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The Ultimate SAT Math
Strategies Guide
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Created by Sherman Snyder
Fox Chapel Tutoring
Pittsburgh, PA
412-352-6596
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Go to Table of Contents
Go to Introduction
Introduction to
The Ultimate SAT Math
Strategies Guide
•Unique math study guide that focuses on math strategies
rather than math content
•Study guide is designed to provide step-by-step
development of math strategies in an easy-to-use format
•Each math strategy is accompanied by examples that
provide opportunity to apply strategy to a variety of
question formats
•Study guide to be used in conjunction with traditional
paperback study guides available in bookstores
Go to Table of Contents
Go to first strategy
Table of Contents
Click on Highlighted Topic
Number and Operations
Algebra
Linear Proportionality
Venn Diagrams
Ratios and their Multiples
Ratios, Proportion, Probability
Counting Problems
The Handshake Problem
Long Division and Remainders
Percent Change
Percentages
Repeating Sequences
Using New Definitions
Elimination of Like Terms and Factors
Equivalent Strategy
System of Equations
Matching Game
Factoring Strategy
Word problems
Basic Rules of Exponents
Additional Rules of Exponents
Absolute Value Inequalities
Creation of Math Statements
Geometry and Measurement
Functions
Dividing Irregular Shapes
Line Segment Length in Solids
Putting Shapes Together
3-4-5 Triangle
30-60-90 Triangle
45-45-90 Triangle
Distance Between Two Points
Midpoint Determination in x-y Coordinate
Midpoint Determination on Number Line
Exterior Angle of a Triangle
Perpendicular Lines
Interval Spacing - Number Line
Triangle Side Lengths
Using Function Notation
Reflections - x axis
Reflections - y axis
Reflections - Absolute Value
Translations - Horizontal Shift
Translations - Vertical Shift
Translations - Vertical Stretch
Translations - Vertical Shrink
Data Analysis, Statistics, and
Probability
Arithmetic Mean
Venn Diagram
18 + 22 + 10 = 50
Total number of students = 50
Strategy: To determine the overlap
(intersection) of members in two groups
(sets), use the following approach:
Step 1: add the number of members of
each group
Step 2: subtract the total number of
members that are in either group or
both groups from the result of step 1
Reasoning: By eliminating the overlap of
members, the sum of three numbers in
the Venn diagram will equal the total
number of members being counted.
Application: Used when members of two
or more groups (sets) have common
members.
Return to Table of Contents
History
Math
18
Number of students
that study math = 40
Step 1
40 + 32 = 72
22
10
Number of students
that study history = 32
Step 2
72 – 50 = 22
Number of students that study
math and history = 22
Number of students
that study math only:
40 – 22 = 18
Number of students
that study history only:
32 – 22 = 10
See example of strategy
The Handshake Problem
n-1=5
handshakes
Strategy: The total number of
handshakes that can be exchanged
within a group of people of size “n” is
equal to ½n(n -1).
Reasoning: For a total of “n” people,
each person can shake hands with “n -1”
other people. However, each handshake
is shared by two people.
Application: Useful for determining the
total number of games played in a sport
league, or the number of lines that can
be drawn between pairs of points on a
plane when no more than two points are
collinear.
Return to Table of Contents
n=6
people
½n(n -1) = ½(6)(5) = 15 total handshakes
shared by a group of 6 people
Alternative Solution: Total number of
handshakes can be found by addition of
the number of handshakes exchanged
by each individual person.
5 + 4 + 3 + 2 + 1 + 0 = 15 handshakes
See example of strategy
Line Segment or Diagonal
Length in a Rectangular Solid
Strategy: To find the length of a diagonal
or a line segment that connects two
edges of a rectangular solid, create a
right triangle within the solid that uses
the unknown segment as the
hypotenuse.
Reasoning: By finding a right triangle
within the solid, Pythagorean Theorem
can be used to find the segment or
diagonal length.
Line Segment
c
a
b
Right Triangle
c 2 = a2 + b2
Application: Any question that asks for
the length of a line segment or diagonal
in a rectangular solid. The information
provided in the question will be sufficient
to apply Pythagorean Theorem.
Return to Table of Contents
Pythagorean Theorem
See example of strategy
Interval Spacing
Strategy: The interval spacing on a
number line is found by a two-step
process:
1. Determine the distance between
two known points on the number line
2. Divide the distance by the number of
intervals separating the two known
points
Reasoning: By design, the number line
has equal distance between each tick
mark on the line
What is
this value?
