Math 6, Unit 11Notes: Integers, Graphs, and Functions

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1 Math 6, Unit 11Notes: Integers, Graphs, and Functions We use positive and negative numbers daily in real life. Positive numbers are those numbers greater than zero. They can be written with a positive (+) sign in front of them, but we usually write a positive number without the sign. Negative numbers are those numbers less than zero. They must be written with the sign ( ). Common situations that use positive/negative numbers are those involving temperature (above/below zero), business (profit, loss), bank accounts (deposits/withdrawals), sea level (above/below), and football (gain/loss of yardage). Using the number line, we can show positives, negatives, zero, and opposites. Note that zero is not positive or negative. negative numbers positive numbers opposite numbers We can now define integers: set of whole numbers and their opposites. (Zero is its own opposite). The number line can also help us when we are asked to compare and order integers. Remind students that numbers increase as we move from left to right on the number line. Particularly troublesome for some students is the comparison of two negatives. (A rule that makes sense to students is that the larger negative number is always the one closest to zero.) Example: Which is larger, 3 or 2? Graphing on a number line, we can see that 3 lies to the left of 2, so 3< 2 or 2 > 3 This makes sense as a person standing at 2 miles above sea level is higher (greater)than a person standing at 3 miles below sea level. Example: Which is larger, 5 or 1? Graphing on a number line, we can see that 5 lies to the left of 1, so 5 < 1 or 1 > 5 This makes sense as 5 degrees below zero ( 5) is colder than 1 degree below zero ( 1) Once a student can compare two integers, ordering is a simple task Math 6 Notes Unit 11: Integers, Graph, and Functions Page 1 of 7

2 Example: Order the integers from least to greatest. 2, 18, 20, 25, 32 The number furthest to the left on the number line would be 25, so it is the smallest number. Negatives are always smaller than positives, so 2 would be the next number. 18, 20 and 32 would follow. 25, 2, 18, 20, 32 The Coordinate Plane Objective: (6.20)The student will locate points, using all four quadrants of a coordinate plane. The way to locate points in a plane is on a coordinate grid, also known as the Cartesian Coordinate System. A coordinate plane is formed by intersecting two number lines at a right angle. The horizontal number line is referred to as the x-axis, and the vertical line is the y-axis. The point of intersection is at zero on both number lines, and it is called the origin. The axes divide the coordinate plane into four quadrants. You can name any point on this plane with two numbers, called coordinates. The first number, the x-coordinate, is the distance from the origin along the x-axis. The second number, the y-coordinate, is the distance from the origin along the y-axis. The pair of numbers is always named in order, first x, then y; hence, they are xy., One way to remember the order is that ordered pairs are listed in called an ordered pair, ( ) alphabetical order: ( xy,, ) (horizontal axis, vertical axis). Quadrant II Quadrant III Quadrant I ( 0,0) origin Quadrant IV Math 6 Notes Unit 11: Integers, Graph, and Functions Page 2 of 7

3 Tables and Functions Objective: (4.1)The student will create a table relating two variables from a given rule. Ordered pairs can be written from a chart or table of values. Example: Write the ordered pairs using the table below. x y The ordered pairs are (1, 6), (2, 7), (3, 8), and (4, 9). These ordered pairs could then be graphed as points on the coordinate Example: Write the ordered pairs using the table. The ordered pairs are (0, 3), (1, 4), (2, 5), (3, 6), and (4, 7). These ordered pairs could then be graphed as points on the coordinate x y plane. plane. A function is a rule that produces exactly one output value for each input variable. The input value (x-coordinate) is the value substituted into the rule; the output value (ycoordinate)is the value that results from the given input. To find the ordered pairs, you substitute any values of x you choose; then using those values, find the corresponding y values. Example: Find three ordered pairs that can be written from the function y = 2x+ 3. We can pick any values we would like for x. 0, 1, and 3 have been chosen. We substitute those values into the rule y = 2x+ 3and find the corresponding y values. x 2x + 3 y ( ) 1 21 ( ) ( ) The ordered pairs are (0, 3), (1, 5), and (3, 9). Math 6 Notes Unit 11: Integers, Graph, and Functions Page 3 of 7

4 Of course your ordered pairs could be different, depending on the values of x you chose. Remember, these ordered pairs can now be graphed on the coordinate plane. Math 6 Notes Unit 11: Integers, Graph, and Functions Page 4 of 7

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Integers and the Coordinate Plane

Integers and the Coordinate Plane Name Date Class 9A Dear Family, A Family Letter: Understanding Integers The student will begin the study of an important set of numbers called integers. Integers are the set of numbers that include all