Transcript Whiteboard Backgrounds for Maths Teachers

Whiteboard Templates for Maths
HW
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Number…
Algebra
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Functions…
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3
Draft Version 17.2 Created :25-1-2017
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Large Grid
Small Grid
Split Screen
Divided Board Plan
Whiteboard
6 Frames
4 Frames
Key Words
How to… Steps
Graphing Functions
Prior Knowledge
Given Couples
Substitution
Trigonometric
Functions &
Calculus
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1
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6
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4
6 Framed Sections
4 Framed Sections
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Topic:
Key Words
Medium Grid with Steps
Steps
How to…
Medium Grid with Prior Knowledge
Prior Knowledge
Graphing a Function using a given Table
Function:
3
Domain:
Evaluating Outputs and Graphing a Function
Function:
3
Domain:
Trigonometric Functions
Sin(A)
Cos(A)
Graphing a Sine Function
Function:
Domain:
sin(x)
1
Graphing a Cosine Function
Function:
1
Domain:
Algebra
Solving Equations
Using Arrays
Linear Patterns
Algebra Tiles
Quadratic Pattern
Quadratic Roots
Binomial Theorem
Algebra Tiles
1
x
x
x2
-x
-x2
-1
-x
Roots of a Quadratic Equation
Roots of the quadratic equation
ax2 + bx + c = 0
p.20
The Binomial Theorem
p.20
Solving Equations & Inequalities
Equations
Inequality
Double inequality
Solving Equations
2
Solving Inequalities
1
Solving Double Inequalities
1
Growing Visual Linear Patterns
Supporting Student Workbook on Linear Patterns Available Online
Click on the Pattern
www.projectmaths.ie
How many dots are there?
How many dots are there?
How many dots are there?
How many dots are there?
How many dots are there?
IMAGINE A SQUARE
WITH 10 DOTS AT
EACH SIDE
Pattern 1
1st Difference
3
Pattern 2
1st Difference
3
Pattern 3
1st Difference
3
Pattern 4
1st Difference
3
Pattern 5
1st Difference
3
Pattern 6
1st Difference
3
Pattern 7
1st Difference
3
Pattern 8
1st Difference
3
Pattern 9
1st Difference
3
Pattern 10
John wants to save up for a school tour which will happen at the end of
the school year. He speaks to his parents about it and they agree that
he can get a part-time job to help him afford the trip. His parents give
him €100 to start him off and each week he saves €20 of his wages.
1st Difference
3
Using Arrays
Quadratic Array
Two Quadratic Arrays
Quadratic Factors
Quadratic Factors
Trial & Error
Cubic Array
Two Cubic Arrays
Quadratic Array for Multiplication and Division
Question:
Answer:
Quadratic Arrays for Multiplication and Division
Quadratic Factorisation using the Guide Method
Question:
Guide Number
Product
Factors
Sum
Answer:
Quadratic Factorisation using the Guide Method
Question:
Guide Number
Product
Factors
Sum
Answer:
Cubic Array for Multiplication and Division
Question:
Answer:
Cubic Arrays for Multiplication and Division
Growing Visual Quadratic Patterns
Pattern 5
Pattern 10
Pattern 1
Pattern 6
Pattern 11
Pattern 2
Pattern 7
Pattern 12
Pattern 3
Pattern 8
Pattern 13
Pattern 4
Pattern 9
Pattern 14
Supporting the Student Workbook
on
Quadratic Patterns
available on our website
Click on the pattern to go to that page.
Quadtatic Pattern
Stage No. (n)
4
No. of
squares (s)
1st
Difference
2nd
Difference
Next, Near, Far, Any?
