From Arithmetic to Algebra: Variables, Word Problems, Fractions and the Rules

Teaching Strategies

English-Math “dictionary”

On the first day of school, I always tell my Math students that Mathematics is a language. And one of its key features is that it is very concise. The compactness of symbolic expressions is an important feature that enables us to work with them efficiently. The trade-off is, the increase in labor needed for translation between symbolic expressions and verbal ones. Numbers and letters, and certain special symbols represent the “words” of this amazing language. When we connect the “words” using signs <, >, +, -, =, etc., we are writing sentences in the mathematical language.

Most of them look confused. Then, I ask them to translate some random English words into Spanish or French. They do it easily and with pleasure. The next assignment is to translate the words “add”, “subtract”, “equal” into mathematical language of symbols. It is important to point out that there are different ways to express in English things that will end up the same symbolically. For example, “the sum of a and b”, “take a and add it to b”, and “a plus b” all result in the expression a + b. This is a starting point of their journey into the wonderful world of mathematical symbols. My statement about Math being a language is becoming clear to them. 

Now it is my students’ turn to give their own examples. They get very excited.  We create our first English-Math “dictionary”. This is how it may look:

My favorite word that I always approach in a special way is the word “zero”. Zero means “nothing”.

I pay special attention to 0, because students do not feel confident with this number, especially when they solve equations and nothing is left on either side of an equation. They are very confused and fail to complete the problem. “0” is a very important and meaningful digit for mathematicians. The very discovery of this digit gave us a possibility to write numbers using a place-value chart. It is an important concept in writing algebraic expressions as well. I sometimes say that the importance of nothing was a major mathematical discovery.

Another chapter in this conversation is superscripts and subscripts.  Superscripts are small characters that are set slightly above the line; subscripts are below the normal line of type.  A superscript or subscript can be a number, a letter or a special math symbol.

Traditionally, students know the following superscripts:

A commonly used subscript denotes base systems, such as log₂10 (logarithm to base 2 of number 10). At some point, students also meet subscripts as indicating terms in a sequence.

The next logical questions would be, “Can we use a different symbol for addition, for example? Why do we need symbols? Why don’t we just use words to describe the methods of solving some math problems?” Also, what is the value of agreeing on a fixed symbol for a particular idea, such as combining numbers by addition?

Algebra is generalized Arithmetic

This is a good time to go back to prehistoric times and briefly follow the footsteps of humankind in its attempt to develop an effective tool to describe calculations and measurements.  

“The modern way to write down the numbers, simple and comfortable, the Europeans borrowed from the Arabs. In their turn, the Arabs borrowed this system from the Indians. Therefore, the Europeans call the modern digits “Arabic”, while the Arabs call them “Indian digits”. It is interesting that the Arabic and Indian versions of these symbols are somewhat different than the standard European ones. The English scientist and traveler Adelard introduced this system to the Europeans approximately in the year 1120. The vast majority of the countries accepted it only by the year 1600.” ² Fibonacci (Leonardo of Pisa) should also be mentioned in this development. So, it took humankind a pretty long time to even symbolize the numbers we use for counting.

The following long quotation captures the idea of my unit very well:

“Although what we now call “algebra word problems” were studied and solved since the days of ancient Egypt and Babylonia, around 4,000 years ago, perhaps more, symbolic algebra is a much more recent invention, pioneered by Francois Vi`ete just before 1600 and developed over the next 50 years. (The use of x for the unknown was popularized by Descartes.) Symbolic algebra in turn was a major enabler of the scientific revolution, in particular calculus. Imagine science without formulas. Imagine trying even to talk about derivatives, let alone compute them, without a compact notation to express the difference quotient.

The emphasis in the last sentence should be on `compact`. Think of a simple expression such as 3x+2. In essence, it is a recipe for computation. It implicitly says: take a number x, multiply it by 3, and add 2 to the result. However, it is substantially shorter! A compound expression such as 4y (3x + 2) - 7 can be translated as: take a number x, multiply it by 3, and add 2 to the result. Take another number y and multiply it by 4. Multiply the first result by the second, and then subtract 7.

