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Mathematics as a Language of Symbols
byIrina AlekseevaContent Objective
While teaching my PreCalculus and AP Calculus students, I have realized that using mathematical symbols to create algebraic expressions and to solve text problems is usually a big problem for them. They struggle to translate a verbal problem statement into symbolic mathematical expressions and equations.
Let’s say students should solve the following problem: “Find the surface area of a sphere at the instant when the rate of increase of the volume of the sphere is nine times the rate of increase of the radius.”^{1} (Actually, this problem statement is not very good. The problem mentions “a sphere’, but it is not about a single sphere. It is about a family of spheres whose radius is varying, in what way or with respect to what variable is not explained.) As you can see, there is not a single math symbol in this problem. It is just one sentence written in English. Students should be able to “translate” it into a mathematical language using formulas and create a model based on the given data. This is an extremely challenging task for my students. Most of them are ready to give up without even trying.
The origin of this struggle goes all the way back to arithmetic and simple algebra. They should have been taught how to write algebraic expressions and create math models of a word problem. These skills are essential for their success in upper level math, physics and chemistry. Unfortunately, most of them have very modest, if any, skills of that kind. So, my goal in designing this unit is to help teachers and students to fill in this gap.
The fear of Mathematics is similar to the fear of speaking a foreign language. Some words in a foreign language we just do not know and, therefore, struggle with translation. To be successful in Math, we should be able to “translate” a problem from English into Mathematical language. Part of this is knowing what each word means. A deeper difficulty is that translation cannot be word for word. It must convey the overall meaning.
Coming back to the above mentioned related rates problem, students will learn, that the word combination “the rate of increase of the volume” is translated into mathematical language in a very simple way: dV/dt. Analyzing the problem, students should come to the conclusion that the volume is changing with respect to time. They should introduce variable t (time) and emphasize, that t was the unspecified variable, as a result of the incompleteness of the problem. The act of introduction of variable t should be made explicit. In similar fashion, the “rate of increase of the radius” is dr/dt.
To translate it further, we will need to create an algebraic expression:
dV/dt = 9 dr/dt
This expression looks complex and “all Greek” to Algebra students. However, the concept we have used while writing this expression is the same we would use to write a very simple algebraic expression “number A is nine times greater than number B”.
A = 9B
So, students should be taught writing algebraic expressions as early as possible.
Background
I teach at a small magnet high school where kids have to apply to be admitted. A big part of the admission policy is to attract students with very different backgrounds. Therefore, the student body is very diverse. We have kids from elite private schools as well as kids from really disadvantaged urban middle schools. As a result, students` skills vary a lot. I have to adjust my teaching style accordingly and approach my students in an individual way. While some of them take the most rigorous AP classes, a lot of them struggle with simple mathematical concepts. My goal, as a teacher, is to address every student`s needs and give all of them an opportunity to succeed.
Teaching Strategies
EnglishMath “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 tradeoff 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 EnglishMath “dictionary”. This is how it may look:
English 
Math 
add 
+ 
subtract 
 
multiply 
× 
divide 
÷ 
equal 
= 
approximately equal to 
≈ 
square root 
√ 
greater than or equal to 
≥ 
less than or equal to 
≤ 
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 placevalue 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:
degree (45 degree) 
45◦ 
percentage 
% 
number squared (exponent) 
x^{2} 
A commonly used subscript denotes base systems, such as log_{2}10_{ }(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.
 By how many dollars is a loaf of bread cheaper than a large pizza?
 How many times is a pizza more expensive than a loaf of bread?
 What is the price of a pizza and a loaf of bread together?
 What is the price of two large pizzas?
 What is the price of five loaves of bread?
 What is the price of two large pizzas and five loaves of bread together?
 By how many dollars are the two large pizzas more expensive than the five loaves of bread?
 How many times more expensive are two large pizzas than five loaves of bread?
Problems I 1) through I 5) are onestep 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.
 By how many dollars is a loaf of bread cheaper than a large pizza?
 How many times is a pizza more expensive than a loaf of bread?
 What is the price of a pizza and a loaf of bread together?
 What is the price of two large pizzas?
 What is the price of five loaves of bread?
 What is the price of two large pizzas and five loaves of bread together?
 By how many dollars are the two large pizzas more expensive than the five loaves of bread?
 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 yx 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. 2y5x 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.
