Addition Properties. Properties something you cannot disprove always true. *You must memorize these properties!

Transcription

1 Addition Properties Properties something you cannot disprove always true. *You must memorize these properties! 1) Commutative property of addition changing the order of addends will not change the sum Ex) 4+5 = = 9 Addends Sum Addends Sum Ex) -4+3[(3+22)3+b-2] = 3[(3+22)3+b-2] +-4 Recognize this as a properties problem. See that addends are the same on both sides. *(Think about the messy part of the problem as one term say negative four plus the mess equals the mess plus negative four) 2) Associative property of addition regrouping the addends will not change the sum. Ex) (3+4)+7 = 3 +(4+7) We did not change the order of addends (that would make it the commutative property) 3) Additive identity property any number (addend) plus zero is that number. The identity stays the same. Ex) - 4+[(3+2 2 ) 3 +b -2 ] 3 +0 = - 4+[(3+2 2 ) 3 +b -2 ] 3 (A mess plus zero equals the mess - its identity didn t change) 4) Additive inverse (opposite) property The sum of a number and its inverse equals zero. (Don t confuse with additive identity property.) Ex) =0 Zero will always be the answer to an additive inverse property problem. Ex) (a 2 +b-c) + - (a 2 +b-c)=0 this would be read the quantity of a 2 +b-c plus the opposite of the quantity of a 2 +b-c equals zero. 3-1

2 Multiplication Properties 1) Commutative property of multiplication changing the order of the factors will not change the product. Ex) ab=ba Two variables next to each other means multiplication (ab means the same as a b) Ex) 4(3)=3(4) 2) Associative property of multiplication - changing the grouping (regrouping) of the factors will not change the product. Ex) (5 4) 6=5(4 6) 3) Multiplicative identity property any number multiplied by the factor 1 is the original number. It keeps its identity. Ex) 6 1=6 Ex) 4[(2+b2)3+7] 1=4[(2+b2)3+7] 4) Multiplicative inverse property any number multiplied by its multiplicative inverse is one (1). This property involves multiplying a number by its reciprocal. 4 Ex) = = sneaky name for one 5) Multiplicative property of zero any factor multiplied by zero is zero. Ex) 7 0=0 Ex) b( ) 0 =0 6) Distributive property a number next to a parenthesis means multiply. Distribute multiplication over parenthesis. Ex) 3(4+-2) = (3 4)+(3-2) Ex) 4(7-6) = (4 7)-(4 6) Ex) (2-5)-(2 3) = 2( - 5-3) 3-2

3 Quadrilaterals A quadrilateral is a closed plane, two dimensional figure with four sides that are line segments. There are no curves. means congruent means congruent (the dashes on the sides of the figure) 3-3

4 Quadrilateral Flowchart Quadrilateral (4 sided polygon ) *All Quadrilaterals diagonals bisect* Trapezoid Quadrilateral with: exactly one pair of opposite, parallel sides. (Isosceles trapezoid 1 set of opposite parallel sides. The other set of opposite sides are congruent lines.) Parallelogram Quadrilateral with: 2 sets of opposite parallel sides Opposite angles are congruent Rectangle Parallelogram with: Four right angles Diagonals are congruent Rhombus Parallelogram with: Four congruent sides Diagonals are perpendicular Square Parallelogram with: Four congruent angles Four congruent sides Diagonals are congruent and perpendicular 3-4

5 21 Similar Figures Shapes that look exactly the same but can be different sizes are similar. Similar symbol ~ Two figures are considered similar if: a) Corresponding angles ( s) are congruent ( ) and b) Corresponding sides are proportional A corresponds to D B E BCA corresponds to EFD corresponds to A C To check if corresponding sides are proportional, first use proportional reasoning. Each side has to increase or decrease by the same number of times. If you can t see proportional reasoning, set up a proportion and use labels. If you don t see the proportional 10 5 reasoning, cross multiply to see if the answers are equal. If they are 7 not, then the figures aren t similar Proportion 15 (big triangle) = 21 (big triangle) X3 5 (small triangle = 7 (small triangle) D F Common mistake to check for corresponding, do not add/subtract. (They must increase or decrease by the same number of times, not the same number.) SOL test example: If the two triangles below are similar, which of the following must be true? X3 A D, B E, C F = =105 so they are similar B E A C D F A B C D AC = AB DE DE AB = BC DE DF AC AB = DF DE ABbig DEsmall = ACbig DFsmall (use labels to check) D is true because AB corresponds to DE and AC corresponds to DF so they must be proportional by definition of similarity. 3-5

6 Similar Figures/Missing Sides Two figures have to be similar before we can find the missing side. If they are asking us to find the missing side, they first tell us the two figures are similar. Ex) X 12 (big triangle) = 20 (big triangle) X4 3 (small triangle = X (small triangle) 12X=60 12X = X=5 X4 Ex) X (big) = 4.3 (big) 5.7 (small) = X (small 10.8X= X = X

