Mathematics Knowledge
solve for x: -4 + 2x = 31 - x^2
-4 + 2x = 31 - x^2 +x^2 -31 = -31 + x^2 x^2 + 2x - 35 = 0 FOIL ( )( ) = 0 ( x )( x ) = 0 ( x + 7 )( x - 5 ) = 0 x + 7 = 0 x - 5 = 0 x = -7, 5
solve for a: -5a > 10
-5a > 10 -5 -5 *flip inequality because dividing by -5 a < -2
Each chef at "Sushi Emperor" prepares 15 regular rolls and 20 vegetarian rolls daily. On Tuesday, each customer ate 2 regular rolls and 3 vegetarian rolls. By the end of the day, 4 regular rolls and 1 vegetarian roll remained uneaten. How many chefs and how many customers were in "Sushi Emperor" on Tuesday?
15 regular rolls & 20 vegetarian rolls per chef 2 regular rolls & 3 vegetarian rolls per customer 4 regular rolls & 1 vegetarian roll left x = number of chefs y = number of customers 15x - 2y = 4 20x - 3y = 1 (15x - 2y = 4)(-3) +(20x - 3y = 1)(2) -45x + 6y = -12 +40x - 6y = 2 -5x = -10 x=2 (still verify) 15x - 2y = 4 15(2) - 2y = 4 30 -2y = 4 -30 -30 -2y = -26 -2 -2 y = 13 (still verify) 20x - 3y = 1 20(2) - 3(13) = 1 40 -39 = 1 1 = 1 x = 2, y = 13
angles that form a strain line add up to blank degrees?
180˚
angles around a point add up to blank degrees?
360
solve for a: 4a + 6 > 2a + 10
4a + 6 > 2a + 10 4a + 6 > 2a + 10 -2a + 6 > -2a + 10 2a + 6 > 10 -6. > -6 2a > 4 2 2 a>2
Area of a triangle is
A = ½(base)(height) A = ½bh
formula for the area of a parallelogram
Area = (base)(height) A = bh
formula for the area of a rectangle
Area = (length)(width) A = lw
formula for the area of a square
Area = (side)² A = s²
formula for the area of a trapezoid
Area = ½(sum of the length of the bases)(height)
Area of a circle (formula)
Area = πr²
solve for x: x^2 - 6x + 9 = 0
FOIL ( )( ) = 0 ( x )( x ) = 0 ( x - 3 )( x - 3 ) = 0 x - 3 = 0 x - 3 = 0 x = 3
solve for y: y^2 + 4y + 3 = 0
FOIL ( )( ) = 0 ( y )( y ) = 0 ( y + 1 )( y + 3 ) = 0 y + 1 = 0 y + 3 = 0 y = -1, -3
What is used to multiply binomials? (y+1)(y+2)
FOIL - first, outer, inner, last
ANSWER FLIP CARD Step 1: Note that the figure is half of a circle sitting on top of a rectangle. Split the figure up and work with the two shapes independently. Step 2: Plug the values from the diagram into the area formulas for a rectangle. Area = lw = 7*4 = 28 Step 3: Since 4 is the diameter of the circle, 2 is the radius. Use that information in the area formula for a circle. Area = πr² = π2² = 4π Step 4: Since it is only half of a circle, you need to divide the area by two. 4π ÷ 2 = 2π Step5: Add the area for the rectangle and the area of the half circle together to get your answer. 28 + 2π
Find the are of the figure above.
ANSWER FLIP CARD Step 1: The question provides a diagram of a box Step 2: The question asks for the surface area of that box Step 3: Use the formula for surface area of a rectangular solid, plug in the values given, and solve: SA = 2lw + 2wh + 2lh SA = 2*8*5 + 2*5*6 + 2*8*6 SA = 80+ 60 + 96 SA = 236 Step 4: Confirm that you plugged in the correct values for the correct variables.
Find the surface area of the box above.
FLIP CARD OVER FOR QUESTION ¾ = 6/s 3s = 4*6 3s = 24 s = 8
Find the value of s in the diagram above.
