# How to work with polynomials in difficult classes?

I am currently teaching a very difficult class, with young people (16 years old) who are deeply unmotivated and restless. I should work with them with polynomials, but I'm having a troublesome time; maybe some game-based activity would be an idea. Do you have some ideas to share?

EDIT:

I am currently teaching in a professional school in Italy. On the average, there are around 20-25 pupils per class, with a 20% (i.e. at least 4 per class) of them having learning disabilities and at least one per class with moderate-to-severe cognitive problems.

I am substituting their teacher for a month, and we are working with simple manipulations of polynomials i.e. adding similar terms and multiplying them toghether. I have the idea that a practical activity would help them find more interest in the process, however I myself am out of imagination on how to apply polynomials to real-life scenarios (apart from simple areas and perimeters of shapes). I'd be glad if you could point me to some game or playful setting that involves simple polynomial manipulation.

EDIT N.2:

Given the two interesting answers below (up to today Feb 3 2016), I feel I should clarify further the situation I'm working in:

1) We don't have access to computers, so only "old school" methods are possible;

2) Student level is extremely basic: they have a lot of difficulties in calculating, for instance, $(x+1)^2$. Some of the students don't yet apply correctly the basics of algebra such as distributive property; others still make errors such as assuming that $x^2=2x$.

• @marcotrevi: You've definitely improved the question with your added information, like being focused on adding and multiplying. Keep in mind that those basic skills are preparatory to solving equations and work in calculus, so it may be unlikely to see any natural (non-busy-work) applications of those basic manipulations. If you're new to this field, and only in for a month, then your best bet is likely to follow the textbook as closely as possible. – Daniel R. Collins Jan 28 '16 at 16:20
• Try "Algebra Tiles". – Amir Asghari Jan 30 '16 at 18:53
• (A link to CPM's Algebra Tiles.) – Benjamin Dickman Jan 31 '16 at 2:03
• This may not help, but fireworks or other projectiles can be fun to simulate: MESE posting on fireworks & water fountains; MO posting on fireworks. – Joseph O'Rourke Jan 31 '16 at 2:20
• You could have them start with the sequence of perfect squares and then analyze the difference between two consecutive ones. Create a table of values and then plot the values with $y=x^2$. You could then talk about shifting it up or down ($y=x^2+c$); then left and right ($y=(x+k)^2$) and extend from there. – Andrew Sanfratello Feb 3 '16 at 21:45

There is a great game about polynomials presented by Rachel Kaplove on the eHow YouTube channel. The rules are very simple:

• Each student gets a card with either expanded polynomial, for example $x^2-81$, or one of its factors, for example $x-9$ and $x+9$.
• If a student has a polynomial, he/she needs to find classmates with the right factors.
• If a student has a factor, he/she needs to find a classmate with the right polynomial.

You don't need computers for this game and students should work in groups to achieve the final goal. But, of course, students need to know the factorization of polynomials for this activity.

Regarding the lack of motivation, I would consider three reasons:

1. Students don't find algebra useful because the tasks are not related to real-world problems.
2. Students aren't excited about algebra because the tasks lack the gaming aspect.
3. Students aren't good with algebra because they don't understand its basic principles.

These issues can be solved with the following approaches:

1. Polynomials in economics (coming a nice example from agriculture!)
2. Playing with maths using computers (there is no excuse not to use them)
3. Didactic vs Project-based learning (explaining the concept of powers)

Now let's cover each point in detail:

1. Polynomials are extremely important in economics to model economical behaviors. Since the author of the question teaches in the agriculture-oriented school, I found the book "Agricultural economics" and the word "polynomial" was mentioned there 21 times! For example, a short-run total cost function of modern production technologies can be realistically described by a third-degree polynomial like this: $TC = 2q^3 - 15q^2 + 50q +50$ (taken from "Maths for Economics (3rd edition)" by Geoff Renshaw, p. 141). The graph of this function shows the most efficient output range (green) where the total cost $TC$ doesn't vary much with the output $q$ as well as the least efficient output range (red) where output increase would cause significant production cost increase. 1. Instead of doing boring calculations by hand, let the students play with maths using computers and programming. Check this inspiring TED talk by Conrad Wolfram about Computer-Based Maths! There are no excuses not to use computers for two reasons:

2. Maths is not fun when you don't know how to use it. Explain basic concepts to students (didactic learning) and then let them explore the topic themselves so they could really understand it (project-based learning). Let's take, for example, the concept of powers. There is some simple code for the Wolfram Programming Lab that can be executed to explain the basics:

3*3
3*3*3
(* Now, how to calculate 3 multiplied by itself 100 times? *)
3^100


After students understand the definition of power, let them explore the following questions:

• How much is 3^0? What about 2^0? Why are both expressions equal to 1? Is it valid for any number x^0? If you want them to tackle really an open question, ask about 0^0.
• How much is the number "googol"? Where can you find it in daily life?

Ideally, students should work in groups of 3-4 people using one computer per group. After their research on the above questions, each group should present their results to the whole class. This method is called Self-Organized Learning Environment (SOLE) invented by Sugata Mitra.

