Following are some activities in algebra.
The items marked with are the contributions of the Summer 2000 participants.

 The Problem of the Full Parking Lot Pascal's Triangle Activity Graphing Irrational Numbers Open Box Construction The Coordinate Plane Property Match

The Problem of the Full Parking Lot

The purpose of this problem is to investigate the process of thinking about the procedure of doing mathematics and explore a problem in several ways. Let's begin!

Full Parking Lot: All 20 parking spaces in my favorite parking lot are filled. Some are occupied by motorcycles, and others by cars. Some people count to 10 when they get angry, but that wasn't nearly far enough. I counted wheels--66 to be exact. How many cars and how many motorcycles have invaded my territory?

You can do this problem, keeping in mind that there is more than one approach. Questions like the parking lot problem are very interesting to mathematicians because there are several very different strategies that lead to a solution. The following strategies might have been used on the given problem.

Guess and test: Guess a solution and test whether the answer matches all the conditions.

What assumptions do you make about the number of wheels? What do you know about the number of vehicles? Take a guess! Now test your potential solution. If you guessed 10 cars and 10 motorcycles and tested these values, you wrote (10 x 4) + (10 x 2) = 40 + 20 = 60 wheels. This value is too low so there must be more cars.

If there are 12 cars and 8 motorcycles, we get (12 x 4) + (8 x 2 ) = 48 + 16 = 64 wheels.

If we test 13 cars and 7 motorcycles, we get the correct solution: (13 x 4) + (7 x 2) = 66 wheels.

Draw a picture: Can you visualize the parking lot? You don't have to draw the vehicles, just the wheels.

You don't have to be an artist to be able to illustrate parking spaces and wheels. You can use the fact that there are 20 vehicles and focus on the number of wheels.

First draw 20 spaces.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Put two wheels in each space.

 1 2 3 4 5 6 7 8 9 10 oo oo oo oo oo oo oo oo oo oo 11 12 13 14 15 16 17 18 19 20 oo oo oo oo oo oo oo oo oo oo

Put two more in each space as long as you can, until you reach 66 wheels.

 1 2 3 4 5 6 7 8 9 10 oooo oooo oooo oooo oooo oooo oooo oooo oooo oooo 11 12 13 14 15 16 17 18 19 20 oooo oooo oooo oo oo oo oo oo oo oo

This last diagram not only represents the problem visually, but also displays the answer of 13 cars and 7 motorcycles.

Notice the reasoning used--since every space was occupied, there were at least 2 wheels in each spot. This uses up 40 wheels. Then adding 2 more to each space is continued until 66 is reached (the number of wheels).

If you initially drew 80 wheels, you need to subtract pairs of wheels in each space until you reach 66 wheels.

Write an equation: Use either the number of cars or the number of motorcycles as a variable to write an equation that represents the problem.

To write an equation, we need a variable. Let's begin by looking at a table to give us a better understanding of how the quantities are changing. We may also be able to see a pattern that will help us to choose what the variable should represent.

 # of Cars Car Wheels Motorcycles Cycle Wheels TotalVehicles # of Wheels 20 4(20) 20 - 20 2(20 - 20) 20 + (20-20) 4(20)+2(0) 19 4(19) 20 - 19 2(20 - 19) 19 + (20-19) 4(19)+2(1) 18 4(18) 20 - 18 2(20 - 18) 18 + (20-18) 4(18)+2(2) 17 16 . . . C

Inspect the information in the table. In the Car Wheels column, the number 4 stays the same. Why? Notice that 4 is multiplied by a different number each time--the number of cars in that row. Look at the other columns and rows. Do you see anymore patterns? Is the same number used across more than one column?

If the letter C represents the number of cars, complete the algebraic expression that describes each column.

We can see that since there are 20 vehicles, if there were 10 cars there would be 10 motorcycles. If ther were 12 cars there would be 20-12, or 8 motorcycles. Thus if C is the number of cars, then 20 - C is the number of motorcycles.

The equation is written using the number of wheels:

4 (C) + 2(20 - C) = 66

Solving the equation leads to a solution of 13. So there are 13 cars and 7 motorcycles.

Write a system of equations: Use the number of cars as one variable and the number of motorcycles as a second variable. Write two equations that represent the relationships between the number of cars and the number of motorcycles.

Sometimes it is easier to have a variable for each unknown quantity. Look at the following table and again look for patterns as you move from column to column and row to row.

 # of Cars Car Wheels Motorcycles Cycle Wheels TotalVehicles # of Wheels 20 4(20) 0 2(0) 20 + 0 4(20)+2(0) 19 4(19) 1 2(1) 19 + 1 4(19)+2(1) 18 4(18) 2 2(2) 18 + 2 4(18)+2(2) 17 4(17) 3 2(3) 17 + 3 4(17)+2(3) 16 . . 0 4(0) 20 2(20) 0 + 20 4(0) + 2(20) C 4(C) M 2(M) C + M 4(C) + 2(M)

If we let C represent the number of cars and M represent the number of motorcycles then we can arrive at two equations.

