Perfect Squares and Perfect Cubes

My algebra 2 students are starting their unit on radicals.  This unit includes a review of square root arithmetic, the quadratic formula, higher order roots and fractional exponents.  It's kind of a dry unit.  I was trying to think of a way to spice it up and also, I was thinking about how in years past, my high school students have not been able to recognize perfect squares or perfect cubes at all because elementary schools around here don't emphasize the multiplication tables.  Then I remembered a comment on a blog (I think it was f(t) but I can't find the actual comment) from a guy who said he got students to recognize perfect squares by coding the alphabet with perfect squares from 1^2 through 26^2 and putting messages up on his board in code.  I thought this was a pretty awesome idea, so I went through and coded messages for my students for each day of the unit.  Because they're in algebra 2 though, I went beyond perfect squares and made more interesting codes.  I'm just awarding candy to those who solve the codes, but I'm also thinking about making it a competition.  The person who decodes the most messages correctly gets some prize.  I did my first one yesterday and my second today and so far, my students are really into it.  I thought I'd share the coded messages here for anyone who was interested.  I got all the messages by googling around for math jokes or math quotes.

Day 1:

“169-1-400-64  81-361  324-1-16-81-9-1-144”

Day 2:

81-400  81-361  196-225-400  400-64-25  100-225-4  225-36  169-1-400-64-25-169-1-400-81-9-1-196-361  400-225  16-225

9-225-324-324-25-9-400  1-324-81-400-64-169-25-400-81-9.   81-400  81-361  400-64-25  100-225-4  225-36  4-1-196-121 

1-9-9-225-441-196-400-1-196-400-361

Day 3:

676  196-676-49-361-484-196-676-49-324-676-169 324-64 676 529-484-25-324-576-484 441-144-81  49-36-81-169-324-169-400  576-144-441-441-484-484  324-169-49-144  49-361-484-144-81-484-196-64

Day 4:

676-289-64-361-49-144-16-361-64-4  64-324  1-16-64-169-36  676-1-121-16  361-196  4-196-400-169-361  400-225  361-196  361-484-16-169-361-576  484-64-361-49-196-400-361  361-676-100-64-169-36  196-25-25  576-196-400-289  324-49-196-16-324

Day 5:

4-144-1-9-121  64-225-144-25-361  324-25-361-441-144-400  36-324-225-169  49-225-16  16-81-484-81-16-81-196-49  400-64-25  441-196-81-484-25-324-361-25  4-625  676-25-324-225

Day 6:

441-324-25-484  144-36-49  144-441  441-144-36-81  121-484-144-121-225-484  361-676-25-484  49-81-144-36-625-225-484  16-324-49-361  441-81-676-576-49-324-144-169-64

Day 7:

3375-1728-64  2197-1-8000-512-125-2197-1-8000-729-27-729-1-2744-6859  2477-125-10648-125-5832  64-729-125  8000-512-125-15625  1000-9261-6859-8000  1728-3375-6859-125  6859-3375-2197-125  3375-216  8000-512-125-729-5832  216-9261-2744-27-8000-729-3375-2744-6859

Day 8:

8191-2-2187-256-32-8192-2-2187-512-8-729  512-729  27-243-9-19683-512-3-128  2187-256-32  8192-9-729-2187  9-4-19683-512-9-6561-729  2187-256-512-3-128  512-3  2197-256-32  4096-32-2-729-2187  9-4-19683-512-9-3561-729  59049-2-531441

Day 9:

3  1296-7776-3-7776-256-7776-256-27-256-3-625  27-3-625  64-3-49-243  64-256-1296  64-243-3-81  256-625  3-625  3125-49-243-625  3-625-81  64-256-1296  4-243-243-7776  256-625  256-27-243  3-625-81  64-243  343-256-25-25  1296-3-16807   7776-64-3-7776 3125-625  3-49-243-216-3-16-243  64-243  4-243-243-25-1296   4-256-625-243

I hope my students learn to recognize perfect squares and perfect cubes after doing these codes.  At the very least they're fun and they don't take away any class time because I'll pass them out on slips of paper at the end of class and students will have to decode in their own time.

If you want an answer key, just leave me a comment.

Quadratic Functions Performance Assessment

My algebra 2 students just turned in a performance assessment over graphing quadratic functions and it turned out really nicely, so I thought I would share it.

First, I asked students to graph several functions I'd put together.  When graphed they form a man with wings.  I then asked them to make their own picture out of functions.  They then traded papers and tried to graph each other's pictures.  Finally, they colored in their pictures and we had an art show.

This project helped them reinforce function transformation rules, especially vertical dilation and some students even created their own functions that we hadn't studied to help them create their picture.  (One student taught himself how to flip an absolute value graph on its side and shift it, then another student had to figure out what he'd done to graph it.)   The best part of the project was when a student thanked me for having fun homework. And of course today was a lot of fun when we got to hang up their pictures, admire them, vote on the best one and eat candy.  The whole project took two and a half periods and I think a lot of learning got done.

Here's my original hand out:

*Oops, there's a typo.  Function 11 should be p(x)=x^2+(y-5)^2.  The squared should be on the outside of the parentheses.  I throw these sheets together too quickly...
  Quadratic Functions Performance Assessment Here are pictures of the students admiring each other's work and voting on the best piece of art:

Here are the pictures they created:

I do have more than 7 students, but two were absent today (I know, 9 students is still an awesome student teacher ratio.)
Here's the third place winner

Here's second place:

And here's first place

Polynomial Division?

