CSET Multiple Subjects Test - Subtest II: Mathematics Study Guide PDF

6/16/23, 9:26 PM Subtest II: Mathematics Study Guide for the CSET Multiple Subjects Test This Study Guide is brought to...

6/16/23, 9:26 PM Subtest II: Mathematics Study Guide for the CSET Multiple Subjects Test This Study Guide is brought to you by + Union Test Prep becomeanything.com https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 1 of 28  6/16/23, 9:26 PM This download is subject to a revocable, non-exclusive, non-transferable, limited right to use this download by You (an individual person) on a single device owned and controlled by You or a single printed copy. You agree not to reproduce, decompile, modify, or otherwise alter this download. You agree that You understand that the download includes Copyrighted material and other intellectual property of Union Media, and that use of the download is strictly in accordance with this limited single-use license. 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Unauthorized copying or reuse of any part of this material is illegal and prohibited. https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 2 of 28  6/16/23, 9:26 PM Table of Contents General Information Number Sense Number Ideas Place Value Number Theory Number Systems Number Order Special Notation Working with Exponents Operations with Positive and Negative Numbers Relationships between Operations Properties Working with Numbers Algorithms The Order of Operations Rounding and Estimating Using Technology for Complex Calculations Algebra and Functions Patterns and Relationships Proportional Reasoning Dependent and Independent Variables Equations and Inequalities Equivalent Expressions Matching Expressions to Situations https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 3 of 28  6/16/23, 9:26 PM Expressing Problems Algebraically Properties of Linear Equations Working with Polynomials Working with Quadratic Equations Interpreting Graphs Measurement and Geometry Geometric Objects Characteristics Object Comparisons The Pythagorean Theorem Parallel Lines Representing Geometric Objects Using Tools Combining and Dissecting Objects Measuring Geometric Objects Formulas for Measuring Comparing Measurement Systems Using Proportions Using Other Measurement Forms Statistics, Data Analysis, and Probability Collecting and Representing Data Representing Data Data Concepts Surveys Analyzing Data Interpretation Finding Patterns https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 4 of 28  6/16/23, 9:26 PM Drawing Conclusions Probability Events Expressing Probabilities Compound Events Math Skills Problem Solving Securing Information Solving Modeling Using Other Information and Strategies Reasoning Reasoning Resources Accuracy Checks Explanation Academic Language Mathematical Notation Relationships https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 5 of 28  6/16/23, 9:26 PM General Information Exactly half of Subtest II of the CSET® Multiple Subjects test is devoted to mathematics. There are a total of 28 questions on math: 26 multiple-choice and 2 that are constructed response, in which you’ll have to write a short, focused answer on your own. The questions assess your skills and knowledge in a variety of math domains, including numbers, operations, algebra, geometry, measurement, statistics and data analysis, and probability. You will have access to an online calculator for these questions. Following is a guide to use as you study. Be sure to access additional resources if you encounter problems with any of this content. Number Sense Those with good number sense are able to make good judgments about how to use numbers to solve a variety of problems. They can wisely choose what mathematical operations to use and whether the answer they get is reasonable. Sadly, there is no short-cut to achieving this. Number Ideas Place Value Starting at the decimal point, whether it is written or implied, each digit to the left represents the next higher multiple of ten, i.e. 10, 100, 1000, 10,000 etc. Each digit to right represents the next lower fractional division of ten, i.e. 1/10 ( 0.1), 1/100 (0.01), 1/1000 (0.001) etc. Here is a table showing the place values for 6,374.502 6 3 7 4. 5 0 2 thousands hundreds tens ones tenths hundredths thousandths Number Theory Number theory is the branch of mathematics that deals with interesting patterns in the set of positive whole numbers, known as the natural numbers. There are different ways to group these numbers, some that are well known such as the even numbers, the odd numbers, and the prime numbers, and some that are more curious, such as square numbers and triangular numbers. Listed below are some terms that are important to know. Greatest common factor—The greatest common factor (GCF) of two (or more) integers is the largest possible integer that can be divided into the two given integers a whole number of times. For example, 12 is the largest integer that can be divided into both 24 and 36. Least common multiple—The least (lowest) common multiple (LCM) of two (or more) integers is the smallest integer that is a multiple of the two given integers. For example, 18 is the smallest multiple of both 6 and 9. https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 6 of 28  6/16/23, 9:26 PM Prime number—If an integer has no factors other than itself and 1, it is a prime number. For example: 11 has no factors other than 1 and 11, so 11 is prime. Prime factorization—When an integer is factored in such a way that all factors are prime numbers, that is prime factorization. For example: 54 = 2 ⋅ 3 ⋅ 3 ⋅ 3 Square numbers—Sometimes called perfect squares, it is worthwhile to know the squares of at least the integers 1 through 12, i.e. