CSET Math Test Prep
Terms for Common Shapes and Behaviors
1) Lines (straight and extends infinitely in both directions) 2) Rays (like a line except it terminates on one end) 3) Segments (a piece of line that terminates on both ends) 4) Parabolas (A curved u-shape that occur when you have a variable squared)
Understanding Ratios and Proportions
A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0. A proportion is an equation in which two ratios are set equal to each other. For example, if there is 1 boy and 3 girls you could write the ratio as: 1 : 3 (for every one boy there are 3 girls)
Exponents
An exponent refers to the number of times a number is multiplied by itself. Ex. 2 to the 3rd power would mean 2x2x2=8
Finding Percent Increase/Decrease
First: work out the difference (increase) between the two numbers you are comparing. Then: divide the increase by the original number and multiply the answer by 100. % increase = Increase ÷ Original Number × 100. If your answer is a negative number, then this is a percentage decrease.
Determining Slope
Slope can be calculated as a percentage which is calculated in much the same way as the gradient. Convert the rise and run to the same units and then divide the rise by the run. Multiply this number by 100 and you have the percentage slope. For instance, 3" rise divided by 36" run = .
Inverse Properties
The inverse property of multiplication states that if you multiply a number by its reciprocal, also called the multiplicative inverse, the product will be 1. (a/b) * (b/a) = 1.
Slope-Intercept Form
The slope intercept form of a line is: y = mx + b. The m stands for the slope of the line and b stands for the y-intercept of the line.
Tables
Used to arrange text in columns and rows and are helpful in presenting, organizing, and clarifying information
Relational and Functional Patterns
relational pattern - A mathematical pattern is a sequence that follows a specific rule such as 1, 2, 1, 2, . . . and 1, 1, 2, 3, 5, 8, etc. A mathematical relationship tells you how to process numbers to get a particular answer. Multiplication and division are examples of mathematical relationships. A relation is any set of ordered-pair numbers. In other words, we can define a relation as a bunch of ordered pairs. functional pattern - In mathematics, a function can be defined as a rule that relates every element in one set, called the domain, to exactly one element in another set, called the range. For example, y = x + 3 and y = x2 - 1 are functions because every x-value produces a different y-value.
Maps and Scale Drawings
-Maps represent real places. -Every part of the place has been reduced to fit on a single piece of paper. -A map is an accurate representation because it uses a scale. -The scale is a ratio that relates the small size of a representation of a place to the real size of a place. -Maps aren't the only places that we use a scale. -Architects use a scale when designing a house. -A blueprint shows a small size of what the house will look like compared to the real house. -Any time a model is built, it probably uses a scale. -The actual building or mountain or landmark can be built small using a scale. -We use units of measurement to create a ratio that is our scale. -The ratiocompares two things. -It compares the small size of the object or place to the actual size of the object or place. -A scale of 1 inch to 1 foot means that 1 inch on paper represents 1 foot in real space. -If we were to write a ratio to show this we would write: 1" : 1 ft-this would be our scale. -If the distance between two points on a map is 2 inches, the scale tells us that the actual distance in real space is 2 feet. -We can make scales of any size. -One inch can represent 1,000 miles if we want our map to show a very large area, such as a continent. -One centimeter might represent 1 meter if the map shows a small space, such as a room.
Applying the Pythagorean Theorem
1. Ensure that your triangle is a right triangle. The Pythagorean Theorem is applicable only to right triangles, so, before proceeding, it's important to make sure your triangle fits the definition of a right triangle. Luckily, there is only one qualifying factor - to be a right triangle, your triangle must contain one angle of exactly 90 degrees.[1] As a form of visual shorthand, right angles are often marked with a small square, rather than a rounded "curve", to identify them as such. Look for this special mark in one of the corners of your triangle. 2. Assign the variables a, b, and c to the sides of your triangle. In the Pythagorean Theorem, the variables a and b refer to the sides that meet in a right angle, while the variable c refers to the hypotenuse - the longest side which is always opposite the right angle. So, to begin, assign the shorter sides of your triangle the variables a and b (it doesn't matter which side is labeled 'a' or 'b'), and assign the hypotenuse the variable c. 3. Determine which side(s) of the triangle you are solving for. The Pythagorean Theorem allows mathematicians to find the length of any one of a right triangle's sides as long as they know the lengths of the other two sides. Determine which of your sides has an unknown length - a, b, and/or c. If the length of only one of your sides is unknown, you're ready to proceed.[3] Let's say, for example, that we know that our hypotenuse has a length of 5 and one of the other sides has a length of 3, but we're not sure what the length of the third side is. In this case, we know we're solving for the length of the third side, and, because we know the lengths of the other two, we're ready to go! We'll return to this example problem in the following steps. If the lengths of two of your sides are unknown, you'll need to determine the length of one more side to use the Pythagorean Theorem. Basic trigonometry functionscan help you here if you know one of the non-right angles in the triangle. 4. Plug your two known values into the equation. Insert your values for the lengths of the sides of your triangle into the equation a2 + b2 = c2. Remember that a and b are the non hypotenuse sides, while c is the hypotenuse.[4] In our example, we know the length of one side and the hypotenuse (3 & 5), so we would write our equation as 3² + b² = 5² 5. Calculate the squares. To solve your equation, begin by taking the square of each of your known sides. Alternatively, if you find it easier, you may leave your side lengths in the exponent form, then square them later.[5] In our example, we would square 3 and 5 to get 9 and 25, respectively. We can rewrite our equation as 9 + b² = 25. 6. Isolate your unknown variable on one side of the equals sign. If necessary, use basic algebra operations to get your unknown variable on one side of the equals sign and your two squares on the other side of the equals sign. If you're solving for the hypotenuse, c will already be isolated, so you won't need to do anything to isolate it.[6] In our example, our current equation is 9 + b² = 25. To isolate b², let's subtract 9 from both sides of the equation. This leaves us with b² = 16. 7. Take the square root of both sides of the equation. You should now be left with one variable squared on one side of the equation and a number on the other side. Simply take the square root of both sides to find the length of your unknown side. In our example, b² = 16, taking the square root of both sides gives us b = 4. Thus, we can say that the length of the unknown side of our triangle is 4.
