Subject: Mathematics and Statistics
Level: Year 10
Title: Probability
Author: Louise Addison
Email address: l.addison@auckland.ac.nz
All Curriculum Support Days resources reflect the work of subject specialists during a two-day forum. You should view
them as ‘work-in- progress’, not as finished units to download and use. They demonstrate a range of ways of thinking
about how you might build the ‘front end’ of the NZ Curriculum (the Vision, Principles, Values, Key Competencies,
Effective Pedagogies and Learning Area Statements) into your existing units of work, by re-focusing how you teach rather
than changing what you teach. The questions and comments recorded in the body of each resource are at least as
important as the unit itself. If for some reason your software does not display such questions and comments, it is likely
that you need to make some technical adjustment to how you are viewing the resource.
1. Why this unit is worth reworking
Key features of this unit that I would like to keep:
Use of data / simulations to challenge initial thinking about uncertain situations.
Students making conjectures and using data to justify / refute their claims.
Multi-levelling across AOs to enable students at a variety of levels to access the work.
Emphasis on making connections between representations.
Use of ICT to simulate situations.
Highlighting of key thinking skills that help develop students’ ability to reason in uncertain situations.
The focus.
2. Re-thinking the unit
I have explored the unit via the following 5 areas of the new curriculum:
A: Learning Area statement (NZC p. 26)
How is the learning essence evident / not evident in the structure of this unit plan? E.g. key ideas of exploring patterns and
relationships; modelling; making connections.
B: Key Competencies (NZC p. 12)
How can the structure be changed to show where specific aspects of the key competencies can be learnt?
C: Effective Pedagogy (NZC p. 34)
How does the structure highlight teacher actions that promote student learning?
D: Vision / Values / Principles (NZC pp. 8-10)
How does this unit plan reflect / not reflect school decisions about the vision, principles and values?
E: Achievement Objectives
How are the AOs expressed? E.g. as curriculum statements, with second tier information, as knowledge / strategy objectives.
Does the unit allow for
- multiple levels?
- multiple strands?
The current unit
TOPIC: YEAR 10 PROBABILITY
Achievement Objectives
Level 4
S4/8: Estimate the relative frequencies of
events and mark them on a scale (p 182)
S4/9: Find all possible outcomes for a
sequence of events, using tree diagrams
(p 182)
Level 5
S5/6: Use data displays and measures to
compare data associated with different
categories (p 188)
S5/9: Determine probabilities of events
based on observations of long-run relative
frequency (p 188)
S5/10: Determine the theoretical
probabilities of the outcomes of an event
such as the rolling of a die or drawing a
card from a deck (p 188)
S5/11: Predict the outcome of a simple
probability experiment, test it, and explain
the results (p 188)
S5/12: Find the probability of a given
sequence of events, using tree diagrams
(p 188)
Level 6
S6/8: Use tables of multi-variate data
from social contexts to find the
probabilities of everyday events or the
proportion of outcomes in a given
category (p 192)
S6/9: Determine the theoretical
probabilities of the outcomes of both
exclusive and independent events such as
the rolling of a die followed by the drawing
of a card from a deck (p 192)
S6/10: Use probability trees to calculate
conditional probabilities (p 192)
Mathematical Processes
Problem Solving
PS1: Pose questions for mathematical
exploration (p 24)
PS3: Devise and use problem-solving
strategies to explore situations
mathematically (p 24)
PS6: Use equipment appropriately when
exploring mathematical ideas (p 24)
Developing Logic and Reasoning
LR1: Classify objects, numbers and
ideas (p 26)
Communicating Mathematical Ideas
C1: Use their own language, and
mathematical language and diagrams, to
explain mathematical ideas(p 28)
LR2: Interpret information and results in
C3: Record information in ways that are
context (p 26)
helpful for drawing conclusions and
making generalisations (p 28)
LR3: Make conjectures in a mathematical C4: Report the results of mathematical
context (p 26)
explorations concisely and coherently (p
28)
LR5: Prove or refute mathematical
conjectures (p 26)
3. The revised unit
TOPIC: YEAR 10 PROBABILITY
Learning Area Statement
Statistics involves identifying problems that can be explored by the use of appropriate data, designing investigations, collecting
data, exploring and using patterns and relationships in data, solving problems, and communicating findings. Statistics also
involves interpreting statistical information, evaluating data-based arguments, and dealing with uncertainty and variation (NZC
p.26).
Focus Achievement Objectives
Level 4
S4-3 Investigate situations that involve
elements of chance by comparing
experimental distributions with
expectations from models of the possible
outcomes, acknowledging variation and
independence.
S4-4 Use simple fractions and
percentages to describe probabilities.
