Qualiﬁed Paraprofessionals Developing Highly

Developing Highly Qualified Paraprofessionals

Assisting the Teacher Module III MATHEMATICS

Port Neches-Groves ISD

Mathematics ObjectiveTo meet the requirements of the No Child Left Behind Act (NCLB) by developing highly qualified paraprofessional who possess

● Knowledge of, and ability to assist in high quality mathematics instruction

● An understanding of key mathematics concepts and how to apply these to instruction

Key Concepts1. Number and Operation2. Algebra Key Concepts to3. Geometry be reviewed4. Measurement during this5. Probability and Data Analysis module6. Underlying processes and Mathematical Tools

( Problem Solving)

Number and Operation● Decimals● Fractions● Percents● Order of Operations● Number Sets

Decimals-Place Value(Greater than 1)

1 ,2 3 4 , 5 6 7.m

illion

s

hund

red-

thou

sand

s

ten-

thou

sand

sth

ousa

nds

hund

reds tens

ones

Decimals-Place Value(Less than 1)

0. 1 2 3 4 5 te

nths

hund

redt

hste

n-th

ousa

ndth

s

thou

sand

ths

hund

red-

thou

sand

ths

Decimals- Place Value

67,890.12345

Decimals-Operations(Addition and Subtraction)● When adding or subtracting with decimal

numbers,ALWAYS align the place values!● Examples:

27 36 27 36

+ 5 9 - 5 9

33 26 21 46



27 36 27 36

+ 5 9 - 5 9

33 26 21 46



27 36 27 36

+ 05 90 - 05 90

33 26 21 46

Decimals-Operations(Multiplication)● When multiplying with decimal numbers,it is NOT

necessary to align the place values● It IS necessary to count the digits that have a

place value Less than one● Example: 4. 1 2

X 5

2 0. 6 0

Decimals-Operations(Addition and Subtraction)

● When multiplying with decimal numbers,it is NOT necessary to align the place values

● It IS necessary to count the digits that have a place value Less than one

● Example: 4. 1 2

X 5

2 0. 6 0

There are TWO digits with a place value LESS than one.

Decimals-Operations(Addition and Subtraction)

● When multiplying with decimal numbers,it is NOT necessary to align the place values

● It IS necessary to count the digits that have a place value Less than one

● Example: 4. 1 2

X 5

2 0. 6 0

There are NO digits with a place value LESS than one.

Decimals-Operations(Addition and Subtraction)● When multiplying with decimal numbers,it is NOT

necessary to align the place values● It IS necessary to count the digits that have a

place value Less than one ● Example: 4. 1 2

X 5

2 0. 6 0 There are TWO digits with a place value LESS than one.

Decimals- Operations( Division)● When dividing a decimal number,maintain the position

of the decimal point● Example:

2.7

6 16.2Dividend: The number being divided.



2.7

6 16.2

Quotient: The answer to a division problem



2.7

6 16.2Quotient: The answer to a division problem

Fractions:Vocabulary

34


34

Numerator

Denominator


34

Fractions: Operations Additions and Subtraction

● In order to add or subtract with fractions,it is first necessary to establish a common denominator

● Establish a common denominator by generating equivalent fractions

● Generate equivalent fractions by multiplying bout the numerator and denominator by the same scale factor

Fractions: OperationsAddition and Subtraction

So what does that all mean? Let’s take a look:



3 2 + 4 3


3 ? + 4 12

3 X 3 = 9

4 X 3 = 12



3 9 = 4 12



2 8 = 3 12



3 2 + 4 3

9 8 + 12 12

Fractions:OperationsAddition and Subtraction

9 8 1712 12 12

+ =


17 5 12 12

= 1


3 2 5 4 3 12

+ =1

Fractions:OperationsMultiplication and Division

● Multiplication of division by fractions do not require a common denominator

● Multiply two fractions by multiplying their numerators together, and then their denominators

● Division by a fraction is the same as multiplying by its reciprocal

Fractions:OperationsMultiplication

Example: 3 2

4 3

x =???