That's a lot of stuff for one answer but I hope that some of the ideas will help you. And don't forget to accept my post as an answer if you liked it ;)

• I guess the student's lack of motivation is due to a mix of all three aspects. Moreover, since this is an agriculture-oriented school, most of the students look forward to practical activities and only a small percentage are drawn to more theorical aspects and reasonings. – marco trevi Feb 4 '16 at 19:31
• This game seems interesting but it involves students moving around the classroom...I am not sure if it's a good idea, since they are already very noisy. – marco trevi Feb 4 '16 at 19:33
• @marcotrevi Maths can be very practical - I will try to find some good examples. Maybe even from agriculture... ;) Don't be afraid of the noisy classroom. This type of education promotes collaborative skills as well as makes the whole process more fun. Check this discussion video about the future of maths education curated by Aoibhinn Ní Shúilleabháin. – Pavlo Fesenko Feb 4 '16 at 20:10
• I don't dislike noisy classrooms either, but I find that in some cases it's important to help some students develop basic social-behavioral rules, since the appear to lack. I'm in this school only since a week, but there are already serious discipline problems (fights, bullying, etc). I have to think about the issue more deeply. – marco trevi Feb 4 '16 at 20:43
• As for approach 2 to issue 2....an excuse might be not having a computer? :) – marco trevi Feb 5 '16 at 21:16

For multiplying polynomials and combining like terms, this website presents a simple level-appropriate tabular method.

For example, to multiply the polynomials $x - 2$ and $2x^2 -3x + 1$ construct the following table: Then fill the table with the products, like the multiplication tables they learned in primary school: Like terms will be located along diagonals (not including the terms of the original polynomials), so students can add them easily.

To gamify this technique, you might distribute a number of versions of worksheets with fourth or fifth degree polynomials to multiply, with the empty tables already constructed and a few lines under each table to add the like terms and write the product. Complete all of the worksheets ahead of time, then cut each table box into an individual piece and deposit all of the pieces into a container. You can pick pieces out one at a time and have students stamp or color the corresponding box on their worksheet to make a BINGO type game.

• Thank you for your contribution, but these students are very, very basic. I'll edit the question to clarify. – marco trevi Feb 3 '16 at 16:53
• Had to downvote because it doesn't address the OP's goal for "simple manipulations of polynomials i.e. adding similar terms and multiplying them together". – Daniel R. Collins Feb 3 '16 at 17:14
• @marcotrevi Is plotting curves by plotting points and connecting-the-dots coloring and counting squares not basic enough? – Andrew Feb 3 '16 at 21:00
• @DanielR.Collins Great, thanks, I will delete that comment. Please review the edited answer. – Andrew Feb 3 '16 at 21:28
• @marcotrevi I appreciate your feedback. Please review the edited answer. – Andrew Feb 3 '16 at 21:30

Well, you could start by explaining what $x^2$ means geometrically.

If I had a square with side length $x$, then the area would be $x^2$. Explain to them how this is completely different from $2x$. Picture often help students.

How to explain to them that $(x+1)^2=x^2+2x+1$? Once again, I would go with a visual: Pictures are often better than numbers, as I have found. Also, using words instead of numbers helps:

$x+2x=3x\to$One apple plus 2 apples equals 3 apples.

So this is how you could apply polynomials into geometry. Furthermore, if you feel confident, volume becomes a $3$rd degree polynomial.

I'm not quite sure how you could gamify this, and I'm not that great with gamifying math in general.

Also, the use of colors helps students understand why which numbers go where, as opposed to strict math tables like the other answers.

As an idea for a game, you could make a rectangle with some numbers (where $x$ should've been) and have each area shaded a different color. Then ask your students what each color represents. Ask them which colors have similar $x$'s, like reducing $x$ and $2x$ into a single area of $3x$, or something along those lines.

EDIT

I noticed you could add on different shapes together to get different polynomials. This should also help out with their algebra I would think.

What would be cool is if you had a lot of squares and rectangles, so you could take away pieces or add on pieces as you wished.

• Volume should be a polynomial if 3rd degree, I think? – Daniel R. Collins Feb 4 '16 at 3:41
• It will have 4 terms, but I agree, it will be a 3rd degree polynomial. – Opal E Feb 4 '16 at 19:09
• @DanielR.Collins Yes, Volume would be a polynomial of the 3rd degree. – Simply Beautiful Art Feb 4 '16 at 21:41

Have you looked into Desmos for possible resources?

In particular: How about the Marbleslides activities?

I'm not sure whether this satisfies the criteria in your case (specifically: I do not know if you have access to the requisite technology) but the activity (it seems to me) can be modified RE:

I'd be glad if you could point me to some game or playful setting that involves simple polynomial manipulation.

It came up in a reddit post (OP) that linked to this site about a month ago.

You can find more on Dan Meyer's blog dy/dan in the post: Marbleslides is here.

Here is a gif directly from that blog post: 