First, state the definitions to students in student friendly language. (source: http://mathworld.wolfram.com/ )

• A point lives in the 0 dimension. It has no size and no shape. It only has a location.
  
• A line lives in 1 dimension. It has no thickness. It extends infinitely in both directions.
  
• A line segment is a finite portion of an infinite line. It has two ending points.
  
• A ray is a half-infinite or "half-line" with a starting point and no ending point.

Next, have students watch the video, The Dot and The Line: A Romance in Lower Mathematics based on the book by Norton Juster. This is an award winning cartoon from Chuck Jones which was completely hand drawn! Tell students that while watching the cartoon, they must determine if the dot is really a point and if the line is really a line. They must use the definitions just presented in class and support their reasoning.

Here's what some of my students came up with for Question 1:
1) According to the mathematical definition of a point, can the dot in the cartoon be called a point? Support your answer.

• No, the dot had a shape.
  
• No, the dot had a shape and it kept moving, not in one location.
  
• Yes, because it was on the line.
  
• Yes, it was a point because it had no dimension.
  
• No, because it had a shape.
  
• No, because it wasn't named.
  
• No, because the dot had dimensions.
  
• No, because points don't move.
  
• No, because a point has no shape or size.
  
• No, because there is no way to know how big or wide it is.
  
• Yes, because points can be anywhere.
  
• No, because it moves all the time.
  
• No, because it didn't belong to a graph.
  
• No, because it is not fixed in one spot.
  

I love this! Every student used some sort of knowledge about the point to support their reasoning. The response "Yes, because it was on a line." was very thought-provoking for me...because that is a way to describe a point. I thought it interesting that many students connected a point with being stationary because the definition includes that it describes a location. I think a follow up question to this can now be: Can a point move? Why or why not?

Question 2 asks:
Is the line depicted in the cartoon actually a line? Justify your reasoning.

Here's what I got:

• No, because it turned into other shapes.
  
• No, it wasn't always a line, sometimes it curved.
  
• No, when it was different shapes it was in different dimensions.
  
• Yes, it is just a line that formed into different shapes.
  
• Yes, even though it changed shapes it was still a line.
  
• No, because lines never end and this one did.
  
• Yes, because no matter what shape it took, it was made up of lines.
  
• Yes, because it has an infinite number of collinear points.
  
• Yes, it was almost always a line because it kept continuing.
  
• Yes, it was always a line but not always straight.
  
• No, because sometimes it stopped at the bottom.
  
• Yes, because it went on forever.
  
• No, because it turned into shapes and connected to itself so it was not infinite.
  
• No, the line because different shapes.
  
• Yes, it was always a line until it became an angle with the dot.
  
• Yes, because it never stopped.
  
• Yes, because never once did it stop. It was so long and it looked like it had two arrows.
  
• No, not always because it had an end at sometimes.
  
• Yes, because a line is an infinite number of collinear points.
  

Again, amazing ways to think about a line! Some were "textbook" answers, completely fine and described infinte number of collinear points. But I think my favorite is "Yes, it was always a line until it became an angle with the dot." I will be presenting these to the class and see what they have to say about the answers. I'm thrilled by the complexity of their reasoning. Can't wait to see what they come up with next!

Dreading something means doing everything possible to avoid it...so change the game. Reward yourself with something positive, let's turn dread to desire.

What does this mean in the math classroom?

Some students just don't like word problems, they dread them and never feel they can understand them. I've started acting them out.  A problem that states "The temperature is -32 degrees and it increases by 20 degrees." is not thrilling at all. How can this be acted out? I placed one student at the left most part of the class and said she needed to act cold because -32 is really cold. Then I pretended to be the sun which was so embarrassing I won't even describe it here. (I'll be honest...with a last name of "Strange" I can get away with being pretty ridiculous in the classroom.) I was bright and shining and generally goofy. The student said "Okay but what does that like as a math problem?" (That is success in math dialogue...students asking questions.)

