4. Number Map
Real Numbers
4
A real number is a number that has a location on the
number line.
On the ACT, imaginary numbers are numbers that involve
the square root of a negative number.
5. Number Map
Rational Numbers vs. Irrational Numbers
5
A rational number is a number that can be expressed
as a ratio of two integers.
Irrational numbers are real numbers – they have
locations on the number line – but they can’t be
expressed precisely as a fraction or decimal.
Examples:
Rational Numbers:
Irrational Numbers:
9. 1. Properties of Numbers:
Parity (Even/Odd) & Sign (Positive/Negative)
To predict whether a sum, difference, or product will be
even or odd, positive or negative just take simple numbers
like 1 and 2 and see what happens.
Example
The product of an even number and an
odd number is…
A. always even
B. always odd
C. always negative
D. always positive
E. sometimes odd
9
12. 1. Properties of Numbers
Converting Improper Fractions to Mixed
Numbers
11
Mixed Number Improper Fraction
1. Divide the
denominator into
the numerator to
get a whole
number quotient
with a remainder
2. The quotient
becomes the
whole number part
of the mixed
number
3. The remainder
becomes the new
numerator with the
same denominator
13. 1. Properties of Numbers
Converting Improper Fractions to Mixed
Numbers
11
Mixed Number Improper Fraction
1. Divide the
denominator into
the numerator to
get a whole
number quotient
with a remainder
2. The quotient
becomes the
whole number part
of the mixed
number
3. The remainder
becomes the new
numerator with the
same denominator
11/4 = 2 Rem. 3
14. 1. Properties of Numbers
Converting Mixed Numbers to Improper
Fractions
Improper fractions: fractions where the numerator is larger
than the denominator
Mixed numbers: numbers with a whole number portion and a
fractional portion
12
15. 1. Properties of Numbers
Converting Mixed Numbers to Improper
Fractions
Improper fractions: fractions where the numerator is larger
than the denominator
Mixed numbers: numbers with a whole number portion and a
fractional portion
12
Mixed Number Improper Fraction
1. Multiply the
whole number part
by the
denominator
2. Then add the
product to the
numerator
3. The result is the
new numerator
(over the same
denominator)
16. 1. Properties of Numbers
Converting Mixed Numbers to Improper
Fractions
Improper fractions: fractions where the numerator is larger
than the denominator
Mixed numbers: numbers with a whole number portion and a
fractional portion
12
Mixed Number Improper Fraction
1. Multiply the
whole number part
by the
denominator
2. Then add the
product to the
numerator
3. The result is the
new numerator
(over the same
denominator)
2 x 4 = 8 8 + 3 = 11
17. 1. Properties of Numbers
Comparing Fractions
Two methods
Re-express them with a common
denominator.
Convert them both to decimals.
Use your calculator!!
13
20. 1. Properties of Numbers
GCF/LCM
Greatest Common Factor Least Common Multiple
15
21. 1. Properties of Numbers
GCF/LCM
Greatest Common Factor
Largest factor common to two or
more numbers
Also called greatest common
divisor (gcd)
Least Common Multiple
15
22. 1. Properties of Numbers
GCF/LCM
Greatest Common Factor
Largest factor common to two or
more numbers
Also called greatest common
divisor (gcd)
How to find it:
Write out a list of factors for
each number – find the largest
in common
Use your calculator:
Least Common Multiple
15
23. 1. Properties of Numbers
GCF/LCM
Greatest Common Factor
Largest factor common to two or
more numbers
Also called greatest common
divisor (gcd)
How to find it:
Write out a list of factors for
each number – find the largest
in common
Use your calculator:
Least Common Multiple
15
MATH NUM GCD(
24. 1. Properties of Numbers
GCF/LCM
Greatest Common Factor
Largest factor common to two or
more numbers
Also called greatest common
divisor (gcd)
How to find it:
Write out a list of factors for
each number – find the largest
in common
Use your calculator:
Least Common Multiple
Smallest number that is a multiple
of two or more numbers
15
MATH NUM GCD(
25. 1. Properties of Numbers
GCF/LCM
Greatest Common Factor
Largest factor common to two or
more numbers
Also called greatest common
divisor (gcd)
How to find it:
Write out a list of factors for
each number – find the largest
in common
Use your calculator:
Least Common Multiple
Smallest number that is a multiple
of two or more numbers
How to find it:
Write out a list of multiples for
each number – find the smallest
number that appears on both
(all) lists
Use your calculator:
15
MATH NUM GCD(
26. 1. Properties of Numbers
GCF/LCM
Greatest Common Factor
Largest factor common to two or
more numbers
Also called greatest common
divisor (gcd)
How to find it:
Write out a list of factors for
each number – find the largest
in common
Use your calculator:
Least Common Multiple
Smallest number that is a multiple
of two or more numbers
How to find it:
Write out a list of multiples for
each number – find the smallest
number that appears on both
(all) lists
Use your calculator:
15
MATH NUM LCM(MATH NUM GCD(
29. 2. Counting Consecutive Integers
To count consecutive integers, simplify the problem to
identify a pattern (or equation!), and then apply it to
the larger context.