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144. Number Systems There are infinitely many numbers in mathematics, and to get a handle on them it helps to separate them into groups that have similar properties. Listed below are the names and descriptions of some common groups. Whole numbers—These are the numbers we all learned in elementary school: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … and so on, never ending, also called natural numbers, sometimes without 0. Integers—Take the set of all whole numbers and add them to the set of their negatives, (except there is no negative zero) and get … -4, -3, -2, -1, 0, 1, 2, 3, 4, … the infinite set of integers. Rational numbers— Any number that can be written as the quotient of two integers is a rational number. For example: 1 5 56 −33 , , , 10 or even 61 3 2 7 ones you’re most likely to run into are 𝝅 and roots that don’t equal a whole number. For example: Irrational numbers— Any number that can’t be written as a quotient of two integers is an irrational number. The √2 ‾, √35 ‾‾ ‾ , but not √36 ‾‾ ‾ because that is 6, a whole number. Real numbers All of the rational numbers and irrationals together make the set of real numbers. For example: 3 −5, , √21 ‾‾ ‾ 8 Number Order You need to be able to put numbers in ascending or descending order with any of the following number types or combinations of types. Quick tip: change all numbers to decimal form first. Whole numbers/integers—The slightly tricky thing about ordering negative integers is that it’s easy to slip up and think -6 is greater than -4, when, in fact, it is less. Fractions and mixed numbers—The easiest way to order these is to first use a calculator to change all numbers to decimal form. Ordering should then be pretty obvious. Other rational numbers—Similarly, to put percents in order with other numbers, first change the percents to decimal form. https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 7 of 28  6/16/23, 9:26 PM Irrational numbers—Although irrational numbers can’t be expressed exactly in decimal form, a calculator will give you an approximate decimal form with enough accuracy to use. Special Notation To deal with very large or very tiny numbers, scientific notation, also called exponential notation, has been developed. Be able to perform operations on these. Scientific notation: addition—This notation is written in the form a × 10r where a is a number between 1 and 10 and r is an integer. You have to adjust the decimal points and r exponents to get both numbers to have the same r values. Then add the r a values together and write × 10. For example: 1.2 × 103 + 5.1 × 104 = 0.12 × 104 + 5.1 × 104 (0.12 + 5.1) × 104 = 5.22 × 104 Scientific notation: multiplication—To multiply two numbers in scientific notation, multiply the a parts and add the r parts.Then adjust the decimal point as needed. For example: (2.3 × 105 )(7.42 × 10−3 ) = (2.3 ⋅ 7.42) × 105+(−3) = 17.066 × 102 To make the a part between 1 and 10, divide 17.066 by 10 and then to compensate for that, add 1 to the r part. This gives: 1.7066 × 103 Working with Exponents Exponents are often positive integers, but can also be negative, and may be positive or negative fractions as well. Negative exponents—A negative exponent is handled by taking the reciprocal of the base and rewriting its exponent as positive. For example: 1 3−2 = 32 x Fractional exponents—A fractional exponent, say a y is handled by rewriting it as a y root of a, raised to the x power, like this: x a y = (√y ‾a)x https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 8 of 28  6/16/23, 9:26 PM Exponents of fractions—Just as integers can be raised to a power, so can fractions. (5) 3 2 2 2 2 8 = ⋅ ⋅ = 5 5 5 125 Operations with Positive and Negative Numbers Operations with positive and negative numbers such as addition, subtraction, multiplication, and division have exact rules you need to know to deal with the signs. Relationships between Operations Each math operation can be thought of as having an opposite operation that “undoes” it: Subtraction is the opposite of addition; Division is the opposite of multiplication; Taking the square root is the opposite of squaring, to name three examples. Properties There are certain properties of number systems that may hold true for some operations and not others. Listed below are some of the common ones. Associative property— For multiplication and addition, how you group (associate) the numbers doesn’t matter. For example: (3 + 9) + 5 = 3 + (9 + 5) Commutative property—For multiplication and addition, the order of the numbers doesn’t matter. For example: 4 ⋅ 6 = 6 ⋅ 4 and 4 + 6 = 6 + 4 Identity property— 0 added to a second number doesn’t change that number’s identity. Likewise, 1 times a second number doesn’t change its identity. 