Oh linear functions–please don’t be boring!


This month I was going to change the world, or at least change the climate in my 2 math 8 classes. It’s been miserable lately. I took responsibility for that and decided to make major changes in class as the second semester began. I changed the daily organization, the look of the room and most importantly, the lessons. Of course, I wanted everything to become perfect all at once, but that didn’t happen. Here’s how change began lesson-wise.

Two weeks ago I started by using a hundreds board and chips to revisit linear equations. We looked for patterns using 5 transparent plastic chips placed in the shape of a plus sign on the numbers on a hundreds board. We added the numbers under the chips; we looked at how we could get the greatest sum in a given pattern, the smallest sum, etc. We moved the plus sign around. We added by decomposing numbers. It was good.


(I modified an old Illuminations lesson that I found in an old notebook and I forgot it at school. Sorry. I’ll copy and edit into here Monday or so.) The lesson was good. I was happier, the kids were happier. Students were noticing patterns. We made a table of ordered pairs by using the middle number in the plus sign as the input and the sum of all five squares as the output. Students graphed individually. Then I gave each student a smiley face sticker to place on a big graph locating one of the ordered pairs they found. The goal was for students to see that even though they each did the activity alone, their points all made a pattern that graphed as a straight line. Since some of the points weren’t too close to the straight line, we also were able to notice parts of a pattern that don’t come out the way we expect. This was a time for self-correction. By giving each student the exact same type sticker, there was no way to single out a particular kid as being the one who didn’t get it. This activity allowed us to work with linear equations –again– without the students realizing it.

The next day we extended the 100s board activity onto a calendar. We did some calendar math using four by four grids. We saw patterns in diagonal sums, four corner sums, and center four sums. This was then extended to the abstract as I had students choose any day in the four by four region to be n. All other days were then defined in terms of that n by adding or subtracting.

2016-02-06 15.53.29

(My students struggle with turning concrete math into abstract math so this was a good exercise.) Sums were then found again. Setting these sums equal to one another or setting a sum equal to the original concrete sum yielded the original date on which the n was placed. I preferred this to choosing the first box as n and simply adding to get subsequent dates in terms of n. Students who persevered were delighted when the solved equation yielded the original n value. It was cool that n was different for various students, but n stilled tied to the original n-date the student picked. That was much more fun when n wasn’t 1! When the students would each get their respective n values, I said to them individually, “where have you seen that number before?” Their eyes lit up when they realized it was their original n value. Any opportunity to link the concrete to the abstract is a win.

Students then worked with squares with only one number and completed the rest of the squares relative to the given square’s value according to patterns that would occur on a calendar.Screenshot 2016-02-06 16.32.17

For the final assessment activity, I had the students solve missing squares for a calendar that was from the fictional planet Crayon…they only have 5 days in a week on Crayon! (I didn’t even steal that part from anywhere, btw!)


That was probably the best two consecutive days if school so far this year in math 8.

I have tear-stained pages of lessons that I tried this week using geo boards to create linear equations and simple systems of equations. That seemed like a great idea. I extended that to equations with one solution, no solution, and infinitely many solutions. The students weren’t nearly as impressed as I was, so I’ll try to polish that up and post geo board-graphing at a later date.


Beyond “Why?”

sbv kid pic
Annoying Kid

I was that annoying kid who always asked questions. Constantly. I especially asked “why” when I was told to do or not do something. If I received an answer that didn’t make sense or that I didn’t agree with, I fell short of compliance. Actually, that still holds today—a point that makes both my husband and principal twitch at times. But that’s a different kind of “why”. A teacher “why” needs to engender deeper thought. A teacher “why” needs to inspire, but that word is not all that inspirational.

In one of my last graduate classes (more than ten years ago) I read an article entitled, “Beyond ‘I like the way you…’.” It was about giving students sincere, positive, specific feedback. When I think about questioning, I envision an article entitled “Beyond, ‘Why?’” that needs to be written. “Tell me more” does no more than “why,” sooooo, I have made a list of some questions and comments I try to make to students as a place to start.

  • How can you check that?
  • Is there another way?
  • Try to find a counter example.
  • Discuss your method with your shoulder partner.
  • Make the abstract concrete and see what you notice.
  • Listen in on the conversation that group is having over there.
  • Think about what you did yesterday. What are the similarities and differences here?

