The IntMath Newsletter - 1 Nov 2008
By Murray Bourne, 01 Nov 2008
In this Newsletter:
1. Math tip - Solving word problems
2. Google's Project 10 to the 100th
3. From the math blog
4. Final thought - The Remainder of 2008
1. Math tip - Solving word problems
Recently, a reader asked me to address the following issue:
I have difficulty converting a text problem into the mathematical equations needed to solve them. I freeze when I see a text problem.
This reader is not alone. Many people suffer from the same thing, and word problems are a key cause of math anxiety.
Our brains get quite a workout when we solve math word problems. How can we reduce this brain stress?
I'm reminded of what happens to athletes when they learn a new movement on the parallel bars or pommel horse. The first time they try it, everything moves really fast and they struggle to complete the move and land properly. But after a few tries, the move slows down in their brain and they can more easily make the small changes necessary to execute a perfect move. The more practice they get, the slower it becomes — and the less stress they feel.
Math word problems are just like this. The more you practice them, the more they slow down in your brain and the more relaxed you feel.
Let's go through an example to give you some ideas on how to approach word problems. I have used a problem that can be solved using basic algebra, but you can use this approach for any kind of math word problem.
Example Problem: At a party last week I overheard a friend being asked how old he is. He answered, "I'm 4 times my daughter's age and eight years ago, I was 12 times her age." I thought it was a clever answer and I was interested to figure out how old he is.
First - can you figure it out? Only read the next bit if you've already had a go at the problem.
Of course, you could do this problem by trial and error, but what if the numbers involved were much larger, or they involved fractions or decimals? It could take you a very long time to solve it.
Algebra to the rescue.
Problems like these are given to you so you can get practice in translating word problems into algebra, and then solving them. So while there may be other ways to solve them, try to do it using algebra. the following strategy is based on George Polya's How to Solve It. Polya was an expert in how to solve math word problems.
A. UNDERSTAND THE PROBLEM
* What is the unknown? The father's age.
* What do we already know? We know that the daughter's age is one quarter (1/4) of the father's age.
We also know that 8 years ago, the father's age was 12 times the daughter's age.
* Introduce suitable notation. Let's call the father's unknown age x. We could use other letters — but x is commonly used as the unknown.
B. DEVISE A PLAN
* Have you seen a similar problem before? This is a key point. The first time we see a problem like this, it seems very difficult. But once we have struggled with it for a while and then had our "a-ha" moment, we have made a vital connection in our brains. The next time we see a similar problem, it is less daunting. The key learning point here is do lots of word problem examples. The more you do, the easier they become.
* Could you restate the problem? This goes a long way to help our understanding of a problem. Tell someone else about the problem — just the act of expressing it in your own words helps you to understand what it is asking and can trigger some ideas on how to solve it.
I do a lot of things on computer and occasionally get stuck by some problem. When I do, I will go to an online forum and start describing my problem. I often don't need to submit my question because just the act of writing the problem clearly — and the conditions under which it occurs — helps me to figure it out for myself. And that situation always achieves the best learning.
* Find the connection between the given information and the unknown.
The given information is:
- The father's age is 4 times daughter's age
- 8 years ago, the father was 12 times the daughter's age
Our plan is to use algebra to express the current ages and the ages 8 years ago. We will need to decide which expression(s) are equal, and solve them to find the unknown.
C. CARRY OUT THE PLAN
The daughter's current age is 1/4 the father's age, so we write it as:
x/4
Eight years ago, the father's age would be written
(x − 8),
and the daughter's age would have been her current age minus 8, written
(x/4 − 8).
We are told that 8 years ago, the father's age was 12 times the daughter's age, so we would write the father's age at that time as
12(x/4 − 8).
So we have 2 expressions for the father's age 8 years ago. One is (x − 8) and the other is 12(x/4 − 8). These must be equal, so we connect them with an equals sign and solve.
x − 8 = 12(x/4 − 8)
Expand the bracket on the right:
x − 8 = 3x − 96
Get the x terms all on the left by subtracting 3x from both sides:
−2x − 8 = − 96
Get the numbers all on the right by adding 8 to both sides:
−2x = − 88
Divide both sides by −2:
x = 44
* Check your answer. Is our answer correct? Does it make sense in the 'real world'? Let's check in the original question.
If the father's age now is 44, the daughter now is 11 years old (both are reasonable ages).
Eight years ago, the father would have been 36 and the daughter would have been 3, which are also reasonable. The father's age is 12 times the daughter's age, which also fits with the information given in the question. We have found the correct answer.
D. LOOK BACK
Before leaving the problem, think about the following:
* Can you derive the solution differently?
* Can you see the answer at a glance?
* Can you use the result, or the method, for some other problem?
This is a vital step. We only really learn things effectively when we think over what we have done, and how we could have done it better.
In this problem, you could have used a different starting point. You could have used the daughter's age 8 years ago. Let's call it y years old. The father would be 12y then. Move forward 8 years, and the daughter will be (y + 8) and the father will be (12y + 8). But since the father is now 4 times the daughter's age, we can write:
12y + 8 = 4(y + 8)
We can solve this to get y = 3 and then work up to the father's present age.
This example is fairly simple so there are not too many other ways to do it. But for more complicated questions, it is a good idea to think over the problem to see how else it could have been solved.
How to Solve Harder Word Problems
What if our math word problem was much more difficult than this one? What should we do to attack it? Here's some more advice based on Polya's ideas:
- If you cannot solve the proposed problem try to solve first some easier related problem.
- Could you solve a part of the problem? Perhaps you could keep only a part of the condition and drop the other part — can you figure out something about the unknown from this?
- Could you think of other data appropriate to determine the unknown?
- Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?
- Did you use all the data? Do you need to use it all?
Above all, when you have a word problem that seems difficult, don't just stare at it! There are many things to try out first, like:
- Do some trial and error (to give you an idea of what the question means),
- Read over your textbook for similar problems,
- Draw a diagram,
- Make sure you understand all the key vocabulary in the problem
Back in July I wrote about what happens in the brain when we solve math word problems. You might also find that article useful.
Final tip: Like the athlete that we talked about at the beginning, your problem solving skill will increase with practice. Aim to do problem solving regularly — at least 3 times a week. It will slow down for you and you will get better and better at it — and your stress will be reduced.
All the best with solving your math word problems!
2. Google's Project 10 to the 100th
Google's corporate motto was "Don't be evil" up until May 2018.
As part of the search giant's 10th birthday, they developed a project where they are trying to do some good. They have tapped on the worldwide global community to come up with really good ideas to improve people's lives.
The categories were: Community, Opportunity, Energy, Environment, Health, Education, Shelter, Everything else.
3. From the math blog
1) Friday Math Movie - Mathmaticious
Mathmaticious is a nerdy parody of Fergalicious.
2) Life expectancy and semi-log graphs
Life tables give us an idea of how much longer we can expect to be around. There are interesting probabilities and semi-log graphs involved.
3) Disrupting Class
Computers in school are disruptive - but thatβs not a bad thing, according to this book.
4. Final thought - The Remainder of 2008
Here's some things to ask yourself as we count down the last 2 months of 2008.
- What specific results have you achieved so far this year?
- What results are you committed to achieve by the end of the year?
- What have you learned this year (in school and outside)?
- On a daily basis, are you committed to the most important things in your life?
- What habits or behaviors do you need to change to ensure better results?
- What is the key issue that gets in the way of your ability to perform at your best?
- What are your aims for 2009?
Until next time, all the best.
See the 8 Comments below.