I told her I didn't know yet what the math problem looked like but it probably has a -32 in it. And I wrote -32 on the board. "Then what happened?" I asked. The student took the marker and wrote -20, looked at it, said "no", then wrote +20 on the board. I told her to turn that into a math problem and she wrote -32 + 20 = -12. This turned into a problem that was not so scary.

(Now...what I should have done next is say to pretend it is -32 degrees and the temp changes by some amount, what would the expression be? Then she might have written -32 + x and we would talk about what positive value for x and a negative value for x would mean. I hate missed opportunities but oh well...next time I'll bring that up.)

So a student who said she didn't do word problems successfully solved one. The next word problem was about football yard gains and losses and she said, can we do that one now? 

I'm now toying with the idea of making a promise to the kids that we will act out EVERY word problem we come across...I'm wondering if I can actually do that. But I'm also wondering if that will help turn their dread of word problems into an anticipation to see what they mean...and thus a method to solve them.

It's the first week of school and getting kids to talk freely about math is tough. Students (especially 9th  and 10th graders) may not know anyone in the class, they are unsure of themselves, they want to make a good impression on the teacher, and they believe everyone is looking at them...(all very normal thoughts for teens). I've used this activity several times but this is the first time I've used it so early in the school year.

My goals were to get kids to talk about math, use correct math vocabulary, minimize "I don't know what to say" and possibly learn from their classmates. I think the activity accomplished all of these goals in some way.

To perform the activity, use any words or concepts that are new to the students. For my geometry class we used points, lines, planes, segments, rays, collinear, coplanar, non-collinear, non-coplanar, parallel lines, parallel planes, skew, opposite rays. Students had learned these concepts over several days, taken some notes, created paper prisms and labeled with points, then identified parallel planes, collinear points, skew lines, etc. So the students had used the words in some fashion. Now it was time to review. Students were given half index cards to make flash cards for all of the words. The front contained the word or phrase, the back was a description, drawing, or other key words that described the concept. It took students about 15 minutes to create them. Then I acted out the activity with the students by borrowing a set of someone's cards. I read the description on the back to the class and had them guess the words. For example, I said "these are lines that don't intersect and they are in different planes." Some students said parallel lines, but (and it is important to note this) many students didn't say anything at all. (Here's where you collect data. Note how many students answered aloud when you asked the class, then compare to the number of students talking when you start the activity. Any increase in the number of students engaged is success!)

Explain that students will be paired up, then each will read the descriptions to their partner and their partner must guess the word or phrase. One of my favorite ways to pair kids up is to distribute a playing card to each student. Then tell them they must get up and find the person with the same number and color. (So a red 2 finds another red 2, a black four finds another black four, etc.) Tell them they have 2 minutes to find their partner and start quizzing each other. Then (and this is important) set a timer for 2 minutes! Walk around the room and listen. First recognize that more students are talking about math than before...so yay it's working! After two minutes, tell students to go find someone with the same number but a different color and set the timer for another two minutes. Next pairing can be an even numbers must find even numbers with same color (odds must find odd numbers with same color). Then switch to even number and opposite color, etc. Depending on the number and difficulty of concepts, I try to do about 4 pairings. If you do too few, they don't get comfortable. If you do too many, they stop talking about math. Because I am circulating and listening in, I often see 100% of kids talking about math. Sometimes I listen and write down words or phrases I hear and share them with the class after the activity. I tell them these are my favorite things. I always include an error from someone and tell them how happy I was to hear that because it allowed the pair of students to have a deeper discussion about math.

Here's why I think this activity gets them talking about math. They have the vocabulary in front of them. They can take turns reading or answering. They only have to say it in front of one other person. Listening helps them with their pronunciation (non-collinear is a mouthful for non-verbal kids). They know the activity will end. (The timer is important.) They believe they can be successful before they start.

Here's my new goal. How do I create all those components in every activity I do? I think these are the key parts to getting kids to talk about math.