Graphing a Line Based on Its Equation
1. Identify the slope and y-intercept from the equation of a line. 2. Plot the y-intercept of a line given its equation. 3. Plot a second point on a line given the y-intercept and the slope. 4. Graph a line given its equation in slope y-intercept form. Often we are given an equation of a line and we want to visualize it. For this reason, it is important to be able to graph a line given its equation. We will look at lines that are in slope intercept form: 𝑦=𝑎+𝑏𝑥y=a+bx where 𝑎a is the y-intercept of the line and 𝑏b is the slope of the line. The y-intercept is the value of 𝑦y where the line crosses the y-axis. The slope is the rise over run. If we write the slope as a fraction, then the numerator tells us how far to move up (or down if it is negative) and the denominator tells us how far to the right we need to go. the main application to statistics is in regression analysis which is the study of how to use a line to make a prediction about one variable based on the value of the other variable.
Roots (aka Radicals)
1. Root of a number The root of a number x is another number, which when multiplied by itself a given number of times, equals x. For example the second root of 9 is 3, because 3x3 = 9. The second root is usually called the square root. The third root is usually called the cube root. Ex. The square root of 81 is 9 because 9 times itself equals 81.
Boxes
A Box is a polyhedron with all its faces in rectangular shape. In general, the shape of a box is similar to that of a cuboid or a rectangular prism.Boxes are also known as cartons, cases, and containers.
Bar Graphs
A bar graph is a chart that uses bars to show comparisons between categories of data. The bars can be either horizontal or vertical. ... A bar graph will have two axes. One axis will describe the types of categories being compared, and the other will have numerical values that represent the values of the data.
Cones
A cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.
Why Quadratic Equations May Have Two Solutions
A quadratic expression can be written as the product of two linear factors and each factor can be equated to zero, So there exist two solution.
Similarities in Area Formulas
Area Formulas Area of a Rectangle = Base × Height. Area of a Square = Base × Height. Area of a Square = s2 Area of Triangle = ½(Base × Height) Area of Parallelogram = Base × Height. Area of Trapezoid = ½(Base1 + Base2) × Height. Area of Circle = π(radius)2 = πr2
Units of Temperature - Metric
Celsius or Centigrade is the metric system measuring unit for temperature. It was originally defined based on the freezing point of water (0°C) and the boiling point of water (100°C), both at a pressure of one standard atmosphere.
Supplementary and Complementary Angles
Complementary Angle- Two angles are called complementary if their measures add to 90 degrees Supplementary Angle- angles are supplementary if their measures add to 180 degrees
Similarity and Congruence
Congruence essentially means that two figures or objects are of the same shape and size. Similarity means that two figures or objects are of the same shape, though usually not of the same size. Two circles will always be similar, for example, because by definition they have the same shape.
Slides, Flips, and Turns
Flips- Flips are also know as reflections. To make a flip pattern, take an exact copy of your shapes and reflect it across an axis of symmetry or mirror line. A flip creates a MIRROR Image of the original shape and can be in any direction. Some shapes look exactly the same when they have been reflected, which can confuse your students into thinking all shapes will look the same when they are reflected! Slides- Slides are also known as Translations. A slide pattern is created by moving your original shape without turning or flipping it. You can move up, down, forwards and back. You can even overlap shapes in a slide transformation. Just make sure that each shape in the pattern is still facing the same direction as the first shape. Turns- Turns are also know as rotations. To make a rotation pattern, turn shapes around a point and copy them into the new position. Turns can be made clockwise and anticlockwise. The rotation can be an infinite number of degrees. Introduce students to quarter turns, 90° and half turns, 180°
Fractional Exponents
Fractional exponents are ways to represent powers and roots together. In any general exponential expression of the form a to the b power, a is the base and b is the exponent. When b is given in the fractional form, it is known as a fractional exponent. Few examples of fractional exponents are 2 1/2, 3 2/3, etc.