7 Area The word cover is a good synonym for area. Area is 2 dimensional (2D) squared, power of 2, covering with2 dimensional (flat) squares. Area of a parallelogram Formula: area = bh (base times height) Steps: 1) Always write the formula first - area=bh 2) Label height and base of the parallelogram base is 6, height is 6 3) Substitute dimensions (numbers) into the formula area = 6 5 4) Label correctly area = 30 in. 2 5 in 6 in The height must be perpendicular to the base in a parallelogram area=bh area = 7 4 area = 28 cm 2 5 cm Common mistake would be to use 5 cm as the height instead of 4cm. 4 cm 7 cm Area of a triangle Formula: area = 2 1 bh (base times height) Use the same steps as for parallelogram. area= 2 1 bh 8(4) area = 2 32 area = 2 area = 16 in 2 8 in 4 in Volume is 3 dimensional (3D) cubed, power of 3, filling with cubes Perimeter is 1 dimensional (1D) go around with line segments 3-7

8 Surface Area (SA) of Rectangular Prisms Each face of a rectangular prism (3D) is a 2D parallelogram. When finding the surface area of a rectangular prism you are covering each face with 2D squares. When you unfold a 3 dimensional figure it is called a net. Formula: SA(surface area) = 2 lw (length times width)+2lh (length times height)+2wh (width times height) Steps: 1) Always write the formula first SA=2lw+2lh+2wh 2) Label* length, width, height of the rectangular prism 3) Substitute dimensions (numbers) into the formula 4) Label correctly *When you label, it does not matter which edge is length, width or height as long as you are consistent when substituting within the formula. Ex) 7 in length SA=2lw+2lh+2wh SA=2(7)(6)+2(7)(5)+2(6)(5) SA= SA=214 in 2 5 in height 6 in width Ex) To find the amount of glass needed for an aquarium with no top, you would alter the formula as follows: SA = lw+2lh=2wh because you don t need to add the surface area for the top. SA=(7)(6)+2(7)(5)+2(6)(5) SA= in height SA=172 in 2 5 in 7 in 35 in 2 5 in 30 in 2 42 in 2 30 in 2 35 in 2 42 in 2 6 in 5 in 6 in Using the Net add: 35 in 2 42 in 2 30 in 2 35 in 2 42 in 2 30 in 2 = 214 in 2 7 in length 6 in width 3-8

9 Surface Area (SA) of Rectangular Prisms Formula Explanation SA=2lw+2lh+2wh Part 1 Part 2 Part 3 SA= 2lw + 2lh + 2wh Bottom and top Front and back sides Part 1 top and bottom faces 1 2 length height width Part 2 front and back faces To get the area for the bottom face and the area for the top face, multiply the length by the width. lw 1 length 2 height width To get the area for the front face and the area for the back face, multiply the length by the height. lh Part 3 side faces 1 2 length height width To get the area for the side faces, multiply the width by the height. wh 3-9

10 Volume of Rectangular Prisms Volume means fill. You are filling with 3 dimensional cubes. Even liquid is measured in 3D cubes. Formula: v=lwh (volume = length times width times height) volume = length times width times height Tells how many cubes cover the bottom face (1 layer) Steps: Tells how many layers 1) Always write the formula first v=lwh 2) Label length, width, height of the rectangular prism 3) Substitute dimensions (numbers) into the formula 4) Label correctly Ex) v=lwh v=7(6)(5) v=210 in 3 Scale Factor: What you are multiplying by. 7 in length 5 in height 6 in width When you change one attribute by any scale factor, the volume will change by the same scale factor. Using the above example, if the width is doubled to 12 the volume is doubled (2 210 in 3 =420 in 3 ) as shown below. v=lwh v=7(12)(5) v=420 in

11 Area and Circumference of Circles Circle terminology: Diameter distance across the center of a circle Circumference distance around the circle (perimeter) Radius distance from the center to any point on the circle the diameter Formulas: Circumference - c=2πr (2 times pi times radius) or c=πd (pi times diameter) Area a=πr 2 (pi times radius squared) π 3.14 (Circumference/diameter = π) Ex) c=2πr c=2(3.14)(7) c cm 7 cm Ex) a=πr 2 a=3.14(5 2 ) a 78.5 in 2 10 in 3-11

12 Surface Area and Volume of Cylinders Surface Area Formula: SA =2πrh+ 2πr 2 (2 times pi times radius times height plus 2 times pi times radius squared. Formula explanation: Part 1 Part 2 Part 3 SA= 2πr X h + 2πr 2 The circumference of a circle which when unfolded becomes the base of a rectangle. Multiply by height to get the area of the rectangle. It is easier to visualize with a net of a cylinder: 2 circles and a rectangle. Ex) SA=2πrh+ 2πr 2 SA=2(3.14)(3)(7)+2(3.24)(3 2 ) SA=2(3.14)(3)(7)+2(3.24)(9) SA= SA in 2 Volume Formula: v=πr 2 h Ex) v=πr 2 h v=(3.14)(3 2 )(7) v=(3.14)(9)(7) v in 3 Add the area of the two circles. 3-12

