Geometry – First Semester Review Worksheet

I know there are some people out there who could use this.  Especially as finals near.  It covers parallel lines cut by a transversal, vertical angles, complementary / supplementary, rhombus, special right triangles, polygon interior angles, and transformations.

This was created by my colleague Dave Casey – who is definitely at the top of my “I wish they had a blog, were on Twitter, and were presenting at conferences” list.      This is just one of his worksheets – you should see his activities, amazing stuff.

The Goods (aka: the pdf)

1st sem review chart

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.

Rotations Worksheet

Some days you just need a worksheet.  So here’s one.  I took all the problems from math-drills and just spaced them out better.  I love using matrix notation so always have my students do it.

The Goods: (aka:  The pdf’s)

RotationsWS

VNPS & VRG Tango!

So Kristen and I are taking dance classes – so any three step process feels like a tango to me.  This activity has nothing to do with dancing (sorry).  This is one way that I use Vertical Non-Permanent Surfaces (VNPS) and Visible Random Groupings (VRG) for problems where we are working on our procedural fluency – whether it being solving equations with fractions, or factoring polynomials.

One down side of VNPS is that students believe they are finished when the correct answer reaches the whiteboard, rather than when everyone in the group understands how to do the problem.  The one student who knows how to solve it just goes up to the board and solves it.  There are a couple methods to help give students less places to hide in a group – here’s one of those methods:

Pick three problems that are similar.  Tell students that there are going to be three rounds.  In the 1st round they will be in groups of 3 (randomly assigned of course).  Then in the 2nd round they will be in groups of 2 (randomly assigned again of course), and lastly they will be back in their seats working alone on paper.  That’s it.  Throw whatever standard you want at it.  I believe it works because the thought of eventually being alone gives them the extra motivation to learn from their group.

Here are the three problems I started with in my geometry class, where my goal was to give them practice multiplying polynomials, solving by factoring, and using pythagorean theorem:

From what I’ve seen – the knowledge that they are eventually going to be alone makes those students who usually look for a place to hide in a group more apt to contribute and learn from the first two rounds.

I love to pose the question by waiting until they are in their groups and standing at whiteboards, then I pose the question on my whiteboard in the center of the room.   It’s a nice math coach moment versus math teacher moment.

Here’s a smattering of whiteboard activity from this day:

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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.

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:

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…