17
30. 2. Counting Consecutive Integers
To count consecutive integers, simplify the problem to
identify a pattern (or equation!), and then apply it to
the larger context.
17
Example 1:
How many even integers are there between 3 and 525?
33. 3. Number Line
Can be asked to locate rational numbers on the number line or find the
distance between points on the number line
19
34. 3. Number Line
Can be asked to locate rational numbers on the number line or find the
distance between points on the number line
Draw/label a number line as you work through these questions!
19
35. 3. Number Line
Can be asked to locate rational numbers on the number line or find the
distance between points on the number line
Draw/label a number line as you work through these questions!
19
Example:
What is a irrational number between points A and B on the
number line?
36. 3. Number Line
Can be asked to locate rational numbers on the number line or find the
distance between points on the number line
Draw/label a number line as you work through these questions!
19
Example:
What is a irrational number between points A and B on the
number line?
0 1 2-1-2
A B C
𝐴. − 3 𝐵. −
1
2
2 C. −2 D. 3 E. 2
40. 4. Radicals
Simplifying Square Roots
1. Write the number under the square root sign as a product of a perfect
square and another number
• Will be beneficial to find the largest perfect square factor
2. Take the square root of the perfect square and take it outside the
radical sign
21
41. 4. Radicals
Simplifying Square Roots
1. Write the number under the square root sign as a product of a perfect
square and another number
• Will be beneficial to find the largest perfect square factor
2. Take the square root of the perfect square and take it outside the
radical sign
21
42. 4. Radicals
Simplifying Square Roots
1. Write the number under the square root sign as a product of a perfect
square and another number
• Will be beneficial to find the largest perfect square factor
2. Take the square root of the perfect square and take it outside the
radical sign
21
If you don’t find the largest
perfect square factor, you can
still get the right answer. It will
just take longer!
43. 4. Radicals
Multiplying and Dividing Roots
Recall that fractional exponents indicate a root! The same
rules apply with fractional exponents.
22
46. I. Number & Quantity
24
A. Real Number System
B. Complex Number System
C. Vectors
D. Matrices
47. The Complex Number System
Imaginary number: any number that involves the square root of a
negative number
Ex: ,
Basic definition to remember:
Complex Plane: a 2D space that represents both real and imaginary
numbers
25
48. The Complex Number System
Imaginary number: any number that involves the square root of a
negative number
Ex: ,
Basic definition to remember:
Complex Plane: a 2D space that represents both real and imaginary
numbers
25
Simplification
49. The Complex Number System
Imaginary number: any number that involves the square root of a
negative number
Ex: ,
Basic definition to remember:
Complex Plane: a 2D space that represents both real and imaginary
numbers
25
Exponent Pattern
Simplification
50. The Complex Number System
Imaginary number: any number that involves the square root of a
negative number
Ex: ,
Basic definition to remember:
Complex Plane: a 2D space that represents both real and imaginary
numbers
25
Exponent Pattern
Simplification
= i
51. The Complex Number System
Imaginary number: any number that involves the square root of a
negative number
Ex: ,
Basic definition to remember:
Complex Plane: a 2D space that represents both real and imaginary
numbers
25
Exponent Pattern
Simplification
= i
52. The Complex Number System
Imaginary number: any number that involves the square root of a
negative number
Ex: ,
Basic definition to remember:
Complex Plane: a 2D space that represents both real and imaginary
numbers
25
Exponent Pattern
Simplification
= –i
= –i
= i
53. The Complex Number System
Imaginary number: any number that involves the square root of a
negative number
Ex: ,
Basic definition to remember:
Complex Plane: a 2D space that represents both real and imaginary
numbers
25
Exponent Pattern
Simplification
= –i
= –i
= i
56. I. Number & Quantity
27
A. Real Number System
B. Complex Number System
C. Vectors
D. Matrices
57. 1. What is a Vector?
