MPM1D
Key
Questions
&
Concepts
Grade 9Math
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Unit I: Rational Numbers
Important Stuff
Mixed Number Operations
Adding and Subtracting à When adding or subtracting, convert to an
improper fraction, find the lowest common denominator, carry out the
operation, then convert back
Lowest Common Denominator à The smallest number which can make
the denominator of two equations the same
Multiplying and Dividing à To multiply and divide mixed fractions, convert
to an improper fraction, carry out the operation, then switch back
BEDMAS
BEDMAS à Gives the order of operations; brackets, exponents, division,
multiplication, addition and subtraction; multiplication and division, as well
as addition and subtraction can be done in order of appearance
Exponents à A number raised to a power indicates a number multiplied
by itself some amount of times (i.e. 23 is 2x2x2)
Rational Numbers à Numbers which are represented by decimals or
fractions; follow same operation rules as integers
Radicals
Radical à Numbers under roots; can be simplified by breaking down into
common factors which have perfect roots
Other Notes
• Remember, always put fractions into improper form first
• Exponents of fractions mean we take both the numerator and
denominator to the outside power
Practice
1. Evaluate
!
!
a. 2 ! + 3 !
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!
b. 2 + 2 !
!
!
!
c. 5 ! − 3 ! + 2 !
2. Evaluate
!
a. 2 !
!
!
3!
!
b. 4 ! ÷ 2 !
3. Evaluate
a. (4 – 3)2
b. 2(3 + 22) – 4
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c.
! !!! !
!"
4. Evaluate
!
a. 2 !
b.
!
−
!
!
!
!
3 ! ÷ (2 !)(3 !)
!
! !
+ 3!
!
5. Simplify the Following Radicals
a. 27
b.
12
c.
72
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Unit II: Powers and Polynomials
Important Stuff
Exponent Laws
Product Rule à If two values with the same base number are being
multiplied, we add the exponents
Quotient Rule à If two values with the same base are being divided, we
subtract the exponents
Power Rule à If we have a base to an exponent to an exponent, the
exponents are multiplied (i.e. (x2)2 means that we have x2x2)
Polynomials
Polynomial à an expression containing variables to a certain power
Distributive Property à If we have a number outside of the brackets of a
polynomial, we multiply each term in the polynomial by the number
outside
Simplifying Polynomial Expressions à Collecting like terms
Important Notes
• Remember, only things with the same base raised to the same
power can be collected
• Sometimes things can’t be simplified any further and you can have
multiple terms
Practice
1. Simplify the following
a. 𝑥 ! ∙ 𝑥 !
b.
!!
!!
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c. 𝑦 !
d.
e.
!
! ! ∙! !
!
!!
!! ! ∙!! !
!
!!
2. Expand and simplify the following
a. 2 𝑥 − 4
b. 3(2x + 3)
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c. 3x(4x – 7)
d.
!
!
𝑥 27𝑥 + 9
3. Expand and Simplify
a. 2 𝑥 − 5 + 3 2𝑥 + 3
b. 4𝑥 3𝑥 + 2 − 2 2𝑥 + 1
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Unit III: Solving Equations
Important Stuff
Algebra
Isolating Variables à The manipulation of equations to solve for unknown
values; done by moving things across equals signs using opposite
operations; addition becomes subtraction, multiplication becomes
division
Word Problems
Solving Word Problems à Determining the equation of a word problem
which allows you to relate variables to each other and solve for unknown
values
Important Notes
• There are three types of word problems: coin problems, number
problems and measurement problems; know how to do all of them
Practice
1. Solve the following
a. 𝑥 + 3 = 5
b. 4 = 𝑥 − 5
c. 3𝑥 = 12
d.
!
!
=3
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e.
!!!
f.
!!!!
!
!
=6
=
!!!!
!
2. There are two chimpanzees, Dexter and Joel. If Dexter is twice
Joel’s age, and the sum of their ages is 30, what are Dexter and
Joel’s ages?
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3. Vanessa is measuring the size of her room so that she can figure out
where to install her own personal frozen yogurt machine. If the
length of her room is 7m more than the width, and the perimeter is
46m, what is the length and width of her room?
4. David is walking around with a lot of change in his pocket: exactly
57 coins; no one is quite sure as to why. If he has $10.80 in dimes
and quarters, determine how many of each type of coin he has.
