MYP Mathematics

361 Pages·2017·20.17 MB·English

MYP Mathematics
A concept-based approach
4&5
Extended
Rose Harrison • Clara Huizink
Aidan Sproat-Clements • Marlene Torres-Skoumal
3
Great Clarendon Street, Oxford, OX2 6DP, United Kingdom
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How to use this book
Chapters
Chapters 1–3 each focus on one of the key concepts:
Form, Relationships and Logic.
Chapters 4–15 each focus on one of the twelve related concepts
for Mathematics:
Representation, Simplication, Quantity, Measurement, Patterns, Space,
Change, Equivalence, Generalization, Justication, Models, and Systems
Creating your own units
The suggested unit structure opposite shows just one way of grouping all the
topics from different chapters, from both Standard and Extended, to create
units. These units have been created following the ofcial guidelines from the
IB Building Quality Curriculum. Each unit is driven by a meaningful statement
of inquiry and is set within a relevant global context.
Each chapter focuses on one of the IB’s twelve related concepts for
●
Mathematics, and each topic focuses on one of the three key concepts.
Hence, these units, combining topics from different chapters, connect the
key and related concepts to help students both understand and remember
them.
Stand-alone topics in each chapter teach mathematical skills and how
●
to apply them, through inquiry into factual, conceptual, and debatable
questions related to a global context. This extends students’ understanding
and ability to apply mathematics in a range of situations.
You can group topics as you choose, to create units driven by a
●
contextualized statement of inquiry.
This book covers the MYP Extended skills framework.
●
MYP Mathematics 4 & 5Standard covers the Standard skills.
Using MYP Mathematics 4 & 5 Extended with an
existing scheme of work
If your school has already established units, statements of inquiry and global
contexts, you can easily integrate the concept-based topics in this book into
your current scheme of work. The table of contents on page vii clearly shows
the topics covered in each concept-based chapter. Your scheme’s units may
assign a different concept to a given topic than we have. In this case, you can
simply add the concept from this book to your unit plan. Most topics include
a Review in context, which may differ from the global context chosen in your
scheme of work. In this case, you may wish to write some of your own review
questions for your global context, and use the questions in the book for
practice in applying mathematics in different scenarios.
ii
Suggested plan
The units here have been put together by the
Access your support website for more
authors as just one possible way to progress
suggested plans for structuring units:
through the content.
www.oxfordsecondary.com/myp-mathematics
Units 1, 5, 6, 9 and 10 are made up of topics solely
from the MYPMathematics 4 & 5 Standard book.
UNIT 2 UNIT 11
43 123 E41 73 91 E91
Topics: Scatter graphs and linear regression, Topics: Circle segments and sectors, volumes
drawing reasonable conclusions, data inferences of 3D shapes, 3D orientation
Global context: Identities and relationships Global context: Personal and cultural expression
Key concept: Relationships Key concept: Relationships
UNIT 3 UNIT 12
111 143 E141 131 132 E131
Topics: Equivalence transformations, inequalities, Topics: Using circle theorems, intersecting chords,
non-linear inequalities problems involving triangles
Global context: Identities and relationships Global context: Personal and cultural expression
Key concept: Form Key concept : Logic
UNIT 4 UNIT 13
51 103 E51 102 113 E102
Topics: Rational and irrational numbers, direct Topics: Algebraic fractions, equivalent methods,
and indirect proportion, fractional exponents rational functions
Global context: Globalization and sustainability Global context: Scientic and technical innovation
Key concept: Form Key concept: Form
UNIT 7 UNIT 14
81 121 122 E81 E61 E101 E121
Topics: Finding patterns in sequences, making Topics: Evaluating logarithms, transforming
generalizations from a given pattern, arithmetic logarithmic functions, laws of logarithms
and geometric sequences
Global context: Orientation in space and time
Global context: Scientic and technical innovation Key concept: Relationships
Key concept: Form
UNIT 15
UNIT 8
E71 E92 E111
44 151 E151
Topics: The unit circle and trigonometric functions,
Topics: Simple probability, probability systems, sine and cosine rules, simple trigonometric
conditional probability identities
Global context: Identities and relationships Global context: Orientation in space and time
Key concept: Logic Key concept: Relationships
iii
About the authors
Marlene Torres-Skoumal has taught DP Rose Harrison is the Lead Educator for MYP
and MYP Mathematics for several decades. Mathematics at the IB Organization. Sheis an
In addition to being a former Deputy Chief MYP and DP workshop leader, a senior reviewer
Examiner for IB HL Mathematics, she is an for Building Quality Curriculum in the MYP, and
IB workshop leader, and has been a member has held many positions of responsibility in her
of various curriculum review teams. Marlene 20 years’ experience teaching Mathematics in
hasauthored both DP and MYP books, international schools.
including Higher Level Mathematics for
Aidan Sproat-Clements is Head of Mathematics
OxfordUniversity Press.
at Wellington College in the UK, an IB World
Clara Huizink has taught MYP and DP School. He has spent his career in British
Mathematics at international schools in the independent schools where he has promoted
Philippines, Austria and Belgium. She has rigorous student-led approaches to the learning
alsobeen through the IB experience herself of Mathematics. He helped to develop the pilot
as a student and she is a graduate of the MYP eAssessments for Mathematics.