 Three times the quantity of the difference between a and b. Find the value if a=5 and b=4.
 The quotient of 25 by the sum of numbers x and y. Find the value if x=3 and y=2.
 Triple number a and add it to b. Find the value if a=6 and b=10.
 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.
 What is the distance between the car and the bicycle one hour after they start?
 What is the speed at which they are moving away from each other?
 What is the distance between the car and the bicycle two hours after they start?
 What is the distance that the car travelled after two hours?
 What is the distance that the bicycle travelled after two hours?
 How much greater is the distance travelled by car than the distance travelled by bicycle after 2 hours?
 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.
 What is the distance between the car and the bicycle one hour after they start?
 What is the speed at which they are moving away from each other?
 What is the distance between the car and the bicycle two hours after they start?
 What is the distance that the car travelled after two hours?
 What is the distance that the bicycle travelled after two hours?
 How much greater is the distance travelled by the car than the distance travelled by the bicycle after 2 hours?
 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.
Given 
Math model 
Translation 
There are a oranges and b bananas in the bowl. 
a + b = 30 
Total number of fruit is 30. 
a = 2b 
The number of oranges is twice the number of bananas. 

a= b + 10 
There are 10 more oranges than bananas. 
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:
 2 × 94+17.
 25 ÷(18÷6+18÷9).
This type of problem is important for developing mathmodeling 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}6t^{2}+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= 3t^{2}12t+9. (11)
Hence, …the angular velocity when t=4 seconds is
ω_{1} = 3× 4^{2}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×t^{3}, (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/sec^{3}. The precise law of the motion is therefore
S= (3 ̸8) ×t^{3} (4)
The desired distance from the starting point at the end of 6 seconds will therefore be
S_{1}=3/8× 6^{3}= 81 feet.
As to velocity, we have from (4)
dS/dt=9/8× t^{2},
So that, in accordance with the explanations of this article, the desired velocity at the end of 6 seconds will be
V_{1}=9/8 × 6^{2}=40 ½ feet per second”.^{8}
Note: Actually, you don’t have to compute k. If the distance from the start is proportional to t^{3}, then since 6 = 3x2, the point will be 3^{3} 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.
Classroom Activities
In this section, I would like to discuss how I plan to use problems like the ones mentioned above to help my students improve their skills in translation and solving.
Lesson1.
Objective: Students will practice writing numeric and algebraic expressions to translate word problems into mathematical models.
The problems in the unit are designed in a way to gradually increase the rigor starting from Problem I, which is relatively simple. Thus, Problem I can be given in class as an example. I plan to facilitate a whole group discussion and help students write down numeric expressions. Students should find the value of each expression without use of a calculator.
Problem II builds on Problem I. Students should complete it independently using the ideas expressed in Problem I. Students may take turns to write the answer to each statement on the board to check the solutions.
I will decide what other problems to use for classwork, and will give similar ones for a homework assignment. However, I do not plan on doing more than five or seven for the first lesson. I also think that there should be several lessons with the same objective.
Lesson 2.
Objective: Students will practice to interpret each mathematical model in terms of the given scenarios.
This type of problem is traditionally complex. Students have difficulty to “decode” mathematical models. I will probably start with Problem VIII. In the beginning the third column, “Translation”, should be blank. A teacher will help students to analyze the mathematical models for each scenario and fill in the column. Students can work on the other problems independently in pairs or groups, followed by whole group discussion.
Again, depending on objectives, student skills and curricular timeframe, I will adjust the particular classwork and homework assignments.
Lesson 3.
Objective: students will solve related rates problems.
The topic is complex. My approach will be to start with Problem X gradually progressing with rigor. Students may work on Problems X and XI independently or in groups, followed by whole group discussion. I will especially pay attention to the units. It will help students understand the problem physically.
Note: The issue of units in Problem X is even more complicated than in XI. The “3”, the “12” and the “9” have different units.
Problem set
Problem I. A pound of strawberries is x dollars. A pound of cherries is y dollars. Write an algebraic expression for each phrase or question below.
 A price for 2 lb. of strawberries.
 A price for 3 lb. of cherries.
 By how much money is a pound of cherries more expensive than a pound of strawberries?
 How many times is a pound of cherries more expensive than a pound of strawberries?
 A price of 1 lb. of strawberries and 1 lb. of cherries together.