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Unit IV: Linear Relations
Important Stuff
Equations of Lines
Slope à Rise/Run; determines how steep a line is; a slope of 0 is horizontal
! !!
and undefined is vertical; found by m = !! !!!
!
!
Y-Intercept à Where a line hits the y-axis
X-intercept à Where a line hits the x-axis
Slope-Intercept Form à Creating an equation in the form of y = mx + b; m
is the slope, b is the y-intercept
Direct Variation à A line passing through the origin
Partial Variation à A line which passes through a different y-intercept than
the origin
Important Notes
• When doing our slope equation, make sure that the coordinates of
x2 and y2 as well as x1 and y1 are coming from THE SAME POINT; if
they’re not, the line won’t work!
Practice
1. Given the following, state the slope, y-intercept, and if it is direct or
partial variation
a. y = 3x – 2
b. y = 2x
c. 3x + y = 0
d. 2x – 3y = 9
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2. Graph the line with the equation of y = 2x -1
3. Find the equation of the line with the following
a. A slope of 3 and a y-intercept of y=2
!
b. A slope of − ! and passing through (2,3)
c. Passing through (5,7) and (8,10)
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4. Determine the equation of a line with the following graph:
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Unit V: Analytic Geometry
Important Stuff
Equations of a Line
Standard Form: ax + by + c = 0; moving all variables in a linear equation to
the same side of an equality; no numbers can be fractions, and the avalue must be positive
Point-Slope Form à (y – y2) = m(x – x2); another way of expressing the
equation of a line
Parallel Lines à Lines which have the same slope; they can either be the
same line or distinct lines
Perpendicular Lines à Lines which have slopes that intersect at a 90o
angle; they are the negative reciprocal (i.e. if one slope is 2, the
perpendicular slope is -1/2)
Important Notes
• Remember, the slope equation from the last unit applies here too!
Practice
1. Determine the equation of a line passing through (-1,5) and (1,1) in
all three forms.
2. Find the equation of a line passing through (4,6) and parallel to
9x + 3y = 6
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!
3. Determine the equation of a line perpendicular to 𝑦 = ! 𝑥 − 5 and
passing through (7,5). Graph both lines.
!
4. Convert the following to standard form: 𝑦 = ! 𝑥 − 5
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5. Joel, in an attempt to get to space, launches a rocket from the top
of a 30m platform. He then starts to rise at a rate of 10m every
second. Determine an equation to model this, and determine his
height after 1 minute.
6. Dexter decides to charge people for advice, though the advice
usually seems insulting. If he charges people a base fee of $10, as
well as $5 every hour, determine an equation to model cost, and
determine how much it would cost to get Dexter’s advice for 10
minutes.
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Unit VI: Measurement and Geometry
Important Stuff
Shape Properties
Internal Angles à Is the angles inside of a polygon; all internal angles of a
triangle sum to 180o; in polygons bigger than a triangle, add 180o for each
side greater than 3
External Angles à All External angles sum to 360o in all polygon
!
Area à The space inside of a shape; for triangle, A=!bh; for rectangle,
LxW; for circle, 𝜋𝑟 !
Surface Area à The area of the faces of a polyhedron; for cone,
S.A. = radius x slant + radius2
Volume à The space inside of a polyhedron; for any prism, Volume = area
of base x height; area of a cone is
Other Notes
• Make sure you take the area of ALL sides of 3D figures for Surface
Area
• When you’re dealing with volume, do the area of your base and
multiply it by height for all prisms, INCLUDING CYLINDERS!
Questions
1. Solve the missing angle
a.
x 64o 42o b.
x 114o 56o www.themathguru.org
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c.
102o 97o 108o 2. Dexter and Vanessa are arguing over who has a bigger box.
Vanessa claims that her rectangular box which is 30cmx20cmx40cm
is bigger than Dexter’s cylindrical box with a diameter of 16cm and
a height of 80cm is bigger. Determine who’s box has a greater
volume so that Dexter and Vanessa stop arguing!
3. David has decided to paint giant cone which he has at the top of
his house for aesthetics…because cones look good people. The
cone is 5m tall and 3m wide. If a litre of paint covers 0.5m2, how
many litres of paint does Nicole need to paint his awe-inspiring
giant cone?
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4. Joel was bored while hiking through the Rouge Valley, and decided
to arrange some rocks into a random shape to confuse people. If
the shape he made looks like the shape below, determine the area
which the shape occupies.
7.3m 3.2m