0 is the additive identity and 1 is the multiplicative identity. Distributive property— a(b + c) = ab + ac Think of a as being distributed to the b first and then c. Working with Numbers Here are a few general things to keep in mind when working with numbers. Algorithms An algorithm is just a series of steps that will solve a certain problem. Remember how to add two two digit numbers? 1. Add the ones column. 2. If the result is over ten, carry a one to the tens column, and so on. That procedure is an algorithm. Standard algorithms exist for adding, subtracting, multiplying, and dividing whole numbers, fractions, and decimals, to name the most common ones that you should know. It’s often said that there’s more than one way to skin a cat, (ew!) and you may see an algorithm that you haven’t seen before. Usually you can use one you do know to check the correctness of the new one. https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 9 of 28  6/16/23, 9:26 PM The Order of Operations Remember to follow the order of operations (PEMDAS): 1. Do everything in parentheses (P), left to right. 2. Evaluate any exponents (E), left to right. 3. Do all multiplication and division MD), in order, left to right. 4. Then do all addition and subtraction (AS), in order, from left to right. Good way to remember: Please Excuse My Dear Aunt Sally Rounding and Estimating Rounding is a way of changing a number to a more approximate version of itself. If you knew that something cost $158.95, you would very likely just think of it as $160 (rounding to the nearest ten dollars). The simplest rule of rounding is to look at the place to which you want to round and, if the digit to the right of it is 5 or greater, add 1 to the targeted place and drop, or turn into zeros, all non-zero digits to its right. For example: Round 409 to the nearest ten. To the right of the tens place is a 9 (bigger than 5), so add 1 to the tens place and change the 9 to a zero, giving 410. You should review rounding decimals as well. Estimating means to use rounded numbers to quickly get an approximate answer. For example: Estimate the product of 53 × 288. Round 53 to 50 and 288 to 300. 50 × 300 = 15, 000 Using Technology for Complex Calculations Scientific and graphing calculators are common, and they make it easier to work with scientific notation, probability, and other calculations. Spreadsheets can help deal with a lot of data, as can a computer program that you write. https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 10 of 28  6/16/23, 9:26 PM Algebra and Functions Algebra is largely about working with quantities that are related to each other. The relationships generally use symbols called variables to write expressions or equations, and there are numerous rules for manipulating them. As in a lot of mathematics, it helps to have a good eye for noticing and remembering patterns. Patterns and Relationships Numerical patterns can range from the obvious to the kind that make you want to pull your hair out. To help decipher them we can use tables, graphs, word rules, and symbolic rules. For example: You are riding in your car at a speed of 60 mph. What is the relationship between the distance you go and the time it takes? You could make a table to show it. d (miles) 15 30 60 150 t (hours 0.25 0.50 1.0 2.5 Or you could graph the values, in this case giving a straight line. State the rule in words: distance ÷ time = 60. State the rule symbolically: d/t = 60. Some relations are called functions and you should know how to recognize one. Proportional Reasoning x 7 An equation like 4 = 12 is called a proportion and you can use it to deal with quantities that are directly proportional to each other. It’s nothing more than a statement of two ratios that are equal, also known as equivalent fractions. The example in the previous section is a perfect example of a direct proportion: d 60 = t 1 If you know t is 3 hours, you can easily calculate d. d 60 = 3 1 d = 180 Any problem involving proportional reasoning can be set up as a proportion and solved. Review similar triangles to get a geometric view of proportional reasoning. Dependent and Independent Variables https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 11 of 28  6/16/23, 9:26 PM Be able to recognize, represent, and analyze relationships between dependent and independent variables. The independent variable is the one you are directly changing, and the dependent variable is the result of that change. Suppose you put a pressure gauge on a tank of nitrogen and put it in the oven at 200 ℉ and read the pressure gauge. Then do the same at 300 ℉ and 400 ℉ and 500 ℉. The pressure readings keep going up each time. You are directly setting the temperature (input) and observing the pressure (output). The temperature is controlling the pressure, or you could say pressure is dependent on the temperature. Pressure is the dependent variable, and temperature is the independent variable. On a graph, the independent variable is on the horizontal axis and the dependent one is on the vertical axis. Equations and Inequalities Equations tell us only one thing: The thing on the left side equals the thing on the right side. Inequalities can tell us one of four things: a < b tells us that a is less than b. a ≤ b tells us that a is less or equal to b. a > b tells us that a is greater than b. a ≥ b tells us that a is greater than or equal to b. Except for one case, solving inequalities is like solving equations. 