Graphing Inequalities
How to Graph a Linear Inequality Rearrange the equation so "y" is on the left and everything else on the right. Plot the "y=" line (make it a solid line for y≤ or y≥, and a dashed line for y< or y>) Shade above the line for a "greater than" (y> or y≥) or below the line for a "less than" (y< or y≤).
Irregular, Isosceles, and Equilateral Triangles
Irregular Triangle- A triangle where all three sides are different in length. Sometimes called an irregular triangle. Scalene triangles are triangles where each side is a different length. Isosceles Triangle- An isosceles triangle is a triangle with (at least) two equal sides. In the figure above, the two equal sides have length and the remaining side has length. . This property is equivalent to two angles of the triangle being equal. An isosceles triangle therefore has both two equal sides and two equal angles. Equilateral Triangle- An equilateral triangle is a triangle with all three sides of equal length , corresponding to what could also be known as a "regular" triangle. An equilateral triangle is therefore a special case of an isosceles triangle having not just two, but all three sides equal. An equilateral triangle also has three equal angles.
Dependent Choices
Permutations are groupings in which the order of items matters. Combinations are groupings in which content matters but order does not. Two events are dependent if the original state of the situation changes from one event to the other event, and this alters the probability of the second event.
Properties of Circles
Some of the important properties of the circle are as follows: -The circles are said to be congruent if they have equal radii -The diameter of a circle is the longest chord of a circle -Equal chords of a circle subtend equal angles at the centre -The radius drawn perpendicular to the chord bisects the chord -Circles having different radius are similar -A circle can circumscribe a rectangle, trapezium, triangle, square, kite -A circle can be inscribed inside a square, triangle and kite -The chords that are equidistant from the centre are equal in length -The distance from the centre of the circle to the longest chord (diameter) is zero -The perpendicular distance from the centre of the circle decreases when the length of the chord increases -If the tangents are drawn at the end of the diameter, they are parallel to each other -An isosceles triangle is formed when the radii joining the ends of a chord to the centre of a circle
Spinners
Spinners fit into a similar catagory to dice. Basically spinners add randomness to the generation of numbers, colours or shapes. Spinners may be substituted for dice and vice versa to generate numbers. Spinners may be used as part of a game scenario in much the same way that dice help to generate moves.
Median
The Median is the "middle" of a sorted list of numbers. To find the Median, place the numbers in value order and find the middle.
Basics of the Pythagorean Theorem
The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by a^2 + b^2 = c^2, where a and b are legs of the triangle and c is the hypotenuse of the triangle. A Pythagorean Triple is a set of three whole numbers a,b and c that satisfy the Pythagorean Theorem, a^2 + b^2 = c^2.
Units of Temperature - U.S. Customary
The U.S. customary measurement unit for temperature is °Fahrenheit. Customary system is called the International System of Units or the Modern Metric System.
Mean
The mean is a type of average. It is the sum (total) of all the values in a set of data, such as numbers or measurements, divided by the number of values on the list. To find the mean, add up all the values in the set. Then divide the sum by how many values there are.
Spinners with Repeated Values
The probability of landing on 8 twice is 164. Explanation: If you spin the spinner once, there are 8 possibilities each time (1-8). The possibility of landing on the number 8 is 1 of these 8 possibilities. Its probability is therefore 18. Each spin is independent. In other words, the spins do not affect each other. The number you land on during the first spin does not impact the second spin's number. The probability of landing on 8 the second time is the same as the first time: 18. Because the two events are independent, we find the probability of them both occurring by multiplying them together. 18⋅18=164 The probability of landing on 8 twice is 164. Another way to think about this is to consider all of the total possibilities. If there are 8 possibilities for the first spin, there are 8 possibilities for the second spin. 8⋅8gives 64 total combinations for the two spins. Below are just some of the possibilities. 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 2,1 2,2 There are 64 total possibilities and 8,8 is just one of them. Some combinations occur multiple times (like 1,2 and 2,1). However, 8,8 is unique, so it is one out of 64 possibilities. Its probability is 164.
General Properties of Triangles
The properties of a triangle are: -A triangle has three sides, three angles, and three vertices. -The sum of all internal angles of a triangle is always equal to 180°. This is called the angle sum property of a triangle. -The sum of the length of any two sides of a triangle is greater than the length of the third side. -The side opposite to the largest angle of a triangle is the largest side. -Any exterior angle of the triangle is equal to the sum of its interior opposite angles. This is called the exterior angle property of a triangle.