As expressions get more complex, the contrast between the lengths of the full set of verbal instructions needed to paraphrase a symbolic expression and the brevity of the expression itself increases dramatically. The compactness of the symbolic form, together with a compact and elegant set of rules (the Rules of Arithmetic) for transforming (often for purposes of simplifying) expressions, allows the practitioner to compose and manipulate expressions whose verbal translations would be unmanageable. This combination of brevity and formality helps make symbolic algebra a powerful tool.

However, as with other topics in mathematics, the same compactness that makes symbolic notation powerful also makes it a challenge to teach. Students, especially students who may come to algebra unsure of the meanings behind numerical notation, and with limited understandings of the operations, will not immediately adapt to symbolic algebraic notation or realize the possibilities that it offers.” ³

“The main job of beginning algebra students is to become comfortable with working with variables, and in particular, working with symbolic expressions – at interpreting them, creating them, manipulating them, and using them to formulate and solve equations, and to interpret the solutions.”⁴

The main premise of this unit is “that teaching symbolic algebra, and its use in solving word problems, might profit from the linguistic perspective; that students might benefit from seeing, studying, discussing and working through the translation into algebra of many examples of verbally formulated situations, including work of solving these problems both with and without algebra, and comparing the algebraic and the arithmetic solutions to these problems. In doing this, they can gradually become acquainted with the language of algebraic notation, the vocabulary and grammar of polynomial expressions, and the rules that allow rephrasing within that language, (i.e., the principles for transforming expressions and equations). Moreover, they can practice translating arithmetic into algebra and algebra into arithmetic. By experiencing the strong connections between the two, they can come to appreciate the maxim that “algebra is generalized arithmetic”, rather than thinking of them as distant lands separated by a vast ocean, which is the situation of too many U.S. students (see, e.g., Lee and Wheeler (1989)).”⁵

The sooner we teachers start this job, the better results students will achieve.

Progressive problem set

For this unit, I have written a collection of problems that progress from very simple single step arithmetic problems, appropriate for first and second graders, to more complex algebraic problems that contain variables.

Numerical and algebraical expressions.

Problem I. A loaf of bread is $2. A large pizza is $20. For each question below, write a numerical expression that gives the answer to the question. Then compute the value of each expression.

1. By how many dollars is a loaf of bread cheaper than a large pizza?
2. How many times is a pizza more expensive than a loaf of bread?
3. What is the price of a pizza and a loaf of bread together?
4. What is the price of two large pizzas?
5. What is the price of five loaves of bread?
6. What is the price of two large pizzas and five loaves of bread together?
7. By how many dollars are the two large pizzas more expensive than the five loaves of bread?
8. How many times more expensive are two large pizzas than five loaves of bread?

Problems I 1) through I 5) are one-step problems for addition/subtraction and multiplication/division. Here it is important that students write down an expression first and not just calculate the answer. For example, in Problem I 3) the expression should look like:

The price of pizza and bread together: 20 dollars +2 dollars =22 dollars. It is important to require students to express the final answer in the correct units, and they should be aware of the unit attached to each number they use. All terms in an addition equation should refer to the same unit.

Problems 6) through 8) are three step problems. As an example, the expression for Problem I 6) is:

The price of two large pizzas and five loaves of bread, in dollars,  = 2 (20) + 5 (2) = 50.

Students should notice that all these expressions only contain numbers and symbols of arithmetic operations. Such expressions are called numerical expressions.

The next step will be to give students the same problem, but replace numerical values of the prices by letters.

Problem II. A loaf of bread is x dollars. A large pizza is y dollars. For each question below, write an algebraic expression that gives the answer to the question.

1. By how many dollars is a loaf of bread cheaper than a large pizza?
2. How many times is a pizza more expensive than a loaf of bread?
3. What is the price of a pizza and a loaf of bread together?
4. What is the price of two large pizzas?
5. What is the price of five loaves of bread?
6. What is the price of two large pizzas and five loaves of bread together?
7. By how many dollars are the two large pizzas more expensive than the five loaves of bread?
8. How many times more expensive are two large pizzas than five loaves of bread?