A quantity that has both a
magnitude and a direction
Vectors have a component
form to indicate the magnitudes
in each direction
Ex: <6, -5>
28
58. 1. What is a Vector?
A quantity that has both a
magnitude and a direction
Vectors have a component
form to indicate the magnitudes
in each direction
Ex: <6, -5>
28
Vector’s X
Magnitude
59. 1. What is a Vector?
A quantity that has both a
magnitude and a direction
Vectors have a component
form to indicate the magnitudes
in each direction
Ex: <6, -5>
28
Vector’s X
Magnitude
Vector’s Y
Magnitude
63. I. Number & Quantity
31
A. Real Number System
B. Complex Number System
C. Vectors
D. Matrices
64. 1. What is a matrix?
A matrix is a rectangular
organization of a set of
numbers into rows and
columns
Nomenclature
m x n matrix where m is
the number of rows and
n is the number of
columns
Think RC Cola – rows
first, columns second
32
This is a 3 x 4 matrix
Column #1
Row #3
65. 1. What is a matrix?
A matrix is a rectangular
organization of a set of
numbers into rows and
columns
Nomenclature
m x n matrix where m is
the number of rows and
n is the number of
columns
Think RC Cola – rows
first, columns second
32
This is a 3 x 4 matrix
Column #1
Row #3
Where are matrices used?
Matrices are used in many mathematical
areas, including the following:
1. Solving complex systems of equations
2. In the Google (Search Engine) PageRank
System
3. In services like Netflix that need to
organize large quantities of data
66. 2. Matrix Addition and Subtraction
Must have two (or more) matrices that are identical in size
Add (or subtract) number in corresponding positions
33
67. 2. Matrix Addition and Subtraction
Must have two (or more) matrices that are identical in size
Add (or subtract) number in corresponding positions
33
68. 3. Scalar Multiplication
Scalar Multiplication: multiplication of an entire matrix by
some factor
Simply multiply each term by the given factor (aka
scalar)
34
69. 3. Scalar Multiplication
Scalar Multiplication: multiplication of an entire matrix by
some factor
Simply multiply each term by the given factor (aka
scalar)
34
72. 4. Matrix Multiplication
Must have two matrices (m x n) and (a x b) such that n
and a are the same
35
Must be the same
73. 4. Matrix Multiplication
Must have two matrices (m x n) and (a x b) such that n
and a are the same
Resulting matrix will be m x b
35
Must be the same
74. 4. Matrix Multiplication
Must have two matrices (m x n) and (a x b) such that n
and a are the same
Resulting matrix will be m x b
Multiply each piece of the first row in Matrix #1 by the first
column in Matrix #2. Find the sum, and this goes in Row
1, Column 1 of the new matrix.
35
Must be the same
75. 4. Matrix Multiplication
Must have two matrices (m x n) and (a x b) such that n
and a are the same
Resulting matrix will be m x b
Multiply each piece of the first row in Matrix #1 by the first
column in Matrix #2. Find the sum, and this goes in Row
1, Column 1 of the new matrix.
35
Must be the same
76. 5. Matrix Determinant
Determinant is a quantity related to a square matrix
Determinant of Matrix A is often called det(A)
Sometimes (but not always!) the definition of determinant
is provided if it’s needed on the test
Generally only used on ACT for 2 x 2 matrices
Example
Find det(A) where .