Related Achievement Objectives
Level 4
S4-2 Evaluate statements made by
others about the findings of statistical
investigations and probability activities.
S4-1 Plan and conduct investigations
using the statistical enquiry cycle:
A determining appropriate variables and
Level 5
S5-3 Compare and describe the
variation between theoretical and
experimental distributions in situations
that involve elements of chance.
S5-4 Calculate probabilities, using
fractions, percentages, and ratios.
Level 5
S5-2 Evaluate statistical investigations
or probability activities undertaken by
others, including data collection methods,
choice of measures, and validity of
findings.
S5-1 Plan and conduct surveys and
experiments using the statistical enquiry
cycle:
Level 6
S6-3 Investigate situations that involve
elements of chance:
- comparing discrete theoretical
distributions and experimental
distributions, appreciating the role of
sample size
S6-3 Investigate situations that involve
elements of chance:
- calculating probabilities in discrete
situations.
Level 6
S6-2 Evaluate statistical reports in the
media by relating the displays, statistics,
processes, and probabilities used to the
claims made.
S6-1 Plan and conduct investigations
using the statistical enquiry cycle:
A justifying the variables and measures
data collection methods
B gathering, sorting, and displaying
multivariate category, measurement, and
time-series data to detect patterns,
variations, relationships, and trends
C comparing distributions visually
D communicating findings, using
appropriate displays.
A determining appropriate variables and
measures
B considering sources of variation
C gathering and cleaning data
D using multiple displays, and recategorising data to find patterns,
variations, relationships, and trends in
multivariate data sets
E comparing sample distributions visually,
using measures of centre, spread, and
proportion
F presenting a report of findings.
used
B managing sources of variation, including
through the use of random sampling
C identifying and communicating features in
context (trends, relationships between
variables, and differences within and
between distributions), using multiple
displays
D making informal inferences about
populations from sample data
E justifying findings, using displays and
measures.
S4-4
S5-2
Prob as a
measure of
how
likelihood of
an outcome
S5-2
S6-3
Equally
likely
Representa
tive
Availability
Proportions
vs counts
S5-3
S5-2
Exp prob
Focus on
distributions idea
Simulate
real
random
situation
Linking
number
ideas with
use in
probability
context
Evaluate
outcomes
Investigate
situations
with
elements of
chance
Describing
distribution
s
Calculate
prob
Explain
results
T: Form
conjectures
T: Classify
objects,
numbers,
and ideas.
M:
Negotiate
meaning
U: Report
findings
T:
Generalise
ideas
T: Find,
use and
justify a
model
What is
probability?
Exploring
probs in the
media –
ranking
likelihoods
Use of FDP
Summary
of findings /
prior
knowledge
Find own
article
Probability
misconcepti
on cards
Sort into
true or false
– explain
and sort by
type of
reasoning
Sharing
ideas
(maybe a
class
histogram)
Take a
claim and
think of at
least two
ways to
prove or
refute
Exp prob
Long-run
frequency
Comparing
languages
worksheet
I notice / I
wonder
Use
encoder
online
Rocket
Launch
simulation
Worksheet
Spreadshe
et
Alternative
simulation
scenarios
Adjust to fit
3+ parts
Ex 30.1
Ex 30.2
Chance
website
TIME
OTHER
ICT
TEXT
EXTN
RESOURCES
PLENARY
Learning Inquiry
CORE
Teaching Inquirry
INTRO
Focusing Inquiry
KEY
COMPETENCIES
TEACHING IDEAS
STRATEGIES
KNOWLEDGE
CURRICULUM
OBJECTIVES
Newspaper
articles
2
Probability
misconcepti
on cards
2
Online
code
decoders
Simon
Singh
Comparing
languages
Rocket
Launch
simulation
Worksheet
4
Simulation
TTRC
PS3
Simulating
situations
Variation
Sample
Size
Distribution
S5-3
S5-4
S5-2
S5-3
S5-4
U: Interpret
information
and results
in context
Probability
experiment
Investigate
probability
games
Exp vs The
(as
expected
prob)
Using
sample
spaces
Expected
Value
Using
sample
spaces
Key
features of
representati
on
Why x
across?
Why +
between?
Theoretical
prob and
conditional
prob
Finding
proportions
to meet set
criteria
Analysing
different
ways of
solving
problems
Variation
Independence
R:
Compare
and
contrast
ideas
M:
Critically
reflect
T: Plan
and carry
out an
investigation
Intro to
other
situations
Use of
TTRC
Hospital
Problem –
Make
prediction
Games of
chance vs
games of
skill
Simulation
cards –
select as
appropriate
for students
Re sort
misconception cards
Explore
simulation
Share
conclusions
about
sample size
Gut
reaction /
randomness
Play and
analyse
game
Design own
game of
chance
Equally
likely
outcomes
Worksheet
Textbook
Compare
and
Contrast
Exp and
The Prob
Invn p 440
Ex 30.3
Expected
value
Student
explain
formula
Textbook
Write own
12
questions
Assumptions of
expected
value?