IBprogram.
Topic opening page
Global context –
some questions in
Objectives – the
E41 How to stand out from this unit are set in
mathematics
this context. You
the crowd
covered in this topic.
may wish to write
Global context: Identities and relationships
your own global
Objectives Inquiry questions
Making inferences about data, given the mean F context to engage
Key concept for and standard deviation deviation of a data set?
Using dierent forms of the standard deviation your students.
the topic. formula C How is the meaning of ‘standard
S Understanding the normal distribution
SHIP MUsaiknign tgh ien sfetarenndcaersd adbeovuiat tnioonrm anald d tihsetr mibueatinons D fCoarmn sualamsp?les give reliable results?
N Using unbiased estimators of the population Do we want to be like everybody else?
O mean and standard deviation Inquiry questions–
TI
A
ATL – the Approach EL the factual,
R
to Learning taught conceptual, and
ATL Communication
in this topic. Make inferences and draw conclusions debatable questions
explored in this
43 topic.
123
Unit plan – shows
E41
Statement of the topics in the
Inquiry for this Standard and
unit. You may wish Statement of Inquiry: Extended books
to write your own. Generalizing and representing relationships that you could
can help to clarify trends among individuals.
teach together as
76
a unit.
iv
Learning features
Each topic has three sections, exploring:
factual
●
conceptual
●
debatable inquiry questions
●
Problem solving – where the method of
solution is not immediately obvious, these are
highlighted in the Practices.
Explorations are inquiry-based learning
activities for students working inidvidually,
in pairs or in small groups to discover
mathematical facts and concepts.
Exploration 4
ATL Using dynamic geometry software, plot the points A (10, 6) and B (2, 8).
M, the midpoint of AB
3 Compare the coordinates of M A and B
Describe anything you notice.
4 Create other pairs of points and nd the midpoint of each pair.
Investigate the relationship between the coordinates of the endpoints ATL highlights an opportunity to develop the
and the coordinates of the midpoint. (Hint: construct a table of
endpoints and midpoints.) ATL skill identied on the topic opening page.
5 Use your ndings in step 4 to predict the coordinates of N, the midpoint
of points C(7, 11) and D (19, 15).
6 Verify your answer using the software.
7 Points P and Q have coordinates P ) and Q d).
Suggest a formula for the coordinates of the midpoint of PQ
8 Verify that your formula gives the correct coordinates for M and N
Reect and discuss – opportunities for small
Reect and discuss 3
group or whole class reection and discussion
Generalization means making a general statement on the basis of specic
examples.
Where in Exploration 4 have you generalized? on their learning and the inquiry questions.
How does generalization enable you to predict?
Why is it important to verify a generalization?
The midpoint of ( ) and ( d) is .
Worked examples show a clear solution and
Example 1
explain the method.
Find the midpoint of A(3, 6) and B (9, 18).
Let M be the midpoint of AB
M 3+96+18 Use the midpoint formula M =
(6, 12)
Practice 3 Practice questions written using IB command
1 Find the midpoint of A(14, 6) and B (18, 20). terms, to practice the skills taught and how to
2 Find the midpoint of C (5, 13) and D (10, 8).
3 a Plot the points A(2, 7), B (3, 10), C (6, 11), and D (5, 8). apply them to unfamiliar problems.
b Find the midpoints of AC and BD
Comment on your answers to part b. Explain what this shows about
quadrilateral ABCD
70 3 Logic
Objective boxes highlight an IB Assessment
Objective and explain to students how to satisfy
the objective in an Exploration or Practice.
Technology icon
Using technology allows students to discover new ideas through examining a wider
range of examples, or to access complex ideas without having to do lots of painstaking
work by hand.
This icon shows where students could use Graphical Display Calculators (GDC),
Dynamic Geometry Software (DGS) or Computer Algebra Systems (CAS). Places where
students should not use technology are indicated with a crossed-out icon.
The notation used throughout this book is largely that required in the DP IB programs.
v
Each topic ends with:
Summary
The standard deviation is a measure of Using notation, the mean of a set of discrete
dispersion that gives an idea of how close the
data values in a frequency table is , the sum
original data values are to the mean, and so
how representative the mean is of the data. of all the x f values divided by the total frequency.
A small standard deviation shows that the
A formula to calculate the standard deviation for
data values are close to the mean. The units of
discrete data presented in a frequency table is: Summary of the
standard deviation are the same as the units
of the original data. key points
x means the sum of all the x values.
