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End of Year Revision Packet
Mathematics-IM1X
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Mathematics
IM1X
End of Year Revision Packet
Chapters 1 – 7 S3
Highlights:
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Identify the elements of different sets of numbers.
Convert numbers from decimal form to fractional form and vice versa.
Test a number for divisibility by a given prime number.
Give the prime factorization of a given number using knowledge of prime and composite
numbers as well as factors and multiples.
Find the greatest common factor and the lowest common multiple of two numbers.
Write a fraction in simplest form.
Express a fraction as a mixed numeral and vice versa.
Add, subtract, multiply, and divide fractions.
Perform operations on directed numbers.
Order directed numbers.
Simplify numerical expressions.
Express numbers in scientific notation, understand significant figures, and round to a
nearest given value.
Find the range of values of rounded numbers and find the error of given measurements.
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Solve a linear equation in one variable and solve word problems leading to linear equations.
Solve a linear equation in more than one variable for one variable in terms of the other.
Solve a linear equation involving absolute values.
Solve a linear inequality in one variable and graph its solution.
Solve word problems leading to linear inequalities.
Solve absolute value inequalities and graph the solution.
Solve word problems leading to absolute value inequalities.
Solve problems involving ratios and proportions.
Solve different types of questions and word problems involving percentages.

Define a relation and determine its range, draw its arrow diagram, list the pairs in it, and
represent it in a coordinate plane.
Define a function and know that it can be represented by a rule, in a coordinate plane, or by
a table.
Analyze graphs of different types of functions.
Recognize the general form of the rule of a linear function and graph it.
Recognize nonlinear functions.
Recognize an arithmetic sequence, find its common difference, apply the rule of its general
term, and graph it.
Use arithmetic sequences to solve problems in context.
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End of Year Revision Packet

Mathematics-IM1X
Page 2 of 15
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Evaluate simple positive and negative powers, solve exponential equations, and evaluate
simple expressions that involve exponents.
Apply different rules related to exponents.
Recognize numbers written in scientific notation and understand what standard form is.
Apply operations on numbers written in scientific notation and solve problems involving
scientific notation.
Graph simple exponential functions, deduce one graph from the other using shifting, and
use exponential graphs to solve problems in context.
Recognize a geometric sequence, find its common ratio, apply the rule of its general term,
and graph it.
Know the general formula for compound interest and solve related problems.
Solve word problems on half-life of radioactive material.
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Work out numerical expressions involving square roots.
Simplify radical expressions and express them using only one radical.
Add, subtract, multiply, and divide radical expressions.
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Graph a linear equation in the coordinate plane.
Find the slope of a line.
Use the slope y-intercept form or the point-slope form to graph a line.
Identify parallel, perpendicular, intersecting, and coincident lines.
Apply the midpoint formula and the distance formula.
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Check if an ordered pair is a solution of a given system of linear equations.
Solve systems of linear equations by graphing, by substitution, or by linear combination.
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End of Year Revision Packet
Mathematics-IM1X
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Vocabulary:

Number sets, Natural numbers, Whole numbers, Set of integers, Set of rational numbers,
Repeating decimals, Terminating decimals, Set of irrational numbers, Set of real numbers,
Divisibility, Factors, Multiples, Prime, Composite, Prime factorization, Greatest common
factor, Lowest common multiple

Equivalent fractions, Simplest form, Mixed number, Improper fraction, Directed numbers,
Ordering numbers
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Numerical expressions, Scientific notation, Standard notation, Significant figures,
Rounding, Range of values
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Linear equations, Solution, Solve, Word problems, Absolute value, Linear inequality, Solve
and Graph
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Absolute value inequality, Ratio, Proportion, Cross multiplication property, Multi-rate
problems, Percentage, Profit, Discount, Cost, Original price, Percentage profit, Percent
change, Percent discount, Percent error, Simple interest, Capital, Interest rate, Mixture
problems

