MYP Next Chapter Guide – Mathematics

In this issue:

Issue 2

l Successfully transitioning students to DP Mathematics

l Preparing Mathematics learners for eAssessment

l Supporting resources

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Transitioning your learners to IB Diploma Mathematics

Rose Harrison is an MYP workshop leader. Here, she discusses strategies that help learners step into IB Diploma level mathematics with confidence.

Developing the key knowledge for DPDeveloping mathematical knowledge and understanding and mastering the different MYP concepts will equip students to manage the academic rigour of the DP Mathematics courses.

For learners transitioning to Mathematical Studies SL, the concepts of representation, simplification, quantity, measurement, patterns and space are paramount for the DP external assessment. Gaining a robust understanding of these concepts at MYP level will build strong foundations for Mathematical Studies SL. In-depth comprehension of statistical analysis through representation will also ensure that students are capable of writing a detailed and appropriate statistically-based project.

For learners transitioning to Mathematics Standard Level, a key challenge of this course is the volume of new material. To help, MYP learners should engage with factual, conceptual and debatable questions for each topic; this more thoroughly engages students in learning, and supports stronger, long-term understanding.

At Higher Level, the two courses require learners to respond to unfamiliar questions; this means students need to display flexibility in their thinking. Concept-based teaching during the MYP years builds students’ holistic understanding of mathematics; it enables them to express knowledge in a number of ways and to transgress different mathematical topics through the same concept. Additionally, students will recognize new material at DP level as connected to one of the familiar MYP concepts, enabling them to more confidently build on their conceptual MYP education.

“Concept-based teaching enables students to transgress different mathematical topics through the same concept...”

Rose Harrison

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Transitioning your learners to IB Diploma Mathematics

The concept-based approach and DP MathematicsA concept-based approach at MYP is integral to enabling students to progress to DP Mathematics with ease. One clear area of benefit is in the DP Internal Assessment, or the Mathematical Exploration.

The DP Exploration provides students with opportunities to increase their understanding of mathematical concepts and processes, and to develop a wider appreciation of mathematics. A firm grounding in conceptual understanding and application of the global contexts studied in the MYP years will put students in a strong position to embark on their Exploration.

For example, an MYP 1 student will have been introduced to representation with the Cartesian coordinate plane. Through real-life examples they will examine the four quadrants and the system developed by correctly giving and naming the coordinates, with added history on the importance of Descartes.

In MYP 2 the student will revisit the plane by plotting coordinates generated by number patterns, furthering their representation knowledge by adding the concept of pattern. Number patterns could be generated in a variety of ways and will expose the students to position-to-term and term-to-term patterns and how these can be represented on the plane.

In MYP 3 these number patterns may be converted into linear functions. How the patterns of these straight lines can be altered by changing the parameters on the plane may be examined through space. MYP 4 and 5 will revisit the same topics and enhance the concepts with a full understanding of the applications by modelling the linear (and other) functions within a specific global context. This enables the student to experience how the concept of model can be accessed.

This five year concept-based approach empowers students with the necessary conceptual understanding of the fundamental underlying principles to embark on any of the DP courses with confidence.

The five year concept-based approach empowers students with the necessary conceptual understanding…

The transition to DP – choosing resourcesResources should cover the prior learning to all four Diploma Programme courses and should closely integrate the key and related concepts. This equips learners with confident conceptual understanding they can apply at DP level.

The MYP Mathematics resources I have co-authored ensure learners revisit each concept 15-20 times through different question cycle topics. This firmly establishes individual understanding of each concept, equipping learners to recognize and work with these concepts in DP Mathematics.

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Preparing learners for the Mathematics eAssessmentMYP educator Aidan Sproat-Clements worked on the pilot MYP eAssessments. Here he discusses how to prepare your mathematics learners for eAssessment.

Aidan Sproat-Clements

The first MYP eAssessments take place in May 2016 and if your school registered before October, students will be able to participate in the assessments and work towards the internationally recognized MYP certificate. But how can you go about optimally preparing learners to perform in the eAssessments?

