Download part 2 of 3 - Auckland Mathematical Association

GENERALLY GENERALISING…
E – Explain the strategy or method used to solve the problem.
G – Give other examples that use the same strategy or method.
G – Generalise – use algebra to show the underlying structure.
Task 1
a) 6 + 6 + 6 + 6 = 4  6
b) 9 + 9 + 9 + 5 + 5 + 5 = 3  9 + 3  5
c) 6 + 4 + 6 + 4 + 6 + 4 = 3  (6 + 4)
d) 9 + 9 + 9 – 5 – 5 – 5 = 3  9 – 3  5
e) 6 – 4 + 6 – 4 + 6 – 4 = 3  (6 – 4)
Task 2
a) 15 + 16 = 15 + 15 +1
= 2  15 + 1
b) 19 + 20 = 20 + 20 – 1
= 2  20 – 1
c) 9 + 10 + 11 = 9 + (9+1) + (9+2)
=39+3
d) 9 + 10 + 11 = (10–1) + 10 + (10+1)
= 3  10
e) 9 + 10 + 11 = (11–2) + (11–1) + 11
= 3  11 – 3
Task 3
12  13 = 12  12 + 12  1
=122 + 12
13  12 = 13  13 – 13  1
=132 – 13
Task 4
7  32 = 7  30 + 7  2
7  39 = 7  40 – 7  1
Task 5
32  42 = 30  40 + 2  40 + 30  2 + 2  2
32  48 = 30  50 + 2  50 + 30  -2 + 2  -2
39  42 = 40  40 + -1  40 + 40  2 + -1  2
39  49 = 40  50 + -1  50 + 40  -1 + -1  -1
Task 6
9  9  9  9  9  9  9 = 97
92 95 = (9  9)  (9  9  9  9  9)
= 97
97
95
=9999999
99999
= 92
(94)3 = 94  94  94
= 912
Our Group:
E – Explain the strategy or method used to solve the problem.
G – Give other examples that use the same strategy or method.
G – Generalise – use algebra to show the underlying structure.
Lightbulb Moments…
Even and Odd Number Proofs
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Show that the sum of two even numbers is even
Show that the sum of an even number and an odd number is odd
Show that the sum of two odd numbers is even
Show that the sum of three odd numbers is odd
Show that an odd number minus an odd number is even
Show that an even number minus an odd number is odd
Show that an even number minus an even number is even
Show that the square of an odd number is odd
Show that the square of an even number is even
Show that the product of an odd number and an even number is even
Show that the product of two even numbers is even
Show that the product of two odd numbers is odd
Consecutive Number Proofs
1. Show that the sum of two consecutive numbers is always odd
2. Show that the sum of two consecutive odd numbers is always even
3. Show that the product of two consecutive even numbers is always a multiple of
4
4. Show that if you add two consecutive odd numbers the result will be a multiple
of 4
5. Show that if you multiply two consecutive odd numbers then add 1, the result
will be a multiple of 4
6. Show that the product of three consecutive numbers is divisible by three
7. Show that the sum of any three consecutive even numbers is divisible by six
8. The product of three consecutive even numbers is a multiple of 8
9. Make a conjecture about the sum of four consecutive numbers and then prove
it
10. Show that if you add the squares of three consecutive numbers and then
subtract 2 you always get a multiple of 3
INFORMATION GIVEN
INFORMATION NEEDED
QUESTION / PROBLEM / CHALLENGE
FURTHER INVESTIGATIONS
PICTURE / MODEL
CONCLUSION(S)
STRATEGY
CALCULATION
SKILLS
Learning to learn in maths
WHAT DO GOOD LEARNERS DO?
 Make connections
 Ask effective questions
 Use problem solving
strategies
 Explain their reasoning
 Apply past knowledge to
new situations
 Show their working
 Think about their thinking
 Check their understanding
 Understand key vocabulary  Learn from their mistakes
 Use conventions correctly
 Generalise
NAME: ____________________________________
My revision plan for this assessment:
Key ideas:
Revision completed:
Self Review:
Questions I am confident about my response to are:
Questions I am a bit unsure about are:
Questions I guessed / could not do are:
How well did your revision fit this assessment?
My learning goal(s) from this assessment are:
Assess how well you use the following ‘learning to learn’ strategies in class:
A – I always use this strategy
M – I mostly use this strategy
S – I sometimes use this strategy
N – I never use this strategy
Check my work
Explain my reasoning
Ask ‘good’ questions
Understand key vocabulary
Show my working
Use algebraic conventions
My learning to learn goals for the remainder of this term are:
Make connections between questions
Apply past knowledge
Think about my thinking
Use problem solving strategies
Learn from my mistakes
Generalise from number
NAME: _____________________________________
1. Factorise 𝑥 2 + 7𝑥 + 10
2. Simplify
𝑥 2 +7𝑥+10
𝑥 2 +2𝑥
3. Show that the square of any prime number cannot be a prime. Give the factors.
4. Expand 2𝑥(3 − 𝑥)
5. Write expressions for the perimeter and area of the following shape:
2𝑥 + 3
𝑥−3
6. Write expressions for the perimeter and area of the following shape:
𝑥
𝑥
3𝑥
𝑥
9𝑥 5
7. Simplify 12𝑥 3
8. Show that the sum of two consecutive numbers is always odd.
9. Factorise 12𝑥 2 𝑦 2 𝑧 − 6𝑥𝑦 2 + 24𝑦 2 𝑧
10. Expand
(7 − 𝑥)(𝑥 + 3)
11. Show that the sum of three consecutive numbers cannot be a prime.
12. Expand
(𝑥 − 4)2
13. Factorise 3𝑥 2 − 5𝑥 − 2
14. Simplify 3 + 5(𝑥 − 8)
𝑥 2 −9
15. Simplify (𝑥+3)
Date
Key Ideas
Examples
Connections
Vocabulary
Learning to learn
1-5
Revision Notes – Give examples for each and list the practice exercises you have completed.
Name: ______________________
Expanding
One bracket
Simplifying
Adding like terms
Factorising
Common factor
Showing
Square of a prime is not
prime
Solving
One step equations
Adding to one bracket
Exponents
Quadratic with 𝑥 2
Sum of two consecutive
numbers is odd
Two step equations
Adding two brackets
Fractions with quadratic
Quadratic with 𝑎𝑥 2
Sum of three
consecutive numbers is
divisible by three
Equations with brackets
Difference of two
squares
Sum of three
consecutive numbers is
not prime
Equations with x on both
sides
Multiplying two brackets
NAME: _____________________________________
Please attach your revision plan to this assessment when you hand it in.
You need to tick the appropriate box on the right for each question.
1. Factorise 𝑥 2 − 7𝑥 + 12
2. Solve 7𝑥 − 5 = 9
3. Simplify
𝑥 2 −7𝑥+12
𝑥 2 −2𝑥−3
4. Expand −𝑥(11 − 2𝑥)
5. Write an expression for the perimeter of the following
shape:
3 -2
6. Write an expression for the area of the above shape:
9𝑥 3
7. Simplify 18𝑥 7
Confident
Not sure
Don’t
know /
Guessed
8. Show that the sum of two consecutive numbers is not
even.
9. Factorise 8𝑥 2 𝑦 3 𝑧 − 12𝑥𝑦 2 𝑧 3 + 36𝑥 2 𝑦𝑧 3
10. Expand
(7 − 𝑥)(𝑥 + 3)
11. Solve 7 − 3𝑥 = 14
12. Expand
(7 − 𝑥)2
13. Factorise 5𝑥 2 + 16𝑥 + 3
𝑥 2 −5𝑥
14. Simplify 𝑥 2 −25
15. Simplify 7 − 3(𝑥 − 2)
16. Solve 7 − 3(𝑥 − 2) = 10
Lesson plan 2 – Quadratic Modelling
Number and Algebra Level 5

Relate tables, graphs, and equations to
linear and simple quadratic relationships
found in number and spatial patterns.
Introducing
Exploring
Find, use, and justify a model (Mo)
Negotiate meaning (NM)
Generalise ideas (GI)
Use generalisations (WS)
Use words and symbols to describe patterns and generalisations (WS)
Use appropriate vocabulary to explain ideas (V)
Compare and contrast ideas (CC)
Critically reflect (CR)
Use technology appropriately (IT)
Number and Algebra Level 6
 Relate graphs, tables, and equations to linear, quadratic and simple
exponential relationships found in number and spatial patterns
To explore the relationship between the table and the equation for quadratic
relationships
Task
Extending
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 Water slide
 What does this quote mean?
“solving a problem simply means
representing it so as to make the
solution transparent.”
 Motorbike example
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Working in teams of 4 to explore
different situations of Task One
How are these ideas the same /
different to linear concepts…
Families of functions
Success
Criteria
Aim
Curriculum
Objectives
Generalising and representing patterns
and relationships.
Key Competencies
Essence Statement
This lesson explores the relationships between different representations of quadratic relationships.
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I understand the key features of quadratic relationships
I can start from any representation (graph, table, equation, context) and connect it to the others
Pedagogy
Making connections to prior learning
and experience:
Focus on why as well as how
Mathematical modelling
Accept all answers, lead academic
discussion
Learning
What are the key features, how are
these shown in each representation?
Facilitating shared learning:
Observation of groups to explore
current understandings and look for
opportunities to develop further
Connections between representations –
how are these the same / different to
linear relationship?
Encouraging reflective thought and
action /
Enhancing relevance of new learning:
Supporting students to make links,
sharing of light bulb moments
Mathematics Level 7
 Form and use linear, quadratic and simple
trigonometric equations.
 Sketch the graphs of functions and their
gradient functions and describe the
relationship between these graphs.
Defining x and y – use of axes to
‘mathematise’
Learning to learn
Use of examples and linking strategies
Use of mathematical language
Identifying key features
Making connections across
representations
Use of geogebra for visualising the
three representations
Looking for underlying generalisations
What effect does each of the
parameters a, b and c have?
Link to key ideas of connection
Discussion of representations, what
does each hide / reveal?
Assessment
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Students can work independently to sketch graphs from a given completed square equation
Students can work from graph to equation
Students can work from a scenario to other representations
Link to AS external through exam questions
REAL WORLD PARABOLAS