Category Archives: algebra

Standard Form Equations With An Open Middle

#CMCN14 was lights out good this year.  Amongst the many things I learned new – were a ton of reminders of things that I used to think about but had let slip.  One of those things was the importance of an open middle, where students have a defined beginning and ending, but how they get there is largely up to them.   During Dan Meyer’s talk he challenged us to find an open middle in the routine, procedural fluency building exercises students get.  Most of the great problems have it – but it is a nice tool for tipping the scale for our procedural problems towards a deeper understanding.

Here’s the typical – pretty much closed middle – version of a problem about standard form:

Find the slope, y-intercept, and x-intercept of the following equation in standard form:  3x – 4y = 20

Here’s my one up

Write the equation of a line in standard form where the both intercepts are integers, and the slope is a fraction.

We could really be here all day playing with these

Write the equation of a line in standard form where the x-intercept is a fraction, the y-intercept is 7, and the slope is a negative fraction.

We can even get at MP3

Explain why it is not possible for the slope and x-intercept of a line to be an integer, but the y-intercept a fraction.

Lastly – the Asilomar conference grounds are so amazingly beautiful.  Each tree, slightly beaten from the ocean breeze, stand in stillness as perfect landmarks to perseverance.  And as the sun begins to set, and that air begins to cool, and those stars begin to show – it’s hard to believe that it’s all just the backdrop to a professional development experience.  It’s humbling to be there – I mean you’re walking from presentation to presentation with a program booklet offering the intellects and energies of 200 amazing educators.   But you only get to pick 5…  good luck with that.

Advertisements

Visual Patterns and VNPS & VRG!

I feel like I’ve been preaching the gospel of vertical non-permanent surfaces and visible random groups everywhere I go these days.  The norm is set in my room – I pose the problem, give them a couple minutes of silent thought, put them in groups, and away they go.

Below is a pattern I made up quickly one morning.  It doesn’t look exciting – but guess what?  It’s doesn’t have to be.  It was close enough to full class engagement for me, which was due to a nice combination of:

1.  They believed they could do it.

2.  Vertical non-permanent surfaces and visible random groupings.

3.  Probably some other things I can’t quite pin down yet.

IMG_2157

I’ve settled on these as my go-to questions for visual patterns.  I know I got the sketch the 10th idea from Fawn’s blog.  I never used to have them do that but when I started requiring it I was impressed with how helpful it was for a lot of my students when they ultimately wrote the equation.

1.  Sketch the 10th

(helps them immensely when writing the equation)

(sketches aren’t exact drawings.  I tell them I should be able to have them sketch the 1,000,000th)

2. How many blocks are in the 49th?

(too big for a table!  For students struggling to write an equation, having them sketch the 49th usually gets them to get it)

3. How many blocks are in the nth?

(I start the year asking it this way:  “Write an equation that relates the step number to the number of blocks in that step.  (another way to ask this question is:  How many blocks are in the nth step?)”

I would literally have that parenthesis in each problem, until I finally got to drop it.)

4. What is the largest step I could build with 1000 blocks?

The first extension.  My true goal here is the equation in #3.

5. How much of the sequence could I build with 1000 blocks?

 A second extension.  It’s quadratic and I haven’t directly covered quadratics, so it will challenge those kids.  We have talked about Gauss addition so it is not completely out of their range.

On the whiteboards below you will see graphs because in this particular case I also asked them to graph the number of blocks per step, and the total number of blocks needed to build the entire sequence per step. I wanted them to have to graph something non-linear.  I think it helps further highlight what makes things linear when they work with things that aren’t.

They don’t go directly to the whiteboards.  I first give them about 5 minutes to develop their own thoughts in quiet.  Then I group them and they do their thing.

After class I always look at every whiteboard and judge how much of the conclusions are in their writing vs my writing.  I’m not sure what I gain from that but it is a research point for me right now.  There is a little bit of my writing on boards 7 and 5, but they are supplementary thoughts and not the main thinking that I wanted to the students to do.  Here are some of the whiteboards after the activity:

IMG_2163

IMG_2164

IMG_2165

IMG_2171

Lastly, after it was finished I had them go back to that paper with their initial thoughts and complete the problem on paper. I give them graph paper and rulers and have them make nice graphs to turn into me.  In some sense, one could think of the paper as the assignment as the whiteboard as a giant scaffolding.  But in another sense the whiteboards could be the assignment, and the paper is something that goes in the notes.  Or in another sense…

Monomial Partners

“What’s your name, what’s your monomial?”

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.

IMG_1868

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.

The Advice

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!!!):

ExplainingExplain2

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

ExplainingExplain1Now all of a sudden my answer doesn’t seem sufficient anymore 😦   Here’s my best guess at a popular student answer:

ExplainingExplain3

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:

1.  It’s hard to explain your mathematical reasoning without access to drawing diagrams.

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:

SGBridge

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!  

TBlock

I asked them two questions:

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:

IBlockMVP

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