INTEGERS, NUMBERS AND GRAPHS
Slide 1. This is a flash animation of an exploding ball of numbers, notice it come out faster on the initial explosion. It keeps repeating itself. From Animation Factory, with permission
Slide 2. The basic theme in this presentation is to make the connection that graphs are two perpendicular number lines (one representing the x axis, the other representing the y axis). It is the intention of this presentation to make the connection that graphs with the x and y coordinates (Cartesian system) can help predict how a variable will react. In algebra the formula with unknowns is very important, but a student in pre-algebra needs to understand the importance the predictive ability of Cartesian graph.
The subtheme of “Rules, Rules, Rules!” is to make students aware that sometimes numbers do not make sense, but since numbers were invented there has to be rules in order to make some operations works.
Slide 3. Go over the definition of integers using a number line. Note that by definition it does not include fractions or mixed numbers. Why? That is one of those rules.
Slide 4. Rules of Integers do not have to make sense. Many students sometimes feel they are missing the point when their teacher tells them that -1 times -1 gives you a positive numbers. It might not make sense, but it is a rule. Ogden Nash (1902-1971) was an American poet noted for his light humor. He understood the point that this is just a rule, and you just accept it.
Slide 5. Definition of positive number can be illustrated by using a number line. They are whole number just right of zero.
Slide 6. Definition of negative number is a whole number less than zero or to the left.
Slide 7. Number lines can be used to distinguish one direction from a point which is called the origin. Sea level for example is just the point of origin. Positive number refer to relief (i.e., a mountain), and below sea level is negative relief called bathymetry.
Slide 8. Opposite numbers are the same distance from the point of origin. However the lines do not have to be equal, only the opposites have to be equal.
Slide 9. This is the answer slide if children are having problems. Show after they do the exercise.
Slide 10. Have the students solve the problem. Some teachers like to equate these to number lines, which is valid as long as it is only refers to subtractions and addition. Just one of those rules!
Slide 11. Integers have rules when you are working out a problem. Again they are the rules. If you do not use these rules you can get a different answer, especially on long equations. This pneumonic device can help your students remember the order: Please excuse my dear Aunt Sally (PEMDAS). Do the operation in the parentheses first, then exponents, multiplication, division, addition and subtraction.
Oh, enough about rules. We just want to learn them because math helps us (especially scientists, economists, and engineers) to predict the world around us. But what math does that. Algebra and graphs.
Slide 12. Students sometimes get confused with the definition of a graph. There are basically two types those that show data, but not important in algebraic theory. However, the Cartesian Coordinate system is fundamental to graphic algebraic formulas and very important. It is here that you must emphasis that Cartesian (x,y axis) system allows you to predict answers. The other graph systems do not.
Slide 13. Cartesian Coordinate System was developed by the famous mathematician Rene Descartes, a Frenchman. (read information on presentation for instructor)
Slide 14. Cartesian Coordinate System allows a user to predict how the numbers will behave. We are using the example x + y = 5. Have the students figure out what could equal five before you click the “play” button. If you plot the information on the x and y axis and connect the lines, every number on that line would be an answer. This gives a linear pattern. Every equation can be graphed. The graphs then can provide you with a way to compare one with another.
Use the angle ruler. Put it backwards with hole at origin. Ask students how many degrees are in first quadrant (0-90 degrees); second quadrant (91-180 degrees); third quadrant (181-270 degrees) and fourth quadrant (271-260 degrees).
Then turn the angle ruler right side and then ask them an angle to show. You would move one of the arms to 30 degrees and read it from the bottom.
If you have time have them put the right angle that they made from Lesson 1 to see the relationship.
Slide 15. Global Positioning System (GPS) uses the Cartesian Coordinate System and trigonometry to figure out where a person is located. The GPS satellites are positioned around the world and can be compared to a grid of longitude and latitude.
Exercise (if time):
Hand out the tuning forks and go over how they should use them. Be careful that they do not hit objects hard as they can misshape the tuning forks. Ask them if they can visualize how a graph might look
Slide 16. Even sound waves can be graphed. Show students the wave front.