Repeated Roll of a Single Die
The set of all the possible outcomes is called the sample space. Thus the numbers on the die become the set. The sample space has six possible outcomes: {1, 2, 3, 4, 5, 6}. Fair Die If the die has been constructed in a perfectly symmetrical manner, then we can expect that no outcome will be preferred over the other. Thus, for a large number of rolls, the relative occurrence of each outcome should be roughly one-sixth 1616. P(F) = No.of favorable outcomes / No. of total outcomes = 1 / 6 Each of the 6 possible outcomes in a fair die has an even chance to appear. The probability of getting any 1 number is 1616.
Determining Y-Intercept
The y-intercept is the point at which the graph crosses the y-axis. At this point, the x-coordinate is zero. To determine the x-intercept, we set y equal to zero and solve for x. Similarly, to determine the y-intercept, we set x equal to zero and solve for y.
Distributive Property
To "distribute" means to divide something or give a share or part of something. According to the distributive property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together. 3(10 + 2) = 3(10) + 3(2) =
Estimating and Rounding
To estimate means to make a rough guess or calculation. To round means to simplify a known number by scaling it slightly up or down. Rounding is a type of estimating.
Defining a Unique Line
What they mean by unique is that there is no line that does not coincide exactly with the first one and passes through the same two points. And if you move your second line it is considered a transformation and it is now a new line distinct from the original line. That is what is meant by unique.
Graphing Parabolas
When graphing parabolas, find the vertex and y-intercept. If the x-intercepts exist, find those as well. Also, be sure to find ordered pair solutions on either side of the line of symmetry, x=−b2a x = − b 2 a . Use the leading coefficient, a, to determine if a parabola opens upward or downward. Vertex: (−1, 3) Y-intercept: (0, 5) X-intercepts: None Extra points: (−3, 11), (−2, 5), (1, 11)
Graphing Patterns
When you describe patterns on graphs you are describing how the dependent variable changes in relation to the independent variable. Example 1. Time spent training. Skill level at football.
Finding Partial Volume of a Box (or Other 3D Figure)
Whereas the basic formula for the area of a rectangular shape is length × width, the basic formula for volume is length × width × height. How you refer to the different dimensions does not change the calculation: you may, for example, use 'depth' instead of 'height'. The important thing is that the three dimensions are multiplied together. You can multiply in which-ever order you like as it won't change the answer. This basic formula can be extended to cover the volume of cylinders and prisms too. Instead of a rectangular end, you simply have another shape: a circle for cylinders, a triangle, hexagon or, indeed, any other polygon for a prism. Effectively, for cylinders and prisms, the volume is the area of one side multiplied by the depth or height of the shape. The basic formula for volume of prisms and cylinders is therefore: Area of the end shape × the height/depth of the prism/cylinder.
Graphing Systems
1. Graph the first equation. 2. Graph the second equation on the same rectangular coordinate system. 3. Determine whether the lines intersect, are parallel, or are the same line. 4. Identify the solution to the system. If the lines intersect, identify the point of intersection. ... 5. Check the solution in both equations.
Comparing Complementary Events Using Spinners
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Using Part of One Shape to Find Part of Another
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Word Problems Using Related Quantities
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Working with Similar Triangles
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Designing Surveys
* Ask questions in logical order * Keep similar items together * If designing the survey yourself, make sure it is neat and uncluttered * Shorter questionnaires have higher completion rates * Use pre-testing to discover flaws
Properties of Squares
-A square is a closed figure of four equal sides and the angle formed by adjacent sides is 90 degrees. -A square can have a wide range of properties. Some of the important properties of a square are given below. -A square is a quadrilateral with 4 sides and 4 vertices. -All four sides of the square are equal to each other. -The opposite sides of a square are parallel to each other. -The interior angle of a square at each vertex is 90°. -The sum of all interior angles is 360°. -The diagonals of a square bisect each other at 90°. -The length of the diagonals is equal. -The length of the diagonal with sides s is √2 × s -Since the sides of a square are parallel, it is also called a parallelogram. -The length of the diagonals in a square is greater than its sides. -The diagonals divide the square into two congruent triangles.
Properties of Parallelograms
-Opposite sides are parallel. -Opposite sides are congruent. -Opposite angles are congruent. -Same-Side interior angles (consecutive angles) are supplementary. -Each diagonal of a parallelogram separates it into two congruent triangles. -The diagonals of a parallelogram bisect each other.
Solving Quadratic Equations Using Factoring
1 . Transform the equation using standard form in which one side is zero. 2 . Factor the non-zero side. 3 . Set each factor to zero (Remember: a product of factors is zero if and only if one or more of the factors is zero). 4 . Solve each resulting equation. Example 1: Solve the equation, x2−3x−10=0 Factor the left side: (x−5)(x+2)=0 Set each factor to zero: x−5=0 or x+2=0 Solve each equation: x=5 or x=−2 The solution set is {5,−2}{5,−2} .
