Activities in Algebra
|The Problem of the Full Parking Lot|
|Pascal's Triangle Activity|
|Graphing Irrational Numbers|
|Open Box Construction|
|The Coordinate Plane|
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.
Put two wheels in each space.
Put two more in each space as long as you can, until you reach 66 wheels.
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.
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:
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.
If we let C represent the number of cars and M represent the number of motorcycles then we can arrive at two equations.
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|
DeMarois P. (1998). Mathematical Investigations. Reading, MA: Addison-Wesley
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
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)
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
Find the binomial expansion.
9) (2x - 1)4
10) (x - 8)3
11) (2x - 3)5
|Contributed by Audrey Smalley|
Contributed by Carol Conrad
Source: Pre-Algebra. New York: Glencoe/McGraw-Hill, 1997.
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:
Build a box, determine the volume, and record the information on the chart on the board or overhead.
|Contributed by Rodney S. Karjala|
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.
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|
Objective: to support and review students’ learning of basic arithmetic properties
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.
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.
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.
|Contributed by Laurie Kiss|