Watkins Mill High School. Algebra 2. Math Challenge

Transcription

1 Watkins Mill High School Algebra 2 Math Challenge

2 "This packet will help you prepare for Algebra 2 next fall. It will be collected the first week of school. It will count as a grade in the first marking period."

3 NAME DATE PERIOD Practice: Skills Squares and Square Roots Find the square of each number Find each square root , , What is the square of 37? 18. Find both square roots of 4, Square Square 4.5. Glencoe/McGraw-Hill 610 Mathematics: Applications and Concepts, Course 2

4 Order of Operations -- PEMDAS Practice Worksheets Remember, PEMDAS (Please Excuse My Dear Aunt Sally) stands for: Parentheses Exponents Multiplication Division Addition Subtraction x x 24 8 x x x x x 19 x x 11 6 x x x 4 x x 7 x x 13 24

5 x ( ) ( 11 4 ) x (17 3 ) x (14 6 ) x ( ) x ( ) x x ( 9 x ) x ( ) 2 + ( ) x ( ) ( 24 4 ) x x (12 x ) ( ) 2 + ( ) 30. (19 8 ) x ( )

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7 Add, Subtract, Multiply and Divide Integers

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11 Add, Subtract, Multiply, Divide Fractions NAME Fractions are just parts of numbers. Usually, they are the most accurate parts of numbers, as they will tell us exactly how much of the number we have. We call these kinds of numbers rational numbers because they are ratios. Multiplying fractions requires multiplying together like parts. The numerator is multiplied by other numerators, and the denominator by other denominators. Example 1: 5 6 = 5 6 = Example 2: 3 1 = 3 1 = 3 = Notice that on the second one, we chose to reduce the fraction. While this does not change the value of the fraction, 1 out of 8 parts is often easier to visualize than 3 out of 24 parts. Now, you try. Reduce when appropriate Dividing fractions requires us to perform an extra step. Instead of multiplying directly across, we must change the second fraction before we multiply, as seen below: Example 1: Example 2: 5 6 = = = = 18 4 = 9 2 We can change these numbers to mixed numbers. Example 1 would be 1 13 and 42 Example 2 would be 4 1 if we did. But, in Algebra 2, we rarely do unless they are whole 2 numbers because of the ease of multiplying fractions as opposed to mixed numbers. Now, you try. Reduce when appropriate

12 Add, Subtract, Multiply, Divide Fractions NAME Adding and subtracting fractions is a very different process than multiplying and dividing fractions. We can agree that 1 out of 2 parts is different than 1 out of 3 or one out of four: As you can see, these are all very different shapes. There s no reason why we should be able to add them together without changing the type of fraction first; that would be like adding circles to triangles. What we do to change fractions into the same size is change their denominators to a common denominator. Let s add We do this by multiplying by different 2 3 versions of the number 1, because any number times 1 is itself = This is true because 3 = 1 and 2 = Notice we use the OTHER denominator to multiply. We multiplied 3 1 because the 3 2 other denominator is 3, and we did 1 2 because the other denominator is 2. This gives 3 2 us a common denominator of 6. From multiplying on the last page, we know this will give us two new fractions, We can add together these fractions. 3 sixths plus 2 sixths = 5 sixths, or = 5 6. Another example: And an example of subtraction: = = 15 and 1 4 = Now we add: = = = Now, you try. Reduce when appropriate

13 Name: Class Pd: Date: Algebra 2 1.4B Worksheet Solving Equations & Algebraic Expressions Write an algebraic expression that models each word phrase. 1. six less than a number 2. the product of 11 and the difference of 4 and a number 3. the quotient of 3 and a number increased by 1 4. the sum of 5 and a number Write a verbal expression that models each algebraic expression or equation. 5. x + 5 = (z 3) 7. 2x 3 5 Solve each equation. x n = y = w =2w s =3s (x +3) +2(x +4) = z + 12 = 5z b 6(11 2b) = k 7 = 2(13 5k)

14 Solving One-Step Equations IMPORTANT: Do each problem on a separate sheet of paper. Then complete the maze by shading in the path with the correct answer. Example:

15 Solving Two-Step Equations IMPORTANT: Do each problem on a separate sheet of paper. Then complete the maze by shading in the path with the correct answer.

16 Substituting for a variable Substituting a number for a variable is one of the most important ideas in all of mathematics. A few things we wouldn t have without this idea: functions, graphs, lines, points, and variable equations. All of algebra depends on being able to perform this function. If we have an equation 2x + 3 = 11, a little algebra determines that x = 4. This means we can replace the letter x with the number 4, and we get 2(4) + 3 = 11. Since = 11, this proves that we got the correct answer to our equation. Functions require us to substitute values multiple times to create different points on a graph. To make a function, you need two or more variables so that as one changes, the other can change. Here s a linear function: y = 2x + 3. Like we saw earlier, if y = 11, we could re-write this as the equation 11 = 2x + 3 because we replace y with the number 11. We just solved above for x = 4. Using the equation y = 2x + 3, find the value of y when: 1) x = 7 2) x = -7 3) x = 10 4) x = 1.5 5) x = -1.5 Now, using the same equation y = 2x + 3, find the value of x when: 1) y = 7 2) y = -7 3) y = 10 4) y = 1.5 5) y = -1.5 Notice starting with y and solving for x takes more work, so try if possible to use x to find y. Substituting for x 2 is a little trickier and is a process often done in error by students, especially with negative numbers. The term x 2 means x times x. Since a negative number multiplied by another negative number is a positive number, x 2 will never give you a negative answer. So if x = 7, x 2 = 7 2 which means 7 * 7 = 49. If x = -7, x 2 = (-7) 2, which means -7 * -7 = 49 also. Whatever is under an exponent, the exponent means to multiply that thing by itself, that many times. x 3 = x * x * x, and (-3) 4 = -3 * -3 * -3 * -3. Using the equation y = 3x 2 4, find the value of y when: 1) x = 7 2) x = -7 3) x = 10 4) x = 4 5) x = -5

17 Graphing Functions Using A Table Now that we understand how to substitute values into a function, we can start working with what we call tables to graph different functions. Some functions you will see in Algebra 2 will allow you to use any value for x or y, while other functions will have some values not allowed. The functions we will see here will allow us to use any value for x. Take the function we saw on the other page: y = 2x + 3. Now, let s look at some x values as x and y coordinates: 1) x = 3 2) x = -6 3) x = 2 4) x = 0 y = 2(3) + 3 = 9 y = 2(-6) + 3 = -9 y = 2(2) + 3 = 7 y = 2(0) + 3 = 3 You can put these points on the graph as (x, y) coordinate points: 1) (3, 9) 2) (-6, -9), 3) (2, 7) 4) (0, 3) Or we could make these into a table of values: x y Notice it is easier figuring out values for y using values for x than the other way around, because it requires less algebra to solve. We also try to organize the table by the lowest x (in this case -6) to the highest x (in this case 3). If you connect the dots, you ll also notice you get a line, which is why we call this a linear graph.

18 Graphing Functions Using A Table Fill in the tables and graph the points on the coordinate plane, then draw the shape of the graph. y = 3x - 4 x y y = 3x 2-4 x y y = -2x + 4 x y