The Cartesian Coordinate System
A Cartesian Coordinate System is a system used to display a specified point defined in a plane by a set of numerical coordinates. The coordinates represent the point's distance from two, fixed perpendicular lines, known as axes (plural of axis).
Word Problems Using a Variable and a Constant
A constant in math is a value that doesn't change. All numbers are constants. Some letters, like e, or symbols, such as π, are also constants. Additionally, a variable can be a constant if the problem assigns a specific value to it. A variable is a letter that represents an unknown number. An expression is a group of numbers, symbols, and variables that represents another number.
Cylinders
A cylinder is defined as a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in a plane not parallel to the given line. Such cylinders have, at times, been referred to as generalized cylinders .
Line Graphs
A line graph is a type of chart used to show information that changes over time. We plot line graphs using several points connected by straight lines. We also call it a line chart. The line graph comprises of two axes known as 'x' axis and 'y' axis. The horizontal axis is known as the x-axis. The vertical axis is known as the y-axis.
Whole Number Lines
A number line is a horizontal line that has points, equally spaced, which correspond to each of the whole numbers. Whole Number Line Each number is greater than all the numbers to its left and less than all the numbers to its right. For example, 5 is greater than 0 and 3, but less than 6 and 8.
Repeated Flip of a Single Coin
A sample space is a set (i.e., collection) of all possible events in a probabilistic experiment. For example, when we flip a coin, we can either get Heads (H) or Tails (T). So the sample space is S={H,T}. Every subset of a sample space is called an event. For a single toss of a coin, we can make four subsets of the sample space, i.e., the empty set Φ, {H}, {T} and the sample space itself {H,T}. The probability of an empty set (i.e., neither Heads nor Tails) is always zero, and the probability of the entire sample space ( i.e., either Heads or Tails) is always 1. For any other given event E (i.e., a subset of S), we can use the following formula P(E)=Number of elements in E / Number of elements in S
Defining Probability of a Single Event
A sample space represents all the possible outcomes of an experiment. A tree diagram is a good way to represent the outcomes. We can also organize the outcomes in a list or chart. A tree diagram is drawn to show all the possible combinations or outcomes in a sample space. Let's look at a very simple tree diagram for a coin toss. There are only 2 possible outcomes. sample space — all possible outcomes The probability of an event occurring, like heads in a coin toss, is the ratio or comparison of desired outcomes to the total number of possible outcomes. If all the outcomes of an event are equally likely to occur, the probability of the event happening is: Event = number of favorable outcomes / total number of possible outcomes So, tossing heads in probability terms would be: Probability (heads) = 12If the probability is not 0 or 1, the probability is expressed in a fraction.By representing probability in fraction form, such as 12 , we are saying there are 1 out of 2 chances that the coin will be tails or 1 out of 2 chances that the coin will be heads. Mathematicians have discovered that if a probability experiment has only a few trials, such as tossing a coin only four times, the results can be misleading. If we only tossed the coin four times, we could end up with heads on all four tosses instead of just two. If, however, we toss the coin 100 times, it is more likely that we will have heads occurring 50 times. In other words, the more times an experiment is done, the closer the experimental probability comes to the probability we figured out mathematically (theoretical probability).
Solving Single Variable Equations
A single variable equation is an equation in which there is only one variable used. (Note: the variable can be used multiple times and/or used on either side of the equation; all that matters is that the variable remains the same.) Ex. each of these have only one variable in the equation. (x+4) / 2 = 12 6x + 3 −2x = 19 4y − 2 = y + 7
Spheres
A sphere is a geometrical object that is a three-dimensional analogue to a circle in two-dimensional space. A sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the center of the sphere, and r is the sphere's radius.
Acute, Obtuse, and Right Angles
Acute Angle- Acute angles measure less than 90 degrees Obtuse Angle- Obtuse angles measure more than 90 degrees Right Angle- Right angles measure 90 degrees
Working with Signed Numbers
Addition- do you have the same sign? If so the answer will too. If not then Find the DIFFERENCE and take the sign of the "larger" (absolute value) number. Subtraction- Add the opposite. ex. (-2-3=) you would keep -2 then put a + and write in -3. (-2 + -3=) or if there are two negatives next to each other they become positive. ex. 3-(-4)= would become 3+4= Multiplication and Division- Are there an EVEN number of negative signs? If YES (even number of negative signs) The answer is positive (+). If no (odd number of negative signs) The answer is negative (-) (-*- = +) (-*+ = -) (+*- = -)
Properties of Rectangles
All the angles of a rectangle are 90° Opposite sides of a rectangle are equal and Parallel. Diagonals of a rectangle bisect each other.
