Negative Sign vs. Subtraction

I had an amazing conversation with my 2^nd hour 8^th graders. They were very confused about why you would subtract the 9 in an inequality like this: 9 – 3x > 42.

When I said you had to look at the sign in front of the 9, not all of them understood. So I showed them that if you add 9, then you have 9 + 9 on the left, which results in 18. But we want 0. So you have to do whatever it takes to get to 0.

Then you bring the -3x down. This is where it got interesting. I said negative three, and they said to me – it’s not negative, that’s a subtraction sign! So I said that I know they think there’s a difference between a subtraction sign and a negative sign, but really they’re the same thing. Think about changing a subtraction sign into “plus a negative.” That’s saying that they’re the same.

One student in particular was tenacious with her questions. She knew exactly what she didn’t understand and she stuck with it. Another seemed stuck in the mindset of “But you didn’t teach this to us” (especially the part about subtracting 9 instead of adding it).

So I put the following problems on the board.

5 + 3x = 24

5 – 3x = 24

3x + 5 = 24

3x – 5 = 24

We talked about the first two and why you subtract the 5.

The one student started to see patterns between them and she tried to say what they were. She said that when there’s addition, you always subtract the number. But then I showed her a new problem where that’s not always true.

-5 + 3x = 24

We also talked about how in the second problem on the board, 5 – 3x = 24, you have to bring the negative with the 3x. I know it looks like a subtraction sign, and it is, but it’s also a negative 3x. They could think of it as changing the subtraction sign to plus a negative, and then voila, it’s negative.

What I loved about this conversation is that it’s a real mind-bender for students. It shows their conceptual understanding maturing from what they learned in elementary school, where a negative is different than subtraction. I told them this – that sometimes learning is confusing. And I had seen them making many mistakes on this and was very glad to be having this conversation.

This conversation worked because they were ready for it. It reminds me of when I tried to teach them about dividing fractions in a conceptual way, rather than the procedure, and they were so confused and didn’t want to hear it. Then later one student was actually curious about it. Why does dividing by a decimal (or a fraction) result in a bigger number? Usually dividing results in a smaller number. His curiosity about this made him ready to talk about it.

I also got to tell them that it doesn’t matter to me at what point they understand this stuff. If they get it yesterday, great; if they get it today, that’s fine too. They can retake if they need to. I don’t want to limit when they can learn it.

Peer Teaching Project

[All documents are linked at the bottom of the page through Scribd]

I am in the midst of a peer teaching lesson on quadrilaterals. I gave students the task of writing and presenting a 20-minute lesson on a quadrilateral. They worked in groups of 3-5, assigned by me. They had to create a number of documents as part of their planning: notes, role division, lesson plan or activity, worksheet with problems and answer key, mini quiz and answer key, and poster. Six groups taught about a quadrilateral: parallelogram, rectangle, rhombus, square, trapezoid, and kite; two groups taught about the relationships between the quadrilaterals by creating a hierarchy and by writing Always, Sometimes, Never True questions.

Below are all of the documents I handed out to students.

This project was a little scary. I don’t know that students have ever been asked to learn something on their own and then teach it to their peers. I chose this chapter for the project because students have been exposed to quadrilaterals for many years now. They basic properties of squares and rectangles are ingrained in their minds, and they can use these properties to extend to other quadrilaterals.

The project generated a TON of work for me. Not only did I have to design the project and how it would play out, with all of the considerations involved and all of the documents; but I had to provide feedback on 30 projects three nights in a row, grade a different mini quiz for each of four classes for five nights in a row, keep track of late and missing assignments and absent students, make copies of worksheets and quizzes for four different classes, and take notes on presentations. I still have to grade them.

One of the decisions I made early on was for students to be graded on their contribution to the project only. I know many students hate group work because of the unequal distribution of work. I don’t want to penalize students for wanting to do well and therefore taking all the work upon themselves. I think this was a good decision, though we’ll see how it works out in the grading.

The quality of the presentations spanned a wide range. Some students have a knack for explaining and some for presenting. Some had no clue how to explain something in front of a class and were unaware of how they presented themselves and the content. I know that students are not trained in teaching, and I did not provide them much training on how to present to a class. To make up for the wide range in quality, I am conducting a day and a half of self-guided review (see below for review). Hopefully this will make up for any gaps in understanding.

I really don’t know what I think of this project. Will I do it again next year? I wonder what students learned, if anything at all. I wonder if they learned more about non-math things, like how they work in a group, what their work ethic is, classroom dynamics, what it’s like to teach in front of their class, their process of getting work done, and public speaking. It’s hard to put my finger on, but I feel like they grew somehow during this process. Or we grew together as a class. Hopefully I will find out more when they fill out a survey about the project.

Changes for next year:

• Provide more structure and requirements for the lesson plan, such as questions they will ask during the lesson, a script, options for teaching methods like using whiteboards or a game.
• Create a checklist that they have to present to me at the end of each class period so I am not running around figuring out who is missing what.
• Create common forms for documents like the lesson plan, role division, worksheet, and mini quiz, so it’s easier to grade and keep track of. Possibly color code each class.
• Grade the documents as they come in so no one loses them before the end of the project and I have less work at the end.
• Do a better job explaining the reason for this project. Figure out the reason.
• Be clearer about the types of questions I want students to be able to do at the end of the lesson.
• Combine the hierarchy and Always, Sometimes, Never topics into one.