Pattern 1
1st Difference
3
2nd Difference
Pattern 2
1st Difference
3
2nd Difference
Pattern 3
1st Difference
3
2nd Difference
Pattern 4
1st Difference
3
2nd Difference
Pattern 5
1st Difference
3
2nd Difference
Pattern 6
1st Difference
3
2nd Difference
Pattern 7
1st Difference
3
2nd Difference
Pattern 8
1st Difference
3
2nd Difference
Pattern 9
1st Difference
3
2nd Difference
Pattern 10
1st Difference
3
2nd Difference
Pattern 11
1st Difference
3
2nd Difference
Pattern 12
1st Difference
3
2nd Difference
Pattern 13
1st Difference
3
2nd Difference
Pattern 14
A tiling company specialises in multi-colour tile
patterns. A small guest house is interested in the
pattern below for its square shaped reception area.
How many tile will be needed in total if the reception
area needs 36 green tiles in the centre pattern?
1st Difference
3
2nd Difference
Statistical Charts
Line Plot
Bar Chart
Stemplot
Histogram
Back-to-Back
Stemplot
Pie Chart
Scatter Graph
Line Plot
Title:
1
Bar Chart
Title:
1
Histogram
Title:
1
Stemplot
Title:
Key:
1
Back to Back Stemplot
Title:
Key:
1
Key:
Scatter Graph
Title:
1
Pie Chart
Title:
1
Number
Number Lines
Fractions
Order of
Operations
Venn Diagrams
Number
Systems
Complex No.
Indices
Logarithms
Sequences…
Financial Maths
Venn Diagrams
1 Set
2 Sets
3 Sets
Venn Diagram with 1 Set
1
Venn Diagram with 2 Sets
1
Venn Diagram with 3 Sets
1
Number Line
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
3
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9 10
9 10 11 12
Using the Calculator
Complex Numbers
Rectangular Form
Polar Form
Argand Diagrams
Number Systems
De Moivre
Complex Numbers: Argand Diagram
Im
Re
Argand Diagram: Complex Numbers in Polar Form
Im
Re
Complex Numbers: De Moivre’s Theorem
p.20
Order of Operations
BEMDAS
BIMDAS
BIRDMAS
Order of Operations - BEMDAS
B
E
MD
left to right
AS
Order of Operations - BIDMAS
B
I
MD
left to right
AS
Order of Operations - BIRDMAS
B
IR
DM
left to right
AS
Probability
Empirical Rule
Probability Tables
Bernoulli Trials
Inferential Statistics
Ordinary Level
Higher Level
The Empirical Rule for Normal Distribution
1
Normal Distribution Tables
standardising formula
p.36-37
p.34
Bernoulli Trials
p.33
Inferential Statistics
LC Ordinary Level – Sample Proportions
Margin of Error
HL
Confidence
Interval
Hypothesis Test
Margin of Error of the Proportion (OL)
Ordinary Level
Approximation for the
95% level of confidence margin of error:
n
Sample size
This formula is not in Formulae and Tables. It
is an approximation relating to the product of
the 95% level of confidence z-value (p.37) and
the standard error of the proportion (p.34)
95% Confidence Interval of the Proportion (OL)
95% Confidence Interval:
p
Population proportion
Sample proportion
n
Sample size
This formula is not in Formulae and Tables.
Hypothesis Test on a Population Proportion (OL)
1. State the Hypotheses
2. Analyse sample data
3. Interpret the results
Step 1 – State the Hypotheses
State the null hypothesis (H0) and
the alternative hypothesis (H1).
p
Population proportion
p0
Hypothesised value of the
population proportion
H0 is a statement of no change. We
express it as a statement of equality.
Note: Stating the alternative hypothesis as “not
being equal” means that it could be either “less
than” or “greater than” a value. This results in a
two-tailed hypothesis test.
Hypothesis Test on a Population Proportion (OL)
1. State the Hypothesis
2. Analyse sample data
3. Interpret the results
Step 2 – Analyse the sample data
using a confidence interval
Construct the 95% confidence
interval
State if p0 is inside or outside the
confidence interval
Step 3 - Interpret the results
If p0 is outside the interval it would
indicate that p ≠ p0 so the result is
significant and we reject H0.
If p0 is inside the interval we fail to
reject H0.
Clearly state your conclusion.