C + M = 20
4C + 2M = 66

There are several ways these equations can be solved, including substitution, elimination, graphing, using matrices, and using determinants. Regardless of which problem solving strategy you use, it is important to understand the problem situation and to organize the information to solve problems sucessfully.

Contributed by Karolee Weller

References:

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Pascal's Triangle

Blaise Pascal was a French mathematician who was born in 1623 and died in 1662. He spent time exploring an arithmetic triangle that was handed down by Chinese mathematicians several centuries earlier. Blaise discovered many new properties of the triangle and used it to solve problems that the triangle has become known as "Pascal's Triangle."

Complete the next three row of Pascal's Triangle.

Pascal noticed that the numbers in this triangle are precisely the same numbers that occur as coefficients in the binomial expansion of (x + y)n.

(x + y)0 = 1
(x + y)1 = x + y
(x + y)2 = x2 + 2xy + y2
(x + y)3 = x3 + 3x2y + 3xy2 + y3

Find the binomial expansion.

4) (x + y)4 =

5) (x + y)5 =

Expand the binomial sum (x + 2)3.

The binomial coefficients in Row 3 of Pascal's Triangle are 1, 3, 3, 1. Notice the top number of the Triangle is Row 0. The expansion is as follows.

(x + 2)3 = (1)x3 + (3)x2(2) +(3)x(22) + (1)(23)
= x3 + 6x2 + 12x + 8

Find the binomial expansion.

6) (x + 1)4

7) (x + 3)3

8) (2x + y)3

Expand the binomial difference (3x - y)4

The binomial coefficients in Row 4 of Pascal's Triangle are 1, 4, 6, 4, 1. The expansion is as follows.

(3x - y)4 = (1)(3x)4 + (4)(3x)3(-y) + (6)(3x)2(-y)2 + (4)(3x)(-y)3 + (1)(-y)4
= 81x4 - 108x3y + 54x2y2 - 12xy3 + xy4

Find the binomial expansion.

9) (2x - 1)4

10) (x - 8)3

11) (2x - 3)5

Contributed by Audrey Smalley

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Graphing Irrational Numbers

MATERIALS:

• compass
• straightedge
PROCEDURE:
1. Draw a number line. At 3, construct a perpendicular line segment one unit in length. Draw the line segment from 0 on the number line to the point one unit up on the perpendicular line in color. Label it c.
2. Using the Pythagorean theorem, you can show that the hypotenuese is sqr(10) units long.
 c2 = a2 + b2 c2 = 12 + 32 c2 = 10 c = sqr(10)
3. Open the compass to the length of the segment you drew in color. With the tip of the compass at 0, draw an arc that intersects the number line at B. The distance from 0 to B is sqr(10) units.

Source: Pre-Algebra. New York: Glencoe/McGraw-Hill, 1997.

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Open Box Construction

This is an activity that can be used at most levels of a middle school/high school math curriculum. It is a hands on activity that incorporates math skills learned at the various levels and encourages the use of technology. It also serves as a method of generating some statistics to study.

DESCRIPTION OF THE PROBLEM:
A manufacturer of boxes wishes to make open-topped boxes by cutting equal squares from the corners of a rectangular sheet of cardboard and then turning up the sides. What sized squares need to be cut from the corners to give the desired volumes.

MATERIALS NEEDED:
Construction paper (all the same size), scissors, ruler, scotch tape

INITIAL PREPARATION:

1. Discuss the process.
2. Discuss the limits to the size of the square.
3. Make a blank table (T-chart) on the board or overhead where the students will record their data.
4. Each student chooses a unique size of square to cut out and records this on the chart.
5. Point out that the cut out squares are to be saved for possible further study.
STUDENT ASSIGNMENT:
Build a box, determine the volume, and record the information on the chart on the board or overhead.

CLOSURE:
When students have completed the task, collect the boxes and display them for all to see. Once again, it depends on the level of student you are working with as to what follows. Following is a list of questions that vary in difficulty from simple to complex that would encompass general math to perhaps calculus. One has to be aware of the skill level of the class members and the subject matter being taught.

1. Which box has the most/least volume?
2. Which box involved the most/least waste?
3. What determined the amount of volume?
4. Is there more than one box with the same volume? Elaborate.
5. Can the relationship between volume and size of square as described on the table be graphed on the coordinate plane?
6. What would the graph look like if you connected the points from left to right?
7. Is this a function by definition?
8. Which dimension on the box determined the volume?
9. Write a function for the volume in terms of the length of side of the square. (Include the restrictions on the size of the square.)
10. Can you determine the volume for a given size of square from the graph? From the function? Elaborate.
11. Given a specific volume, can you determine the size of square to cut out from the graph? From the function? Elaborate.
12. If you actually graphed the function, with no restrictions, on a graphics calculator or computer, how would the graph compare to the one from the table with the attached restrictions? Elaborate.
13. Would there be an area function for the waste material? Extend.
14. What observations could be made if the graphs of the area function and the volume function vere placed on top of each other?
Contributed by Rodney S. Karjala

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The Coordinate Plane

The activity chosen is typically for a pre-Algebra or Algebra I class. However, depending upon the school district’s curriculum, it could be used at a lower level. It would be used when introducing graphing on the coordinate plane. The activity will better define how the coordinate plane came into existence.