36
81. List of Terms
Real & Complex Number Systems
Real numbers
Complex numbers
Integers
Whole numbers
Natural/Counting numbers
Complex Plane
Properties of Numbers
Prime numbers
Least Common Multiple/Least Common Denominator
Greatest Common Factor
Matrices
Determinant
Scalar
Matrix product
39
Editor's Notes
Teacher Notes
The Number and Quantity section focuses on basic numerical properties and definitions.
**We recommend that students take notes in the form of an outline, beginning with Roman numeral two (I) for Number and Quantity and on the next slide, letter A**
Teacher Notes
This map is a visualization of the terminology used to define different types of numbers. The following slides reference this image. It may be beneficial for students to copy this image into their notes or for the teacher to print handouts with this image.
While number terminology will not play a major role on the ACT, there occasionally are questions that rely upon knowledge of the number groups.
Teacher Notes
To be clear, “imaginary numbers” (referred to in the second bullet point) fall outside the realm of the real number system.
Teacher Notes
Numbers that have decimals (ex: 3.1234) are rational numbers but not integers.
https://s-media-cache-ak0.pinimg.com/236x/58/89/59/5889591c71971dde42b85ede5546ff66.jpg
Teacher Notes
There are rules, such as “odd times even is even,” but there’s no need to memorize them. This is a great time to introduce the “plug numbers in and see what happens” strategy. Students can pick simple representative numbers (i.e. 1 or 2) and perform the specified operation to see what the type of output is.
Teacher Notes
Remember the calculator function that can find the remainder of a division problem. As a reminder, see this site: https://epsstore.ti.com/OA_HTML/csksxvm.jsp;jsessionid=e97aae727ce38b3f802fb28d05abf0bbc8c275822615389e92e76e236f843743.e34TbNyLbhiKai0TbNb0?jttst0=6_23871%2C23871%2C-1%2C0%2C&jtfm0=&etfm1=&jfn=ZGEB0E420970AE9FB4A4C7A37DA43C2BB66FA36F4E94A752345E2FDD58CBE718E5AACA3DA32351273A7D224750326A890DBF&lepopus=e1pNTSAT6R8FscJ99uLtnjcx5f&lepopus_pses=ZG8341D44132C32ACC39AFEE51100F2A4450EB1D08F79C11BECB1E0622D31F17EF65D38D9769113FF3A683F9DFC6BDE1A5FAA8753C9B501858&oas=lGFdyzPfpInH52AI-bKccQ..&nSetId=144288&nBrowseCategoryId=10433&cskViewSolSourcePage=cskmbasicsrch.jsp%3FcategoryId%3D10433%26fRange%3Dnull%26fStartRow%3D0%26fSortBy%3D2%26fSortByOrder%3D1
This is also a good time to remind students what the “quotient” is (the result of a division operation).
Teacher Notes
Remember the calculator function that can find the remainder of a division problem. As a reminder, see this site: https://epsstore.ti.com/OA_HTML/csksxvm.jsp;jsessionid=e97aae727ce38b3f802fb28d05abf0bbc8c275822615389e92e76e236f843743.e34TbNyLbhiKai0TbNb0?jttst0=6_23871%2C23871%2C-1%2C0%2C&jtfm0=&etfm1=&jfn=ZGEB0E420970AE9FB4A4C7A37DA43C2BB66FA36F4E94A752345E2FDD58CBE718E5AACA3DA32351273A7D224750326A890DBF&lepopus=e1pNTSAT6R8FscJ99uLtnjcx5f&lepopus_pses=ZG8341D44132C32ACC39AFEE51100F2A4450EB1D08F79C11BECB1E0622D31F17EF65D38D9769113FF3A683F9DFC6BDE1A5FAA8753C9B501858&oas=lGFdyzPfpInH52AI-bKccQ..&nSetId=144288&nBrowseCategoryId=10433&cskViewSolSourcePage=cskmbasicsrch.jsp%3FcategoryId%3D10433%26fRange%3Dnull%26fStartRow%3D0%26fSortBy%3D2%26fSortByOrder%3D1
This is also a good time to remind students what the “quotient” is (the result of a division operation).
Teacher Notes
Work through the example as you explain this step by step. For whatever reason, the ACT tests this concept often, and many students forget how easy it is.
