Common Core Math Tutorial and Practice

Common Core Math Tutorial and Practice

TABLE OF CONTENTS Chapter One Number and Numerical Operations Number Sense...4 Ratios, Proportions, and Percents...12 Comparing and Ordering...19 Equivalent Numbers, Fractions, Decimals, and Percents...25 Repeating Decimals and Irrational Numbers...34 Progress-monitoring Quiz #1...39 Pencil and Paper Computations...43 Exponents...52 Square and Cube Roots...56 Proportions and Percents...60 Progress-monitoring Quiz #2...64 Order of Operations...68 Estimating Square and Cube Roots...72 Estimation...75 Accuracy of Estimates...79 Progress-monitoring Quiz #3...83 Chapter Two Geometry and Measurement Lines, Angles, and Planes...87 Pythagorean Theorem...100 Polygons...105 Progress-monitoring Quiz #4...112 Similar Figures...120 Geometric Logic...128 Constructions...131 Nets...141 Transformations...146 Geometric Patterns...152 Coordinate Geometry...158 Transformations on Coordinate Grids...166 Progress-monitoring Quiz #5...173 Standard and Metric Equivalents...179 Compound Measurement Units...184 Perimeter and Area...188 Progress-monitoring Quiz #6...198 Surface Area and Volume...204 Volume of Pyramids and Cones...218 Volume and Surface Area of a Sphere...222 Progress-monitoring Quiz #7...226

Chapter Three Patterns and Algebra Patterns...230 Graphing...238 Modeling Situations...247 Progress-monitoring Quiz #8...252 Graphing on Number Lines...256 Solving Equations...261 Solving Inequalities...267 Progress-monitoring Quiz #9...272 Evaluating and Simplifying...276 Progress-monitoring Quiz #10...281 Chapter Four Data Analysis, Probability, and Discrete Mathematics Representing Data...285 Analyzing Data...294 Lines of Best Fit...300 Surveys...302 Progress-monitoring Quiz #11...304 Probability...310 Compound Events, Modeling, and Analyzing Probabilities 318 Conditional Events...326 Progress-monitoring Quiz #12...330 Permutations, Factorial Notation, and Combinations...336 Venn Diagrams...343 Systematic Listing, Counting, and Reasoning...348 Vertex Edge Graphs and Algorithmic Thinking...354 Progress-monitoring Quiz #13...362

Probability You have probably heard the word probability in your math class before. Probability is a ratio that shows how likely an event is to happen. For example, if you throw a six-sided number cube once, the probability that you will roll a 4 or a 5 is 2/6. This ratio is found by the rules below. probability = (number of desired outcomes) (number of possible outcomes) In this case, there are 6 possible outcomes since the number cube has 6 sides. So the denominator will be 6. Since there is one 4 and one 5 on the number cube, there are two desired outcomes, so the numerator is 2. Therefore, the probability of rolling a 4 or 5 is 2/6 or 1/3. Just like other ratios, probabilities can be expressed in a number of different ways. Words one in three Ratio 1:3 Fraction 1/3 Percent 33 1/3% Now that you know how probabilities are found, you can practice applying your skills to many different types of problems. Always take the time to count the total number of possible outcomes and identify the desired outcomes.

Example 1: Dot pulled one marble out of the bag below. What is the probability that Dot will draw a marble that is white? Step 1: Identify the number of possible outcomes in the bag. There are a total of 15 marbles in the bag, so the denominator must be 15. Step 2: Identify the number of desired outcomes. There are 3 marbles that are white. So, the numerator will be 3. Step 3: Use the information from Steps 1 and 2 to create a probability ratio. Simplify the ratio, if possible. Probability = 3/15 (3 3)/(15 3) = 1/5 The probability of drawing a white marble is 1/5.