3x+4< 28 Given problem 3x< 24 Subtracted 4 from both sides x< 8 Divided both sides by 3 The exception is that when you multiply or divide by a negative number, you have to reverse the inequality sign. 5-2x< 25 Given problem -2x< 20 Subtracted 5 from both sides x> -10 Divided both sides by -2 Equivalent Expressions Equivalent expressions are expressions that express the same quantity, though they may look different. For example: The expressions x(3x − 2) + 3x and x(3x + 1) are equivalent because they are both the same as 3x 2 + x when they are simplified. Simplifying is the key If you completely simplify equivalent expressions, they will end up being identical. Matching Expressions to Situations Be able to translate symbolic expressions such as 7x − 4 into words such as 4 less than 7 times x or turn words https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 12 of 28  6/16/23, 9:26 PM into a symbolic expression. From geometry, for example, we know that in a right triangle the sum of the squares of the legs is equal to the square of the hypotenuse. Noticing key words like sum, square and equal to will help you to end up with a2 + b2 = c2. Expressing Problems Algebraically Word problems can strike fear into the hearts of math students, and for good reason. They can be hard. Watch for key words or phrases such as added to, increased by, less, less than, of, times, and others to help you on your way. For example: Suppose Max had a certain number of links on his website and Claire had 11 fewer links. If together they have 25 links, how many does Max have? First, assign variables: Max’s links = x Fewer means less, so Claire’s links = x − 11 Together implies added and they have implies equal to, so: (x) + (x − 11) = 25 These kinds of problems also can show up in geometry, dealing with topics like area, perimeter, similar triangles, and volumes. Properties of Linear Equations Linear equations are pretty easy to recognize: You will see no powers of x and y higher than 1. An easy example is y = 2x − 5. Also, their graphs are always straight lines. Be able to determine the slope of a graph by looking at it (rise/run) or from the equation (get the equation in the form y = mx + b and m is the slope). In our easy example above, the slope would be 2. If two lines are parallel, they will have the same slope. If the two slopes are negative reciprocals of each other, they are perpendicular. 1 A graph perpendicular to our easy example above would have a slope of − 2. Working with Polynomials An expression with two or more terms is a polynomial, although the term polynomial is often used to mean an expression with more than two terms, such as 4x 3 − 3x 2 − x + 7. You should know how to multiply, divide, and factor them. Working with Quadratic Equations Any equation that fits this pattern y = ax 2 + bx + c is a quadratic equation. a, b, and c are constants that could be positive, negative or zero. Here are a few examples: 2 https://uniontestprep.com/pdf/study_guide?study_guide_id=397 y=2 + 5x + 6 Page 13 of 28  6/16/23, 9:26 PM y = 2x 2 + 5x + 6 y = x2 − 4 y = −2x 2 There are a few different techniques to use in solving quadratics. Factoring— Follow these steps: 1. Factor the quadratic: x 2 + 5x + 6 = 0 gives (x + 3)(x + 2) 2. Set each factor equal to zero: x + 3 = 0 and x + 2 = 0 3. Solve each little equation: x = −3 and x = −2 Completing the square—If you can’t solve the quadratic by factoring the equation, you can complete the square, but it’s unlikely you will have to. Go to the next method instead. The Quadratic Formula—Look back to the general form of a quadratic.equation above: y = ax 2 + bx + c. Unlike factoring, the quadratic formula can solve any quadratic equation. Unlike completing the square, it doesn’t take a lot of steps to get to the answer(s). This is it: −b ± √‾b‾‾‾‾‾‾‾ 2 − 4ac x= 2a Solve: x 2 + 4x + 1 = 0 a = 1, b = 4, c = 1 Substitute into the quadratic formula. −4 ± √‾ 4‾‾‾‾‾‾‾‾‾‾ 2 − 4 ⋅ 1 ⋅ 1‾ x= 2⋅1 Simplifying will give you −2 ± √3 ‾ Interpreting Graphs Be able to interpret graphs of: https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 14 of 28  6/16/23, 9:26 PM Linear equations (straight lines) Quadratic equations (parabolas) Inequalities (straight line + all area to one side of the line) Systems of equations (two plots on the same axes, often intersecting) In systems of equations, the point where the graphs meet is the solution to the system. https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 15 of 28  6/16/23, 9:26 PM Measurement and Geometry Geometric Objects Geometric objects include 2-dimensional (flat) figures known as plane figures, and 3-dimensional