I made for the class:

The individual time gives students a chance to access their own knowledge. The group time gives students a structured way to collaborate, and it also prevents them from asking me for help before they have thoroughly discussed a problem. I am freed up to be able to walk around and see what's happening, listen to their discussions, and point things out to them when needed. I've done this activity in the past where it did not work exactly as it's supposed to (i.e. students start collaborating right away, or they look at the answer right away), but my class today did an outstanding job with it.Showdown Setup:

- Each group selects a team captain who will set the pace of work.
- Each student needs a handful of answer slips (there are 16 problems in total).
- Each group needs a team answer sheet.
Process:

- The captain shows problem 1. All team members work individually and silently on the problem. Write answers on answer slips, turning them upside-down when done (or totally stuck).
- When the captain sees all slips are upside-down, he/she calls “Showdown”. Group members show and compare their answers, explain their work, and come up with a group answer. The captain writes the group answer on the team answer sheet.
- The captain turns the problem card over and compares the given answer to the team’s answer. If the answers are different, discuss. At this point, you can ask me for help if you can’t figure out how to get the answer.
- Move on to the next problem and repeat.

## 6 comments:

How do you teach them how to factor trinomials in the form ax^2 + bx + c when the gcf = 1 ?

In the next week or so I will post about the factoring unit we've written (and used for 5 years).

In short, we factor things like

10ab - 15b + 6ac - 9

a whole lot, first, then we move to "breaking the middle"

What do you use? What have you tried and rejected?

Jonathan

n Algebra 1, we don't really deal with that situation much (the book we have has some crazy scheme to list lots of factor pairs and guess and check stuff); we hold off for Algebra 2.

I taught Algebra 2 for the first time last year (I plan the honors class, and another teacher plans the regular class - we use a shared prep model at our school). He came up with a "magic square" technique, where you find the factors of

acthat sum tob, and then use those as the "cornerstones" of the magic square, and then do some other mystical stuff.So I had to use that in my regular classes, and I agreed to try it in honors also. It was a good experiment, but it did not work out so well. Students didn't understand why it worked and couldn't use it consistently. And then, when I taught "completing the square" in honors, they kept mixing up the terms and I heard nightmarish phrases like "oh, don't we have to use the completing the magic square thingy?"

This year, we decided to do the more traditional factoring by grouping, which I believe is exactly what you are referring to. It seems to be going pretty well so far, but I think my students still need more practice at it.

I also wanted to get them started on factoring by grouping now, so later in the semester they can handle factoring expressions like x^3+5x^2-9x-45 into (x^2-9)(x+5) into (x-3)(x+3)(x+5).

I think we use a version of the magicky square thing. Example:

34x^2 + 13xy - 15y^2

We need to break 13xy into the sum of two terms. Their sum is 13 (I just said that), their product will be (-15)(34) = -510

Search:

(14)(-1) = -14

(17)(-4) = -68

(23)(-10) = -230

(30)(-17) = -510

Break the Middle term:

34x^2 -17xy + 30xy - 15y^2

Factor by grouping

17x(2x - y) + 15y(2x - y)

(2x-y)(17x+15y)

Why does it work?

(ax + by)(cx + dy) =

acx^2 + (ad + bc)xy + bdy^2

So ad + bc will break the middle and allow us to factor by grouping. But ad + bc is the middle coefficient and (ad)(bc) = (ac)(bd) = the product of the first and last coefficient.

Since all the math teachers in my school are on the same page, we reinforce this nicely

Jonathan

When I teach this I do something similar to the "magic square" but with less mysticism involved.

Consider: 2x^2 + 7x + 5 We take the 2 and multiply it to the 5, and then pretend the number infront of the x^2 is a 1 when finding factors of (the new) c that add (or subtract) to be b. However, when we create 2 binomials the 2x^2 gets broken into 2x and 2x. This is clearly wrong as 2x * 2x = 4x^2, but since we broke the rules in our first step we need to undo it. We can do this by dividing one of the binomials by what we multiplied in the start. So...

2x^2 + 7x + 5 becomes

2x^2 + 7x + 10 and factors of 10 that add to be 7 are 2 and 5 so

(2x + 2)(2x + 5) But then we divide the first binomial by 2 and get

(x + 1)(2x + 5)

Sometimes you have like 6x^2 in the start and will have to divide one binomial by 2 and the other by 3 to get a total of 6.

What is the captain doing while everyone else is writing?

The captain is doing the work as well. Their additional job is to pay attention to when the others seem to be finished or stuck, and to handle the cards. But it is not assumed that they are better at the skills than the other students in the group.

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