 A price of 2 lb. of cherries and 3 lb. of strawberries.
 By how much money are 2 lb. of cherries more expensive than 3 lb. of strawberries?
 How many times are 2 lb. of cherries more expensive than 3 lb. of strawberries?
To find the values of algebraic expressions we should know the value of each variable. If we know a price of 1 lb. of strawberries and cherries, we can find the value of each expression in Problem II.
Problem II. Let in Problem I 1 lb. of strawberries be equal to 2 dollars, and 1 lb. of cherries be equal to 6 dollars.
Now, find the value of each expression you have created to Problem I.
Problem III. Write a numerical expression for each phrase and find its value:
 A product of number 100 and the sum of numbers 8 and 7.
 A product of the difference of numbers 57 and 42 and number 1000.
 A quotient of the sum of numbers 32 and 24 by number 7.
 A quotient of number 81 by the difference of numbers 77 and 68.
Problem IV. Write an algebraic expression for each phrase:
 A product of number x and the sum of numbers y and z.
 A product of the difference of numbers a and b and number c.
 A quotient of the sum of numbers t and w and number q.
 A quotient of number f and the difference of numbers g and h.
Problem V. Write a numerical expression for each phrase and find its value:
 The sum of the product of numbers 15 and 2 and the quotient of number 42 by 6.
 The difference of the quotient of number 270 by 3 and the product of numbers 25 and 3.
 The sum of the product of numbers 17 and 3 and the product of numbers 4 and 13.
 The difference of the quotient of number 45 by 3 and the quotient of number 64 by 32.
Problem VI. A car and a bus have started their trips from the same point, going in opposite directions. The speed of the car is 60 mi/h. The speed of the bus is 50 mi/h. For each question below, write a numerical expression that gives the answer to the question. Then compute the value of each expression.
 What is the distance between the car and the bus after one hour?
 At what speed are they moving away from each other?
 What is the distance between the car and the bus after 2 hours?
 What is the distance travelled by the car in two hours?
 What is the distance travelled by the bus in two hours?
 How much larger is the distance travelled by the car than the distance travelled by the bus after 2 hours?
 How many times larger is the distance travelled by the car than the distance travelled by the bus after 2 hours?
Problem VII. A car and a bus have started their trips from the same point in opposite directions. The speed of the car is x mi/h. The speed of the bus is y mi/h. The car runs faster than the bus. For each question below, write an algebraic expression that gives the answer to the question.
 What is the distance between the car and the bus after one hour?
 At what speed are they moving away from each other?
 What is the distance between the car and the bus after 2 hours?
 What is the distance travelled by the car in two hours?
 What is the distance travelled by the bus in two hours?
 How much larger is the distance travelled by the car than the distance travelled by the bus after 2 hours?
 How many times larger is the distance travelled by the car than the distance travelled by the bus after 2 hours?
 What do expressions xy and 2x3y mean?
Problem VIII. The price for one tulip is x dollars, a rose is y dollars more expensive. Write an algebraic expression for each phrase:
 The price of a bouquet made of 5 tulips and 4 roses is $25.
 Three roses are 10 dollars more expensive than five tulips.
 The price of seven tulips is less than 20 dollars.
 The price of seven roses is more than 20 dollars.
 What do expressions 7x+3(x+y) and 12 (x+y) – 8x mean?
Note: I will ask my students to simplify these expressions, and state what the simplified expressions mean, and if this makes sense in the context of the problem; that is, is it clear that the simplified expressions should have the same values as the originals?
Problem IX. A pound of apples is m dollars; a pound of pears is 4 dollars more. Write an algebraic expression for each phrase:
 The price of 2 lbs. of apples and 3 lbs. of pears is 17 dollars.
 The price of 7 lbs. of pears is 30 dollars more than the price of 5 lbs. of apples.
 The price of 2 lbs. of pears is less than 12 dollars.
 The price of 4lbs. of apples is more than 3 dollars.
 What do expressions 3m+2(m+4) and 4(m+4)3m mean?
Problem X. Explain the following mathematical models related to the given data:
Given 
Mathematical model 
There are a chairs and b tables in a restaurant. 
1) a=4b 2) a=b+30 
Problem XI. Interpret each mathematical model with regards to the given scenarios.
Scenario 
Mathematical model 
There are a cows and b horses in a herd. 