ACTIVITY:
Have students get into pairs with their backs to each other. (Perhaps the students would need to be assigned by the teacher). Give one of the students of each pair a sheet of plain paper with a dot, a sticker, or some kind of mark on it. Give the other student of the pair a plain sheet of paper. Have one student describe to the other one the location of the mark, dot, or sticker while the second student tries to put an “X” on the location on the plain sheet from the description given. Repeat the activity with the other student doing the describing.

When completed, discuss how easy or difficult the task was. Point out that a way of doing this activity with more success and accuracy was developed by Rene Descartes, a French mathematician and philosopher who lived from 1596-1650. A story goes that once, while sick in bed, he noticed a fly on the ceiling. It came to him that there should be a way to accurately describe the location of the fly to someone else, like his nurse. Supposedly, from this thought he developed the cartesian coordinate plane with coordinates to determine the exact location of points on a plane.

Discusss the coordinate plane and its associated vocabulary. Be sure to point out the meaning of origin, intersecting lines, positive/negative directions, order, ordered pairs, abscissa, ordinate, graph, etc.

Collect all the papers with the stickers or marks on them, shuffle, and redistribute them. Also give each student a sheet of paper, or transparency, with the cartesian plane on it so they can lay it over their paper. Repeat the activity and have the students give the exact location.

Other activities could follow such as connect the dots, plotting some statistics, students finding their own examples, maps, teacher bringing in related examples, etc. The hope is that the activity will make students realize the importance of something that has been around for a very long time and how much it is used. One sees examples such as the population growth, stock market fluctuation, agriculture prices, interest rate changes, the cost of a college education, etc. This activity will serve as a basis for better understanding the concepts in future math courses.

Contributed by Rodney S. Karjala

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Property Match

Objective: to support and review students’ learning of basic arithmetic properties
 Properties may include: Additive Identity a + 0 = 0 + a = a Multiplicative Identity a1 = 1a = a Multiplicative Property of Zero a0 = 0a = 0 Other properties: Multiplicative Inverse, Reflexive Property of Equality, Symmetric Property of Equality, Transitive Property of Equality, Substitution Property of Equality, Distributive Property (of Multiplication over Addition and over Subtraction), Commutative Property, Associative Property

Level:
General math, pre-algebra, and algebra 1

Materials:
For the walls (or stations): one sign for each property covered, listing the name of the property and showing the property in symbolic notation.

For the students: each student or pair will need a list containing one numerical example to demonstrate each property. To avoid students following each other around the room, these lists may be prepared and cut into individual strips of paper, each containing one example. These strips can be bundled and stapled in advance, varying the order in each bundle so that students who have the same property on one strip will not match up again on the next. Students should also carry pencils to label each strip as the correct property name is found.

Historical notes:
Description of the associative property is attributed to Sir William Rowan Hamilton, an Irish mathematician who lived from 1805 to 1865. It was included in a paper published in 1848. That paper included work from 1843 to 1847, and it is thought that Hamilton recognized the importance of the associative law while working on Graves’s octaves in 1844.

The French mathematician Francois Joseph Servois, who lived from 1768 to 1847, described commutative and distributive properties. His memoir, published in 1814, described distribution over addition, and commutativity for multiplication.

It is less clear when the other properties that we take for granted or as truth came to common acceptance. Still, a discussion of the relatively recent date of the formalization of these properties could bring students to better understanding of the increasing and changing nature of mathematics. Properties that may now seem obvious or true have not always been assumed. Accordingly, topics that may seem mystifying to your students now may become their assumed, background knowledge in a short time.

Set-up:
Prepare the student materials as described above. Place property signs in visible locations around the room. Students will need to move to each sign. Students may work individually, or be assigned to work in pairs. Each student or pair should get one bundle of property strips before beginning.

Activity:
Students should start from their desks. Then when the teacher says “Begin,” each student finds the sign corresponding to the example on his or her first property strip, and goes to that sign. Students may compare strips when they arrive, to make sure that each student at their sign belongs there. Students may write the correct property name on the first strip. The teacher should be prepared to mediate in case of disagreement. If an aide is in the room, that person might circulate to verify student responses.

Next, the teacher can say, “OK, turn to the next strip. Ready, and move.” The procedure repeats until each student has identified each property example.

Discussion:
Discuss which properties were difficult, which got confused with one another, and why. Verify that all students got all of the properties labeled correctly. These property strips may serve as a study guide, or as a testing aid.

Variations:

1. Set up workstations with each property sign. Then after students identify the correct property, they would have a small number (4-6) of problems related to that property to complete. These problems could be prepared in advance, copied onto half-sheets of paper, and stacked at each station.

2. On each identifying sign list only the name of the property, giving no symbolic representation. This encourages students to learn the proper terminology for steps they take in their work. It may also help them to start recognizing properties in more general situations, thereby increasing the properties’ usefulness to them.

Contributed by Laurie Kiss

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