I'm trying to put together lesson plans for my Algebra 2 students' unit on polynomials and rational expressions.  The curriculum we're using doesn't have any lessons on factoring polynomials beyond quadratics, but our state standards do require factoring higher order polynomials.  I'm kind of torn as to how to teach this because I can't think of very many compelling reasons students would want to factor higher order polynomials without spending a lot more time on this topic than I want to.  I feel like it's a pretty useful skill for students to have- I know at least my calculus student has encountered polynomial division a few times this year, but I just feel a little tepid on the topic.  I don't know how important the broader math world regards this topic.  Is it worth spending more than a day on in Algebra 2?  Is there any way to make it fun and or applicable?  I've been searching around and I can't find anything at all that other teachers have put together other than drill and kill worksheets.  Anyway, here are two worksheets I've thrown together to help students factor higher order polynomials.  I think they should provide two relatively straight forward lessons albeit a little boring.  I would love any advice on spicing this topic up, or maybe compelling reasons to either go more in depth into the topic or to drop it all together.

Factoring With Pascal's Triangle Polynomial Division Exploratory Ws

One other note: Maybe this is something novices do, but I am a little torn about font choice.  I know in the larger scheme of things font matters so little, but I've heard so many people making comments about Comic Sans MS being unprofessional yet my students love it.  My first year teaching I was just playing with fonts, switching back and forth and not really caring, but my algebra 1 students who have had terrible experiences with math latched onto the assignments I printed up in Comic Sans as being friendlier and easier to understand.  They said it made math less scary.  Since then I've been using exclusively Comic Sans.  Another educator told me to stop using it because it was unprofessional, but when I polled all my students, they insisted that I stick to Comic Sans font.  Notice I did decide to switch fonts for the second worksheet.  I tried to find something that was equally friendly but didn't have the same stigma.  Sadly it's tiny.  What a silly thing to obsess over yet shouldn't I listen to my students?

Complex Fractions in Algebra 2

I just came up with this warm up to spice up my lesson on simplifying complex fractions:


Suppose a mother and daughter decide to make a pie. 

(a)   A pecan pie to be specific.  The slave over it all day, and they are so engrossed by their pie baking that they forget to eat.  By the time the pie is ready, they’re both starving.  Suppose the mom eats twice as much as her daughter.  They’re both so hungry that they eat the whole pie.  There is 1 pie to be divided by 1 adult appetite and the little girl has ½ of an adult appetite. 


      (1)     Write a fraction that represents this situation
      (2)       How much of the pie will the mom get?  How much of the pie will the little girl get?

(b)  On another day, the mother daughter pair make an apple pie.  The father LOVES apple pie.  He comes home and shares in the eating with his wife and daughter.  So now there’s 1 pie, 2 adult appetites and a ½ an adult appetite.


   (1)       Write a fraction that represents this situation
   (2)       How much of the pie will the mom get?  How much will the dad get?  How much will the little girl get?

(c)   The mom and the dad have decided to have another baby.  This time, it’s a baby boy.  The baby boy also loves pie, but because he’s growing, the fraction of the pie that he can eat keeps changing, so let’s just let the fraction that he can eat be represented by 1/x.  The mom and the daughter again go back to pie baking and one day, they make a blueberry pie.  All four family members split the pie to match their appetites.


        (1)      Write a fraction that represents this situation
        (2)        Write an expression for how many pieces they’ll need to cut the pie into.

I like this warm up in that I think it shows pretty clearly why we would want to simplify complex fractions, but I'm thinking that it's going to be pretty time consuming, and won't even begin to address how to simplify complex fractions.  I so badly want to show students the why of math, but the how of math is all we have time for and that's what they're tested on.  


Note: I know this problem is a little sexist.  I thought I could insert students actual names when it came to teaching this so that it's not so blatantly- mom and daughter bake while father and son eat.

Vertical Dilation

My algebra 2 students are learning about various function transformations and I was trying to think of a way to make this topic a little more engaging.  My students are all wild about silly bands (I'm not sure why... They're still pretty excited about the fact that the silly bands always maintain their shape no matter how stretched they become) so I thought it would be pretty fun to see if we could explore vertical dilation through silly bands.  This led to kind of a neat little warm up, but I think it was maybe a little more prep than the educational content was worth.

I cut out squares of cardboard and taped graph paper to them.  Then I let the students choose their own silly bands out of a selection:

The students had a lot of fun picking out their silly bands at least.  The robot shaped ones were the most popular.

I then had them put their silly bands against the graph paper and draw a coordinate axis.  They were told to plot at least 10 points that their silly bands went through.  I asked them to double all their y-values and plot the points on their cardboard graph paper with thumb tacks.  They then tried to stretch their silly bands over the tacks to see how the shapes of their rubber bands changed:

This was a cup cake shaped silly band.

It was a nice visual because in the past, I've had students get confused because a parabola that's been stretched vertically by a factor of 2 looks skinnier, so students thought that it must have been multiplied by 1/2 instead of 2.  This activity I think showed them that multiplying by a number bigger than 1 made the object taller and skinnier, while a multiplication by 1/2 would make it shorter and fatter.

It was kind of anti-climactic however when we just launched right into normal classwork after this activity.  I would love to find a way to pull this idea into a larger lesson plan rather than just a warm up problem.