Ex 30.4
More
complex
tree
situations
Ex 30.5
Tree
Diagrams
Textbook
Two Way
tables
Cond Prob
False + / -
Combining
all three
elements of
probability
Worksheet
Key
features of
tree
diagrams.
When?
Key
features of
two way
tables
Summary
of topic
Use of
Excel /
Fathom as
appropriate
Design own
simulation
card
Simulation
Cards
Hospital
Problem
Specificity
and
Sensitivity
Own
Murphy’s
law invn
Homer
Simpson
PowerPoint
Monopoly
probability
Arwen wins
Spin the
Wheel
6
Two way
tables ppt
Two way
tables
worksheet
nzmaths
Rock Paper
Scissors
Murphy’s
Law
4
Heuristics
Refute
conjecture
T: Prove or
refute
P: Cont
own ideas
to
discussion
Relook at
misconcepti
on cards
Presentation of
proofs
Summary
of ideas
Investigate
other
misconceptions
4. Reflecting on the process
I found the questions we formulated in Section 2 to be a useful way of looking at current unit planning and looking at ways to adapt
it to better incorporate the new curriculum. I certainly don’t think it is necessary for all of the areas suggested to be explicit across
all areas of a unit plan – I just saw them as ways of getting me thinking about my current structure / planning and possible
alterations that could be made to better reflect the new curriculum.
I think it is worth looking at the structure rather than the detail initially, as this will hopefully transfer across all unit planning in the
department and provide a focus for where the department is heading. Completing the analysis as a department would be a good
way of sharing examples of current good practice and deciding where there are gaps in either teacher or student
knowledge/understanding. These could then be the focus of future unit plans (for students) or department meetings (for teachers).
The revised structure is highly dependent on my initial starting point. For some departments it may be better to start from scratch
and then include elements from their current planning. I would also expect that this structure would continue to change as it was
taught and/or other units were written. I have kept the time column – this is part of a guideline for teachers (this is a four week unit).
They certainly will not be able to do all of the items in the lesson plan during this time but will pick and mix depending on their class.
I see this plan as a way of sharing lots of the great ideas that other teachers may be using whilst teaching this topic.
Learning Area Statement – I decided to include this as part of my unit plan, and highlight key areas that I saw as particularly
relevant and worth emphasising throughout the plan. I thought this idea could be extended by using the “What is maths and stats
about?” and “Why study maths and stats?” sections of the statement in a similar way at the year plan level, where certain aspects
were identified for highlighting across certain units across that Year level/course.
Key Competencies – I have kept the meanings of the abbreviations in this document as a way of helping me to remember them!
Maybe they could be abbreviated in the future – we as a department had already done quite a bit of work around thinking skills so
this aspect was good to include.
Effective Pedagogy – I like the idea of using the columns we already had and linking them to the teaching as inquiry cycle. Maybe
as a department a certain aspect of this could be decided as part of PD while this unit was being taught? This may be particularly
useful for topics that have been identified as weaknesses for students through e-AsTTle or the like.
Achievement Objectives – I have summarised these at the beginning of the unit. Again they could be written in full during the plan
to help me get used to them (I used to do this in previous plans). Further work is required to break down the key Knowledge and
Strategies as related to the new objectives. I think I have a starting point that as I learn more about teaching of ideas such as
distribution/variation, it can be easily adapted – and will also work across a range of topics and levels.
NB: If you would like copies of the resources from this unit I will make them available on aucksecmaths.wikispaces.com. Please
note none of these activities have at present been updated for the new curriculum, though many of them readily could be – please
feel free to adapt and alter as you wish.
