MATHEMATICS PD ASSIGNMENT FALL 2013 - Richard Ploetz

PART 1: PLANNING

The Problem

At the Dreyer movie theater, adult tickets cost twice as much as children’s tickets. Ramon and Hector (who are adults) took three children to see a movie. The total price for all the tickets was $28.00. What was the price of each child’s ticket?

 

I solved the problem using algebra:

X = price of child ticket

2X = price of adult ticket (twice as much as a child’s)

There are 3 children times X dollars = 3X dollars for total price of child tickets

There are 2 adults times 2X dollars = 4X dollars for total price of adult tickets

THEREFORE: 3X plus 4X represents total price of all tickets, $28

7X = $28; X = $4, or price of each child’s ticket

This may be checked: price of an adult ticket is twice a child’s, or $8. 3 children times $4 is $12; 2 adults times $8 is $16. Add these two numbers and get $28.

 

Alternative solutions

1.      Trial and Error: knowing there are 3 kids and 2 adults, and adult tickets are twice the kids’ . . . you can draw pictures of 3 kids and 2 adults and assign ticket prices for kids and double the ticket price for adults, multiply number of kids and adults by their respective ticket prices, and see if your total comes out to $28. If it doesn’t, adjust the ticket prices up or down until you hit the magical numbers.

2.      Instead of pictures, you could draw up a table, and use this to keep track of your trials and errors.

Using trial and error, you can ask students if they see a pattern developing (their totals going up or down as they adjust the individual ticket prices), that leads them toward a solution.

 If students seem stuck? I might ask them to say what they are looking for (what does the question ask). Then what are the elements given in the problem (number of children and adults, different prices of tickets). If no light bulbs go off, I might get more specific: “What if a kid’s ticket were $8: how much would an adult ticket cost?” And further: “How would you find the total cost of kids’ tickets?” Hoping that I am not giving too much away. These leading questions on an individual basis.

What math practices would I hope to be developing using this problem? 1. Reading carefully and getting the lay of the land before jumping in: what question is being asked? 2. What information is provided? 3. How might I organize (or examine) this information? 4. Do I have any thoughts as to how I might proceed? 5. Do I have previous experience with this sort of problem that might aid me here? 6. If I get discouraged, can I try another approach – “trial and error”? 7. Can I talk to my neighbor – see if they have an idea? 8. Get something down on paper – see what it looks like. 9. Don’t erase any work – it’s a starting place I may build on – or return to.

 I would hope by working on this problem, sharing approaches and solutions, that students would see there are a variety of (all equally valid) ways of doing it. I would hope their self-confidence as mathematicians would increase and affect they way they approach future problems.

 

PART 2: STUDENT WORK: Low/Intermediate GED Math Class

1.      OVI. He started something and erased it. Then wrote the number 28. Under this he wrote 2x2 = 4; then added 3 to 4 to get 7. Then he took the 7 and divided it into 28 and got 4. This I think he identified as the price of a child’s ticket. Next to this, knowing adult tickets cost twice a child’s, he wrote 4x2 = 8. As the solution to his problem he wrote “each = $8.00”.

I asked him: “Is this what the problem asks for?” requesting he read the problem carefully. Eventually he got to $4 for the price of the child’s ticket. My sense is Ovi tried a few approaches, without a particular method, hit on some numbers, checked the parameters of the problem (price of individual tickets), and fit his numbers in to come out with a correct answer. He didn’t go back to check his answer (3 kids plus 2 adults, total prices of $28), which I asked him to do – to be SURE he got the right answer. When we looked at other students’ work who approached the problem in a more organized way, I hoped Ovi saw a useful way to organize his rather random calculating.

 

2.      LATISHA. She first divided 28 by 3, getting 9 plus. Then she listed  $9 three times and added this to $27. Next to this she added three 8’s to get 24. She added 4 to this to get the required total sum of $28. The number 3, however, would indicate these were three child tickets. At this point I think she gave up and listed the answer to the problem as “$9”.

I tried to get at this by asking her if her answer of $9 checked out? It took a while but eventually she realized the problem asked for price of child tickets – AND child tickets were half the price of adult tickets – which made adult tickets $18 each. If that were the case, the 2 adult tickets alone would cost $36. We wound back to her original listing of three 8 dollar tickets adding to $24, with the addition of $4 totaling to the required $28. Would there be a way of rethinking these numbers to make the problem come out? In other words, could she play with the price of tickets, holding to the fixed number of children (3) and adults (2)?  Eventually Latisha got the idea of $8 for adults and $4 for kids, and this worked out for a total of $28. I think she got a feeling there was a “method” somewhere here, but we were so long wandering in the trees it was difficult to clearly see the forest (method).