Relation, Ordered pair, Abscissa, Domain, Ordinate, Range, Arrow diagram, Function,
Element, Rule of a function, Tables, Graph of a function, Speed

Function notation, Variable, Independent variable, Dependent variable, Linear functions,
Non-linear functions, Sequence, Arithmetic sequence, Common difference, General term,
nth term, Patterns
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Integer exponents, Exponential expressions, Base, Power, Exponential functions, Horizontal
shift, Vertical shift, Geometric sequence, Common ratio, Compound interest, Half-life of
radioactive material
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Square root, Radical sign, Radicand, Principle square root, Radical expressions, Rationalize,
Unlike terms, Pythagorean theorem
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Linear equation, Slope of a line, Horizontal lines, Vertical lines, Undefined slope, Rate,
Intercepts, Slope y-intercept form, Point-Slope form, Parallel lines, Perpendicular lines,
Intersecting, Coincident, Midpoint formula, Distance formula
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System of linear equations, Solution of a system, System with no solution, System with a
unique solution, System with infinitely many solutions
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Solve by graphing, Graphically, Substitution, Approximate values of the solution

Adding a linear equation to another, Adding a multiple of a linear equation to another,
Linear combination
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End of Year Revision Packet
Mathematics-IM1X
Page 4 of 15
Revision Exercises:
Ch. 1
Section 1
Basics
Number Sets
1. For each of the numbers listed below, identify the smallest set from among the sets , W, ,
,  that contains it.
22
9


,  6, , 7.2,  , 8.00,  4.75
2.202002...,
7
3


Section 2
Factors and Multiples
1. Find the greatest common factor of each pair of numbers.
a) 2940 and 2520
b) 4840 and 528
c) 1401 and 4056
2. Find the least common multiple of each set of numbers.
a) 108, 90, and 35
b) 7, 100, and 200
c) 48, 32, and 64
3. Jimmy changes the password on his laptop on a regular basis. He always chooses the password
to be the sum of all the prime numbers which are less than or equal to the number of the day in
the month.
a) List all days of the month on which Jimmy cannot change his password.
b) What will his password be if he changes it on the 19th of October?
Section 3
Fractions
3 1 1
3
1. Locate the fractions 5 , 2 , 5 , and 10 on the number line then write them in increasing order.
1
0
2. Multiply. Express the answer as a fraction in simplest form or as a mixed number.
3 5

25
2
a)
1 10

b) 5 33
5 9

27
25
c)
5 7

d) 4 4
2 15

15
2
e)
32 96

f) 48 64
15
6
h) 54
1 3

i) 2 5
g)
42 
6
7
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Mathematics-IM1X
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3. Add. Express the answer as a mixed number.
1
3
5 2
a) 3 4
b)
1
1
2 7
d) 5 2
3
3
7 8
e) 8 7
5
1
3
5
1 2 3
c) 4 5 7
2
7
2
25 10
3
1
10
1 2 5
f) 7 14 21
4. Subtract. Express the answer in simplest form.
3
4
2
11 11
6
3
5 2
a) 7 7
b)
1
1
6 2
d) 4 3
4
7
3 2
e) 7 8
7
7
5
6 4
c) 8 12
2
3
5 3
f) 7 8
5. Divide. Express the answer in simplest form as a fraction or as a mixed number.
1
1
1 2
a) 2 3
4
3
3 5
b) 7 7
2
3
5 4
c) 3 4
 5
8  1 
d)  4 
 2
6  1 
 5
e)
5  3
2 
f) 7  4 
Section 4
Directed Numbers
1. Refer to the number line below to state the coordinate of the specified point.
A
B C
D E
F G H
I
J
K
L M N P Q
R
8 7 6 5 4 3 2 1
O
1
2
3
8
4
5
6
7
a) The point between C and P that is twice as far from C as it is from P.
b) The point between C and P that is five times as far from P as it is from C.
c) The point to the left of L that is twice as far from R as it is from L.
Section 5
Simplifying Numerical Expressions
1. Simplify.
a) 8 + 10  5
b) 90  10  2
c) 100 + 20  5
d) 8  10  5
e) 90  10  2
f) 100  20  5
1
2
3
z
x
y
4.
3,
5 , and
2. Evaluate each expression for
x y
x
2x  3y
1  z
y
y