Teacher Notes
Work through the example as you explain this step by step. For whatever reason, the ACT tests this concept often, and many students forget how easy it is.
Teacher Notes
Work through the example as you explain this step by step. For whatever reason, the ACT tests this concept often, and many students forget how easy it is.
Teacher Notes
Remember, factors are “few” and multiples are ”many.” Factors are numbers that can be multiplied to produce the specific number, and multiples are numbers that can be divided into a specific number with a remainder of zero.
Make sure to separate two numbers using a comma. The functions only work to find the lcm or gcd of two numbers. To find the lcm or gcd of three numbers (x, y, z), use the following syntax:
lcm(x,lcm(y,z)) or gcd(x,gcd(y,z))
Teacher Notes
Remember, factors are “few” and multiples are ”many.” Factors are numbers that can be multiplied to produce the specific number, and multiples are numbers that can be divided into a specific number with a remainder of zero.
Make sure to separate two numbers using a comma. The functions only work to find the lcm or gcd of two numbers. To find the lcm or gcd of three numbers (x, y, z), use the following syntax:
lcm(x,lcm(y,z)) or gcd(x,gcd(y,z))
Teacher Notes
Remember, factors are “few” and multiples are ”many.” Factors are numbers that can be multiplied to produce the specific number, and multiples are numbers that can be divided into a specific number with a remainder of zero.
Make sure to separate two numbers using a comma. The functions only work to find the lcm or gcd of two numbers. To find the lcm or gcd of three numbers (x, y, z), use the following syntax:
lcm(x,lcm(y,z)) or gcd(x,gcd(y,z))
Teacher Notes
Remember, factors are “few” and multiples are ”many.” Factors are numbers that can be multiplied to produce the specific number, and multiples are numbers that can be divided into a specific number with a remainder of zero.
Make sure to separate two numbers using a comma. The functions only work to find the lcm or gcd of two numbers. To find the lcm or gcd of three numbers (x, y, z), use the following syntax:
lcm(x,lcm(y,z)) or gcd(x,gcd(y,z))
Teacher Notes
Remember, factors are “few” and multiples are ”many.” Factors are numbers that can be multiplied to produce the specific number, and multiples are numbers that can be divided into a specific number with a remainder of zero.
Make sure to separate two numbers using a comma. The functions only work to find the lcm or gcd of two numbers. To find the lcm or gcd of three numbers (x, y, z), use the following syntax:
lcm(x,lcm(y,z)) or gcd(x,gcd(y,z))
Teacher Notes
Remember, factors are “few” and multiples are ”many.” Factors are numbers that can be multiplied to produce the specific number, and multiples are numbers that can be divided into a specific number with a remainder of zero.
Make sure to separate two numbers using a comma. The functions only work to find the lcm or gcd of two numbers. To find the lcm or gcd of three numbers (x, y, z), use the following syntax:
lcm(x,lcm(y,z)) or gcd(x,gcd(y,z))
Teacher Notes
Remember, factors are “few” and multiples are ”many.” Factors are numbers that can be multiplied to produce the specific number, and multiples are numbers that can be divided into a specific number with a remainder of zero.
Make sure to separate two numbers using a comma. The functions only work to find the lcm or gcd of two numbers. To find the lcm or gcd of three numbers (x, y, z), use the following syntax:
lcm(x,lcm(y,z)) or gcd(x,gcd(y,z))
?????
This was the note:
Teacher Notes
Why subtract one? When we are counting the numbers between 4 and 10, we are including 4 and 10. If we count the numbers (4, 5, 6, 7, 8, 9, 10), there are 7. However, when you find the difference between 10 and 4, it is 6. We need to add one because the operation of subtraction cuts off the lowest number that we want to include (4), so we must add one to our “count” of integers.
Example 1: 56, Example 2: 11
?????
This was the note:
Teacher Notes
Why subtract one? When we are counting the numbers between 4 and 10, we are including 4 and 10. If we count the numbers (4, 5, 6, 7, 8, 9, 10), there are 7. However, when you find the difference between 10 and 4, it is 6. We need to add one because the operation of subtraction cuts off the lowest number that we want to include (4), so we must add one to our “count” of integers.