2.5
3
18 23
(18 - 3) = 2.5
6
18 + 2(2.5) = 23
Application: Used to identify an
unknown coordinate on number line. Also
used to identify the value of specific term
in an arithmetic sequence.
Return to Table of Contents
See example of strategy
Triangle Side Lengths
Strategy: The 3rd side of any triangle is
greater than the difference and smaller
than the sum of the other two sides
3 < x < 15
6
9
Reasoning: A side length of 15 would
require the formation of a line, not a
triangle. A side length of 3 would also
require the formation of a line, not a
triangle
15
6
9
3
Application: Given two sides, choose
the smallest or greatest integer value of
third side. Given three sides as answer
choices, which will not form a triangle.
Return to Table of Contents
6
9
See example of strategy
Using Function Notation
Strategy: Replace the variable in the
function expression (right side of equal
sign) with the value, letter, or expression
that has replaced the variable (usually x)
in the function notation (left hand side of
equal sign)
Reasoning: Function notation is a road
map or guide that directly connects the
“x” value for a given function with one
unique “y” value.
Application: Function notation can be
applied in many different ways on the
SAT. See examples for details. Function
notation is commonly used to describe
translations and reflections of functions.
See Table of Contents for additional
strategies that use function notation.
Return to Table of Contents
Introduction
Function notation such as f(x), g(x), and
h(x) is a useful way of representing the
dependent variable “y” when working
with functions. For example, the function
y = 2x + 5 can be written as f(x) = 2x + 5,
g(x) = 2x + 5, or h(x) = 2x + 5.
Important Note: Function notation is
not a mathematical operation.
See example of commonly made
mistake.
See example of strategy
Function Translations
Horizontal Shift
Strategy: A horizontal shift of a function
y = f(x) is easily performed by sliding the
function right or left parallel to the x-axis
a specified distance. Using function
notation, a shift to the right of 2 units
can be communicated as y = f(x-2). A
shift to the left of 4 units can be
communicated as y = f(x+4)
y = f(x)
2
2
y = f(x+4)
Reasoning: A horizontal shift described
by y = f(x-2) has the same y-value at x =
2 as the original function f(x) at x = 0.
Application: Horizontal shifts can be
performed for any function using the
strategy described above.
Return to Table of Contents
y = f(x-2)
Venn Diagram
Example 1
Baseball
Football
Question: The Venn diagram to the right
shows the distribution of students who
play football, baseball, or both. If the
ratio of the number of football players to
the number of baseball players is 5:3,
what is the value of n?
28
n
14
Solution Steps
What essential information is needed?
Connection between the number of
players in each sport to “n”, the number
of players that participate in both sports.
What is the strategy for identifying
essential information?:Use the
properties of Venn diagrams and ratios to
find the value of “n”
Return to Table of Contents
1) Create a proportion of the number
of football players to baseball players
n + 28 5
=
n + 14 3
2) Solve for “n” using cross
multiplication: 5n + 70 = 3n + 84
Return to strategy page
2n = 14
n=7
See another example of strategy
Venn Diagram
Example 2
Question: The 350 students at a local
high school take either math, music, or
both. If 225 students take math and 50
take both math and music, how many
students take music?
Solution Steps
1) Create an appropriate Venn diagram
to help visualize the given information.
Music
Math
What essential information is needed?
Connection between the multitude of
given information and the unknown
quantity.
What is the strategy for identifying
essential information? Use the
properties of Venn diagrams to help
“visualize” the given information.
175
50
m
2) Find the value of m, the number of
students that take music only
175 + 50 + m = 350
m = 125
3) Find the value of m + 50, the
number of students that take music
m + 50 = 125 + 50 = 175
Return to Table of Contents
Return to strategy page
Return to previous example
The Handshake Problem
Example 1
Question: In a baseball league with 8
teams, each team plays exactly 4 games
with each of the other 7 teams in the
league. What is the total number of
games played in the league?
Solution Steps
1) Find the number of games played
between the 8 teams
What essential information is needed?
How many games are played between
the eight teams.
½(8)(7) = 28 individual games
played without repeats
What is the strategy for identifying
essential information?: Find the
number of games played between the 8
teams using the handshake problem
strategy. Multiply the result by 4 to
account for the fact that each team
plays exactly 4 games with each of the
other 7 teams.
2) Multiply by 4 to account for the fact
that each team plays exactly four
games with each of the other 7 teams
Return to Table of Contents
Return to strategy page
Total number of games played:
28 x 4 = 112 games
See another example of strategy
The Handshake Problem
Example 2
Question: How many diagonals can be
drawn inside a regular polygon with 6
congruent sides.
What essential information is needed?