solid figures. Characteristics Know the characteristics of common plane and solid geometric figures. Some, but not all, are listed below. Also, see the term congruence in the next section. Isosceles triangle— a triangle with two congruent sides (also two congruent angles) Right triangle— a triangle with a right angle Sphere— informally, a ball-shaped object; formally, the set of all points a given distance in space from a fixed point (the center) Quadrilateral— a plane figure with four sides Square—a quadrilateral with four congruent sides (also four congruent angles) Parallelogram— a quadrilateral with both pairs of opposite sides parallel Rectangular prism— a solid figure with six rectangles for faces Cube— a rectangular prism with all square faces Object Comparisons The list below shows different ways geometric figures can be compared to each other. Congruence is by far the most used of these concepts in geometry. Congruence—Congruent means identical in every way. Congruent segments or angles have the same measure. Congruent figures fit exactly on top of each other. Congruent Figures: Similarity—Two figures are similar if they have the same shape, though one may be bigger. For example, all squares are similar. Similar Figures: https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 16 of 28  6/16/23, 9:26 PM Symmetry—The three types of symmetry are line symmetry, point symmetry, and rotational symmetry. Line symmetry is very common. Your face is a pretty good example. If you draw a line down the center of your face, the left side and right side match up, point for point. Of course it won’t be perfect on a real face, but picture it done to a capital A. An A has line symmetry, and the line you draw is called the line of symmetry.. Translation—Picture a paper triangle, or other figure, sliding across your desk to a new location without any rotation. That movement is called a translation. Rotation—Picture a paper triangle, or other figure, on your desk. Stick a pin through it into your desk. Spin it. The triangle can’t go anywhere but it can rotate. That’s rotation. Easy. The point where you stuck the pin is the center of rotation. In the diagram below, D is the center of rotation. https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 17 of 28  6/16/23, 9:26 PM Reflection— Picture a trapeze performer doing a handstand on a trapeze bar.and then swinging down to hang below the bar. That kind of a flip is called a reflection. If you can picture a triangle doing the same kind of flip over a line, you have the right idea. The Pythagorean Theorem In a right triangle with legs a and b and hypotenuse c , it is true that: a2 + b2 = c2 Normally, you would be given two of the sides and asked to find the third one. The converse is also true. If you have some triangle with sides a,b, and c, and you can show that a2 + b2 = c2 is true, then you know the triangle is a right triangle. Parallel Lines Parallel lines will never intersect. If a third line (a transversal) intersects them, eight angles will be formed. In the figure below, some pairs of angles are defined like this and are congruent https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 18 of 28  6/16/23, 9:26 PM In the figure above, these things are true: A and C (as well as B/D, E/G, and F/H) are vertical angles A and E (as well as D/H, B/F, and C/G) are corresponding angles D and F, as well as C and E are alternate interior angles A and G, as well as B and H are alternate exterior angles Other pairs of angles are not congruent, but have a relationship that helps you determine their measure. The measures of these pairs of angles add to 180 degrees. A and B (as well as C/D, E/F, G/H, A/D, E/H, B/C, and F/G): adjacent angles A and H (as well as B/G): exterior angles on the same side of a transversal D and E (as well as C/F): interior angles on the same side of a transversal Representing Geometric Objects Strictly speaking, geometric objects have a perfection that can exist only in the mind. However, our minds need a lot of help visualizing them, so we resort to drawing lines on paper to approximate the figures or using an x − y coordinate system to define them. Sometimes, physical models can help, especially with solid figures. Using Tools Common tools of geometry include a compass and a straightedge. With these, many geometric figures can be exactly copied using construction techniques, and certain operations can be performed, such as bisecting angles or line segments to name a couple. For approximately measuring angles and segments, a protractor and ruler are used. Combining and Dissecting Objects Sometimes it’s helpful to imagine dissecting figures into smaller simpler figures. Imagine splitting a rectangle into two triangles, for example, or dissecting a hexagon into six triangles to find its area. Or, you could go the other way and imagine combining simpler figures into larger ones. A rectangle with identical right triangles added to each end could be a parallelogram or a trapezoid, depending on how you do it. https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 19 of 28  6/16/23, 9:26 PM Measuring Geometric Objects An