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

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.

1. A price for 2 lb. of strawberries.
2. A price for 3 lb. of cherries.
3. By how much money is a pound of cherries more expensive than a pound of strawberries?
4. How many times is a pound of cherries more expensive than a pound of strawberries?
5. A price of 1 lb. of strawberries and 1 lb. of cherries together.
6. A price of 2 lb. of cherries and 3 lb. of strawberries.
7. By how much money are 2 lb. of cherries more expensive than 3 lb. of strawberries?
8. 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:

1. A product of number 100 and the sum of numbers 8 and 7.
2. A product of the difference of numbers 57 and 42 and number 1000.
3. A quotient of the sum of numbers 32 and 24 by number 7.
4. A quotient of number 81 by the difference of numbers 77 and 68.

Problem IV.  Write an algebraic expression for each phrase:

1. A product of number x and the sum of numbers y and z.
2. A product of the difference of numbers a and b and number c.
3. A quotient of the sum of numbers t and w and number q.
4. 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:

1. The sum of the product of numbers 15 and 2 and the quotient of number 42 by 6.
2. The difference of the quotient of number 270 by 3 and the product of numbers 25 and 3.
3. The sum of the product of numbers 17 and 3 and the product of numbers 4 and 13.
4. 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.

1. What is the distance between the car and the bus after one hour?
2. At what speed are they moving away from each other?
3. What is the distance between the car and the bus after 2 hours?
4. What is the distance travelled by the car in two hours?
5. What is the distance travelled by the bus in two hours?
6. How much larger is the distance travelled by the car than the distance travelled by the bus after 2 hours?
7. 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.

1. What is the distance between the car and the bus after one hour?
2. At what speed are they moving away from each other?
3. What is the distance between the car and the bus after 2 hours?
4. What is the distance travelled by the car in two hours?
5. What is the distance travelled by the bus in two hours?
6. How much larger is the distance travelled by the car than the distance travelled by the bus after 2 hours?
7. How many times larger is the distance travelled by the car than the distance travelled by the bus after 2 hours?
8. What do expressions x-y and 2x-3y 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:

1. The price of a bouquet made of 5 tulips and 4 roses is $25.
2. Three roses are 10 dollars more expensive than five tulips.
3. The price of seven tulips is less than 20 dollars.
4. The price of seven roses is more than 20 dollars.
5. 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:

1. The price of 2 lbs. of apples and 3 lbs. of pears is 17 dollars.
2. The price of 7 lbs. of pears is 30 dollars more than the price of 5 lbs. of apples.
3. The price of 2 lbs. of pears is less than 12 dollars.
4. The price of 4lbs. of apples is more than 3 dollars.
5. 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:

Problem XI. Interpret each mathematical model with regards to the given scenarios.

Note: Students should interpret all five mathematical models for each scenario.

Problem XII. Create scenarios that can be described by the following mathematical models.

1. 100 – 3×15.
2. 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.