Example: 3 2

4 33 2 6

4 3

x =???

x =


Example: 3 2

4 33 2 6 3 2 6

4 3 4 3 12

x =???

x = x =


3 2 14 3 2

x =

Fractions:Comparing and Ordering

● To Compare and order fractions,first convert the fractions to decimals by dividing the numerator by the denominator

● Example:

1 0.25 1

4 4 1.00 4= 0.25

Fractions: Comparing and Ordering

Put the following fractions in order from

least to greatest value:

2 2 5 3 3 5 8 7

Fractions: Comparing and Ordering2 2

3 5

5 3

8 7

=

=

=0.66 0.4

0.625 ∬ 0.429


3 5

5 3

8 7

=

=

=0.66 0.4

0.625 ∬ 0.429

Least value(smallest number)


3 5

5 3

8 7

=

=

=0.66 0.4

0.625 ∬ 0.429


Second greatest value

Fractions: Comparing and Ordering 2 2

3 5

5 3

8 7

=

=

=0.66 0.4

0.625 ∬ 0.429



Third greatest value

Fractions: Comparing and Ordering

2 2

3 5

5 3

8 7

=

=

=0.66 0.4

0.625 ∬ 0.429



Third greatest value

Greatest value( largest number)

Fractions: Comparing and OrderingIn order from least to greatest:

2 3 5 25 7 8 3

Percents● Percent is always out of 100 (per-cent)● To find the percent of a number, convert

the percent value to decimal value and the multiply

● Example: What is 6% of $ 13.95?

PercentsExample: What is 6% of $ 13.95?


6% = = 0.06 6 100


6% = =0.06 6 100

0.06 x 13.95 = 0.837


6% = =0.06

So 6% of $13.95 is 84￠

6 100

0.06 x 13.95 = 0.837

Order of Operations● When multiple operations are included in a

problem,there is a specific order in which those operations are to be performed

● This is called the Order of Operations

Order of OperationsP E MD AS = Order of Operations

P-Parentheses

E-Exponents

M-Multiplication D-Division

A-Addition S-Subtraction

Order of OperationsExponents● An exponent is used to denote how many

times a number is multiplied by itself● Examples:

3² = 3 x 3 = 9

3³ = 3 x 3 x 3 = 27

3⁴ = 3 x 3 x 3 x 3 = 81

Order of OperationsExponents● An exponent is used to denote how many

times a number is multiplied by itself● Examples:

3² = 3 x 3 = 9

3³ = 3 x 3 x 3 = 27

3⁴ = 3 x 3 x 3 x 3 = 81

Note that 3² is NOT the same as 3 x 2!

Order of OperationsSquare Roots● Finding the square root of a number is the

opposite of finding the square of a number● Examples:

3² = 9 4² = 16 5² = 25

√9 = 3 √16 = 4 √25 = 5

Order of Operations

Simplify the following expression using the correct order of operations:

4² ÷ 8( 7- 3)

Order of Operations

4² ÷ 8( 7- 3) 4² ÷ 8(4)

Order of Operations

4² ÷ 8( 7- 3) 4² ÷ 8(4)

16 ÷ 8(4)

Order of Operations

4² ÷ 8( 7- 3) 4² ÷ 8(4)

16 ÷ 8(4)

2 (4)

Order of Operations4² ÷ 8( 7- 3)

4² ÷ 8(4)

16 ÷ 8(4) 2 (4) 8

Number Sets

● Rational vs. irrational numbers● Prime vs. composite numbers● Integers● Counting numbers

Algebra● Proportional relationships● Functional relationships● Variables and equations

Proportional Relationships● A ratio is a comparison of two values● Two equivalent ratios form a proportion● Proportionality is one of the most critical

components of the mathematics standards● Example:

On a map, 1 inch represents 15 miles. If the distance between two cities on the map is 7 inches, what is the actual distance between those two cities?

Proportional Relationships Example:On a map, 1 inch represents 15 miles. If the distance between two cities on the map is 7 inches, what is the actual distance between those two cities?

1 inch 7 inches

15 miles ??? miles=

Proportional Relationships Example:On a map, 1 inch represents 15 miles. If the distance between two cities on the map is 7 inches, what is the actual distance between those two cities?

1 inch 7 inches

15 miles 105 miles=

Functional Relationships● A functional relationship can exist when one

quantity depends on another● Examples of functional relationships:

● The amount of my paycheck depends on the number of hours I work.

● The distance I am able to drive in my car depends on the amount of gas in the tank.

Variables and Equations● A variable is a letter or symbol that is used

to represent a changing value● Variable are used in formulas and algebraic

equations● Examples:

● d = r t● y = 3x - 9● 3(6x - 5) + 9 = 120

Variables and Equations Let’s try solving an equation:

3(6x - 5) + 9 = 120

18x-15 + 9 = 120

18x -6 = 120

18x =126

X = 7

+ 6 +6

18 18

Substitution 3(6x - 5) + 9 = 120

3[6(7)-5] + 9 = 120

3( 42 - 5 ) + 9 = 120

3( 37 ) + 9 =120

111 + 9 = 120

120 = 120

Substitute 7 in place of X

Follow PEMDAS to simplify left side

It checks!