Identity Properties
An identity is a number that when added, subtracted, multiplied or divided with any number (let's call this number n), allows n to remain the same. ... In multiplication and division, the identity is 1. That means that if 0 is added to or subtracted from n, then n remains the same. 32 x 1 = 32
Factor Trees
Factor trees are a way of expressing the factors of a number, specifically the prime factorization of a number. Each branch in the tree is split into factors. Once the factor at the end of the branch is a prime number, the only two factors are itself and one so the branch stops and we circle the number. Ex. 48 8 * 6 2*4 2*3 2*2*2*2*3
Using a Compass and Straightedge
Compasses are used to draw precise circles and arcs, leading to making many geometric figures. Straightedges are used to make straight lines that are exact measurements. There is a need for students to understand and be able to construct geometric figures using a compass and straightedge.
"Completing the Square"
Complete the square formulaIn mathematics, completing the square is used to compute quadratic polynomials. Completing the Square Formula is given as: ax2 + bx + c ⇒ (x + p)2 + constant. The quadratic formula is derived using a method of completing the square. Completing the square means writing a quadratic in the form of a squared bracket and adding a constant if necessary. For example, consider x2 + 6x + 7. Start by noting that. (x + 3)2 = (x + 3)(x + 3) = x2 + 6x + 9.
Using the FOIL Method to Distribute in Quadratics
In other words, we can use the FOIL method when polynomials or quadratic equation or expression is in its factored form. This example is a quadratic expression in its factored form. We perform FOIL by multiplying the terms in this order: First, Outside, Inside, Last. (2x + 3)(4x +7) = 8x^2 + 14x + 12x + 21
Using Proportions to Solve Word Problems
Many "proportion" word problems can be solved using other methods, so they may be familiar to you. For instance, if you've learned about straight-line equations, then you've learned about the slope of a straight line, and how this slope is sometimes referred to as being "rise over run". But that word "over" gives a hint that, yes, we're talking about a fraction. And this means that "rise over run" can be discussed within the context of proportions.
Choosing the Best Measure of Central Tendency
Mean is the most frequently used measure of central tendency and generally considered the best measure of it. However, there are some situations where either median or mode are preferred. Median is the preferred measure of central tendency when: There are a few extreme scores in the distribution of the data. (NOTE: Remember that a single outlier can have a great effect on the mean). There are some missing or undetermined values in your data. There is an open ended distribution (For example, if you have a data field which measures number of children and your options are 00, 11, 22, 33, 44, 55 or "66 or more," than the "66 or more field" is open ended and makes calculating the mean impossible, since we do not know exact values for this field). You have data measured on an ordinal scale. Mode is the preferred measure when data are measured in a nominal ( and even sometimes ordinal) scale.
Analyzing Sources of Bias
Selection bias Self-selection bias Recall bias Observer bias Survivorship bias Omitted variable bias Cause-effect bias Funding bias Cognitive bias
Multiplying and Dividing Fractions
Multiplying fractions is easy: you multiply the top numbers and multiply the bottom numbers. For instance: 2/3⋅4/15 = (2⋅4) / (3⋅15) = 8/45 then simply if needed. Dividing fractions is just about as easy as multiplying them; there's just one extra step. When you divide by a fraction, the first thing you do is "flip-n-multiply". That is, you take the second fraction, flip it upside-down (that is, you "find the reciprocal"), and then you multiply the first fraction by this flipped fraction. 3/5 / 9/4 = would be 3/5 x 4/9 = 3*4 / 5*9 = 12 / 45 then simplify by dividing both by 3 to get 4/15.
Solving Inequalities
Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign. Solving inequalities is very like solving equations, we do most of the same things, but we must also pay attention to the direction of the inequality. These things do not affect the direction of the inequality: Add (or subtract) a number from both sides Multiply (or divide) both sides by a positive number Simplify a side But these things do change the direction of the inequality ("<" becomes ">" for example): Multiply (or divide) both sides by a negative number Swapping left and right hand sides We can often solve inequalities by adding (or subtracting) a number from both sides Everything is fine if we want to multiply or divide by a positive number If we multiply or divide by a negative number, be sure to flip the sign in the inequality.
Order of Operations
PEMDAS (parenthesis, Exponents, Multiplication, Division, Addition, Subtraction) All done from left to right when getting to the MDAS part.
Base 10 Number System and Place Value
Simply put, base-10 is the way we assign place value to numerals. It is sometimes called the decimal system because a digit's value in a number is determined by where it lies in relation to the decimal point.
