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We aren’t afraid of it anymore: word problems

Ordinary people rarely associate word problems with mathematics. Really, the name seems to be misleading enough, so one might think of something closely connected with grammar, linguistics or something like this. But here we are going to consider word problems exactly in the light of mathematics.

In mathematical education, word problems normally refer to mathematical exercises where crucial background data on the issue is presented in the form of text, without the use of specialized mathematical notation.

Let’s have a look at a typical math word problem:

Solve for J:

J = A − 20

J + 5 = (A + 5)/2

John’s 20 years younger than Amy, and being aged five he’ll be half her age. What’s his current age?

That’s a typical representative of simple word problems. John is 15 years old and that’s the answer to the word problem, while the answer to the mathematical problem would be J = 15.

In fact, word problems can be solved on three levels:

  • The verbal formulation
  • The underlying mathematical relations
  • The symbolic mathematical expression

Evidently, linguistic properties might come with such variables as the mean sentence length or the number of words in the problem. The logic-mathematical properties can be classified in many ways. For instance, one can classify the quantities in the mathematical problem right into known quanitites.

The most common types of word problems include age problems, distance problems, percentage problems, work problems, numbers problems, mixture problems and so on.

The use of word problems

In mathematics, word problems are traditionally utilized to mode various mathematical questions, where information regarding a particular system is provided and a student is urged to develop a certain model.

For instance, if the water level in a cylinder with a 2m radius is surging at a 3m rate per second. The learner is expected to figure out the rate of increase of the water volume.

The given example isn’t only intended to involve learners into developing various mathematical models by themselves, but can also be used to promote sincere mathematical interest as well as understanding by simply relating the subject to real-life situations. Of course, the relevance of these situations to the learners is varying.

Word problems is an excellent way to train understanding of key concepts in a descriptive problem. That’s far more progressive compared to this worn-out testing the learner’s capability to perform algebraic manipulation as well as other fully mechanical skills.

The history of word problems

Today’s common notation enabling mathematical ideas to be symbolically expressed was developed in Europe in the 16th century. By the way, before this, all the mathematical solutions and problems were expressed solely by words. Therefore, the more complicated the mathematical problem, the more convoluted and laborious the verbal explanation was.

The first examples of word problems date back to Babylonian times. Besides a few procedure texts such as finding square roots, the vast majority of ancient Babylonian problems dealt with measurement of everyday activities and objects. For instance, at that time learners had to figure out lengths of canals dug, lengths of broken reeds, weights of stones, numbers of bricks in the construction, areas of fields, etc.

Ancient Egyptian mathematics also boasts vivid examples of word problems. A typical old Egyptian word problem states that there’re seven houses and in every house seven cats live. Every cat kills up to seven mice. Every mouse has already consumed seven grains of barley. Every grain would have produced up to seven hekat. The question – what’s the total sum of all the things mentioned above? You can look for more ancient word problems online.

Let’s handle word problems in the right way

Perhaps, the toughest thing regarding word problems is taking the English words and then translating them into math. Once a math equation is ready, you can congratulate yourself. However, calculating the actual equation might seem almost impossible. Fortunately, you can overcome this if you have enough practice.

When it comes to solving word problems, the first thing to do is to read it attentively. You shouldn’t get down to tackling it if you’ve only read half a sentence. Instead, you’d better try to get a clear feel for the whole issue.

Secondly, you require working in an organized manner. You should find out what you really need, but avoid naming things. You need to choose variable to designate the unknows, strictly labeling these variables with what they actually stand for. After this you require drawing and labeling pictures. You’re expected to explain your reasons clearly. That’s a common approach to word problems. Do you still want to learn more about word problems ?

Having done this, you need to seek keywords. Yes, they must be there. Here, we mean specialized words, standing for certain mathematical operations. We can’t do without them in word problems. Let’s cycle through them:

Subtraction

  • fewer than
  • less than
  • decreased by minus

Addition

  • added to
  • plus
  • increased by
  • more than
  • sum
  • total of

Multiplication

  • multiplied by
  • multiplication
  • factor of

Division

  • divided by
  • per
  • out of
  • ratio of
  • percent

Equals

  • yields
  • will be
  • were
  • was
  • are
  • is
  • sold for
  • gives

Now let’s try extracting the keywords from the word problems here below.

  • Convert the sum of y and 8 into a clear algebraic expression. We’ll get 8 + y.
  • Convert four less than x into an algebraic expression. Here we have x – 4.
  • Convert x multiplied by 13 into an algebraic expression. There’s nothing difficult here. We’ll obtain 13x.
  • Convert the quotient of x and 3 into an algebraic expression. We’ll cope with it too - x/3.
  • Convert the difference of five and y into an algebraic expression. Sure, you’ve guessed it. That’s 5-y.
  • Convert the ration of nine more than x to x into an algebraic expression. Let the magic be, we’ll get (x+ 9) / x.
  • Convert nine less than the total of a number and two into a clear algebraic expression and then have it simplified. As a result, we’ll get (n + 2) – 9, which could be simplified into n – 7.

Well, we’ve just dealt with relatively simple word problems. It’s high time to try more complicated ones.

The overall length of this particular football field exceeds its width by 30 yards. You require expressing the length of this field in terms of the width w. Regardless of the width w, the length is definitely 30 more than it. Plus would be a good substitution here. So, we’ll get w + 30.

Twenty gallons of oil were poured right into two containers, having different size. You should express the total amount of oil poured into the smaller container via the amount g delivered into the larger one. Well, we have 20 gallons. We’ve already poured g gallons of this commodity. Respectively, the expected answer is 20 – g.

That’s what you should know about word problems.

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