Peer Teaching Project Description 

Group Evaluation Form 

Grading Rubric 

 

Standards:

• G.3.1 – Describe, classify, and understand relationships among the quadrilaterals square, rectangle, rhombus, parallelogram, trapezoid, and kite.
• G.3.3 – Find and use measures of sides, perimeters, and areas of quadrilaterals. Relate these measures to each other using formulas.

Transitioning to Abstract Thinking

In 8th grade, we're learning how to solve one and two step equations. The students are fairly good with the solving, though some are still rusty with their adding and subtracting integer skills. As an extension of the one and two step equations solving, we did word problems. They're nothing special; I just pulled them straight from the book. But what I've noticed is that students know how to solve them, but they have trouble seeing how to write the problem as an equation. I don't fault them for this - it's a big leap conceptually. And we haven't even done any work on what equations really mean. My struggle is that I don't know how to help them on their journey from knowing how to answer the problem to knowing how to write the problem as an equation. They're focused on the answer, so the extra step of writing it as an equation seems like a burden.

Here are some examples of their work:

 

I showed a couple of students how to work backwards from their solution to writing an equation. For example, if they know that to get the answer they need to subtract 5 and then divide by 2, I said that the equation will have the opposite - it will have addition and multiplication. That was enough to help a couple of them figure out how to write the equation. I know there's an inverse relationship between the solving and the equation, but I don't know how to teach it without confusing the heck out of some of them. The few that I showed this to, I showed during the quiz, actually. (I know many might disagree, but I often use assessment time as teaching time. I have their full attention then and they get the one-on-one instruction that many of them need.)

Now that I've graded their quizzes, I know they still need work on this. Tomorrow they're going to go through and correct their mistakes, and it might be a good time to show them the relationship between their intuitive steps for solving a problem and the written expression of the equation.

Rorschach Inkblot Test

While studying reflections and reflectional symmetry in 8th grade math, I showed the class images of the Rorschach Inkblot Test. I also showed the test to one of my geometry classes, and they had a blast with it! They were bursting to share their answers. A couple of times students saw the same image without talking about it first. Some of the images they saw:

 

two pigs

two pigs carrying a beetle

Patty Cake explosion

two Asian mean giving high-fives

two men pushing a fat woman down a well

Miley Cyrus looking in the mirror

Two people twerking

 a bat

a moth

 an iguana breathing fire

 two lion fish (in blue)

My First Favorite Proof

I have been struggling with finding good proofs for my geometry classes - ones that aren't obvious and that are within reach of my students' abilities. Then I found this problem on Don Steward's blog.



It fit well with the work we were doing in class about parallel lines and transversals, so I presented it to the students. The directions I gave were:

• Draw two parallel lines (maybe using both sides of your ruler).
• Draw a line that cuts the two parallel lines (called a transversal). Measure the two consecutive interior angles.
• Bisect the angles that the line makes with the two parallel lines. Measure one of each of the bisected angles.

Every student created the same construction, though their transversals all met the parallel lines at different angles.

Then I asked: What is the angle at their intersection? Why? 

I had them think about the problem silently for about 5 minutes. Then we talked. Not many students got it the first time around. I explained it, as well as had other students explain it. Then I gave them the assignment to write 5 sentences explaining why the two bisectors intersect at a right angle. 

The next day, very few people had completed the assignment, presumably because they didn't understand why it happens. Or they didn't know how to communicate it in written word. Or some other reason. So we spent about 15 minutes the second day talking over the problem. I had as many students as I could in that time give an explanation. That way, students could hear explanations from multiple perspectives and different wordings, and they could practice communicating verbally. I have found that just because students understand a problem doesn't mean they can explain it. I tested the waters by asking how many understand it, how many can explain it to someone else, and how many have no clue. There were students in all categories. So I had students keep explaining until they got tired of it or no one else was willing to explain. I also showed them an applet I made in GeoGebra of the problem to make it more dynamic.

This is the very first proof that I felt comfortable presenting my students. I wish I could find more!

Test Taking Environment

Just wanted to share something I've noticed in the classroom. When I turn off a set of lights during a test-taking situation, the room feels calmer. Students seem to prefer a set of lights off when they're taking a test or quiz, and I get the sense that it relaxes them. I originally only did it in one of my eighth grade classes when they asked, but then I started doing it in all of my classes. I ask them if it's okay to turn off a set of lights, and they all say Yes! Some want all the lights out, but I think that would put them to sleep. So I turn a set off, and it instantly feels calmer in there, which is great for taking a test. Of course I wonder if it enables easier cheating, but I don't think it does. I also give multiple versions which I hope prevents a lot of cheating.

I'll let the pictures speak for themselves

 

"Adjacent" angles

 

Camo Day (that's a Ghillie suit)