MPM 1D: Summative By: Fay Dickinson

Number Sense and Algebra: Number Sense Multiplying Exponents: If base number is the same add the exponents together. Example: 2^3+2^6=2^9 (Add 3 and 6 to get 9) This works because this can also be written as 2*2*2*2*2*2*2*2*2 which is also 2^9. Dividing Exponents: If base number is the same subtract the exponents from each other. Example: 2^6/2^3=2^3 (Subtract 3 from 6 to get 3) This works because this can be written as 2*2*2*2*2*2 which is also 2^3. 2*2*2 2^32^6 2^3

Number Sense and Algebra: Algebra Purpose: Find the value of a variable. To do this get the numbers on one side and the variable on the other. In certain cases this will not be possible Example: 2x+2=2x+1 becomes 2=1 These exceptions are easily identified when there is the same amount of a variable on both sides and nothing else but unequal numbers, however sometimes this is not clear. Sometimes you will be asked to solve an equation with no numbers, in other words rearrange a formula. To do this simply isolate the variable you have been told to find. Example: d=m/v (find m) m=dv You find the value of m by isolating it by multiplying both sides by v. Not true

Number Sense and Algebra: Algebra Solving Equations:  What is done to one side must be done to the other.  To get rid of something perform the opposite action: addition for subtraction, multiplication for division, and vice-versa  Simplify whenever possible Example: 6x instead of 2x+4x  Keep equals sign aligned Example: 2x+5=3x+4 2x+5-2x=3x+4-2x 5=x+4 5-4=x+4-4 1=x  Proper formatting for answers is to have to variable on the left Example: x=2 instead of 2=x Proper formatting would mean the final answer is written as x=1

Number Sense and Algebra: Algebra Solving Problems: 5x+2=2x+17 5x+2-2x=2x+17-2x 3x+2=17 3x+2-2=17-2 3x=15 3x/3=15/3 x=5 Or show it using algebra tiles: Get rid of 2x from right side by subtracting 2x Simplify Get rid of 2 by subtracting 2 Simplify Get rid of 3x by dividing by 3 Simplify

Linear Relations Relation: a description of how two variables, the independent (x) and the dependant (y) variables, are connected. A relation is linear when it constantly increases or decreases by the same amount. Relations can be:  Discrete (can’t be broken down into smaller parts, no fractional values: Ex. Number of People (you can’t have ½ a person)) which is shown by connecting the points with a dotted line.  Continuous (can be broken down into smaller parts, can have fractional values: Ex. Time (you can have ½ an hour)) which is shown by connecting the points with a solid line.  Deterministic (every time you plot a graph of the information it will look the same: Ex. Money made if you charge $3 per window + $5)  Non Deterministic (every time you plot a graph of the information it will look different: Ex. Results of rolling a dice)  Partial (doesn’t pass through point (0,0), has a constant in the equation)  Direct (passes through point (0,0), doesn’t have a constant in the equation)

Linear Relations  Positive (points tend to go up as you go to the right)  Negative (points tend to go down as you go to the right)  Strong (points are close to the line of best fit)  Weak (points are far away from the line of best fit) Relations can be shown in many different ways: T-charts, graphs, cubes, etc. Example: XY Y=2x

Analytic Geometry Formulas for lines (formulas with exponents are non-linear):  Slope-y-intercept Form: y=mx+b m=slope b=y-intercept (value of y when x=0) x-intercept (value of x when y=0)=-b/m  Standard Form: Ax+By+C=0 m (slope)=-A/B b (y-intercept)=-C/B A must be positive A, B, or C can be 0 but not both A and B  x=a (undefined slope)  y=b (slope=0) Calculating Slope:  m=rise/run  m= ∆y/∆x  m=(y 2 – y 1 )/ (x 2 – x 1 ) x-intercept y-intercept y x *Slope (m)=the amount y increases as x increases by 1

Analytic Geometry Types of Slope:  Positive  Negative  0  Undefined Point of Intersection:  The point where two lines meet  This is where “interesting stuff” happens One company becomes cheaper than another Where profit begins to be made Best price point Etc. y x Point of Intersection

Measurement and Geometry Composite Area and Perimeter: When a complex shape is broken down into several different, simpler shapes which you can then use to calculate the area or perimeter of the entire shape.  Example: Surface Area: The total area of all the faces on a 3-D figure.  Example: Pyramid=Area of Base+((number of sides on base)Area of Triangle Example: Square Pyramid=Area of Square (l*w)+4(Area of Triangle (1/2(b*h)) For the area of a triangle in a pyramid you really need the slant height (square root of h 2 +r 2 ). This is really Pythagorean Theorem. 9m 4m 10m 2m 9m 5m 10m 11m Find areas of all rectangles for total area: Rectangle 1: 9m(4m)=36m Rectangle 2: 19m(2m)=38m Rectangle 3: 10m(5m)=50m Total=124m Slant Height Height Radius

Measurement and Geometry Comparing Volumes:  3 of a “pointy”=“un-pointy”  1/3 “un-pointy”=“pointy ” 3 (volume of cone (pointy))=volume of cylinder (un-pointy) 3 (volume of square pyramid (pointy))=volume of cube (un-pointy) 3 (volume of triangular pyramid (pointy))=volume of triangular prism (un- pointy) Optimization: Maximizing Area or Minimizing Perimeter  When given a perimeter you’re trying to find the greatest area.  When given an area you’re trying to find the minimum perimeter.  Result is usually a square.  If it is only 3 sides the result will have one side double the other. Example: Cottage 5m 10m 5m is ½ of 10m and the cottage makes up one side, this is the maximum area.

Measurement and Geometry Angles:  Supplementary Angles: Two angles that equal 180 Example:  Complimentary Angles: Two angles that equal 90 Example:  Acute: Less than 90 Example:  Right: 90 Example:  Obtuse: More than 90 Example: o o o 12 2 o 1 o

Measurement and Geometry Parallel Lines: Lines with the same slope, they will never meet.  Example:  Opposite angles and parallel angels are congruent Angles 1, 3, 5, and 7 are congruent Angles 2, 4, 6, and 8 are congruent Triangles: 3 sides, 3 angles, interior angles=180  Right  Isosceles  Equilateral  Scalene  Obtuse Parallel Lines o These triangles are all acute. An acute triangle is a triangle with all acute angles

Measurement and Geometry Quadrilaterals: 4 sides, 4 angles, interior angles=360  Square (Equilateral Quadrilateral)  Rectangle  Parallelogram  Rhombus  Trapezoid  Diamond  Kite Interior Angles (convex polygons):  A=180n-360 or A=(n-2)(180) n stands for number of sides Example: Quadrilateral A=180(4)-360 A=360 Exterior Angles (convex polygons):  Total=360 o Sum of interior angles in a quadrilateral=360 o o

Measurement and Geometry Conjunctures: A hypothesis, a rule you believe is true. If it is proven wrong it is not true.  Example: If a quadrilateral’s diagonals are at 90 it is a square. o Angle=90, it’s a square o Angle=90, it’s a rhombus therefore the conjuncture isn’t true o