1) a+b=30 2) a=3b 3) a=b+15 4) ab=17 5) a:5=b 
A hiker walked a miles and paddled b miles. 

A person paid a dollars for apples and b dollars for oranges. 

There are a girls and b boys in class. 
Note: Students should interpret all five mathematical models for each scenario.
Problem XII. Create scenarios that can be described by the following mathematical models.
 100 – 3×15.
 48÷ (10÷2+24÷8).
Related Rates Problems.
A list of Related Rates problems and solutions is offered by KhanAcademy (see Resources). Here teachers and students can find all major types of the problems that appear on the AP Calculus AB test. A web tutor will guide you through the whole process of problem solving.
Resources
Related Rates Problems
https://www.khanacademy.org/math/apcalculusab/derivativeapplicationsab/relatedratesab/e/relatedrates
Appendix
Implementing Mathematics Standards of Learning for Virginia Public Schools
Grade Four
Computation and Estimation
4.4 The student will
e) create and solve singlestep and multistep practical problems involving addition, subtraction, and multiplication, and singlestep practical problems involving division with whole numbers.
Grade Five
Patterns, Functions, and Algebra
5.19 The student will
a) investigate and describe the concept of variable;
b) write an equation to represent a given mathematical relationship, using a variable;
c) use an expression with a variable to represent a given verbal expression involving one operation; and
d) create a problem situation based on a given equation, using a single variable and one operation.
Grade Six
Patterns, Functions, and Algebra
6.14 The student will
a) represent a practical situation with a linear inequality in one variable; and
b) solve onestep linear inequalities in one variable, involving addition or subtraction, and graph the solution on a number line.
Grade Seven
Patterns, Functions, and Algebra
7.11 The student will evaluate algebraic expressions for given replacement values of the variables.
Grade Eight
Patterns, Functions, and Algebra
8.14 The student will
a) evaluate an algebraic expression for given replacement values of the variables; and
b) simplify algebraic expressions in one variable.
Expressions and Operations
A.1 The student will
a) represent verbal quantitative situations algebraically; and
b) evaluate algebraic expressions for given replacement values of the variables.
Above I have listed relevant standards. As you can see, starting from Grade Four, students should be able to translate a practical situation into a mathematical model that involve writing down a numerical or an algebraic expression and evaluate it. They should also be competent in writing and solving equations to represent a given mathematical relationship. Thus, elementary and middle school teachers will be able to use problems from my unit in their classrooms. After students master the content within my unit, they will be better prepared for upper level mathematics and physics where they have to translate word problems into mathematical models using mathematical language. In my classroom the unit will help me remediate my students who plan to take AP Calculus in the future.
Bibliography
Bashmakova, Isabella, Galina Smirnova The Beginnings and Evolution of Algebra. Cambridge: Cambridge University Press, 2000.
Bluman,Alan. Math Word Problems Demystified. New York: McGrawHill, copyright 2005.
Ford, Walter Burton. A First Course In The Differential And Integral Calculus. New York: Henry Holt And Company, copyright 1928.
Lam, Lay Yong, Tian Se Ang. Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China. Singapore: World Scientific, copyright 2004.
Ma,William. 5 steps to a 5 AP Calculus AB 2017. New York: McGrawHill, 2016.
Roger Howe, From Arithmetic to Algebra. Beijing: Mathematics Bulletin, 2010.
Vilenkin, Naum. Mathematics 5 th grade. Moscow: Mnemozina, 1997.
Zubareva, Irina, Alexander Mordkovich, Mathematics 5 th grade. Moscow: Mnemozina, 2004.
Zubareva, Irina, Alexander Mordkovich, Mathematics 6 th grade. Moscow: Mnemozina, 2004.
Endnotes
 William Ma, 5 steps to a 5 AP Calculus AB 2017, 178.
 Naum Vilenkin, Mathematics 5 th grade, 44 (translated from Russian by me).
 Roger Howe, From Arithmetic to Algebra, 1.
 Roger Howe, From Arithmetic to Algebra, 2.
 Roger Howe, From Arithmetic to Algebra, 2.
 Irina Zubareva, Alexander Mordkovich, Mathematics 6 th grade, 62 (translated from Russian by me).
 Walter Burton Ford, A First Course In The Differential And Integral Calculus, 107.
 Walter Burton Ford, A First Course In The Differential And Integral Calculus, 105.
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