ANSWER FLIP CARD V = lwh V = 8*5*6 V = 240
Find the volume of the box in the figure above.
ANSWER FLIP CARD V = πr²h V = π*4²*5 V = 20π
Find the volume of the cylinder in the figure above.
FOIL
First, Outer, Inner, Last
ANSWER FLIP CARD Step 1: The question gives you two circles and there radius measures. Step 2: You need the ratio of the area of A to the area of B. The answer choices are all expressed as fractions, so it will look like this: (Area of A / Area of B) Step 3: Calculate the area of each. Area of A = π1² = π Area of B = π2² = 4π Area of A = π = 1 Area of B 4π 4 Step 4: Choice a) is correct. Briefly double-check your calculations and confirm
In the figure below, the radius of circle A has a length of 1 unit, and the radius of circle B has a length of 2 units. What is the ratio of the area of A to the area of B? a) ¼ b) ½ c) 1/π d) π/2
obtuse angle
Measures more than 90 degrees and less than 180 degrees
IF PR = 12 and QR = 4, PQ = ? | | | p q r
PR - QR = PQ 12 - 4 = PQ 8 = PQ
formula for the perimeter of a rectangle
Perimeter = 2(length + width) P = 2(l+w)
formula for the perimeter of a rhombus
Perimeter = 4*sides P = 4s
formula for the perimeter of a square
Perimeter = 4*sides P = 4s
Factor: x^2 - 4x - 21
Reverse FOIL To begin build parentheses with the First terms that multiply to x^2 (x )(x ) What Last term will multiply to produce -21? (factorization, factors) (1x-21, 3x-7, 7x-3, 21x-1) Which of these pairs will add to produce -4? Plug these into the parentheses (x+3)(x-7)
surface area of a rectangle or box formula
SA = 2lw + 2wh + 2lh
surface area of a cylinder formula
SA = 2πr² + 2πrh
Find the value of a using combination: 4a + 2b =44 6a - 2b = 46
Step 1: The question give you two linear equations with two variables Step 2: You need to solve for one variable, a Step 3: If you add the two equations together, you will eliminate b. This can be done by using combination. 4a + 2b = 44 +[6a - 2b = 46] 10 a = 90 a = 9 choice b) is correct Step 4: Briefly confirm that you performed the calculations correctly
Find the value of x using substitution: 6x + y = 15 2x - 3y = -5 a) -2½ b) 5/4 c) 2 d) 3
Step 1: The question give you two linear equations with two variables Step 2: You need to solve for one variable, x Step 3: It is easy to isolate y in the first equation, so use substitution to do so: 6x + y = 15 y = 15 - 6x Plug in 15 - 6x for y in the second equation: 2x - 3y = -5 2x -3(15 - 6x) = -5 Solve for x: 2x - 45 +18x = -5 20x - 45 = -5 20x = 40 x = 2 Choice c) is correct Step 4: Briefly confirm that you followed PEMDAS and performed the calculations correctly
ANSWER FLIP CARD Step 1: The question supplies this information A = 64π d = x Step 2: You're asked for the measure of the diameter, or x. Step 3: Work backwards from what you know. A = 64π = r²π r² = 64 r = 8 x = diameter = 2r = 16 Step 4: Choice d) is correct. Briefly double-check your math and confirm.