z
a)
b)
c) 2 y  5 z
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1
d)
xz
y
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Section 6
Mathematics-IM1X
Page 6 of 15
Estimation
1. Round 3,409,981.348 meters to the nearest.
a) meter.
b) 10 meters.
c) kilometer (1,000 meters).
2. The dimensions of a rectangular sheet of paper are measured to the nearest inch as 11 in. by
17 in.
a) What is the least possible value of the area of this sheet of paper?
b) The cost of printing colored images is $0.05 per square inch. What is the range of the cost
for printing a colored image on this sheet of paper?
Chapter Summary
Chapter Test
Ch. 2
Section 1
TB pages 33 – 35
TB pages 36 – 38
Linear Equations and Linear Inequalities
Linear Equations
1. The greatest of four consecutive integers is 8 less than twice the least. Find the integers.
2. Jose is saving to buy a coat that costs $85. He has $20 already saved. Every week he plans to
add $5 more to his savings. Based on this plan, how many weeks will it take Jose to save
enough money to buy the coat?
Section 2
Linear Equations Involving Absolute Value
1. Solve. If the equation has no solution, state so.
a)
1 2
|a| 
3 3
Section 3
b)
|b|
7
4
1
9
9
c)
3
1
3  | c | 2
4
4
Linear Inequalities
1. Solve each inequality and graph its solution set on a number line.
a)
3  4x
3
1
x
2
b) 13  4(x + 2) + 3x  2  0
2. An employee is paid $20 for the first working hour and $12 for every additional hour. How
many hours should the employee work to earn more than $80?
Section 4
Absolute Value Inequalities
1. Solve and graph.
a)
x2 5

3
6
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b) 8 | a  8 |  4
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Section 5
Mathematics-IM1X
Page 7 of 15
Ratios and Proportions
1. A cable is 144 cm long. It is cut into two pieces whose lengths are in the ratio of 3 to 4. How
long is each piece?
2. A plane travels a distance of 1,000 km in 2.5 hours while flying against the wind and in 2
hours while flying with the wind. Find the speed of the wind.
Section 6
Percentages
1. May paid a 5% sales tax on her new car. What was the total price of the car if the sales tax was
$425?
2. Roberto invested $2,000 for two years at a simple interest rate of 8.5% per year. How much
interest did he earn?
Chapter Summary
Chapter Test
Ch. 3
Section 1
TB pages 73 – 74
TB pages 75 – 77
Functions
Relations
1. The range of a relation is {0.5, 1, 2, 3.5, 5.5}. Consider the relation that pairs each number in
the domain with its half in the range.
a) Give the domain of this relation.
b) Draw an arrow diagram that joins each pair in this relation.
c) List the set of ordered pairs of the relation.
d) Graph the set of ordered pairs in the relation in a coordinate plane.
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Section 2
Mathematics-IM1X
Page 8 of 15
Functions
1. The table below represents a function.
x
y
1
4
2
8
3
12
4
16
5
20
a) Give the domain and range of this function.
b) List the set of ordered pairs in the function.
c) Graph the function in a coordinate plane.
d) Give a rule that can be used to define this function.
e) Use the rule to find the value of b if the ordered pair (8, b) were also an element of the
function.
2. Determine whether the graph is the graph of a function. Explain.
y
a)
O
d)
O
x
e)
y
O
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x
c)
y
b)
O
x
f)
y
O
y
x
x
y
O
x
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Section 3
Mathematics-IM1X
Page 9 of 15
Linear Functions
1. Consider the function f given by f(x) = 2x  2.
a) Find f(0) and f(1).
b) Draw the graph of f.
c) What is the value of a for which f(a) = 4?
d) Is there a value for x in the domain of f for which f (x) = x?
2. a) Find the range of g(x) = x – 5 defined over the domain D = {x: –2  x  5}.
b) Find the range of h(x) = 2x defined over the set of whole numbers.
c) Find the range of f (x) = 10x if the domain of f is the set of real numbers between 0 and 2
inclusive .
Section 4
Arithmetic Sequences
1. The 2nd and the 7th terms of an arithmetic sequence are –2 and –17, respectively.
a) Is the sequence increasing or decreasing?
b) Find the values of the 4th and the 40th terms.
2. The first five terms of an arithmetic sequence are
plotted on the coordinate grid to the right.
9
a) What are the coordinates of the point that
corresponds to the 6th term of this sequence?
7
b) What is the rule for this sequence?
5
y
8
6
4
3
2
1
0
Chapter Summary
Chapter Test
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TB page 99
TB pages 100 – 102
1
2
3
4
5
6
x
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Ch. 4
Section 1
Mathematics-IM1X
Page 10 of 15
Exponential Functions
Exponential Expressions
3x
3x
3
1. Express 81 as a power of 3 and solve 81 .
2. Simplify using only positive exponents.
a)
 4a 2b 4 

