Algebra Activity
*Each student needs pencil, ruler, graphing calculator, one piece of notebook paper to show work, and one piece of graphing paper (preferably a copy with the x/y axis already on it, numbered and labeled).
Instructions:
1. Choose one single digit whole number, then choose a second single digit whole number, and write the two numbers down in the order you choose them.
(Teacher will assign each student a number of 1, 2, 3, or 4, and this will be the quadrant where they will graph their first point.)
2. The first number you chose will be the xcoordinate, and the second number you chose will be the ycoordinate of the first point you will graph. If necessary, make one or both of your numbers negative, so that they will represent a point in the quadrant you have been assigned. Now, graph your point and label the point with its coordinates (x,y).
3. Now, choose a positive integer less than 10, and graph it as a point on the yaxis. Label the point with its coordinates (x,y).
4. Next, choose a negative integer greater than 10, and graph it as a point on the yaxis. Label the point with its coordinates (x,y).
5. With a ruler, connect your first point and your second point to form a straight line. Extend the line across the graph paper, placing arrows at both ends to indicate the line continues on forever in both directions.
6. With a ruler, connect your first point and your third point to form a straight line, and extend the line across the graph paper, placing arrows at both ends to indicate the line continues on forever in both directions.
7. Use the slope formula (refer to reference sheet, if necessary) to find the exact slope of the first line you graphed. Show your work, and express the slope in simplified fraction form.
(If you need help with this, use the calculator, and press
“MATH”, “ENTER”, “ENTER.”)
Note: Make sure if your line slopes upward as you look from left to right, that the slope of the line is positive; and if your line slopes downward as you look from left to right, that the slope of the line is negative.
8. Since you now know the slope of your first line, and you already know the yintercept of the line, write the correct equation of your first line in slopeintercept form (y = mx + b). Write the equation of the line next to one of the two arrows of your first line on your graph.
9. Use the slope ratio (m = rise/run) to find the exact slope of the second line you graphed. On your graph, show dashes or dots as you count the run and then the rise from your first point to your third point (the yintercept with negative value). Then, measure the slope of your second line with the slope ratio from the third point to the first point. Show them both as simplified fractions.
(If you need help with this, use the calculator, and press
“MATH”, “ENTER”, “ENTER.”)
Note: Make sure if your line slopes upward as you look from left to right, that the slope of the line is positive; and if your line slopes downward as you look from left to right, that the slope of the line is negative.
10. Since you now know the slope of your second line, and you already know the yintercept of the line, write the correct equation of your second line in slopeintercept form (y = mx + b). Write the equation of the line next to one of the two arrows of your second line on your graph.
11. Now, take the coordinates of the very first point you graphed, and substitute them into the equation of your first line. Then, simplify both sides to verify the ordered pair really is a solution to your first equation. (Remember, this means both sides of the equal sign should end up with the same number, making the equation a true statement.)
12. Next, take the coordinates of that same point (the very first point you graphed), and substitute them into the equation of your second line. Then, simplify both sides to verify the ordered pair really is a solution to your second equation. (Remember, this means both sides of the equal sign should end up with the same number, making the equation a true statement.)
13. Take the coordinates of your second point (the yintercept with positive value), and substitute those coordinate values into the equation of your first line. Then, simplify both sides and state whether or not the ordered pair is a solution to your first equation.
*Write a sentence explaining why it is or is not a solution based on your results. Then, write a second sentence explaining why it is or isn’t a solution based on how the graph of the point and the graph of the line are related.
14. Take the coordinates of your second point (the yintercept with positive value), and substitute those coordinate values into the equation of your second line. Then, simplify both sides and state whether or not the ordered pair is a solution to your second equation.
*Write a sentence explaining why it is or is not a solution based on your results. Then, write a second sentence explaining why it is or isn’t a solution based on how the graph of the point and the graph of the line are related.
15. Take the coordinates of your third point (the yintercept with negative value), and substitute those coordinate values into the equation of your first line. Then, simplify both sides and state whether or not the ordered pair is a solution to your first equation.
*Write a sentence explaining why it is or is not a solution based on your results. Then, write a second sentence explaining why it is or isn’t a solution based on how the graph of the point and the graph of the line are related.
16. Take the coordinates of your third point (the yintercept with negative value), and substitute those coordinate values into the equation of your second line. Then, simplify both sides and state whether or not the ordered pair is a solution to your second equation.
*Write a sentence explaining why it is or is not a solution based on your results. Then, write a second sentence explaining why it is or isn’t a solution based on how the graph of the point and the graph of the line are related.
17. Rewrite your first equation, and substitute only the ycoordinate value of your first point into the equation. Then, solve the equation for x. Show all of your work. (Before you start, think about what the solution will have to be based on the results of your previous work.)
18. Rewrite your second equation, and substitute only the ycoordinate value of your first point into the equation. Then, solve the equation for x. Show all of your work. (Before you start, think about what the solution will have to be based on the results of your previous work.)
19. Now, write a new onevariable equation by using the “substitution method” (for solving a system of linear equations). To do this, look at each of your linear equations, and notice the right side of each equation is an expression “in terms of x,” and each expression is equal to the same variable “y.” So, set these two expressions equal to each other. (Before you start, think about what the solution will have to be based on the results of your previous work.)
Note: Remember, in algebra and in every aspect of life (at least I can’t think of an exception), when two different things are both equal in value to a third thing, then the two different things must be equal in value to each other.
20. Graph a third line that is parallel to your first line.
21. Next to your third line, write the correct linear equation for that line. What do you notice, when you compare the slopes of your first line and your third line?
22. Now, graph a fourth line that is parallel to your second line.
23. Next to your fourth line, write the correct linear equation for that line. What do you notice, when you compare the slopes of your second line and your fourth line?
24. Finally, determine exactly where your third line will intersect your fourth line.