The diameter of the circle below has measure x. The area of the circle is 64π. what is the value of x? a) 4 b) 2π c) 8 d) 16
Answer flip card Step 1: You're given two shapes, a square and a trapezoid. Area of square = 9 Area of trapezoid > 2 * Area of square Area of trapezoid > 18 Step 2: You're asked for the value of x Step 3: There's one unknown in the question and numbers in the choices, so use Back-solving. Try choice b): if x = 3, then Area of trapezoid = ½ (4+5)(3) = 13.5 That's too small. Eliminate a) and b). Try choice c): if x = 4, then Area of trapezoid = ¼ (4+5)(4) = 18 That's also too small, since the are of the trapezoid is supposed to be greater than 18. Eliminate c); d) must be correct Step 4: Briefly check that you answer makes sense and move on to the next question
The trapezoid below has an area that is greater than twice the area of the square below. Which of the following could be the value of x? a) 2 b) 3 c) 4 d) 5
Answer flip card Step 1: The area of the trapezoid is a multiple of five. So, ½( y + 11 ) 2 = ( y + 11 ) is a multiple of five. Step 2: You're asked for a possible value of y. Step 3: Start by using Strategic Guessing Using Logic. Multiple of 5 always end in 5 or 0. To produce a number that ends in 5 or 0, y would have to end with 4 or 9. Only choice c) fits Step 4: Briefly confirm: ½ (14 + 11)2 = (14 + 11) = 25, which is a multiple of 5
The value of the are of the trapezoid below is divisible by 5 with no remainder. Which of the following is a possible value of y? a) 11 b) 12 c) 14 d) 18
volume of a cylinder formula
V = πr²h
FLIP CARD OVER FOR QUESTION A = bh A = 2*4 A = 8
What is the area of the parallelogram?
FLIP CARD OVER FOR QUESTION A = ½bh A = (½)(7)(4) A = 14
What is the area of the triangle?
FLIP CARD OVER FOR QUESTION 5 + 8 + 3 + 7 = 23
What is the perimeter of the quadrilateral?
FLIP CARD OVER FOR QUESTION Step 1: The figure shows three intersecting lines that form six angles, which all together equals 360˚. An unlabeled angle is 90˚. Three labeled angles are x, y, and z Step 2: The questions asks for the sum of the three labeled angles. Step 3: Because the angle labeled y is a vertical angle to the right, y must equal 90˚. Because the angles labeled x and z form a straight line with the right angle, together they equal 180˚ - 90˚ = 90˚ y + (x + z) = 90˚ + 90˚ = 180˚ Step 4: Choose option c). Confirm that the answer includes the correct three angles and that your arithmetic is correct.
What is the value if x + y + z ? a) 90 b) 120 c) 180 d) 270
FLIP CARD OVER FOR QUESTION Step 1: The figure shows two parallel lines, m and n, crossed by transversal p Step 2: Find the sum of the four labeled angles Step 3: Two of the labeled angles are acute, and two are obtuse. Every pair of acute + obtuse angles formed by a transversal crossing parallel lines equals 180˚ Two acute + obtuse angle pairs = 2 x 180˚ = 360˚ Choice d) is correct Step 4: Confirm that the answer includes the correct four angles and that your arithmetic is correct
What is the value of a + b + c + d ? a) 120 b) 180 c) 200 d) 360
FLIP CARD OVER FOR QUESTION x + 50 + 100 = 180 x + 150 = 180 - 150 = - 150 x = 30
What is the value of x?
FLIP CARD OVER FOR QUESTION y = 40 + 90 y = 135
What is the value of y?
diameter
a circle is a chord that passes through the circle's center all diameters are the same length and are equal to twice the radius
circle
a figure in which each point is an equal distance from its center
quadrilateral
a figure with four sides
coordinate grid
a grid composed of a horizontal x-axis and a vertical y-axis
a line on the coordinate grid can be represented by the blank
a line on the coordinate grid can be represented by the slope-intercept form of a linear equation
chord
a line segment that connects any two points on a circle
transversal line
a line that intersects two or more parallel lines
expression
a mathematical phrase that contains operations, numbers, and/or variables.