Geometry and Measurement● Vocabulary● Coordinate system - graphing● Transformations● Angles● Polygons● Circles● Perimeter / Area

Vocabulary ● Congruent(≅)-same size and same shape● Similar (∼) - same shape but not necessarily

the same size● Parallel Lines- are always the same distance

apart from each other; will never intersect● Perpendicular- form right angles(90°)● Regular Polygons-all angles are equal and

all sides have same length

GraphingGraphing on a coordinate plane requires the working knowledge of certain vocabulary terms.

GraphingGraphing on a coordinate plane requires the working knowledge of certain vocabulary terms.

y-axis

x-axis

originPoint with coordinates (3 , 2)

3 is the x-coordinate and2 is the y-coordinate

(3 , 2) is the ordered pair that locates the point in the coordinate plane

Transformations

● Reflection - mirror image

● Rotation - turn

● Translation - slide

● Dilation - change in size

Angles● Right angle - measure is exactly 90°

● Acute angle - measure is less than 90°

● Obtuse angle - measure is greater than 90° but less than 180°

TrianglesTriangles can be classified by their angle measures:

● Right Triangle - one right angle

● Acute Triangle - all acute angles

● Obtuse Triangle - one obtuse angle

TrianglesTriangles can also be classified by their side lengths:

● Scalene Triangle - no sides are the same length

● Isosceles Triangle - at least two sides are the same length

● Equilateral Triangle - all three sides are the same length

Other Polygons● Quadrilaterals have four sides and include:

○ Squares--all four sides are the same length and all angles are right angles

○ Rectangles--all angles are right angles, but all four sides are not necessarily the same length

● Pentagon--five sides● Hexagon--six sides● Octagon--eight sides

Circles Circumference: the distance around a circle

C = 2 ℼ r OR C = ℼ d

Area of a circle:

A = ℼ r²

Pi is the ratio of the circumference to the diameter of

a circle: ℼ = 3.14

radiusdiameter

center

Perimeter and Area12 feet

7 fe

et

Applying SimilarityWhat is the length of the larger rectangle?

2m 4m

5m ???

4 m

2 m 5 m

=

Probability and Data Analysis

● Independent vs. dependent events

● Measures of central tendency

● Reading and interpreting various displays of

data

Independent /Dependent Probability

● Independent event-one in which the outcome of one event DOES NOT depend on the outcome of another event

● Dependent event-one in which the outcome of one event DOES depend on the outcome of another event

Independent /Dependent ProbabilityAmeena has a bag full of jelly beans in her backpack. There are 6 blue jelly beans, 9 red jelly beans, 4 green jelly beans, and 5 pink jelly beans.

● What is the probability of randomly choosing a blue jelly bean from the bag, replacing it, and then choosing a green one?

● What is the probability of randomly choosing a blue jelly bean from the bag, eating it, and then choosing a green one?

Independent ProbabilityAmeena has a bag full of jelly beans in her backpack. There are 6 blue jelly beans, 9 red jelly beans, 4 green jelly beans, and 5 pink jelly beans.

What is the probability of randomly choosing a blue jelly bean from the bag, replacing it, and then choosing a green one?

6 4 24 1

24 24 576 24 x = =

Independent ProbabilityAmeena has a bag full of jelly beans in her backpack. There are 6 blue jelly beans, 9 red jelly beans, 4 green jelly beans, and 5 pink jelly beans.

What is the probability of randomly choosing a blue jelly bean from the bag, eating it, and then choosing a green one?

6 4 24 1

24 23 552 23 x = =

Measures of Central Tendency● Mean: average

● Median: middle

● Mode: most frequently occurring

Measures of Central Tendency78, 72, 75, 79, 72, 73, 76

Mean: 525 ÷ 7 = 75


72, 72, 73, 75, 76, 78, 79


72, 72, 73, 75, 76, 78, 79

Median


72, 72, 73, 75, 76, 78, 79

Mode


72, 72, 73, 75, 76, 78, 79

Range: 79 - 72 = 7

Reading and Interpreting Various Displays of Data● Lists

● Tables / Charts

● Graphs○ Circle graphs ( Pie Graphs)○ Bar graphs

Manipulatives

● Base-ten blocks● Color tiles● Number lines● Number cubes● Counters ● Unifix cubes

Resources● http://www.tea.state.tx.us

○ Texas Education Agency● http://www.nctm.org

○ National Council of Teachers of Mathematics

● The Mathematics Dictionary and Handbook○ Nichols Schwartz Publishing○ ISBN: 1-882269-09-8

http://www.tea.state.tx.us

http://www.nctm.org

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Qualiﬁed Paraprofessionals Developing Highly