For example, the solution for Problem II 1) is:

A loaf of bread is y-x dollars cheaper than a large pizza.

Here it is important to teach kids that this problem can be phrased in a different way. We may ask: How much more expensive in dollars is a large pizza than a loaf of bread. Students should notice that the solution to the problem remains exactly the same.

Problem II 7) is more complex. Students should learn that the term for the more expensive product is always the first term in the expression.

The two large pizzas are more expensive than five loaves of bread. 2y-5x represents the difference in price. Again, kids should be able to redesign the problem to replace “more expensive” by “cheaper”. Thus, the problem will be: How much cheaper are five loaves of bread than two large pizzas, in dollars?

As in Problem II 1), the solution will not change.

Students should answer the question:  What is the difference between Problem I and Problem II? They should point out that the expressions in Problem II contain not only numbers and symbols of arithmetic operations, but also letters that represent numbers, that is, variables. Such expressions are called algebraic expressions.

Problem III. Write an algebraic expression, and then find its value given the value of each variable.

1. Three times the quantity of the difference between a and b. Find the value if a=5 and b=4.
2. The quotient of 25 by the sum of numbers x and y. Find the value if x=3 and y=2.
3. Triple number a and add it to b. Find the value if a=6 and b=10.
4. The difference of 72 and twice c. Find the value if c=20.

Problem III echoes Problem II, but this time there are no real life scenarios. Students should “translate” each algebraic expression from English using mathematical symbols, and then evaluate them. The solution for Problem III 3) may look like:

3a +b

3(6) + 10 = 18+10 = 28

It is important that problem sets include different scenarios, like Problems IV and V below. Kids can envision themselves in this real life situation, which makes the whole idea of solving word problems more relevant for them. These two problems also give an opportunity for a teacher to emphasize that mathematics is also the language of science. Physics teaches that to find distance we should multiply speed by time. If we replace each word in this rule by letters, we can describe it in mathematical language. “An equality that represents the rule for calculating the value of some variable is called a formula.”⁶

Problem IV. A car and a bicycle start at the same point, but they move in opposite directions. The speed of the car is 60 mi/h. The speed of the bicycle is 10 mi/h. For each question below, write a numerical expression that gives the answer to the question. Then compute the value of each expression.

1. What is the distance between the car and the bicycle one hour after they start?
2. What is the speed at which they are moving away from each other?
3. What is the distance between the car and the bicycle two hours after they start?
4. What is the distance that the car travelled after two hours?
5. What is the distance that the bicycle travelled after two hours?
6. How much greater is the distance travelled by car than the distance travelled by bicycle after 2 hours?
7. How many times greater is the distance travelled by the car than the distance travelled by the bicycle after 2 hours?

Problem V. A car and a bicycle start at the same point, but they move into opposite directions. The speed of the car is x mi/h. The speed of the bicycle is y mi/h. For each question below, write an algebraic expression that gives the answer to the question.

1. What is the distance between the car and the bicycle one hour after they start?
2. What is the speed at which they are moving away from each other?
3. What is the distance between the car and the bicycle two hours after they start?
4. What is the distance that the car travelled after two hours?
5. What is the distance that the bicycle travelled after two hours?
6. How much greater is the distance travelled by the car than the distance travelled by the bicycle after 2 hours?
7. How many times greater is the distance travelled by car than the distance travelled by bicycle after 2 hours?

A numerical and algebraic expression written with numbers, letters and symbols of arithmetic operations is a “translation” of real life events from English into mathematical language. As you have seen, several different scenarios may be described using exactly the same mathematical models. For this reason, mathematics gets used in construction, agriculture, medicine, engineering and many other fields of human life. The next two problems underline this important feature of mathematics.

Problem VI. It takes a car 2 hours to travel the distance of 180 km, while it takes 3 hours for a truck to cover the same distance. When will the car and the truck meet if the distance between them is 300 km and they start driving toward each other?