Inferential Statistics
LC Higher Level
Proportions
Margin of Error
Sample Size?
Confidence
Interval
Hypothesis Tests
Means
CI for Population
Mean
OL
CI for Sample
Mean
Hypothesis Tests
95% Margin of Error for a Population Proportion (HL)
Higher Level
95% level of confidence margin of error:
p
Population
proportion
Sample
proportion
n
Sample size
Standard error of
the proportion
(p.34)
This formula is not given in Formulae and Tables.
1.96 is the z-value relating to a 95% level of
confidence (p.37).
Sample Size for a required Margin of Error (HL)
Use the approximation for the
95% level of confidence margin of error
for a Population Proportion:
n
Sample size
Margin of error for 95%
L.o.C
This formula is not in Formulae and Tables. It
is an approximation relating to the product of
the 95% level of confidence z-value (p.37) and
the standard error of the proportion (p.34)
95% Confidence Interval for a Population Proportion - HL
95% Confidence Interval:
This formula is not in Formulae and Tables. It is a
confidence interval based on the standard
deviation of σ “p hat.”
p
Population
proportion
Sample
proportion
n
Sample size
Standard error of
the proportion
(p.34)
Hypothesis Test on a Population Proportion - HL
1. State the Hypotheses
2. Analyse sample data
3. Interpret the results
Step 1 – State the Hypotheses
State the null hypothesis (H0) and
the alternative hypothesis (H1).
p
Population proportion
p0
Hypothesised value of the
population proportion
H0 is a statement of no change. We
express it as a statement of equality.
Note: Stating the alternative hypothesis as “not
being equal” means that it could be either “less
than” or “greater than” a value. This results in a
two-tailed hypothesis test.
Hypothesis Test on a Population Proportion - HL
1. State the Hypothesis
2. Analyse sample data
3. Interpret the results
Step 2
Analyse the sample data using either a:
• Confidence interval
• Z-value
• P-value
Hypothesis Test on a Population Proportion - HL
1. State the Hypothesis
2. Analyse sample data
3. Interpret the results
Step 2 – Analyse the sample data
using a confidence interval
Construct the required confidence interval for
a proportion
p.34
State if p0 is inside or outside the confidence
interval
Step 3 - Interpret the results
If p0 is outside the interval it would indicate
that p ≠ p0 so the result is significant and we
reject H0. If p0 is inside the interval we fail to
reject H0.
Clearly state your conclusion.
Hypothesis Test on a Population Proportion - HL
1. State the Hypothesis
2. Analyse sample data
3. Interpret the results
Step 2 – Analyse the sample data
using a z-value
We can standardise our p0 value using:
This formula is not in Formulae and Tables.
Step
3 - Interpret
the resultsof the
It is derived
from an understanding
standardising
andthat
the p ≠ p0 at
If
|z|>1.96 itformula
would (p.34)
indicate
standard
error
the proportion
(p.34).
a 5% level
of of
significance
so we
reject H0.
If |z|<1.96 we fail to reject H0.
Clearly state your conclusion.
1
Hypothesis Test on a Population Proportion - HL
1. State the Hypothesis
2. Analyse sample data
3. Interpret the results
Step 2 – Analyse the sample data using
a p-value
Samples are normally distributed around p so
we can standardise p0 using:
Find P(Z ≤ |zp|)
(p.36-37)
Our p-value = 2(1 - P(Z ≤ |zp|)
This formula is not in Formulae and Tables. It
is derived from an understanding of the
standardising formula (p.34) and the
Step
3 - error
Interpret
the results
standard
of the proportion
(p.34).
If p-value < 0.05 it would indicate that p ≠ p0
at a 5% level of significance so we reject H0.
If p-value > 0.05 we fail to reject H0.
Clearly state your conclusion
4
95% Confidence Interval for a Population Mean - HL
95% Confidence Interval:
µ
Population mean
Sample mean
n
Sample size
σ
Standard deviation
If σ is unknown use s
This formula is not given in Formulae and Tables.