Prime Numbers and Rational Numbers
Prime Number: A prime number is any whole number greater than 1 that only has the number 1 and itself as factors. {1,3,5,7....} Rational Number: A rational number is a number that is of the form p/q where p and q are integers and q is not equal to 0. {1/2, 5/6, 5/1, 6/2....}
Real Numbers, Integers, and Whole Numbers
Real Numbers: The real numbers is the set of numbers containing all of the rational numbers and all of the irrational numbers. {1, 12.38, −0.8625, 3/4, π (pi), 198, √2} Real Numbers can also be positive, negative or zero. Integers: The integers are the set of real numbers consisting of the natural numbers, their additive inverses and zero. {...,−5,−4,−3,−2,−1,0,1,2,3,4,5,...} Whole Numbers: The whole numbers are the part of the number system in which it includes all the positive integers from 0 to infinity. {0,1,2,3,4,5......1489....903837...}
Fractional Number Lines
Representing fractions on a number line means that we can plot fractions on a number line, which is similar to plotting whole numbers and integers. Fractions represent parts of a whole.
Scientific Notation
Scientific notation is a way of writing very large or very small numbers. A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10. For example, 650,000,000 can be written in scientific notation as 6.5 ✕ 10^8.
Using the Rule of Threes
The Rule of Three is a Mathematical Rule that allows you to solve problems based on proportions. By having three numbers: a, b, c, such that, ( a / b = c / x), (i.e., a: b :: c: x ) you can calculate the unknown number. The Rule of Three Calculator uses the Rule of Three method to calculate the unknown value immediately based on the proportion between two numbers and the third number. The working of the Rule of Three Calculator can be expressed as follows: A ---> B X---->Y ---------> Y = B*X / A
Associative Property
The associative property is a math rule that says that the way in which factors are grouped in a multiplication problem does not change the product. addition (A + B) + C = A + (B + C) multiplication (A × B) × C = A × (B × C)
Commutative Property
The commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer. The property holds for Addition and Multiplication, but not for subtraction and division. a + b = b + a
Frequency Tables and Scatter Plots
The frequency of a particular data value is the number of times the data value occurs. For example, if four students have a score of 80 in mathematics, and then the score of 80 is said to have a frequency of 4. The frequency of a data value is often represented by f. A frequency table is constructed by arranging collected data values in ascending order of magnitude with their corresponding frequencies. A scatter plot (aka scatter chart, scatter graph) uses dots to represent values for two different numeric variables. The position of each dot on the horizontal and vertical axis indicates values for an individual data point. Scatter plots are used to observe relationships between variables.
Solving Systems Using the Substitution Method
The method of substitution involves three steps: Solve one equation for one of the variables. Substitute (plug-in) this expression into the other equation and solve. Resubstitute the value into the original equation to find the corresponding variable. 3x+y=−3 x=−y+3 you plug x= -y+3 into the first equation to find y
The Metric System
The metric system has meter, centimeter, millimeter, and kilometer for length; kilograms and gram for weight; liter and milliliter for capacity; hours, minutes, seconds for time.
Mode
The mode is the value that occurs most often. The mode is the only average that can have no value, one value or more than one value. When finding the mode, it helps to order the numbers first.
Rolling Two Dice Together
The number of total possible outcomes is equal to the total numbers of the first die (6) multiplied by the total numbers of the second die (6), which is 36. So, the total possible outcomes when two dice are thrown together is 36.
Finding Percent
The percentage is the result when a specific number is multiplied by a percent. Most of the time, percentages are smaller than the number since a percentage is a portion of a number or quantity. But there are cases that the percentage is greater than the number. This would happen if the percent is greater than 100%. What percent of 72 is 18. 18 is the percentage. 72 is the quantity. Percent is asked. Dividing the percentage by the quantity; 18 / 72 = 0.25 Multiplying the product by 100 and place a percent symbol (%) after; 0.25 x 100 = 25% Therefore, 18 is 25% of 72.
Properties of Trapezoids
These are the properties of a trapezoid that make it stand out from other quadrilaterals: -The bases (the top and bottom) are parallel to each other -Opposite sides of a trapezoid (isosceles) are of the same length -Angles next to each other sum up to 180° -The median is parallel to both the bases -Median's length is the average of both the bases i.e. (a +b)/2 -If both pairs of the opposite sides are parallel in a trapezoid, it is considered a parallelogram -If both pairs of the opposite sides are parallel, all sides are of equal length, and at right angles to each other, then a trapezoid can be considered as a square -If both pairs of the opposite sides are parallel, its opposite sides are of equal length and at right angles to each other, then a trapezoid can be considered as a rectangle
Converting Improper Fractions to Mixed Numbers
To convert an improper fraction to a mixed fraction, divide the numerator by the denominator, write down the quotient as the whole number and the remainder as the numerator on top of the same denominator. 13/4 is an improper fraction. You divide 13 by 4 which is 3 remainder 1. The remainder is put on top of the denominator to get 3 1/4.