 

3. BRIAN. His page was a model of simplicity: three 4s added to 12; two 8s added to that, which totaled $28. “I just saw it,” he said.

 

PART 3: REFLECTION

 Following a period of students working alone, together, and one on one with me, the whole class came together and I asked students with a fairly coherent ways of solving this problem to demonstrate on the board and talk about how they worked on it. This was helpful to others who had struggled – and those who only came to solutions with much coaching.

I tried to clarify (simplify) the “trial and error” methods students used by showing how an outcome table could be handy in keeping track of numbers. I also demonstrated the algebra solution (which no one had used). The simplicity of it was pleasing, but (at this level) algebra can seem daunting (really needs groundwork laid).

 I felt this was a good problem for students: a balance between too hard and too easy. At first glance it seemed pretty clear and straightforward – got them into it – and then got them thinking. In other words an “engaging” problem, not one they looked at and got right away - or threw up their hands. There was a path in, and they could start playing.

 What did I learn? There is no one way to “teach” this kind of problem – it breaks down into individuals working alone, together, and with help from me. Different levels of math skills requiring intensive managing (oversight – no mathematician left behind!). I learned to hold back and try to let students arrive at their own solutions. Backseat driving: the old Socratic method – help them get to where you’d like them to go.

I learned to trust students’ willingness to engage and work at a “thinking problem”, and not feel I had to lead them by the nose. That the frustration and the hard work was a real way of owning the learning. Using this problem, and similar ones, on a regular basis engages students as active learners – thinkers – in a way that teacher-led classes do not. What you earn yourself stays with you.

Did all my students get what I wanted from the problem? It felt like an engaging class, so in that sense, yes. I felt they came out of it with a confidence they have skills and can find their ways into a problem whose solution may not be immediately obvious. Not give up – come back and try again.

The “highlight” of the class was seeing a variety of small breakthroughs – a student getting one step along toward the solution of the problem, and being encouraged to try another step. Students being willing to ask questions instead of sitting there in silent frustration. We were all in this together. It was a social activity with the kind of effective learning that happens when you work on a problem with your neighbor. A process-oriented class rather than one just looking for the “answer”.

What would I do differently when I teach this problem again? Introduce them to similar “Hmmm . . . problems” – as warm ups before each class. So they get used to this thinking approach. For example: the turkey and foxes in a pen – so many animals and so many feet – can you figure out how many of each kind of animal? Or, the ferryboat that transports cars and trucks across the river. The boat is full when it has 10 cars on board; also full when it has 6 trucks on board. It never carries cars and trucks at the same time. Yesterday it made 7 trips across the river and was full each time. It carried a total of 50 vehicles across the river. How many trucks did the boat carry across the river? Or: 6 students running a race, using clues to determine the order in which they finished.

 I have used these (and other) problems in the past and they are engaging and fun for students to work on. And the more exposure to this way of approaching math the better they get at it. And the more they enjoy it. And the better they . . .

 

Student Reflection

I didn’t have time to request written reflections, but at the end of the class we discussed the experience. These are what I remember of responses:

1.      WHAT DID YOU LEARN: “Be patient – try another way if first doesn’t work – don’t give up.” “I know more than I thought I did.” “It was fun helping Kevin – and seeing he got it.”

2.      DESCRIBE WHAT WAS EASY AND WHAT WAS HARD: “What was hard, at first, was understanding the problem – how to get a grip on it.” “It took me a while to keep track of what was going on. The numbers changing.” “I liked that it seemed real – I could see the people and the theater – and kids tickets are cheaper. I actually made a picture.”

3.      WHAT WAS ONE THING YOU REMEMBER? “I remember thinking after Richard read the problem and we looked at it, that it wasn’t so easy. That it would take some work.” “I was bummed at first – I like math that doesn’t take much thinking – get the answer and move on. This one – I ran into a wall. A couple walls!” “I remember liking it – it wasn’t like a math class for a change.” “Will we get a problem like this on the GED? It took like an hour!” “I wanted to give up – I tried this and that but nothing worked – but everyone kept going, and finally I got some help and sort of got it.” “It seemed easy, but when I started working on it it wasn’t so easy.” “I felt pretty good after working through this. I got it.”