There are three parts to the eAssessments in mathematics:

1. Short and medium response questions covering a range of mathematical techniques

2. Exploration of real-life problems centered on one of the global contexts

3. An investigation

Those familiar with the MYP Assessment Objectives will recognize these as responding to the needs of assessment criteria A, D and B respectively. Objective C (Communication) is assessed throughout the exam.

A number of strategies are crucial when preparing for the eAssessments. The most important is that students are familiar with the exact meanings of the IB command terms. For example, they must understand that find and write down have different expectations, or know the difference between verify and justify.

To prepare students for the real-life aspects of the eAssessment, it is important that they are used to seeing questions asked in a wide variety of contexts. Everyday learning should aim to reintroduce mathematical material in a range of global contexts – this is crucially important. Learners need to be flexible and able to respond to a variety of different contexts and interpret them mathematically.

The final component of the eAssessment is an investigation. This is something students typically find difficult in examination situations. You can help your learners tackle this section by ensuring robust understanding of the related concepts of pattern, generalization and justification. By working through a range of investigative tasks linked to these concepts, students gain an understanding of what it means to investigate mathematically: to gather information, to form conjectures, to verify ideas and to justify claims.

“By working through a range of explorative tasks, students understand what it means to investigate mathematically…”

The eAssessment – choosing resourcesResources should clearly explain the IB command terms. You should also look for thorough integration of the global contexts, with contextual problems spread across learning material. This will enable students to more confidently respond to a mix of contexts and relate them to mathematical principles.

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Many teachers mention that it’s hard to see how published resources can help you deliver a truly concept-based approach that concretely prepares learners for eAssessment and establishes a firm base for DP Mathematics. Resources for MYP Next Chapter Mathematics can provide this level of support – look for resources that:

Our MYP Mathematics specialists

Rose Harrison is an MYP workshop leader with nearly 20 years’ teaching experience. She is also an IB curriculum review consultant.

Marlene Torres-Skoumal has taught IB Mathematics for over thirty years. Marlene is an IB workshop leader and curriculum review consultant.

Aidan Sproat-Clements is a Head of Mathematics who consulted on the pilot MYP eAssessments. He also manages mathematical outreach projects.

Clara Huizink has taught MYP Mathematics at international schools in the Philippines, Austria and Belgium. She has a Master’s degree in Communication Systems Engineering.

Recommendations from our panel of MYP Mathematics specialists

Resources and support

1 Fully integrate concept-based learning

This sets learners up for a strong start at IB Diploma, equipping them with the fundamental understanding and flexibility to apply their knowledge to new mathematical principles, as well as preparing them for the eAssessment.

2 Blend contextual problems across learning material

Thorough integration of the global contexts builds the adaptability learners need to successfully tackle the real-life aspects of the eAssessment.

3 Are structured by related concept

This both gives you the flexibility to introduce material in the order that works best for you, but also increases conceptual understanding, supported by factual knowledge and skills. This framework establishes a clear foundation for steadily linking in new mathematical objectives in later units.

4 Cover all the DP Mathematics prior learning

This ensures learners are fully prepared to confidently leap into DP Mathematics.

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Develop a strong mathematical foundation through clear conceptual connections

l Build mathematical confidence – extensive practice refines and progresses skills and understanding

l Transition to a concept-based approach – structured by related concept for an easy shift to concept-based teaching

l Prepare for eAssessment – concept-based approach equips learners to recognize and manipulate new principles

l Progress learners to IB Diploma – integrated global contexts strengthen understanding and adaptability

l Fully matched to the Next Chapter curriculum and supports the Common Core

Written by MYP Mathematics specialists

Rose Harrison

Marlene Torres-Skoumal

Aidan Sproat-Clements

Clara Huizink

NEW

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A clear pathway that builds mathematically confident, inquiring thinkers

Fully prepare learners for eAssessment and IB Diploma

8

Equivalent chordsMusicians use “pitch class analysis” to analyse atonal music, such as Schoenberg’s compositions. Chords are described by the number of semitones between the notes, and two chords are equivalent if one is a transposition of the other. In this way, the chord G-B-F is equivalent to the chord C-D-F#.

