After working on a series of word problems (from the previous lessons), we are ready to start creating our own! This process can really make for a great lesson. It brings creativity and humor into an otherwise dreadful topic. Most students whince at the thought of a word problems, but something amazing happens when they are given the chance to create a word problem. When they become the story tellers, the assemble complex word problems that they enjoy sharing and solving.
I start the lesson with what I consider to be the toughest Scientific Notation problem to ever appear in middle school: Tough SN Problem This problem was featured on the Engage New York site, but in case the link ever breaks, I included the entire pdf in the resource bin: math-grade-8
Now this problem may not seem like a super tough problem or even in fact be a super tough problem (I am not trying to objectively define the "hardest" problem ever), but it is a problem that requires proportional reasoning and an awareness of a minuscule time scale.
Here is the problem: Tough SN Problem
The problem is on the board as students enter. I ask one student to read the problem and then ask them some questions to get them focused on solving the problem.
"What information is valuable here?"
"What units do they give us?"
"What do we need to find?"
"How might we start solving this?"
"What is a general way to approach this problem?"
"Is 4.5 x 10^-6 seconds bigger or smaller than 1 second? Does this make sense?"
"Do we need to assume that this relationship is proportional?"
I give students 3 minutes to work on this problem and try and find a student that at least made a big dent in solving the problem. Then we share their approach (even if they didn't finish).
The most popular approach is to use part to part ratios. The second most popular approach is to set up a proportion and work through some approach of "the product of the means = the product of the extremes." They might also use ratio tables or double number lines.
Here is what you might see with the part to part approach:
"They tell us that a computer does 1000 operations in 4.5 x 10^-6 seconds, so I write this as a ratio."
1000 operations: 4.5 x 10^-6 seconds
The units are key here. I ask other students to revoice what this ratio represents, something like "to complete 1000 operations it takes a total of 4.5 x 10^-6 seconds." Or they might also rephrase it in a much more comprehensible way, "It takes a computer 4.5 x 10^-6 seconds to complete 1000 operations."
I encourage students to rewrite the ratio and ask them if this is equivalent:
4.5 x 10^-6 seconds : 1000 operations
Then I ask the student to continue explaining their algorithm (or what they might do next if they didn't finish).
"I want to find how many operations it can do in 1 second."
I tell them that this sounds tough, but wonder if can represent our approach with a much simpler ratio. We come up with something like:
.25 seconds : 10 operations
1 second : 40 operations
Now my questions revolve around this example, which helps them realize that in this problem they simply need to find out how many times 4.5 x 10^-6 seconds goes into 1 second and then multiply this by 1000 operations (to keep the ratio balanced.)
1 second = 10^0 seconds
10^0 seconds / (4.5 x 10^-6 seconds) = (1/4.5) x 10^6 = .2222222222 x 10^6 = 2.222222 x 10^5
Then multiply by 1000, since each group of seconds includes another 1000 operations:
1000 = 10^3
10^3 x 2.22222222 x 10^5 = 2.2222222 x 10^8
Here the decimal is repeating, so we can discuss rounding, etc. Also, I show them the process with exponents to verify that this can all be solved mentally (except perhaps 1/4.5 which required me to use an algorithm).
So our answer is 2.2 x 10^8 operations
(I place the vinculum over the 2 in the tenths place to indicate a repeating decimal)
I encourage students to push their thinking around the word problem process through a variation of the classic Mad Libs game. In the game Mad Libs, you plug random nouns, adjectives, etc. into a story. The words you gave fill in all the blanks and they are essentially random. The silly stories created by the random choices can be really funny. I demo the game with the class and run through a Mad Lib with them. I popcorn around the room asking for nouns, etc and then display the final result. The game is a blast.
In our version, we might also ask for parts of speech, but we also ask for certain math numbers. I have the students work in partners. One partner asks for the parts of speech and numbers, while the other give the values. Then they read and solve the problem they created together.
As I circulate, I ask questions like, "what choices alter the way you might approach the problem?" I want them to recognize that the parts of speech are somewhat irrelevant. The numbers will certainly change the answer, but if they played the same puzzle several times, they might find that their algorithms were still the same.
When students are finished, I ask them to make their own math lib to share with the class.
These are two math libs that I ask all students to work on: Math Libs
This is an opportunity to laugh and share their Math Libs. The underlying goal is to review successful algorithms, but the hook is in giving them the opportunity for them to share their experiences in solving the problems. What was fun? What was challenging? Across the class, how do their experiences compare? I encourage kids to make the connection that even though they chose different parts of speech, numbers and units, they ultimately faced the same problem. The "structure" of every word problem is beneath all of the specific parts of speech and number values. The structure is purely mathematical and always be accessed if we see past the plethora of information thrown at us.
I like to finish with an example:
Pluon, Duon and Luon purchased 5 x 10^6 gluzboras. They want to know how many trempties of they can buy with 5 gluzborras? 1 gluzborra = 3 x 10^4 trempties.
Together we can translate this:
Three people purchased 5 million of something. They want to trade that something for something else. 1 something = 3 x 10^4 something else.
Students might decide to use variables in place of the words "something" and "something else". Then it becomes more abstract but easier to work with the mathematics:
Three people purchased 5 million of a. They want to trade a for b. 1 a = 3 x 10^4 b.
The goal is for students to see how complicated the subjects and words of a question can make word problem. Our goal is to see past that. This exercise it meant to help them in this process.