term
a number, a variable, or one or more numbers multiplied by one or more variables 4, 3x, 7ab^2, (6z)/4
rectangle
a parallelogram containing four right angles opposite sides are equal
origin (coordinate grid)
a place where the axes meet (0,0)
coordinate
a point on the coordinate grid with an x-coordinate and a y-coordinate (x,y)
parallelogram
a quadrilateral with two sets of parallel sides opposite sides are equal as well as opposite angles
equilateral triangle
a triangle in which all three sides are equal and all three angles are also equal all angles measure 60 degrees
right triangle
a triangle with a right angle and two acute angles
isosceles triangle
a triangles that has two equal sides and the angles opposite the equal sides are also equal
exterior angles of a triangle
an exterior angle of a triangle is equal to the sum of the two non-adjacent (opposite) interior angles of the triangle
interior angles
angles on the inside of parallel lines cut by a transversal d˚, c˚, e˚, f˚
corresponding angles
angles with the same relationship to the transversal line a˚ = e˚, b˚ = f˚, d˚ = h˚, c˚ = g˚
parallel lines
are lines that have the same slope as on another
Pythagorean Theorem
a² + b² = c²
One leg of a right triangle is 2 and the other leg is 3. What is the length of the hypotenuse?
a² + b² = c² 2² + 3² = c² 4 + 9 = c² 13 = c² c = √13
rhombus
has four equal sides, with the opposite sides parallel to one another
line segment
is a piece of a line, and it has an exact measurable length
trapezoid
is a quadrilateral with one pair of parallel sides the two parallel sides are called the bases
square
is a rectangle with four equal sides
x-intercept
is the point where the line crosses the x-axis; the y-coordinate is always zero at the x-intercept
y-intercept
is the point where the line crosses the y-axis; the x-coordinate is always zero at the y-intercept
When multiplying or dividing an inequality by a negative number, what happens to the inequality signs?
it changes direction
Kevin is 2 times as old as Gabriela. 12 years ago, Kevin was 6 times as old as Gabriela. How old is Gabriela now?
k = 2g k-12 = 6(g-12) 2g - 12 = 6g - 72 60 = 4g g=15 (now we verify) k = 2g k = 2(15) k = 30 k-12 = 6(g-12) 30-12 = 6(15-12) 18 = 6(3) 18 = 18 (IT VERIFIES!!) Gabriela is 15 years old
perpendicular lines
lines that intersect to form right angles
bisect
means to cut in half, so the midpoint of a line segment bisects that line segment
right angle
measures 90 degrees and is usually indicated in a diagram by a little box
acute angle
measures less than 90 degrees
monomial
one term
slope
represents the steepness of the line and the direction it goes can be thought of as "rise over run," or "change in y over change in x"
slope formula
slope = change in y / change in x
base angles (isosceles)
the angles opposite the equal sides of an isosceles are also equal
circumference (and formula)
the distance around it 2πr or πd
midpoint
the point exactly in the middle of a line segment, halfway between the endpoints
what are the sides of a right triangle called
the sides opposite the acute angles are called the legs, and the side opposite the right angle is called the hypotenuse
radius
the straight-line distance from its center to any point on the circle, all radii of a given circle have equal lengths
perimeter
the sum of the length of the sides
perimeter of a quadrilateral
the sum of the measure of all four sides
interior angles of a triangle
the three angles add up to 180˚
similar triangles
triangles that have the same shape, but may be different sizes; their corresponding angles are equal, and their corresponding sides are proportional
polynomial
two or more terms in an expression
binomial
two terms
volume of a rectangle or box formula
v = l * w * h
vertical angles
when lines intersect, angles across the vertex are equal to each other a˚ = c˚ b˚ = d˚
supplementary angles
when two lines intersect the adjacent angles ass up to 180 degrees
combination (systems of equations)
when you have two variables and two equations you can combine the two equations by adding or subtracting them in such a was that you eliminate one of the variables
Simplify: (x+y)(x+y) or (x+y)^2
x^2 + 2xy + y^2
Simplify: (x-y)(x-y) or (x-y)^2
x^2 - 2xy + y^2
Simplify: (x+y)(x-y)
x^2 - y2
slope-intercept form of a linear equation
y = mx + b
Ben is 39 years old and Ishaan is 3 years old. How many years will it take until Ben is only 4 times as old as Ishaan?
y is the number of years 39+y = 4(3+y) 39+y = 12 + 4y 27 = 3y y=9