Saturday, November 6, 2010
One Math Period, One Problem, No Problem
How is it possible to spend an entire math period (a shortened one for that matter) on only one math problem? It is quite simple.
A well designed problem requiring a variety of math skills, allowing the students to collaborate with one another with built in differentiation. That is how a period can be spent on one problem.
Many times we start off math class with a math message problem to get the students thinking about the lesson of the day or connect to a topic or concept we have already studied.
For the past week we have had 20 minute math periods for early dismissal days. The problem takes up a good percentage of the period.
The problem posed on Monday was: My haircut at Supercuts typically costs $14.00. Yesterday I had a coupon for $2 off a haircut and then Supercuts gave an extra dollar discount for "Football Sunday". I gave a $5 tip.
The three questions to answer were: How much was my haircut with the discount? How much did I pay with the tip? About what percentage off the regular cost of a haircut did I receive with the coupons?
Now the first two questions were not a problem. With quick subtraction and addition skills the students realized that the haircut was $11 and the total cost with the tip was $16. The real discussion and thinking came with the last question.
The students worked with a partner to discuss about what the discount was. We had already begun discussing last week how to find 10% of a number, a skill not yet mastered, but a very important skill connected with comparing and calculating, fractions, decimals, and percents.
We heard many different responses ranging from 5% to $3.00. We were able to then use this information to systematically ask questions to lead to "about" the right answer.
The first question: Should the answer be in dollars or percent?
The second question: Is the answer close to 100%? Why or why not?
The third question: Is the answer close to 0%? Why or why not?
The fourth question: Is the answer close to 50%? Why or why not?
Each of these questions makes the student think about the discount in comparison to the original price. They knew the discount was $3 in comparison to a $14.00 haircut. Some of the students at this point had a reasonable estimate. Some of them had a reasonable estimate before the questioning. The questioning and ensuing discussion allowed for them to confirm their thinking or change their answer. It also helped to push their mathematical thinking in new ways. Listening to the responses of their peers is always valuable. The important thing I have learned is to make sure I restate and try to show what they say and validate their thinking, or it will be lost on most students accept for the one making the claim.
At this point we are able to then narrow it down to less than 50% and greater than 0%.
What would 50% be? Half of $14.00 is $7.00.
This is when the best part of the discussion happens. We know it is less than 50% but then what?
Here is a list of possibilities:
I know 25% is half of 50% and $3.50 is half of $7.00, so it will be close to 25%.
I know 10% would be $1.40 because you move the decimal point once to the left to find 10% of any number. Double 10% is 20% and double $1.40 is $2.80. $2.80 is close to $3.00, so the answer is close to 20%
Since $2.80 is 20% and $3.50 is 25%, the discount must be between 20 and 25%.
Now we have it really narrowed down, but the questioning does not stop.
Would the percent discount be closer to 20% or 25%?
20% of course.
Then more answers are given comparing 20 cents to 50 cents and how much further away $3.50 is than $2.80. Then the students are able to realize that right between 20% and 25% is 22.5% and that would be $3.15 because the difference between $2.80 and $3.50 is .70. Half of 70 cents is 35 cents.
Now they realize that it is between 20% and 22.5%. And that is close enough for now.
I could have just taught them to divide $3.00 by $14.00 to find the answer. That is an important skill that will be coming soon along with the many ways to break down a problem like this. Right now the important skill is to learn how to reason and work through a problem through logical steps, discussion, and writing. This lesson required lots of questioning, thinking aloud, and talking with others. Learning how to reason and estimate will help the students so much more when they need to apply algebraic skills to solve a problem like $3.00 is what percentage of $14.00?
There was no need yet to find the exact answer of 21.428571%.
By doing a haircut problem, all of the students could connect in some way. They all have gotten a haircut either at a shop or by a parent. Either way they could understand the financial aspect involved and the social part of getting a haircut. They quickly become interested for the most part because it is a story they understand with a main character they know. So many book problems are quite unrealistic or do not draw the reader in. Isn't that what reading or any subject is all about, to draw the student in, to build that bridge, to connect them (yet another future blog topic), to transform their learning? I was able to tell them how the hairdresser was wearing a Tom Brady jersey, so as a Jets fan I was a bit nervous as my eyes were glued to the Jets-Packers game on the television in the corner. I explained that I paid with a $20 bill and we were able to discuss a bit how much change I received and how much of that went toward the tip. We were able to hear about some haircut stories, and how a girl I know who does cut mens' hair charges almost 5 times my regular haircut. I needed to go to Trader Joes, get home to mow the lawn and get my kids ready for trick or treating. How much time did I have to get everything done?All of these things help to draw students in.
I did get lots of compliments on my about 22.5% discounted haircut!
The rest of the week centered on finding 10% (sure to be a blog topic in the future) and the patterns that follow. As we head toward decimals and percentages, the question to ponder this weekend is Why are gas prices expressed to the thousandths place? Why on the sign do they use both decimals and fractions
Now the final question is: How many math skills were used or brought up in a problem like this?
It took longer than twenty minutes to type this. Period over. See you tomorrow.