amazing number of things in this world can be, and are, measured. Research scientists, for example, go to extraordinary lengths to accurately measure things like time, distance, force, temperature, mass, and a lot more. In geometry, we stick to length and angle measures, and use those to calculate these: perimeter area surface areas of solids volumes of solids Especially when sketching your own geometric figures, you should be able to estimate angle measures and comparable lengths. Formulas for Measuring Perimeter—There are formulas for some figures, but basically you just add up all the side lengths. Area— Some common area formulas: Rectangles and parallelograms: A = hw Triangles: A = 12 bh Circles: A = πr 2 Surface Area of Solids—Some common surface area formulas: Rectangular prism: S.A. = 2lw + 2hw + 2hl Cube: S.A. = 6s2 Cylinder: S.A. = 2π r 2 + 2πrh Sphere: S.A. = 4πr 2 Volume— Some common volume formulas: rectangular prism— V = lwh cone—V = 13 π r 2 h cylinder—V = πr 2h sphere— 43 π r 3 Comparing Measurement Systems The metric system relies on base units, prefixes and powers of ten to identify its various units of length, mass, and volume. Complete listings showing all prefixes can easily be found, and here we will just show a few basic ones being used with the gram unit. https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 20 of 28  6/16/23, 9:26 PM milligram centigram decigram kilogram megagram mg cg dg g kg Mg 10−3 g 10−2 g 10−1 g 100 g 103 g 106 g Converting between units in the metric system is just a matter of moving the decimal. If you are going to a bigger unit, the decimal point moves to the left. If you are going to a smaller unit, the decimal point moves to the right. Subtract the powers of ten to get the number of places to move (bigger unit minus smaller unit). For example: change 5 kg to cg. Centigrams are smaller than kilograms so the decimal goes right. Subtracting exponents: 3 − (−2) = 5 places to move the decimal. 5kg = 500,000 cg U.S. customary units have no overall system of converting between, say, pounds and ounces or miles and feet. See the next section for using proportions to do those conversions To estimate rough conversions between the two systems these may help. 1 liter is close to a quart. 1 meter is close to a yard. 1 kg is a little over 2 pounds. 5 8 mile is close to a kilometer. Using Proportions Remember that a proportion is a statement of equality between two ratios. Such as: x4 = 7 12 A common use of proportions is to convert from one measurement unit to another. Say you need to convert 30 inches to feet. You would use this proportion: 1 30 12 = x Solving this would give you x = 2.5 inches. In the case of making scale drawings, like blueprints, or models, you would do the same sort of thing. 1 For example, using a scale of 48 how long would you draw a 80 foot wall? x 1 80 = 48 would give you a length of 1 23 ft, or 20 inches. Using Other Measurement Forms Basic measurement units can be combined to give useful new units. For example, take a length unit divided by a time unit and get speed: miles hour https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 21 of 28  6/16/23, 9:26 PM or meters second Weight (force) divided by area gives a pressure unit: pounds square inch (psi for those who check their tire pressures) https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 22 of 28  6/16/23, 9:26 PM Statistics, Data Analysis, and Probability Collecting, interpreting, analyzing, and representing data are the operations of statistics. Probability can be used to predict the likelihood of some event occurring. Collecting and Representing Data Anywhere you find things being observed, counted, and recorded, those counts or measures makeup what are known as data. The population in California, for example, has been recorded for decades. This set of data can be represented in different ways for different purposes that we will go into below. Representing Data Data is commonly set forth in a table and from that it can be represented in different ways. To help visualize trends, some sort of a graph (sometimes called a chart) is usually drawn. Histograms, pie graphs, bar graphs, scatter plots, and line graphs are good ones to know. Data Concepts Know how to find the following. mean— The mean is what is commonly called the average. It’s the sum of all the data values divided by the number of values. median—The median is the value that is in the center of a set of values that are arranged in numerical order. If there are an even number of values, there will be two values in the middle. The mean of those two is the median. mode—The mode is the value that shows up most often. There can be more than one mode if there are ties for the most often used value. range—The range of data values is the difference between the highest value and the lowest. You can think of it as how much the data is spread out. Surveys Many times, data is collected by surveys, and it’s good to know something about survey design. Things to consider are sample size (how many took the survey), having unbiased questions, and randomizing those being surveyed. The bigger the sample size, the