1.96 is the z-value relating to a 95% level of
confidence (p.37). The margin of error is the product
of the z-value multiplier and the standard error of
the mean (p.34):
2
95% Confidence Interval for a Sample Mean - HL
95% Confidence Interval:
µ
Population mean
Sample mean
n
Sample size
σ
Standard deviation
If σ is unknown use s
This formula is not given in Formulae and Tables.
1.96 is the z-value relating to a 95% level of
confidence (p.37). The margin of error is the product
of the z-value multiplier and the standard error of
the mean (p.34):
2
Hypothesis Test for a Sample Mean - HL
1. State the Hypothesis
2. Analyse sample data
3. Interpret the results
Step 1 – State the Hypotheses
State the null hypothesis (H0) and
the alternative hypothesis (H1).
H0:µ = µ0
H1: µ ≠ µ0
µ
Population mean
µ0
Hypothesised value of the
population mean
H0 is a statement of no change. We
express it as a statement of equality.
1
Note: Stating the alternative hypothesis as “not
being equal” means that it could be either “less
than” or “greater than” a value. This results in a
two-tailed hypothesis test.
Hypothesis Test for a Mean - HL
1. State the Hypothesis
2. Analyse sample data
3. Interpret the results
Step 2
Analyse the sample data using either a:
• Z-value
• P-value
Hypothesis Test for a Mean HL
1. State the Hypothesis
2. Analyse sample data
3. Interpret the results
Step 2 – Analyse the sample data
using a z-value
Use the one-sample z-test (p.35):
If σ is unknown
use s
Step 3 - Interpret the results
If |z|>1.96 it would indicate that µ ≠ µ0 at
a 5% level of significance so we reject H0.
If |z|<1.96 we fail to reject H0.
Clearly state your conclusion.
1
Hypothesis Test for a Mean - HL
1. State the Hypothesis
2. Analyse sample data
3. Interpret the results
Step 2 – Analyse the sample data using
a p-value
Apply the one-sample z-test (p.35) :
If σ is unknown
use s
Find P(Z ≤ |z|)
(p.36-37)
Our p-value = 2(1 - P(Z ≤ |z|)
Step 3 - Interpret the results
If p-value < 0.05 it would indicate that p
≠ p0 at a 5% level of significance so we
reject H0.
If p-value > 0.05 we fail to reject H0.
2
Clearly state your conclusion.
Financial Maths
Compound Interest
Depreciation
Sum of Present
Values (t1 = 0)
Sum of Present
Values (t1 ≠ 0)
Sum of Future
Values
Amortisation
Financial Maths: Compound Interest
Compound interest
F = final value, P = principal
p.30
Present
1
Future
Present value
P = present value, F = final value
p.30
Financial Maths: Depreciation
Depreciation
- Reducing balance method
F = later value, P = initial value
p.30
Present
1
Future
Series of equal withdrawals from fixed amount (Sum of Present Values)
Present value
P = present value, F = final value
p.30
Immediate first withdrawal
Present
4
Future
where:
a is the first term
r is the common ratio
p.22
Series of equal withdrawals from fixed amount (Sum of Present Values)
Present value
P = present value, F = final value
p.30
Delayed first withdrawal
Present
4
Future
where:
a is the first term
r is the common ratio
p.22
Series of equal installments (Sum of Future Values)
Compound interest
F = final value, P = principal
p.30
Present
4
Future
where:
a is the first term
r is the common ratio
p.22
Financial Maths: Amortisation
Amortisation – mortgages and loans
(equal repayments at equal intervals)
A = annual repayment amount
P = principal
A=P
i(1+1)t
(1+ i)t -1
p.31
Functions
Mapping Functions
Table and Mapping
Table (and Graph)
Mapping Diagrams
Graph given Table