Example 2: Wayne put the cards below into a bag and drew one out without looking. N E W B R U N S W I C K What is the probability that Wayne will draw a card with the letter N? Step 1: Identify the number of possible outcomes in the bag. There are 12 index cards in the bag, so there are 12 possible outcomes. Step 2: Identify the number of desired outcomes. Two of the cards in the bag have an N on them. So, there are 2 desired outcomes. Step 3: Use the information from Steps 1 and 2 to create a probability ratio. Simplify the ratio, if possible. Probability = 2:12 (2 2):(12 2) = 1:6 The probability of drawing an N out of the bag is 1:6.

Example 3: Leon had a bag with different-colored pieces of paper. He drew a piece of paper out of the bag, recorded the color, and replaced it. He performed this experiment 100 times. The table below shows his results. Color Number of Times Drawn Orange 31 Blue 9 Yellow 48 Pink 12 Using the information above, most of the pieces of paper are probably what color? Step 1: Analyze the table to find the color that was drawn the most times. The color yellow was drawn the most times. Step 2: Understand the probability results. Since the yellow pieces of paper were drawn most, there are probably more yellow pieces of paper than any other color. Two specific types of of probability you will encounter are experimental and theoretical probability. Experimental probability, like probability, is a ratio. Experimental probability is very specific because it refers to the results of a particular experiment. It is the ratio of the number of times an event occurs to the total number of trials.

Another way to say this is: experimental probability = (number of times desired outcomes occurred) (number of trials) You can find experimental probability only after actually carrying out the experiment. Theoretical probability, unlike experimental probability, is not based on the actual results of an experiment, but on the possible results of the experiment. Theoretical probability is the ratio of the total number of ways an event, or a desired outcome, can occur, to the total number of possibilities. So unlike experimental probability, you can figure out the theoretical probability before you actually perform the experiment. Example 4: Srich decided to flip a quarter 100 times and to record the results of his experiment. What is the theoretical probability of Srich s quarter landing on tails? Step 1: Identify the total number of possible outcomes i.e., how many ways the coin can land. There are only two ways the coin can land heads up, or tails up. Step 2: Identify the total number of ways the desired outcome i.e., the coin landing on tails can occur. There is only one way the coin can land tails up tails up! Step 3: Use the information from Steps 1 and 2 to create a theoretical probability ratio. theoretical probability = 1 2 The theoretical probability of Srich s coin landing on tails is 1/2, 1 out of 2, or 50%.

Example 5: Srich has completed his coin-flipping experiment and recorded his results in the table shown below. Coin-tossing Experiment Heads Tails 38 62 Based on the information in the table, what is the experimental probability of Srich s coin landing on tails? Step 1: Identify the total number of trials. Srich tossed the coin 100 times. Step 2: Identify the number of times the coin landed on tails. The coin landed on tails 62 times. Step 3: Use the information from Steps 1 and 2 to create an experimental probability ratio. Simplify the ratio if possible. experimental probability = 62 = 31 100 50 The experimental probability of Srich s coin landing on tails is 31/50, or 31 out of 50, or 62%.

Practice 4 5 1 A B 3 1 7 3 1 4 7 8 2 5 6 3 6 2 3 5 4 Express all the following probabilities as a percent, a ratio, and a fraction. What is the probability that in one spin... 1. an even number will come up on spinner B? 2. a three will come up on spinner A? 3. a number greater than 5 will come up on spinner B? 4. a number greater than 5 will come up on spinner A? 5. an odd number will come up on spinner A?

Name Date Practice 4 5 1 A B 3 1 7 3 1 4 7 8 2 5 6 3 6 2 3 5 4 Express all the following probabilities as a percent, a ratio, and a fraction. What is the probability that in one spin... 1. an even number will come up on spinner B? 50%, 1:2, 1/2 2. a three will come up on spinner A? 25%, 1:4, 1/4 3. a number greater than 5 will come up on spinner B? 37.5%, 3:8, 3/8 4. a number greater than 5 will come up on spinner A? 16.6%, 1:6, 1/6 5. an odd number will come up on spinner A? 66.6%, 2:3, 2/3