better. Randomizing helps to insure that a wide cross-section of the population is included, which reduces bias. Designing the questions themselves so as to not induce bias is a whole topic of its own and way beyond the scope of this guide. Analyzing Data Collecting, organizing and, maybe, graphing data is a first step, but interpreting it all is just as important. Interpretation Know how to interpret graphs or tables, noticing upward, downward or cyclic trends. Just what do all the numbers and lines mean? https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 23 of 28  6/16/23, 9:26 PM Finding Patterns While noticing trends in the data, look for details. Does the graph show a straight line pattern, or maybe one that curves up or down? How steep is it? In a scatter plot, does the line seem to closely follow the majority of points (goodness of fit)? In a frequency plot, do you see an asymmetric curve indicating a left or right skewing of the data? Drawing Conclusions Drawing conclusions involves evaluating all of the above, and there are specific statistical calculations to help decide the likely correctness of your conclusion, though they are beyond the scope of this guide. Be able to identify potential sources and effects of bias, such as small sample size, poor randomizing, and poorly worded questions. Probability In a number of equally likely events (outcomes), some of them will satisfy a certain requirement. If we call those winners, probability is the ratio of winners to the number of possible events. The set of all possible events is called the sample space. Suppose you want to draw an ace from a deck of 52 cards. There are 52 distinct events that could occur when you draw one card, but only 4 of them will be aces. The probability of drawing an ace in this scenario is 4 1 or 13 52 Events Suppose you have a 4-sided die numbered 1, 2, 3, and 4. The event you want is to roll a 3. 1 The probability of that happening is 4. What is the probability of not rolling a 3? That event would be rolling a 1, 2, 3 or 4, i.e. 4. Those are two complementary events. Notice two things here: 1. First, the two probabilities add up to 1. 2. Second, they are mutually exclusive, meaning there is no overlap of events. If you roll a 3, you can’t have rolled a 1, 2, or 4.. Now, let’s say you have 2 blue and 2 red marbles in a bag and you pull out two, one after the other. What is the probability of getting a blue marble on the second draw? If you pull out a blue marble first, you will have 1 blue and 2 reds left. The probability of picking out a blue then will be 13. If you pull out a red marble first, you will have 2 blues and one red left. 2 The probability of picking out a blue then will be 3. Notice how the probability of drawing a blue changed, depending on which marble you pulled out first.That makes getting a 2 on the second draw a dependent event. Events where this isn’t true are called independent. (In this example, the events would be independent if the drawn marble was replaced in the bag after each draw.) https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 24 of 28  6/16/23, 9:26 PM Expressing Probabilities Probabilities can be written in a variety of ways, including fractions (ratios), proportions, decimals, and percents. For example: 1 = 0.077 = 7.7% 13 Compound Events A compound event consists of more than one simple event. Imagine we have two four-sided dice, one red and one green. We will roll the red one, then the green one. What is the probability of rolling a sum of 5? We can use an outcome table to find out. Here is a table showing all possible outcomes of the two rolls, red across the top and green on the left side. Each outcome is a complex event. 1 2 3 4 1 (1, 1) (1, 2) (1, 3) (1, 4) 2 (2, 1) (2, 2) (2, 3) (2, 4) 3 (3, 1) (3, 2) (3, 3) (3, 4) 4 (4, 1) (4, 2) (4, 3) (4, 4) You can see there are 16 possible outcomes, so that is the sample space. How many of those will give us a sum of 5? These four: (4,1), (3,2), (2,3), and (1,4). This gives us a probability of 4 1 16 = 4 = 0.25. It would also be good to know about using a tree diagram to do a similar thing. The tree diagram below represents two spins of a spinner that has only numbers 1, 2, and 3 on it. If you spin it twice, what is the probability of the spins adding up to 4? Count up all the paths. From the top down, it shows a path from 1 to1, then 1 to 2, 1 to 3, 2 to 1, 2 to 2, 2 to 3, 3 to 1, 3 to 2, and last, 3 to 3. The number of paths is 9 and only 3 of them add up to 4: 1 to 3, 2 to 2, and 3 to 1. 