The total number of diagonals drawn
from the 6 vertices of the polygon.
What is the strategy for identifying
essential information? Use the
handshake problem with modifications.
Polygons have sides that do not require
lines connecting adjacent vertices. To
account for this, multiply the total
number of vertices “n” by “n - 3” rather
than “n - 1”. Total number of diagonals
is ½n(n - 3).
Return to Table of Contents
Solution Steps
n=6
sides
n -3 = 3
diagonals
½n(n - 3) = ½(6)(6 - 3) = 9 diagonals
can be drawn in a regular polygon
with 6 sides
Return to strategy page
Return to previous example
Line Segment Length in Solid
Example 1
Solution Steps
Question: What is the volume of a cube
that has a diagonal length of 4√3?
What essential information is needed?
Side length of the cube is needed to find
the volume.
What is the strategy for identifying
essential information?: Use the
properties of a cube, the diagonal length,
and Pythagorean theorem to find the
side length.
1) Establish relationships between
cube diagonal length and side length
using properties of a cube
•Let “a” be the side
length of cube
a 4√3
•The longer side length
a
of right triangle found
using properties of
a
45-45-90 triangle
a√2
2) Apply Pythagorean theorem to find
side length a2 + (a√2)2 = (4√3)2
a=4
Volume = a3 = 43 = 64
Return to Table of Contents
Return to strategy page
See another example of strategy
Line Segment Length in Solid
Example 2
E
A
Solution Steps
B
D
C
Question: In the figure above, if AB =
24, BC = 12, and CD = 16, what is the
distance from the center of the
rectangular solid to the midpoint of AB?
1) Diagonal BD is the hypotenuse of
right triangle BCD. Find the length of
E
BD.
A
24
B
12
16
What essential information is needed?
A connection between given side lengths,
the center of solid, and the midpoint of AB
What is the strategy for identifying
essential information? Half the length
of diagonal BD is equivalent to the
desired distance. Use Pythagorean
theorem.
Return to Table of Contents
D
C
Can easily find the length of BD by
recognizing that triangle BCD is a
multiple of the 3-4-5 triangle. The
length of BD is 20. (12-16-20)
2) Half the length of diagonal BC is
20/2 = 10 (shown in white on diagram)
Return to strategy page
Return to previous example
Interval Spacing
Example 1
Question: The value of each term of a
sequence is determined by adding the
same number to the term immediately
preceding it. The value of the third term
of a sequence is 4 and the value of the
eighth term is 16.5. What is the value of
the tenth term?
What essential information is needed?
The common value added to each term of
the sequence.
Solution Steps
1) Find the common value.
16.5 - 4
12.5
= 2.5
=
5 intervals 5 intervals
2) Add twice the common value of 2.5
to the eighth term value of 16.5.
What is the strategy for identifying
essential information? Use interval
spacing strategy to identify the common
value. Add twice the value to the eighth
term to find value of tenth term.
Return to Table of Contents
Return to strategy page
Tenth term = 16.5 + 2.5 + 2.5
Tenth term = 21.5
See another example of strategy
Interval Spacing
Example 2
Solution Steps
2n+1
P
2n+2
Question: On the number line above,
what is the value of point P?
a) 2n+½
b) 2n+¾ c) 3·2n
d) 3·2n+1 e) 3·2n+2
18
1) Find the interval spacing
2n+2 - 2n+1 Expand the powers
2n ·22 - 2n ·21 Common factor is 2n
What essential information is needed?
The interval spacing can be used to find
the value of “P”.
2n (22 - 21) Simplify 22 - 21
2n (2) Divide by six intervals
2n (2) 2n Interval
=
36
3 spacing
What is the strategy for identifying
essential information? Find the interval
spacing by dividing the difference of the
two endpoints by the number of intervals
(six). Multiply the interval spacing by
three and add to the value of the left
endpoint.
2) Find the value of “P”
n
2n+1 + (3) 2 = 2n+1 + 2n Expand the
3
powers and
2n ·21 + 2n = 2n (21 + 1) factor
3∙ 2n
Value of point “P”
Return to Table of Contents
Return to strategy page
Return to previous example
Triangle Side Lengths
Example 1
Question: If the side lengths of a triangle
are 8 and 23, what is the smallest integer
length of the third side?
a) 14
b) 15
c) 16
d) 30
e) 31
Solution Steps
1) Find the smallest possible length of
the third side
What essential information is needed?
The smallest possible length of the third
side of the triangle
What is the strategy for identifying
essential information?: The third side
of a triangle must be greater than the
difference of the given two sides of the
triangle.