Problem VII. It takes the first team of tractor drivers 2 days to plow up 180 acres. The second team can do the same job in 3 days. How many days will it take to plow up 300 acres, if the two teams work together?

To solve Problems VI and VII students should set up and find the value of the same algebraic expression:

300÷(180÷2+180÷3)

These two problems will emphasize the idea that completely different real life situations can be described in the same way using mathematical language. This single numerical expression is a mathematical model of both of these real life situations. Note, however, that in Problem VI, the units are hours, while in Problem VII, the units are days.

Reverse translation.

It is extremely important for students to be able to do a reverse “translation”. They should understand what kind of real life situation a given math model describes.

Problem VIII. Look at the table below. Explain how you understand it.

As an example, students should mention that in the first column there is some data. The second column provides us with mathematical models based on the data and some new information. The third column gives us an idea of how algebraic expressions in the second column should be “translated” into English.

Problem IX. Create scenarios that can be described by the following numerical expressions:

1. 2 × 94+17.
2. 25 ÷(18÷6+18÷9).

This type of problem is important for developing math-modeling skills. Scenarios will vary. Students may struggle with these problems. I plan to lead a class discussion of this sort of problem, and will do some examples before asking my students to create their own scenarios for expressions. Afterwards, I will ask for volunteers to share their scenarios with the whole class.

Related Rates.

The problems in this section are designed for calculus students. To succeed in this topic they should easily “translate” word problems from English into mathematical “language”, feel confident setting up and evaluating algebraic expressions and working with formulas. This is the highest level of Mathematics available for schoolchildren. Regarding this topic, my students always mention that it makes them understand why they spend so much time mastering their skills on algebraic expressions. The “whole math” now makes sense to them.

As an example, let`s look at this relatively easy problem:

Problem X. “A point moves about on a circle of radius 6 inches, the law of its motion being

Ɵ =t ³-6t²+9t,  (10)

where t (time) is measured in seconds and Ɵ (the counterclockwise angle the radius to the point makes with the x – axis) in radians. Find (a) the angular velocity ω when t=4 seconds. Solution. (a) From (10) we obtain

ω = dƟ/dt= 3t²-12t+9.  (11)

Hence, …the angular velocity when t=4 seconds is

ω₁ = 3× 4²-12×4+9=9 radians per second.”⁷

Disregard the issue of computing the derivative; students must be able to evaluate expression (11). This problem is a good example to emphasize the importance of teaching algebraic expressions in middle school and early high school. 

The next problem is more rigorous.

Problem XI. “A point moves along a straight line in such a manner that the distance passed over (means the distance from the position when t = 0) varies as the cube of the time. If the point is observed to be 3 feet from the starting point at the end of 2 seconds, what will be its distance from the starting point and its velocity at the end of 6 seconds?

Solution. The general law of the motion when stated in the form of equation becomes

s=k×t³,  (3)

where k is some constant, and s stands for the distance travelled. Moreover, by hypothesis s=3 when t=2.

Hence (3) gives 3=k×8 so that k=3/8 ft/sec³. The precise law of the motion is therefore

S= (3 ̸8) ×t³  (4)

The desired distance from the starting point at the end of 6 seconds will therefore be

S₁=3/8× 6³= 81 feet.

As to velocity, we have from (4)

dS/dt=9/8× t²,

So that, in accordance with the explanations of this article, the desired velocity at the end of 6 seconds will be

V₁=9/8 × 6²=40 ½ feet per second”.⁸

Note: Actually, you don’t have to compute k. If the distance from the start is proportional to t³, then since 6 = 3x2, the point will be 3³ x 3 = 81 feet away.

As you can see, in the very first step students have to “translate” the first sentence in the problem from English into mathematical language. The relationship between distance and time should be expressed as a formula. If this skill has not been developed, they are going to fail the problem right away. This problem clearly illustrates the importance of teaching students how to translate word problems into mathematical models starting as early as is feasible.