by Josh Rappaport
I’ve started using a new technique to help students solve word problems, and every day I’m amazed at how well it works.
The new technique helps students overcome their confusion with word problems. Using this approach, students first replace the “messy numbers” in a word problem with what I call “friendly numbers.” Then, by figuring out how to solve these problem with the “friendly numbers,” children figure out how to solve the original problem, even with its “messy numbers.” Essentially, “friendly numbers” are numbers that are easy to think about because they are simple, round numbers.
Here’s how this problem-solving process works.
Example Word Problem: Of the people who voted, 90 percent voted for Sammy. If 1930 people voted, how many of them voted for Sammy.
I was tutoring a student who needed to solve this problem. Her response after reading the problem: “Huh?”
Then I told her that it’s ok — temporarily — to replace the numbers in the problem with “friendly numbers,” to make the problem easier to grasp. I showed her that in this problem, for example, she could temporarily replace the 90% with 50% and replace the 1,930 figure with a nice round number, like 600.
The girl picked up her pencil and re-wrote the problem as follows:
Of the people who voted, 50 percent of them voted for Sammy. If 600 people voted, how many of them voted for Sammy.
When I asked this girl if she could understand the problem, she said it now made sense. She went on to reason that if 50 percent of the people voted for Sammy, that means that half of the 600 people voted for Sammy. So that means that 300 people voted for Sammy.
Then I asked her if she could come up with an equation to solve this problem. With a bit of help, she came up with this:
.5 x 600 = # of people voting for Sammy
She then solved this using decimal multiplication and got the right answer: 300 voted for Sammy.
But — you’re ready to object — this is not the problem the student needs to solve. How does this “friendly number” process help her solve that problem?!
Don’t worry. At this point I steered the child back to the original problem.
First I asked if she could make a similar equation for the original problem, prompting her with these questions:
Q#1: What number in the original problem corresponds to the 50% amount in the “friendly number” problem?
A#1: 90%
Q#2: What number in the original problem corresponds to 600 figure in the “friendly number” problem?
A#2: 1,930
Once the student spotted these correspondences, I had her first write her equation for the “friendly number” problem, then, just below, write a corresponding equation for the original problem. Her work looked like this:
“Friendly Number” Equation: .5 x 600 = # voting for Sammy
Original Problem Equation: .9 x 1,930 = # voting for Sammy
I then asked her to solve the second equation, using decimal multiplication, and she got the correct answer, 1,737.
After guiding students through this process I often ask them what made the original problem seem hard. Often they’ll just shrug their shoulders and say that for some reason, they couldn’t understand it when they first read it.
Based on my work with “friendly numbers” I’ve come up with a bit of a theory. I believe that for many students, merely looking at “messy numbers” sets in motion a “psych-out” factor. When kids get “psyched out” by those “messy numbers,” they go into a minor mental panic. And in that panic they just lose touch with their intuition; they lose their intuitive sense of what they need to do.
When this happens, all we educators need to do is help the student re-cast the problem with “friendly numbers.” When children do that, the “psych-out” factor vanishes, and because the “mental air” becomes clear, students suddenly realize what they need to do to solve the problem. Then, with a little bit of prompting, students can generally transfer their sense of what needs to be done with the “friendly number” problem to the original problem. At that point they’re on their way toward solving the original problem.
I encourage you to teach your children how to use “friendly numbers” when solving word problems. If this technique works as well for you as it does for me, you’ll be pleasantly surprised, and those scary word problems will look a good deal less daunting.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on the Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com
I’ve started using a new technique to help students solve word problems, and every day I’m amazed at how well it works.
The new technique helps students overcome their confusion with word problems. Using this approach, students first replace the “messy numbers” in a word problem with what I call “friendly numbers.” Then, by figuring out how to solve these problem with the “friendly numbers,” children figure out how to solve the original problem, even with its “messy numbers.” Essentially, “friendly numbers” are numbers that are easy to think about because they are simple, round numbers.
Here’s how this problem-solving process works.
Example Word Problem: Of the people who voted, 90 percent voted for Sammy. If 1930 people voted, how many of them voted for Sammy.
I was tutoring a student who needed to solve this problem. Her response after reading the problem: “Huh?”
Then I told her that it’s ok — temporarily — to replace the numbers in the problem with “friendly numbers,” to make the problem easier to grasp. I showed her that in this problem, for example, she could temporarily replace the 90% with 50% and replace the 1,930 figure with a nice round number, like 600.
The girl picked up her pencil and re-wrote the problem as follows:
Of the people who voted, 50 percent of them voted for Sammy. If 600 people voted, how many of them voted for Sammy.
When I asked this girl if she could understand the problem, she said it now made sense. She went on to reason that if 50 percent of the people voted for Sammy, that means that half of the 600 people voted for Sammy. So that means that 300 people voted for Sammy.
Then I asked her if she could come up with an equation to solve this problem. With a bit of help, she came up with this:
.5 x 600 = # of people voting for Sammy
She then solved this using decimal multiplication and got the right answer: 300 voted for Sammy.
But — you’re ready to object — this is not the problem the student needs to solve. How does this “friendly number” process help her solve that problem?!
Don’t worry. At this point I steered the child back to the original problem.
First I asked if she could make a similar equation for the original problem, prompting her with these questions:
Q#1: What number in the original problem corresponds to the 50% amount in the “friendly number” problem?
A#1: 90%
Q#2: What number in the original problem corresponds to 600 figure in the “friendly number” problem?
A#2: 1,930
Once the student spotted these correspondences, I had her first write her equation for the “friendly number” problem, then, just below, write a corresponding equation for the original problem. Her work looked like this:
“Friendly Number” Equation: .5 x 600 = # voting for Sammy
Original Problem Equation: .9 x 1,930 = # voting for Sammy
I then asked her to solve the second equation, using decimal multiplication, and she got the correct answer, 1,737.
After guiding students through this process I often ask them what made the original problem seem hard. Often they’ll just shrug their shoulders and say that for some reason, they couldn’t understand it when they first read it.
Based on my work with “friendly numbers” I’ve come up with a bit of a theory. I believe that for many students, merely looking at “messy numbers” sets in motion a “psych-out” factor. When kids get “psyched out” by those “messy numbers,” they go into a minor mental panic. And in that panic they just lose touch with their intuition; they lose their intuitive sense of what they need to do.
When this happens, all we educators need to do is help the student re-cast the problem with “friendly numbers.” When children do that, the “psych-out” factor vanishes, and because the “mental air” becomes clear, students suddenly realize what they need to do to solve the problem. Then, with a little bit of prompting, students can generally transfer their sense of what needs to be done with the “friendly number” problem to the original problem. At that point they’re on their way toward solving the original problem.
I encourage you to teach your children how to use “friendly numbers” when solving word problems. If this technique works as well for you as it does for me, you’ll be pleasantly surprised, and those scary word problems will look a good deal less daunting.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on the Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com