Converting Fractions, Decimals, and Percentages
To convert from percent to decimal divide by 100 and remove the % sign. An easy way to divide by 100 is to move the decimal point 2 places to the left To convert from decimal to percent multiply by 100% An easy way to multiply by 100 is to move the decimal point 2 places to the right. To convert a fraction to a decimal divide the top number by the bottom number. To convert a decimal to a fraction First, write down the decimal "over" the number 1. Multiply top and bottom by 10 for every number after the decimal point (10 for 1 number,100 for 2 numbers, etc) This makes a correctly formed fraction, then simplify. To convert a fraction to a percentage divide the top number by the bottom number, then multiply the result by 100% To convert a percentage to a fraction, first convert to a decimal (divide by 100), then use the steps for converting decimal to fractions
Finding Common Factors
To find the common factors of two numbers, you first need to list all the factors of each one and then compare them. If a factor appears in both lists then it is a common factor. Find the common factors of 20 and 36. Solution We need to list the factors of 20 and 36 separately; Factors of 20 = 1, 2, 4, 5, 10 and 20. Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, and 36. Therefore, we can observe the common factors of 20 and 36 are 1, 2 and 4
Using a Graph to Determine the Equation of a Line
To find the equation of a graphed line, find the y-intercept and the slope in order to write the equation in y-intercept (y=mx+b)form. Slope is the change in y over the change in x. Find two points on the line and draw a slope triangle connecting the two points.
Reducing Fractions
To reduce a fraction, divide both the numerator and denominator by the GCF. 6/12 reduces when you divide both the top and bottom number by 6. You then get 1/2.
Simplifying Expressions
To simplify any algebraic expression, the following are the basic rules and steps: Remove any grouping symbol such as brackets and parentheses by multiplying factors. Use the exponent rule to remove grouping if the terms are containing exponents. Combine the like terms by addition or subtraction. Combine the constants.
Solving Systems Using the Addition/Subtraction Method
To use the addition/subtraction method, do the following: Multiply one or both equations by some number(s) to make the number in front of one of the letters (unknowns) the same or exactly the opposite in each equation. Add or subtract the two equations to eliminate one letter. Solve for the remaining unknown. Solve for the other unknown by inserting the value of the unknown found in one of the original equations.
Independent Choices
Two choices are independent if they do not affect one another's outcomes; if each may be made without influence from the other. The licence plate being used as an example has the digits 723. The fact that the first digit is a 7 still leaves all ten digits, 0-9, available for the second digit. If we look at the choices of three digits available they go from 000 to 999 with one thousand total three-digit choices possible. Why? Because of the law of counting for independent choices: Independent choices multiply. There are ten choices for the first digit, ten choices for the second digit, and ten choices for the third digit. The total number of possible outcomes is therefore 10×10×10=1000. When Occupy Math computed the number of licence plates with four letters and three digits, that thousand was multiplied by 26 four times — once for each 26-way choice of a letter.
Properties of Parallel Lines
Two lines in a plane are said to be parallel if they do not intersect when extended infinitely in both directions. Two straight lines are said to be parallel if their slopes are equal and they have different y-intercepts.
Types of Measurements
Types of data measurement scales: nominal, ordinal, interval, and ratio. nominal- A Nominal Scale is a measurement scale, in which numbers serve as "tags" or "labels" only, to identify or classify an object. This measurement normally deals only with non-numeric (quantitative) variables or where numbers have no value. Below is an example of Nominal level of measurement. Please select the degree of discomfort of the disease: 1-Mild 2-Moderate 3-Severe In this particular example, 1=Mild, 2=Moderate, and 3=Severe. Here numbers are simply used as tags and have no value. ordinal- "Ordinal" indicates "order". Ordinal data is quantitative data which have naturally occurring orders and the difference between is unknown. It can be named, grouped and also ranked. "How satisfied are you with our products?" 1- Totally Satisfied 2- Satisfied 3- Neutral 4- Dissatisfied 5- Totally Dissatisfied interval- The interval scale is a quantitative measurement scale where there is order, the difference between the two variables is meaningful and equal, and the presence of zero is arbitrary. It measures variables that exist along a common scale at equal intervals. The measures used to calculate the distance between the variables are highly reliable. ratio- Ratio scale is a type of variable measurement scale which is quantitative in nature. It allows any researcher to compare the intervals or differences. Ratio scale is the 4th level of measurementand possesses a zero point or character of origin. This is a unique feature of this scale. For example, the temperature outside is 0-degree Celsius. 0 degree doesn't mean it's not hot or cold, it is a value. Following example of ratio level of measurement to help understand the scale better. Please select which age bracket do you fall in? Below 20 years 21-30 years 31-40 years 41-50 years 50 years and above
Pie Charts
What is a pie chart? A pie chart shows the relationships of parts to the whole for a variable. How are pie charts used? Pie charts help you understand the parts-to-a-whole relationship. Pie charts are often used in other situations, even if bar charts or line graphs might be a better choice. What are some issues to think about? Pie charts are used for nominal or categorical data. When there are many levels to your variable, a bar chart or packed bar chart may provide a better visualization of your data.