Example 1: 56, Example 2: 11
Teacher Notes
There are other ways of simplifying square roots, but this tends to be the most efficient strategy.
This is usually not a problem by itself but rather is part of another problem where the answer is a square root that must be simplified.
The first line shows how finding the largest perfect square factor can make the process simpler. However, if the largest perfect square factor is not found (see line 2), the correct answer can still be found.
Teacher Notes
There are other ways of simplifying square roots, but this tends to be the most efficient strategy.
This is usually not a problem by itself but rather is part of another problem where the answer is a square root that must be simplified.
The first line shows how finding the largest perfect square factor can make the process simpler. However, if the largest perfect square factor is not found (see line 2), the correct answer can still be found.
Teacher Notes
There are other ways of simplifying square roots, but this tends to be the most efficient strategy.
This is usually not a problem by itself but rather is part of another problem where the answer is a square root that must be simplified.
The first line shows how finding the largest perfect square factor can make the process simpler. However, if the largest perfect square factor is not found (see line 2), the correct answer can still be found.
Teacher Notes
There are other ways of simplifying square roots, but this tends to be the most efficient strategy.
This is usually not a problem by itself but rather is part of another problem where the answer is a square root that must be simplified.
The first line shows how finding the largest perfect square factor can make the process simpler. However, if the largest perfect square factor is not found (see line 2), the correct answer can still be found.
Teacher Notes
Remind students that they may need to simplify the answers after they perform multiplication or division!
Teacher Notes
This is another topic that will be re-visited later in the slides. Ensure understanding on the examples by using additional problems (if necessary).
Answer in the example: 4i sqrt(2)
Teacher Notes
This is another topic that will be re-visited later in the slides. Ensure understanding on the examples by using additional problems (if necessary).
Answer in the example: 4i sqrt(2)
Teacher Notes
This is another topic that will be re-visited later in the slides. Ensure understanding on the examples by using additional problems (if necessary).
Answer in the example: 4i sqrt(2)
Teacher Notes
This is another topic that will be re-visited later in the slides. Ensure understanding on the examples by using additional problems (if necessary).
Answer in the example: 4i sqrt(2)
Teacher Notes
This is another topic that will be re-visited later in the slides. Ensure understanding on the examples by using additional problems (if necessary).
Answer in the example: 4i sqrt(2)
Teacher Notes
This is another topic that will be re-visited later in the slides. Ensure understanding on the examples by using additional problems (if necessary).
Answer in the example: 4i sqrt(2)
Teacher Notes
This is another topic that will be re-visited later in the slides. Ensure understanding on the examples by using additional problems (if necessary).
Answer in the example: 4i sqrt(2)
Teacher Notes
Vectors are a more advanced concept that typically appear in Pre-Calculus, so many students will not be familiar with them. As such, the following slides may be confusing for some students.
Teacher Notes
There are multiple representations for vectors. For example, the vector <4, 3> can also be written as 4i + 3j. Both representations refer to a horizontal and vertical component.
Again, this topic will serve as a reminder for students who have studied this concept in Physics or Pre-Calculus, but students who have not seen it before should just focus on the basics (how vectors are represented and combined by adding or subtracting the components).
Teacher Notes
Matrix addition and subtraction is very simple! Even students who have never encountered this topic before likely will not have much difficulty with this concept.
Teacher Notes
Matrix addition and subtraction is very simple! Even students who have never encountered this topic before likely will not have much difficulty with this concept.
Teacher Notes
Scalar multiplication is also very simple! Even students who have never encountered this topic before likely will not have much difficulty with this concept.
Teacher Notes
Scalar multiplication is also very simple! Even students who have never encountered this topic before likely will not have much difficulty with this concept.
Teacher Notes
1. This is a much more complex topic than the previous two.
Teacher Notes
1. This is a much more complex topic than the previous two.
Teacher Notes
1. This is a much more complex topic than the previous two.
Teacher Notes
1. This is a much more complex topic than the previous two.
Teacher Notes
1. This is a much more complex topic than the previous two.
Teacher Notes
1. This is a much more complex topic than the previous two.