Using this notation, the mean of a set of values The normal distribution is a symmetric
is Σx, the sum of all the values divided by the distribution, with most values close to the mean
n and tailing o evenly in either direction. Its
number of values. frequency graph is a bell-shaped curve.
Mixed practice
Number of strawberries on each plant fed with
In questions 1 and 2 the data provided is for special strawberry plant food: Mixed practice –
the population.
14 15 17 17 19 19 12 14 15 15 summative
1 Findthe mean and the standard deviation of Analyze the data to nd whether there is an assessment of the
each data set:
eect of using the special strawberry plant food.
facts and skills
a 2, 3, 3, 4, 4, 5, 5, 6, 6, 6
3 Bernie recorded the weight of food in grams his
b 21 kg, 21 kg, 24 kg, 25 kg, 27 kg, 29 kg hamster ate each day. Here are his results: learned, including
c problem-solving
x f Day 1 2 3 4 5 6 7 8 9
questions, and
3 2 Food (g) 55 64 45 54 60 50 59 61 49
4 3 a Find the mean and standard deviation of questions in a
5 2 the weight of food the hamster ate over the range of contexts.
9days.
d Interval Frequency b Bernie assumes that his hamster’s food
−
Review in context
The ancient art of origami has roots in Japanese The resulting creases are illustrated in this diagram.
culture and involves folding at sheets of paper A B C
to create models. Some models are designed to
resemble animals, owers or buildings; others E
just celebrate the beauty of geometric design.
Review in context –
summative
D
F assessment questions
within the global
H context for the topic.
G Z
The artist thinks lengths FH and DY are equal.
b Explain why EX 1EG.
2
Reect and discuss Reection on the
statement of
How have you explored the statement of inquiry? Give specic examples. inquiry
vi
Table of contents
Access your support website:
www.oxfordsecondary.com/myp-mathematics
Form Space
E1.1 W hat if we all had eight ngers? • E9.1 A nother dimension •
Dierent bases 2 3D Orientation 167
E9.2 M apping the world •
Relationships
Oblique triangles 183
E2.1 M ultiple doorways •
Composite functions 21
Change
E2.2 D oing and undoing • E10.1 Time for a change •
Invserse functions 35 Log functions 202
E10.2 M eet the transformers •
Logic
Rational functions 216
E3.1 H ow can we get there? •
Vectors and vector spaces 54
Equivalence
E11.1 U nmistaken identities •
Representation
Trigonometric identities 233
E4.1 H ow to stand out from the crowd •
Data inferences 76
Generalization
E12.1 G o ahead and log in •
Simplication
Laws of logarithms 249
E5.1 Super powers •
Fractional exponents 98
Justication
E13.1 A re we very similar? •
Quantity
Problems involving triangles 264
E6.1 I deal work for lumberjacks •
Evaluating logs 108
Models
E14.1 A world of dierence •
Measurement
Nonlinear inequalities 282
E7.1 S lices of pi •
The unit circle and
Systems
trignometric functions
E15.1 B ranching out •
124
Conditional probability 299
Patterns
Answers 315
E8.1 M aking it all add up •
Arithmetic and geometric sequences 145 Index 350
vii
E11
What if we all had eight
.
ngers?
Global context: Scientic and technical innovation
Objectives Inquiry questions
● Understanding the concept of a number F ● How have numbers been written in
system history?
Counting in dierent bases What is a number base?
● ●
Converting numbers from one base to How can you write numbers in other
● ●
M another bases?
R Using operations in dierent bases
O ● C ● How are mathematical operations in
other bases similar to and dierent from
F
operations in base 10?
D ● Would you be better o counting in
base 2?
How does form inuence function?
●
ATL Communication
Use intercultural understanding to interpret communication
2
NUMBER
You should already know how to:
use the operations of addition, 1 Calculate these by hand:
●
subtraction and long multiplication a 10442 + 762 b 10887 7891
in base 10, without a calculator
c 27 × 43 d 14078 × 71
understand place value 2 Write down the value of the 5 in:
●
a 351 b 511002
c 15 d 1.5
F Numbers in dierent bases
How have numbers been written in history?
●
What is a number base?
●
How can you write numbers in other bases?
●
Humans have used many dierent ways to
record numbers. How would you write down the
number of green bugs in this diagram?
You might have written a symbol 5, or the word
“ve”. You could have used a word in a dierent
language, or maybe even a tally: . Each of
these represents the number in a dierent way,
but they all represent the same quantity.
Dierent cultures use dierent forms of notation
to represent number. The ancient Egyptian
hieroglyphic number system was an additive
system. Each symbol has a dierent value and
you nd the total value of the number by adding
the values of all the symbols together.
Egyptian numerals use these symbols:
Green shield bugs
are sometimes
called green stink
stroke heelbone coiled rope
bugs, as they
1 10 100
produce a pungent
odor if handled or
disturbed.
lotus ower pointed nger tadpole scribe
1000 10000 100000 1000000
The number 11 is written , and 36 is written .
E1.1 What if we all had eight ngers? 3