13 Coordinate Plane Horizontal line Vertical Line Point of origin is where the horizontal and vertical number lines meet at 0,0. This forms four quadrants. The horizontal number line is the x-axis (move left and right) Vertical number line is the y-axis (move up and down) Coordinate plane extends forever in all directions. Quadrants extend forever but have 2 boundaries the x and y- axis. Whenever we draw a point on the coordinate plane it is called plotting a point. The coordinates of the point are a pairs of numbers with an x value and a y value (x,y). The x-axis is always first. Quadrant I both coordinates are positive (+,+) Quadrant II - x is negative, y is positive (-,+) Quadrant III - both coordinates are negative (-,-) Quadrant IV x is positive, y is negative (+,-) Ex) On this coordinate plane what are the coordinates for points A, G, F, K? A F G A (-4,3) G (5,4) F (1,0) K (-2,-3) K 3-13

14 Transformations (Movements) Movement of a figure or a point on a coordinate plane is a transformation. There are four types of transformations: 1. Translations slide 2. Reflections flipping over the x or y-axis 3. Rotations turn 4. Dilations enlarging/shrinking (similar figures on a coordinate plane) Translations 1-3 change the location. Transformation 4 (dilations) change the size. Of these, rotations and dilations are the hardest to understand. Pre-image original point or figure (given figure) on a coordinate plane Image new point or figure after the transformation (movement) Points and figures (after transformation) are marked with the prime symbol. ( ) Translation do one point at a time. = Arrow notation Ex) Translate ABCD 6 units to the right and 4 units down. The translation rule in arrow notation form for this example is: A(-5,5) (-5+6, 5-4) You can find A by just doing the addition or subtraction of the x and y. A (1,1) because -5+6=1 and 5-4=1 Coordinates: Pre-image (original) A(-5,5) B(-3,5) C(-3,1) D(-5,1) Image (new) A (1,1) B (3,1) C (3,-3) D (1,-3) 3-14

15 Transformations (Movements) cont d. Rotation 90 clockwise to the right 90 counter clockwise to the left 180 can be in either direction. The point of rotation can be any point on the figure or the point of origin (0,0) Rotation steps: SOL step one: using graph paper draw x, y axis then redraw the shape on the graph paper (include original coordinates) Step two: list the original coordinates Step three: predict what quadrant your image will be in Step four: turn the paper in the opposite direction of the rotation Step five: re-plot the original points (looking at the new x,y axis) Ex) Rotate ABCD 90 clockwise. List Coordinates: Pre-image (original) A(-5,5), B(-3,5), C(-3,1), D(-5,1). Turn the paper in the opposite direction of the rotation. In this example turn the paper counter clockwise since the rotation is clockwise. Re -plot the original points (looking at the new x,y axis). Turn the paper back. List the image (new) coordinates. A (5,5) B (5,3) C (1,3) D (1,5) 3-15

16 Transformations (Movements) cont d. Reflections You will be told to flip over either the x or y axis. Do one point at a time. Whatever axis you are reflecting over, the value for that axis in the coordinate will stay the same. The other value will be the inverse. Reflection steps: SOL step one: using graph paper draw x, y axis then redraw the shape on the graph paper (include original coordinates) Step two: list the original coordinates Step three: predict what quadrant your image will be in Step four: (one point at a time) Whatever axis you are reflecting over, count how many spaces you are away from that axis. Plot the same number of spaces on the other side of the axis. Points need to be lined up. Step five: list the new coordinates. You can check the reflection by folding the graph paper. Ex) Reflect ABC over the y axis Pre-image coordinates Image coordinates A(-3,0) A (3,0 ) B(-2,-2) B (2,-2)) C(-1,2) C (1,2) 3-16

17 Transformations (Movements) cont d. Dilations Dilations are similar figures on a coordinate plane. Dilations can enlarge or shrink the pre-image. Scale factor tells how much to shrink or enlarge the pre-image. Dilations are all multiplications even if by a fraction. SOL NOTE: tests usually include the scale factors 41, 21, 2, 3, 4 for dilations because they fit on a coordinate plane easily. Dilation steps: SOL step one: using graph paper draw x, y axis then redraw the shape on the graph paper (include original coordinates) Step two: list the original coordinates Step three: look at the scale factor and determine if the image will be bigger or smaller than the pre-image. Step four: multiply the scale factor by the x and y coordinates for each point Step five: plot and list the new coordinates Ex) Dilate ABC by a scale factor of 2 Pre-image coordinates Multiply by scale factor Image coordinates A(-3,0) -3x2, 0x2 A (-6,0) B(-2,-2) -2x2,-2x2 B (-4,-4) C(-1,2) -1x2, 2x2 C (-2,4) 3-17