3 1 That gives a probability of 9 or 3. https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 25 of 28  6/16/23, 9:26 PM Math Skills Problem Solving While you have no doubt done a lot of math problems in your life, to a math person the word problem implies reading a description of a situation, understanding the given information, knowing what you are looking for, figuring out the steps to find it, and doing the calculation(s). In short, it is a “word problem”—a fearsome term to every math student. This kind of problem is very common in most math subjects, as they are, in a sense, the intersection of math with the real world, and the very reason we learn math. Securing Information Problem solving essentially boils down to three main ideas: What do you have? What do you want? How do you get there? Be organized about this. List the relevant given facts with their units. In a math class, everything given is generally relevant, but you never know. Watch for unnecessary information. Write down what you are looking for. Include units. Know that there may be relevant information not given and it may take an extra step or two to come up with that. Solving It may take a lot of thinking to make sense of a problem, but persevere and very often you will be rewarded with a sweet “aha moment” when it all becomes clear. Complex problems can often be broken down into two or three simpler problems that you will recognize. If you can’t figure out the final answer, see if there is something you can figure out. That may well be the missing link to finding the answer. A trick that can sometimes help if your problem has weird numbers is to change them all to simple numbers. That way, your brain doesn’t get distracted by the numbers and can better cope with the rest of the problem. Sometimes trying to work backward works. You might think “If only I knew fact 2, I could see how to get to fact 3, (the answer). OK, how can I get fact 2?” Probably from fact 1. Modeling Models in math can mean different things. An equation using words or symbols can be a model. Labeled diagrams are models that can be a huge help to make sense out of a problem. Is something falling off a cliff? Draw it, using words and/or symbols to label distances, times, and velocities. Brains work much better with pictures of some sort than they do with just numbers and words. Sometimes, an actual hold-in-your-hand model will be useful, especially with three dimensional problems. Software is available to model 2-D and 3-D shapes. Using Other Information and Strategies A conjecture is a statement made without complete information. For example, draw a lot of examples of different looking right triangles. Look for patterns. Try to come up with an idea about them that you think may be true, such https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 26 of 28  6/16/23, 9:26 PM as A right triangle may be an isosceles triangle. That’s an example of a conjecture. It doesn’t have to be true to be a conjecture, but the hope generally is that it will turn out that way. Reasoning Reasoning is a combination of recognizing patterns, using problem-solving strategies, being creative, and above all using logic to draw correct conclusions. Reasoning Resources These are the resources in your head that make use of essentially all the skills you have learned in mathematics. Abstract and quantitative reasoning in all areas of mathematics should be in your skill set for solving math problems. Being able to recognize types of problems and knowing the tools needed to solve them is important, and always try to see if there is some alternate way to go. Accuracy Checks Checking your results can be done a few different ways. 1. Simply repeat the solving steps you used to see if you get the same answer. This will often catch simple mistakes. 2. Try rounding off all values in the problem to get a quick estimate and see if it is close to your answer. 3. Substituting your answer into the original equation to see if it works can often tell you if you are right or wrong. 4. Sometimes just having good number sense can help you decide if your answer is reasonable. If you’re trying to come up with the radius of the earth and your answer is 520 m, that tells you that you may have messed up a unit conversion somewhere. It’s common in geometry to ask if a statement is always, sometimes, or never true. For example: A right triangle is an equilateral triangle. (Answer: never) Explanation Explanation is generally carried out using the well-organized modeling techniques listed above to make your reasoning clear to others. You want to show a step-by-step listing of your thinking. Adding comments can usually make it even clearer. An ultimate example of this is a formal mathematical proof. Academic Language When making comments, you’ll need to use appropriate language for the situation, showing an understanding of mathematical terms and procedures. You’ll also need to apply this language to your evaluation of a given argument…did the creator do it well? If not, be able to point out where the argument went wrong. It’s also related to the mathematical notation section that comes next which cautions you to be careful to use accepted, well-defined terms in your arguments. Mathematical Notation In short, be very careful and precise in what you write and say. One incorrect or misplaced symbol changes the https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 27 of 28  6/16/23, 9:26 PM numerical representation. Relationships Explain how a result is related to other ideas. Mathematics is logically built, layer by layer. It’s important to understand how each idea is supported by the ideas below it, and how, in turn, it supports ideas above it. https://uniontestprep.com/pdf/study_guide?study_guide_id=397 Page 28 of 28