2 3 
 16a b 
d)
 3a 2b 4 


2
 2a b 
Section 2
1
2
b)
 2a 2 b 5 

6 7 
 10a b 
e)
 2a 3b 2 

1 0 
 3a b 
3
c)
 6a 4 b8 

2 
 15a b 
5x y z 
25  x y z 
2
3
3
f)
0
3
2
2 1 2
Operations With Scientific Notation
1. Simplify. Express your answer in decimal notation.
a) 5101 + 6102 + 4103
b) 110 + 2102 + 4104
2. The half-life of radioactive Iodine-129 is 1.56×107 years and the half-life of radioactive
Nickel-59 is 7.5×104 years. How many times greater is the half-life of Iodine-129 than the
half-life of Nickel-59?
Section 3
Exponential Functions
1. The graph of y = 3x is shown to the right.
y
By using the given graph, sketch the
graph of y = 32x.
10
Explain your reasoning.
5
x
5
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0
5
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Section 4
Mathematics-IM1X
Page 11 of 15
Geometric Sequences
1. The nth term of a pattern is given by the formula nbelow:
1
an     1
2
a) Find a1, a2, and a3.
b) Find the sum of the first four terms of the pattern.
c) Show that
an1 
1
 an  1
2
.
2. The half-life of radioactive Carbon-11 is 20 minutes. Suppose there were 200 grams of
Carbon-11 present.
a) How many grams were present 40 minutes earlier ?
b) How many grams were present 1 hour later?
Chapter Summary
Chapter Test
Ch. 5
Section 1
TB pages 128 – 129
TB pages 130 – 132
Radical Expressions
Square Roots
1. Find the square root of each of the following numbers.
a) 1,521
Section 2
b) 1,225
d) 3,025
Multiplying and Dividing Radical Expressions
1. Simplify.
a)
c) 2,025
5 x   4
b)
10
1

3
5
2
2. Simplify.
a)
4  25 


5  64 
Section 3
x4 y 2

49 x 2
25 y 4
b)
Adding and Subtracting Radical Expressions
In 1 – 4, simplify the expression.
3
1.
3.
7
9
4 28  2 7  21
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2
3