Solve problems in
unfamiliar situations
(PS)
Pose questions for
exploration (Q)
Plan and carry out an
investigation (PI)
Choose and use
appropriate
representation (Re)
Interpret information
and
results in context (In)
Use words and symbols
to describe patterns and
generalisations (WS)
Actively listen (AL)
Adapt to different roles
(AR)
Participate actively in a
collaborative team or
community (CT)
Understand thinking of
others (UT)
Work independently
(WI)
Empower and enable
others in a team (EE)
Negotiate meaning
(NM)
Set achievable goals
(SG)
Show awareness of the
needs of others (AN)
Compare and contrast
ideas (CC)
Accept challenges and
take informed risks (AC)
Contribute own ideas to
discussions (CD)
Find, use, and justify a
model (Mo)
Use spatial visualisation
(SV)
Make connections (MC)
Use appropriate
vocabulary to explain
ideas (V)
Demonstrate resilience
and perseverance (RP)
Classify objects,
numbers, and ideas (Cl)
Devise and follow a set
of instructions (DF)
Manage time effectively
(MT)
Form conjectures (FC)
Record information in
systematic, concise and
coherent ways (Ri)
Critically reflect (CR)
Generalise ideas (GI)
Report findings (Rf)
Seek assistance and
guidance (SA)
Prove or refute (Pr)
Use technology
appropriately (IT)
Locate resources (LR)
Key Competency breakdown courtesy of NCEA Alignment Group
LR1
LR2
LR5
Probability
as
measure
of how
likely it is
that an
event will
occur
Represent
ative /
availability
heuristics
Linking
number
ideas with
use in
probability
context
Exploring
math
conjectures
Making
conjecWhat is
tures /
probability
classifying ?
ideas
Exploring
probs in
the media
– ranking
likelihoods
Use of
fractions /
%/
decimals /
ratios
Find own
article
Justifying
Probability
claims /
misconcep
classifying
tion cards
ideas
Sort into
true or
false –
explain
and sort
by type of
reasoning
Sharing
ideas
(maybe a
class
histogram)
Take a
claim and
think of at
least two
ways to
prove or
refute
Ex 30.1
Chance
website
TIME
OTHER
ICT
TEXT
EXTN
Big Ideas
Generalisations
Connections
RESOURCES
PLENARY
Learning
Experinences
CORE
Questions
Prompts
Current thinking
INTRO
THINKING
TEACHING IDEAS
STRATEGIES
KNOWLEDGE
CURRICULUM
OBJECTIVES
Newspape
r articles
2
Probability
misconcep
tion cards
2
Exp prob
Long-run
frequency
Making
predicttions using
long run
frequency
Use it
Simulate
real
random
situations
Simulation
techniques
S5/9
PS1 Simulation
PS6
PS3
LR3
Simulating
techniques
PS6
S5/1
0
S5/1
2
C1
C3
C4
Exp vs
Theory
Prob
Expected
probability
Expected
Value
Comparing
languages
Adjust to
fit 3+ parts
Rocket
Launch
simulation
Worksheet
Design
own
simulation
card
Use of
Excel /
Fathom as
appropriate
Simulation
Cards
Exp prob
Long-run
frequency
I notice / I
wonder
Use
encoder
online
Explain
results
Rocket
Launch
simulation
Worksheet
Spreadsh
eet
Alternative
simulation
scenarios
TTRC to
explain
Intro to
other
situations
Use of
TTRC
Simulation
cards –
select as
appropriat
e for
students
Re sort
misconcep
tion cards
Hospital
Problem –
Make
prediction
Explore
simulation
Share
conclusions
about
sample
size
Games of
chance vs
games of
skill
Gut
reaction /
randomness
Play and
analyse
game
Design
own game
of chance
Venn
diagram
Exp vs
Theory
Invn p 440
Write 12
key
questions
Assumptio
ns in
calculating
Variation
Sample
Size
Probability
exp
Online
code
decoders
Simon
Singh
Comparing
languages
worksheet
Investigat
e
probability
games
Using
sample
spaces
Compare
and
Contrast it
Equally
likely
outcomes
Worksheet
Textbook
Using
sample
spaces
Question
it
Expected
value
Students
explain
formula
Ex 30.2
4
Hospital
Problem
Homer
Simpson
ppt
Ex 30.3
Monopoly
probability
Arwen
wins
Spin the
Wheel
6
Ex 30.4
Textbook
Key
features of
representa
tion
Why x
across?
Why +
between?
Theoretica
Finding
S5/6 l prob and proportion
S6/8 conditional s to meet
prob
S5/9 Variation
S5/1
0
S5/1
1
S5/1
2
LR5
Independe
nce
Heuristics
set criteria
Analysing
different
ways of
solving
Refute
conjecture
Use it
Justify it
Tree
Diagrams
Explore
represent
ations
Two Way
tables
Connect it
Key ideas
of it
Combining
all three
elements
of
probability
Relook at
misconcep
tion cards
expected
value?
Key
features of
Textbook
tree
diagrams.
When?
Key
Cond Prob
features of
/ False +
two way
and tables –
Worksheet
Presentation of
proofs
More
complex
tree
situations
Ex 30.5
Specificity
and
Sensitivity
Two way
tables
PPT
Two way
tables
worksheet
Summary
of topic
Own
Murphy’s
law invn
Nzmaths
for
Murphy’s
law
Rock
Paper
Scissors
Murphy’s
Law
Summary
of ideas
Investigate
other
misconcep
tions
4