Comparing Functions
Composite Functions
Types of Functions
Calculus
Differentiation
Integration
Function Machine with Mapping Diagram
Function:
Function machine
1
Mapping diagram
Function Machine with Table and Mapping Diagram
Function:
Function machine
Table
Mapping
Diagram
1
Functions: Table and Graph
Function:
Table
4
Domain:
Function machine
Mapping Diagrams
3
Investigating Functions with a Mapping Diagram and Graph
Function:
4
Domain:
Graphing a Function from a Table
Function:
4
Domain:
Comparing Functions - Simultaneous Functions
Function:
4
Function:
Domain:
Composite Functions
Function:
Mapping
Diagram
Table
2
Function:
Function machines
Differential Calculus
p.25
Integral Calculus
p.26
Fraction Wall – Comparing Fractions
5
Length, Area and Volume
Parallelogram
Trapezium
Circle/Disc
Arc/Sector
Triangle
Cylinder
Cone
Sphere
Frustum of Cone
Prism
Pyramid
Trapezoidal Rule
Length & Area: Parallelogram
p.8
Length and Area of a Trapezium
p.8
Trapezium
Length of a Circle and Area of a Disc
p.8
Length of an Arc and Area of a Sector
p.9
Length and Area of a Triangle
p.9
Surface Area and Volume of a Cylinder
p.10
Surface Area and Volume of a Cone
p.11
Surface Area and Volume of a Sphere
p.9
Surface Area and Volume of a Frustum of Cone
p.12
Surface Area and Volume of a Prism
p.12
Surface Area and Volume of a Pyramid
p.12
Area Approximations: Trapezoidal Rule
p.13
Trigonometry
Pythagoras
Trig Ratios
Sine Rule
Cosine Rule
Area of a Triangle
Identities…
Unit Circle
Angle Ratios
Compound Angle
Double Angle
Product & Sums
Trig. Functions
Pythagoras’ Theorem
p.16
Trigonometric Ratios
p.16
The Sine Rule
p.16
sine rule
The Cosine Rule
p.16
cosine rule
Area of a Triangle
p.16
area
Compound Angle Formulae
p.14
Double Angle Formulae
p.14
Trigonometric Products and Sums/Differences
p.14
Trigonometry: Identities & Definitions
p.13
sec2 A = 1 + tan2 A
Trigonometry: The Unit Circle
p.13
Trigonometric Ratios of certain angles
p.13
Geometry
Synthetic Geometry
Co-ord. Geometry
Synthetic Geometry
Theorems
Constructions
Enlargements
Axial Symmetry
Central Symmetry
Rotations
Geometry Theorems
Theorem ____:
1. Diagram:
4. To Prove:
5. Proof:
2. Given:
3. Construction:
Geometry Construction
Construct:
Geometry Construction
Construct: The perpendicular bisector of [AB]
A
B
Geometry: Enlargements
Centre of
Enlargement
Centre of
Enlargement
3
Geometry Transformations: Axial Symmetry
Axis of
symmetry
Axis of
symmetry
3
Axis of symmetry
Geometry Transformations: Central Symmetry
Centre of Symmetry
Point of Reflection
1
Geometry Transformations: Rotations
Point of Rotation
3
Co-ordinate Geometry of the Line
Line Formulae
Area of Triangle
Distance to line,
Angle & Division
Co-ordinate Geometry of the Circle
Centre (h, k)
Centre (-g, -f)
Co-ordinate Geometry of the Line
Line
1
p.18
Co-ordinate Geometry: Area of a Triangle
p.18
1
Co-ordinate Geometry of the Line (LCHL)
p.19
1
Co-ordinate Geometry: Circle
p.19
1
Co-ordinate Geometry: Circle
p.19
1
Number Systems Venn Diagram
p.23
3
Indices
p.21
Logarithms
p.21
Sequences and Series
Arithmetic
Geometric
Arithmetic Sequence and Series
p.22
In the following Tn is the nth term,
and Sn is the sum of the first n terms
where a is the first term and d is
the common difference.
Geometric Sequence and Series
p.22
In the following Tn is the nth term,
and Sn is the sum of the first n terms
where a is the first term and r is
the common ratio.
Homework Assignment and Notes
Date:
Maths Homework:
Due:
Notes: