MATHEMATICS PD ASSIGNMENT FALL 2013 - FARRELL

Mathematics Professional Development Assignment for Fall 2013

 

Planning
 

1. How does this problem meet our criteria for cognitively challenging math tasks?

 

Having students draw their answers as opposed to writing the answers challenges students to think about the problem in different ways. Students will examine different strategies/ drawings of what 1/6 of ½ looks like. Students will be able to make sense of the problem and persevere in solving this problem. They know that 'of' translates to multiplication, but will they be able to show that visually? They may be able to proceed directly towards a solution by multiplying the fractions arithmetically. But applying a drawing will be a perplexing situation for the students at hand.

 

2.  Describe how you solved the problem.

 

I drew a circle and I cut it in half. I then cut each half into 6 equal pieces, giving me a total of twelve pieces. I colored in one of the one sixths on the halved section.

 

3. What are other ways to solve the problem?

 

I wasn't sure of another way to show half of a circle and then cutting it into 6 equal pieces. I guess I will soon see how the students come up with their solution to find another way of demonstrating what one sixth of 1/2 looks like.

 

4. What are some questions you can ask to help students unpack the problem and/or describe this solution?

 

# What does half look like?

# Think about half of a Hershey's bar

# Use the graph paper to help you figure this out.

#t Think about cutting an orange in half.

#  What does it mean to take 1/2 of something? What does it mean to take one third of something?

# What does it mean to take 1/4 of something?

5. Identify three challenges you think students will have in solving this problem. For each challenge describe what questions you would ask to support the problem solving efforts of those students without giving too much away.

 

Some of the problems that I expect students to have is not knowing how to begin or where to begin. They may be able to solve it arithmetically but may have problems producing a picture of what it looks like. I can guide them by using the graph paper and I can guide them by showing them circles.

 

6. Which math practices are you hoping to develop by using this problem?

 

Mp1 and mp2. Under the depth of knowledge 3 - I want them to strategically think about what they’re doing. They will also be able to analyze similarities or differences between the procedures of solutions they came up with for the drawings. They will translate rote arithmetic to drawings.

7. What do you want students to get from working on this problem?

 

I want my students to relate arithmetic to real life. I want them to be able to draw pictures of what they're actually doing. I want them to see what algebra looks like and what fractions look like. And if they're able to draw the arithmetic, then maybe they can make a sense of what operations they are really performing. For example, what does adding look like when drawn. What does multiplication look like when you are drawing it?

 

Student work

 

1. Describe the specific problem solving strategies of each student. What do you appreciate about each student's method? How could each student’s method be improved?

Some students rewrote the problem using circles under each by showing what 1/6 was and then drawing what ½ looked like and then showing a circle with 1/12 shaded. That was interesting.  I had to really delve into them producing a single work of art what 1/6 of ½ looks like.

 
2. Briefly describe some of the challenges your students had while working on the problem.

• Some dribbled down shapes with no direction of showing 1/6 and half.

• Also they were not able to show the relationship of multiplying fractions.

• Some students waited to draw anything and looked at each other’s paper to get a clear idea of where to begin. So, knowing where to start was a challenge.

 

Reflection

 

1.How did it feel doing this problem with your students?

 

I knew how to do the problem using circles, but was flabbergasted by students revealing the answer using squares. That was interesting to see. I did give out graph paper to assist them though. One student had shown half of 1/6 and sectioned off 72 squares. That opened up the parameters for me. I’ll continue to use these types of problems. I enjoyed them when I saw that they were part of the “redefining what it means to do math in the common core era” packet. 

 

2.What did you learn from using this problem with your students (about mathematics, about individual students, about your class, about student thinking in general)?

I always knew that there could be three to four ways to solve any mathematical problem. Being able to justify that publicly was rewarding.

 

3. Briefly comment on how your class or individual students or as a whole benefited from their work on this problem.

 

Well, as mentioned before, being able to present work mathematically visually is stimulating. It forces you to think. Arithmetic looks so rote! When displaying it without using symbols, it gives a little bit more meat to what those actual mathematical symbols means.

 

4. Did all of your students get what you wanted from the problem? How do you know?

 

 Showing what halving something looks like was gratifying.  Showing what taking 1/6 of something explicitly adds depth to the whole lesson of multiplying fractions.  Out of 15 students, 10 seemed to get a grasp by witnessing me looking at their work.

 

 
5. Describe 1 highlight from the class discussion of the problem and solution methods

 

First, I received a little bit of hesitation and “push back” , such as
• “Why are we doing this?”
• Why are we drawing things now?
• Do you draw on the GED exam?
• I already know how to multiply fractions. Just go straight across, NEXT!
• Draw 1/6 of ½, how? Where do I begin?

 

While giving the problem out, I had to introduce it properly and message to them why it was important and why it was relatable to the GED exam and what they could benefit from when doing a problem like this. I also explained what DOK 1, 2, 3 and 4 was and what problems may look like in the future. With this discussion, they loosened up and were ready to receive.

 

One highlight was seeing the 72 squares sectioned off on the coordinate paper and then seeing that ½ of 1/6 was 1/12. That underscored the lesson of showing math visually and producing different routes of coming up with the answer.

 

6. What if anything would you do differently if you use this problem again?

 

 Nothing.

7. What advice or message do you have for teachers considering using this problem with their class?

 

Get ready for questions such as in #5. Messaging to students WHY teachers are teaching in a specific way is important. They then see the connection and understand that a learning progression is taking place. We have to go step by step.

 

My advice to teachers would be to allot 15 minutes for them to answer. Use it as a warm- up. Guide them with questions. Use graph paper also when handing this out. It helps.

 

Student reflection: At the end of doing the activity with your students please have them reflect on their experience by writing a response to one of the following the prompts:

What did you learn from working on this problem? What was difficult about this question? What are you still wondering about?