qualified paraprofessionals developing highly
TRANSCRIPT
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Developing Highly Qualified Paraprofessionals
Assisting the Teacher Module III MATHEMATICS
Port Neches-Groves ISD
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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
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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)
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Number and Operation● Decimals● Fractions● Percents● Order of Operations● Number Sets
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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
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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
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Decimals- Place Value
67,890.12345
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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
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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
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Decimals-Operations(Addition and Subtraction)● When adding or subtracting with decimal
numbers,ALWAYS align the place values!● Examples:
27 36 27 36
+ 05 90 - 05 90
33 26 21 46
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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
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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.
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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.
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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.
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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.
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Decimals- Operations( Division)● When dividing a decimal number,maintain the position
of the decimal point● Example:
2.7
6 16.2
Quotient: The answer to a division problem
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Decimals- Operations( Division)● When dividing a decimal number,maintain the position
of the decimal point● Example:
2.7
6 16.2Quotient: The answer to a division problem
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Fractions:Vocabulary
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Fractions:Vocabulary
34
Numerator
Denominator
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Fractions:Vocabulary
34
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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
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Fractions: OperationsAddition and Subtraction
So what does that all mean? Let’s take a look:
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Fractions: OperationsAddition and Subtraction
So what does that all mean? Let’s take a look:
3 2 + 4 3
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Fractions: OperationsAddition and Subtraction
3 ? + 4 12
3 X 3 = 9
4 X 3 = 12
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Fractions: OperationsAddition and Subtraction
So what does that all mean? Let’s take a look:
3 9 = 4 12
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Fractions: OperationsAddition and Subtraction
So what does that all mean? Let’s take a look:
2 8 = 3 12
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Fractions: OperationsAddition and Subtraction
So what does that all mean? Let’s take a look:
3 2 + 4 3
9 8 + 12 12
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Fractions:OperationsAddition and Subtraction
9 8 1712 12 12
+ =
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Fractions:OperationsAddition and Subtraction
17 5 12 12
= 1
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Fractions:OperationsAddition and Subtraction
3 2 5 4 3 12
+ =1
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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
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Fractions:OperationsMultiplication
Example: 3 2
4 3
x =???
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Fractions:OperationsMultiplication
Example: 3 2
4 33 2 6
4 3
x =???
x =
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Fractions:OperationsMultiplication
Example: 3 2
4 33 2 6 3 2 6
4 3 4 3 12
x =???
x = x =
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Fractions:OperationsMultiplication
3 2 14 3 2
x =
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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
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Fractions: Comparing and Ordering
Put the following fractions in order from
least to greatest value:
2 2 5 3 3 5 8 7
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Fractions: Comparing and Ordering2 2
3 5
5 3
8 7
=
=
=0.66 0.4
0.625 ∬ 0.429
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Fractions: Comparing and Ordering2 2
3 5
5 3
8 7
=
=
=0.66 0.4
0.625 ∬ 0.429
Least value(smallest number)
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Fractions: Comparing and Ordering2 2
3 5
5 3
8 7
=
=
=0.66 0.4
0.625 ∬ 0.429
Least value(smallest number)
Second greatest value
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Fractions: Comparing and Ordering 2 2
3 5
5 3
8 7
=
=
=0.66 0.4
0.625 ∬ 0.429
Least value(smallest number)
Second greatest value
Third greatest value
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Fractions: Comparing and Ordering
2 2
3 5
5 3
8 7
=
=
=0.66 0.4
0.625 ∬ 0.429
Least value(smallest number)
Second greatest value
Third greatest value
Greatest value( largest number)
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Fractions: Comparing and OrderingIn order from least to greatest:
2 3 5 25 7 8 3
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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?
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PercentsExample: What is 6% of $ 13.95?
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PercentsExample: What is 6% of $ 13.95?
6% = = 0.06 6 100
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PercentsExample: What is 6% of $ 13.95?
6% = =0.06 6 100
0.06 x 13.95 = 0.837
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PercentsExample: What is 6% of $ 13.95?
6% = =0.06
So 6% of $13.95 is 84¢
6 100
0.06 x 13.95 = 0.837
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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
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Order of OperationsP E MD AS = Order of Operations
P-Parentheses
E-Exponents
M-Multiplication D-Division
A-Addition S-Subtraction
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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
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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!
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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
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Order of Operations
Simplify the following expression using the correct order of operations:
4² ÷ 8( 7- 3)
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Order of Operations
4² ÷ 8( 7- 3) 4² ÷ 8(4)
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Order of Operations
4² ÷ 8( 7- 3) 4² ÷ 8(4)
16 ÷ 8(4)
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Order of Operations
4² ÷ 8( 7- 3) 4² ÷ 8(4)
16 ÷ 8(4)
2 (4)
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Order of Operations4² ÷ 8( 7- 3)
4² ÷ 8(4)
16 ÷ 8(4) 2 (4) 8
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Number Sets
● Rational vs. irrational numbers● Prime vs. composite numbers● Integers● Counting numbers
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Algebra● Proportional relationships● Functional relationships● Variables and equations
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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?
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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=
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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=
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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.
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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
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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
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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!
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Geometry and Measurement● Vocabulary● Coordinate system - graphing● Transformations● Angles● Polygons● Circles● Perimeter / Area
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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
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GraphingGraphing on a coordinate plane requires the working knowledge of certain vocabulary terms.
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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
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Transformations
● Reflection - mirror image
● Rotation - turn
● Translation - slide
● Dilation - change in size
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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°
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TrianglesTriangles can be classified by their angle measures:
● Right Triangle - one right angle
● Acute Triangle - all acute angles
● Obtuse Triangle - one obtuse angle
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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
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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
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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
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Perimeter and Area12 feet
7 fe
et
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Applying SimilarityWhat is the length of the larger rectangle?
2m 4m
5m ???
4 m
2 m 5 m
=
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Probability and Data Analysis
● Independent vs. dependent events
● Measures of central tendency
● Reading and interpreting various displays of
data
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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
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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?
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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 = =
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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 = =
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Measures of Central Tendency● Mean: average
● Median: middle
● Mode: most frequently occurring
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Measures of Central Tendency78, 72, 75, 79, 72, 73, 76
Mean: 525 ÷ 7 = 75
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Measures of Central Tendency78, 72, 75, 79, 72, 73, 76
72, 72, 73, 75, 76, 78, 79
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Measures of Central Tendency78, 72, 75, 79, 72, 73, 76
72, 72, 73, 75, 76, 78, 79
Median
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Measures of Central Tendency78, 72, 75, 79, 72, 73, 76
72, 72, 73, 75, 76, 78, 79
Mode
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Measures of Central Tendency78, 72, 75, 79, 72, 73, 76
72, 72, 73, 75, 76, 78, 79
Range: 79 - 72 = 7
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Reading and Interpreting Various Displays of Data● Lists
● Tables / Charts
● Graphs○ Circle graphs ( Pie Graphs)○ Bar graphs
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Manipulatives
● Base-ten blocks● Color tiles● Number lines● Number cubes● Counters ● Unifix cubes
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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