3
4
2.
4.
2 200  3 250  4 300
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Section 4
Mathematics-IM1X
Page 12 of 15
Applications
1. Which of the following is an irrational number?
A. 4.2356
B.
80
C.
5
7
D.
144
2. A vertical pole broke at a height 5 feet off the ground. Its tip
landed on the ground 12 feet away from the bottom of the
pole. What was the original height of the pole?
5 ft
12 ft
3. Find the value of x in the figure below.
3
x
3
2
2
Chapter Summary
Chapter Test
Ch. 6
Section 1
TB page 148
TB pages 149 – 150
Linear Equations in Two Variables
The Graph of a Linear Equation in Two Variables
1. Graph each equation on a separate coordinate plane.
a)
x 
y
 2
4
b) y  6x = 1
2. Find the value of m for which the point with the given coordinates lies on the graph of the
given equation.
a) 2mx + 7y = 27; C(2, 1)
b)
x
m
y2
2
; D( 6, 2 )
3. Graph the two equations on the same coordinate plane then determine from the graph the point
of intersection of the two graphs.
a) 4x + y = 7; 2x  y = 1
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b) 4x  y = 6; y  3x = 1
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Mathematics-IM1X
Page 13 of 15
4. Supply the missing values and then graph the given equation.
1
x  x 1
2
x
y
Section 2
2
2
The Slope of a Line
1. Determine the value of p so that the two given points lie on the line whose slope is given.
a) (0, 3) and (5, 8); slope is 2p + 3
b) (2, p) and (9, 7); slope is 1
2. Determine whether the line passing through the given points is vertical, horizontal, slants
upward to the right, or downward to the right.
a) R(7, 7) and S(9, 9)
Section 3
b) E(1, 1) and F(0, 1)
The Slope y-Intercept Form of a Linear Equation
1. Express each equation in the slope y-intercept form and deduce the value of its slope and yintercept.
a) 0.6x + 0.4y = 1.2
b) x = 4
2. The equation below gives the total amount y Tania gets paid when she works x hours of
overtime.
y = 20x + 600
a) What is the y-intercept of the line represented by this equation? What does it represent
relative to Tania’s salary?
b) What is the slope of the line represented by this equation? What does it represent relative to
Tania’s salary?
c) How much will Tania get paid at the end of the month, if she works 9 hours of overtime?
Section 4
The Point-Slope Form of the Equation of a Line
1. Write the equation of the line with an undefined slope and passing through the point E(5, –3)
in the slope y-intercept form and draw its graph.
2. Write the equation of the line with undefined slope and that has the same xintercept as the
line whose equation is given by 4x  3y = 12.
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Section 5
Mathematics-IM1X
Page 14 of 15
Parallel and Perpendicular Lines
1. Write the equation of the line parallel to n: y = 5x + 2 and having a yintercept of (0, 7).
2. Write the equation of the line passing through A(1, 2) and perpendicular to m: 3x + 6y = 4.
Section 6
The Midpoint and Distance Formulas
1. Find the coordinates of B given that M is the midpoint of AB .
5 2
 5 2
A ,  , M   , 
a)  7 9   7 9 
b) A(n+3, 1), M(5n, 1)
2. ABCD is a parallelogram with A(3, 5), B(7, 5), C(5, 2), and D(1, 2). What is the perimeter
of ABCD?
Chapter Summary
Chapter Test
Ch. 7
Section 1
TB page 181
TB pages 182 – 184
Systems of Linear Equations and Linear Inequalities
Graphical Solutions
 by2 graphing.
1. Solve
 y  x7
a)

3

 y   1 x 1

3
b)
 x  3 y  1

3x  9 y  8
2. Find the point on the graph y  2x =  3 where the xcoordinate is twice the ycoordinate.
Section 2
The Substitution Method
1. Solve using the substitution method.
x  y  8

a)  y  x  8
b)
x y
7

 3
 2 x  3 y
2. Determine whether the lines 2x + 3y = 13, 3x  2y = 0, and x + y = 5 are concurrent. (Three
lines are concurrent if they meet at the same point.)
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Section 3
Mathematics-IM1X
Linear Combinations

1. Solve using the linear combination method.    1
x y
3
a)
3x  4 y  7  0

4x  3 y  8  0
2

1  1  5

b)  x y 6
2. The price of a sandwich and a soda is $3.60,
the price of a sandwich and a bag of chips is $3.14, and
the price of a soda and a bag of chips is $1.84
What is the price of the soda?
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