To help improve my why-type questioning, I am going to try to keep a tally of my questioning next week and focus on replacing “why” with a better response. For each “why”, I will put one tally. For each why-replacement, I will put a star. My students deserve better so I am going to work toward being a better why-replacer.


Favorite Things 2016

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Favorite Things of this Math Teacher

Domains and ranges and intercepts and zeros,
students that don’t lose odd negatives are heroes,
linear functions each graphed with green strings, (I prefer spaghetti but it didn’t rhyme)
these are a few of my favorite things.

New sharpened pencils and eager new faces,
field trips and class trips and visiting new places,
Garfield’s proof and the ah-hah it brings, (Seriously!)
these are a few of my favorite things.

Expanding binomials with gigantic powers,
cheering on students at sports after hours, (LOVE!!)
functions that model bizarre plats and dings,
these are a few of my favorite things.

When the office calls,
when email stings, (man, do I have a few of those!)
when I’m feeling sad,
I simply remember my favorite things,
I teach math and I’m so glad.


Sorry to do this to you. I couldn’t help myself. Just don’t tell my family! They say “life is not a musical,” but I choose not to believe.

I really thought about 1)seating charts, 2)step function grading, or 3)fun math songs that lead to no real learning, but this came out of my fingers.


One good thing a day

For my first official blog entry for the 2016 MTBOS Blogging Initiative, I’m taking a tour through my week by way of ‘one good thing a day.‘

Monday—I finally got through the re-tests for rational functions that students took the day before Christmas break. I know, that’s not so good given the date is January 11, but that’s not thing. Here it is: I actually met one-on-one with each student that retook the test and reviewed it with them. That was good and needs to happen more.

Tuesday—I had the best idea today and I grabbed my curriculum facilitator (CF) to sell it to her. There is this empty classroom next to mine. There are four walls with two huge white-boards. The other two walls are prime real-estate for more white-boards. There are a couple of tables, chairs and desks in the room, but it is mostly empty except for those glorious walls. [I installed 360 degrees of white boards in my room earlier this year,

360 degrees of White Boards!

but they don’t wipe clean as easily as I had hoped.] Students can get from my room to the proposed white-board room, math annex, think tank, construction zone, melding pot or whatever I decide to call it and can be up working and thinking in less than one minute. My CF is sold and is helping me find additional white-boards that are not being used. Next stop, getting my principal’s blessing. I really just want to go to the hardware store though.

Wednesday—After school a former student comes to visit. Wonderful person! I asked former student (whose first AND last name I remembered on the spot) whom she had for math this year at the high school. She told me she had Miss D, a first year teacher. That made me so happy because Miss D was one of my favorite Algebra 1 students my very first year teaching. She too is a wonderful person and now she teaches math at the high school right next door to my school!

Thursday—Math II students were working on constructions as we had done most of the week. I love constructions, except for constructing parallel lines. That one always messes me up! Anyway, a student in the back of the room excitedly calls me over to his desk. He says he’s got it. He shows me a set of circles and two lines that he believes are parallel. I look at it and then I look at it again and ask him some questions about what he had done and why he thinks it works. He replies, “let me think about it.” Before class is over he tells me sadly, “never mind, it didn’t work after all.” The good thing is that he tried and tried again and he learned from his efforts. He was critical of his own work. He may not have succeeded in coming up with a new way of creating parallel lines yet, but he put himself out there and that is a very good thing indeed.

Friday—Attendance at my school wanes of Fridays. This coming Monday is a day of service and remembrance of MLK in our state so I figured many kids would be out starting the long weekend early. Not true. Attendance was very good. Actually, I think it was the best day all week. That’s a good way to end a week!

That’s my ‘one good thing a day’ for this week. Thanks for stopping by. Oh, if you have any suggestions for naming the new white-board room, please let me know. I’m thinking simply The BoardRoom.

Talkin’ bout a Resolution


Photo on 8-6-15 at 10.25 PM
Sara B. Vaughn

I, Sara B. Vaughn, resolve to blog in 2016 in order to open my classroom up and share my thoughts with other teachers. I hope to accomplish this goal by participating in the January Blogging Initiation hosted by Explore MTBoS.

You, too, may join in on this exciting adventure. All you have to do is dust off your blog and get ready for the first prompt to arrive January 10th! Whether or not this is your year to expand your learning community by participating in this initiative, please follow along and give me feedback as I give it a whack. Perhaps you will want to get on board in the near future.

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