Return to Table of Contents
Length of third side > 23 - 8
Length of third side > 15
2) Determine the smallest integer
length of third side of triangle
Return to strategy page
Smallest integer length is 16
See another example of strategy
Triangle Side Lengths
Example 2
Question: Each choice below
represents three suggested side lengths
for a triangle. Which of the following
suggested choices will not result in a
triangle?
a) (2, 5, 6) b) (3, 7, 7) c) (3, 8, 12)
d) (5, 6, 7) e) (6, 6, 11)
What essential information is needed?
The range of possible triangle side
lengths for each answer choice.
What is the strategy for identifying
essential information? Evaluate the
first two numbers of each answer choice
using triangle side length strategy. Test
the third number of each answer choice
by comparing to range of possibilities
based on first two numbers.
Return to Table of Contents
Solution Steps
1) Determine range of possible side
lengths using first two numbers
a) 5 - 2 < x < 5 + 2
3<x<7
b) 7 - 3 < x < 7 + 3
4 < x < 10
yes
yes
c) 8 - 3 < x < 8 + 3
5 < x < 11
no
d) 6 - 5 < x < 6 + 5
e) 6 - 6 < x < 6 + 6
1 < x < 11
0 < x < 12
yes
yes
2)Test third number of each answer choice
a) (2, 5, 6)
b) (3, 7, 7)
d) (5, 6, 7)
e) (6, 6, 11)
Return to strategy page
c) (3, 8, 12)
Correct answer choice is “c”
Return to previous example
Using Function Notation
Example of Common Mistake
Question: At a certain factory, the cost
of producing control units is given by the
equation C(n) = 5n + b. If the cost of
producing 20 control units is $300, what
is the value of “b”?
Common mistake: Function notation
should not be used as a math operation.
C(n) should be replaced with 300 when
n = 20. Do not multiply 300 and 20 as in
a math operation.
Solution Steps for
Commonly Made Mistake
1) Replace “C” with 300 and replace
“n” with 20
C(n) = 5n + b
300(20) = 5(20) + b
Correct use of function notation: C(n)
is replaced with 300 when n is replaced
with 20 in the function equation.
Return to Table of Contents
Return to strategy page
6000 = 100 + b
b = 5900 (incorrect answer)
Correct Solution Steps
C(n) = 5n + b
300 = 5(20) + b
300 = 100 + b
b = 200 (correct answer)
See example of strategy
Using Function Notation
Example 1
Solution Steps
Question: If f(x) = x + 7 and 5f(a) =15,
what is the value of f(-2a)?
1) Find the value of “a”
Given 5f(a) = 15 Divide both sides by 5
What essential information is needed?
The value of “a” is needed to determine
the value of f(-2a).
Result f(a) = 3
Given f(x) = x + 7 Evaluate f(a)
f(a) = a + 7 = 3
Result: a = -4
What is the strategy for identifying
essential information?: Use the given
information and properties of function
notation to identify the value of “a”. Use
this value to evaluate f(-2a).
Return to Table of Contents
2) Use a = -4 to find f(-2a)
f(-2a) = f[(-2)(-4)] = f(8) Evaluate f(8)
Return to strategy page
f(a) = a + 7 = 8 + 7
f(a) = 15
See another example of strategy
Using Function Notation
Example 2
Question: The graph of y = f(x) is shown
to the right. If the function y = g(x) is
related to f(x) by the formula g(x) = f(2x)
+ 2, what is the value of g(1)?
-2
2
-2
What essential information is needed?
The math expression g(1) from which the
value of g(1) can be determined
What is the strategy for identifying
essential information? Find the
expression for g(1) by substitution and
the value of g(1) using the graph of y =
f(x).
y = f(x)
2
Solution Steps
1) Find the expression for g(1)
g(x) = f(2x) + 2
g(1) = f(2) + 2
2) Find the value of f(2) from the
graph of y = f(x)
f(2) = 2
g(1) = 2 + 2
Return to Table of Contents
Return to strategy page
g(1) = 4
See another example of strategy
Using Function Notation
Example 3
Question: Using the table to the right, if
f(3) = k, what is the value of g(k)?
What essential information is needed?
The value of “k” is needed to find g(k).
x
f(x)
g(x)
1
3
8
2
4
10
3
5
8
4
6
6
5
7
4
Solution Steps
What is the strategy for identifying
essential information? Use the table of
function values to find “k”. Once known,
find g(k) using the table of function
values.
1) Find the value of “k” using table.
f(3) = k
f(3) = 5
2) Find the value of g(5) using table.
g(5) = 4
Return to Table of Contents
Return to strategy page
Return to example 1