Tag Archives: algebra

Monomial Partners

This is a great activity that was inspired by Matt Vaudrey’s Equation Speed Dating.  In this lesson each student gets to create their own monomial – which I constrained to having to be even and with a variable.  Then they break up their paper into three columns:  Partner / Our Binomial / Our Rectangle.  The students pick a partner and join each others monomials together to create “Our Binomial”.  Then they factor their binomial and represent it as a rectangle by labeling it’s dimensions and indicating the area.  I circulate the room and once it appears every group is finished, I have everyone get up and find a new partner.  I’m demanding here that all students get up out of their seats and move somewhere new.

After a couple rounds I started having them draw their monomial and their partners monomial as separate rectangles, and then draw them together.

I have been focusing on a geometric approach to factoring, so the rectangle column was a great addition to previous times when I have done this activity but only asked for the solution.

The column “Our Binomial” does a nice job reinforcing that a binomial is the combination of two monomials.

Don’t require them to say “what’s your monomial?”, “do you agree that our binomials is….  “, but inspire them to say it by modeling it.  A lot of my students were saying it because I was giving them messages that anytime they get the chance to say “monomial” or “binomial” they need to take it.

Tell the students not to move onto a new partner when they are finished.  They need to wait until you tell them to switch partners.

Remind them that you are really counting on the partners to catch any errors! Because you can’t do the problems on the board since every pair is working a different problem.  “And yes, you are the partner I am counting on for someone else.”

“What’s your name, what’s your monomial?”   No that’s not a pickup line for Speed Dating…  or is it?

Explaining “Explain”

Here is a released question from Smarter Balanced (I even answered it!!!):

Ok I lied.   That was an edited version of a Smarter Balanced question – here’s the original:

Now all of a sudden my answer doesn’t seem sufficient anymore 😦   Here’s my best guess at a popular student answer:

This word “explain” is keeping me up at night lately.  In this problem I’m not sure adding the word explain to the end gains us enough to warrant it.  To achieve Common Core we can’t just throw the word “explain” after every problem we did last year and call it a day.  By the way I’m not saying that’s what the Smarter Balanced Consortium did on this particular problem.  But this use of the word “explain” does bring two things to mind:

2.  If we ask students to explain something – it should be something worth explaining.

With respect to #1 – my focus this year has been on explanations through multiple representations.  Basically I have students make connections between diagrams, tables, graphs, mathematical symbols, and written descriptions.  I feel underwhelmed asking students to explain with just a typed explanation.  I want explanations to look like this:

In the student work above – image if it was only the conclusion.  Look at how much would be lost.

There are certainly better answers to the rectangle problem from Smarter Balanced than I offered up here.  I actually really like the problem itself, I just do not think having them explain it gains us much versus just solving it.

It’s hard to explain the word explain.  It’s a word that only makes sense to me until I try to explain it.

T-Block Visual Pattern

I created this visual pattern as a followup to the I Rule! exercise from MVP.  It is intended to be more difficult than I Rule!.  When I gave this to my students, I included a linear T-Block just like MVP does for I Rule!

1.  How many squares are in the 10th sequence

2.  How many squares are in the nth sequence

Only a couple of my students actually got to the right answer, but the effort was tremendous.  I had students coming to me during lunch and saying they had asked all their friends and they couldn’t figure it out.  Students were telling me they worked with their parents and couldn’t get it.  I had a student (who failed first semester mind you) tell me that her and her two math tutors stayed 45 minutes after their session working on it and couldn’t figure it out.  She had two pages of work.  I have a couple students who get 100% on everything they touch, and they didn’t figure it out.  So yay me!  I challenged them 🙂

Here’s the I Rule! pattern:

Check out many more visual patterns at visualpatterns.org – a site created and curated by my conference buddy Fawn Nguyen (@fawnpnguyen)

The Goods:

Here is the worksheet I used, not sure if I will include the linear T next year.

TBlockWithLinear

The Handshake Problem

The Overview:

I had a lot of fun with the Handshake Problem this year, so I figured I would write about it.  My goal was to use less structure (meaning no worksheet – especially one with pre-staged t-tables).

I revealed the question in three parts, each time raising the number of people shaking hands:

– 5 people go to a party and shake every one’s hand once.  How many handshakes are there?

– If everyone in the class shakes everyone else’s hand, how many handshakes would that be?

– If everyone in the school shakes everyone else’s hand, how many handshakes would that be?

Lastly, in a rather last second “I want them to process this more” moment,  I had them create solution guides for it.

The Description:

I began with the following two questions as warmup problems:

1.  5 people go to a party and shake every one’s hand once.  How many handshakes are there?

2.  If a 6th person shows up to the party, how many handshakes will they give?

After the warmup I raised the bar a little bit by increasing the amount of people who shake hands:

“If everyone in the class shakes everyone else’s hand, how many hand shakes would that be?”

The class problem raises the bar a little, but still leaves the door open for the students to add up each individual scenario.  For example, they noticed that the 31st student would shake 30 hands, and the the 30th student in the room would shake 29 hands and so forth.  So the final answer would be 30 + 29 + … + 2 + 1.  No equation needed, no additional math tools needed.  But I still took this moment to show them a new math tool that would have made there job easier – the summation!  I pulled up the Wolfram Alpha Summation calculator and let them know that a math symbol will do all that addition for you.

Then I asked them – How many handshakes would there be if everyone in the school shook eachothers hand?  Now they are dealing with a numbers that is far too big for them to simply do the summation on their own.  But luckily they had this new math tool I just gave them.  I kept that summation calculator up on my computer for them to use.  Pattern found – now execute.  And some of them literally ran to my computer to find the new sum.  Now I can ask them whatever the hell I want – If everyone on earth shook each others hand?  The size of the number is irrelevant now!  They got this.

Finding the above pattern was the most popular solution method.  But other students created a t-table and looked for a pattern to model with an equation.  They pretty quickly noticed that the equation that describes this situation had to be quadratic because we’ve looked at quadratic patterns before.  From there they made things fit and discovered the equation:  (x^2-x)/2.  I told the students that they just had to make the numbers fit.  Which was fine for the students who are good at creating equations like that.  They know what they want the function to equal, they know it’s quadratic – go to work.  The rest of the class was not amused.  And that’s when a student walked up to the whiteboard and amazed all of us.

Jose came up to the board and said, “we know that it is quadratic from the t-table.  So let’s assume there are 3 people at the party.  3 squared would be 9 handshakes, which accounts each person shaking the other two peoples hands, and their own hand.”

“But they can’t shake their own hands, so we have to get rid of those three handshakes”

“Here we are still double counting each handshake.  So then we must divide by 2 in order to only count each handshake once.”

Oh my God Oh my God Oh my God!!!  That was soo excellent!  I had never thought about the problem like that!

And with this equation in hand, when I raised the bar to how many handshakes there would be if every student in the school shook hands, they saw how quickly they could answer it by plugging in the school’s population.

I had students just work in their notebooks, but afterwards I had them formalize their work and create a solution guide.  I will probably write about these solutions guides next – but for the time being, here are some nice ones:

The Reflection:

I am most excited about the idea of these solution guides.  It was kind of a last minute idea I threw together, but turned out kind of gold for me.  Ended up doing a gallery walk and have great classroom talks about what THEY liked and disliked about each others guides.  I”m looking forward to see the quality of the next round of them!  Bring on the “Guess What I Heard?” problem!!!

Real World Math: Project Manager

When I first started this blog the idea was to categorize assignments based on a series of twitter style hashtags, which would ultimately allow a teacher to quantify how differentiated their lessons had been – in a macro sense at least.  I have not really stuck to that idea, but one of the original hashtags I had was #industry.  The purpose of #industry was pretty simple – someone had to do this in their job.

I was hoping that #industry would end up as a collection of problems that come directly from people’s  work experiences.  These experiences would be served to student’s unedited from the workplace to the classroom.  Inherent in this hashtag would be the answer to the question “why?” because presumably anybody doing something for their job would have a clear reason as to why they were doing it. (presumably?)

So here’s the problem:  My fiance is a project manager and she had one site that was 3/4 an acre and a price from that site for \$54,000 for some work. Then she had another site that was 2.5 acres and needed to know how much that same work would be for the larger site.  That was the first thing she needed to calculate, but she ended up just wanting to know how much 1 acre was worth, so she could scale it to all her other jobs.

Here’s an error analysis angle to this question –  To scale the cost for 1 acre she had initially multiplied 54,000 by 1.25 and she was genuinely curious about why that did not work.  Hhhhmmmmm…