Equivalence

Does equivalence mean equal?Statements, quantities and expressions are equivalent when they are identically equal or interchangeable.

You have been working with the concept of equivalence since you learned that a numerical value can be expressed in different, but equal, ways. For example, –1 + 6 = 3 + 2, or 6

8 = 3

4 = 0.75

An equation is an algebraic statement that shows two expressions are equal, or equivalent.

For example 2x + 5 = 11, or x6 = –3. You can solve

an equation to find the value of the unknown. It is true for particular values of x.

➤ Given that the scale is perfectly balanced, how much does one box weigh?

Euclid’s ElementsAbout 300 BC Euclid, a Greek mathematician, wrote the most famous mathematics book of all time, The Elements. In it he described axioms (statements whose truth we accept without proof), and theorems (statements that are proven). His first axioms dealt with equality:

1 A thing is equal to itself.

2 Things equal to the same thing are equal to each other.

3 Things equal to equal things are equal to each other.

➤ Complete these examples to illustrate the three axioms:

1 8 = □

2 3 + 5 = 10 – □

3 3x + 2 = 8 and 4x = 8, so □ + □ = 4x

▲ Things equal to the same thing are equal to each other.

In 1847 Englishman Oliver Byrne

published an edition of Euclid where

he used colour in diagrams to prove

theorems, instead of text and labels.

First, transpose the F down an octave, to make the chord as condensed as possible. Then, transpose each note down 5 semitones. You might think that these chords sound very different, but because they have the same structure, a pitch class analyst would call them equivalent. This chord would be called {0, 2, 6}: can you see why?

100g

G

B

F

FB

GC

FD

➤ What theorem(s) do you think

might be depicted here?Did you know that 0.99999…. ≡ 1 ?Here is the proof – make sure you can follow it.

Let N = 0.999…

Then 10N = 9.999…

9N = 10N – N = 9.999… – 0.999… = 9

So N = 1

Euclid’s 2nd equality axiom tells you that ‘things equal to the same thing are equal to each other’, so as 0.999… = N and N = 1, then 0.999… ≡ 1.

➤ Use the same method to prove that

0.333… is equal to 13

0.2424…. is equal to 833 (hint: when two values are repeating,

multiply both sides of the equation by 100 instead of 10).

Having multiplied both sides by 10

The decimal parts cancel each other

Having divided both sides by 9

32

Linear EquationsExploration 1 Start with the equation x = 5.

Multiply both sides by the same number.

Add the same number to both sides.

Subtract x from both sides.

Divide both sides by the same number.

Compare your equations with others in your group. Are all your equations equivalent?

You know how to solve linear equations, but do you know the mathematical principles that you use to do this?

To solve 3(x + 2) – 6 = 4(2x – 3) + 1 you might follow these steps:

3(x + 2) – 6 = 4(2x – 3) + 1

(1) 3x + 6 – 6 = 8x – 12 + 1

(2) 3x = 8x – 11

(3) 11 = 5x

(4) x = 115 = 2.2

Solution Check – substitute the value of x into each side of the original equation.

Left hand side (LHS): 3(2.2 + 2) – 6 = 3(4.2) – 6 = 6.6

Right hand side (RHS): 4(2 × 2.2 – 3) + 1 = 4(1.4) + 1 = 6.6

LHS = RHS, so the solution is correct.

You may have used quite informal language to answer these two questions. In the rest of this question cycle you will learn the formal terms of the mathematical principles involved in solving equations.

F

Reflect and Discuss 11 Which mathematical principles do you think are being used in each

solution step?

2 Suggest an explanation for why all the equations in Exploration 1 (the original and the following four) are considered equivalent.

Reflect and Discuss 21 Why don’t we need a subtraction principle?

2 Why don’t we need a division principle?

Comm

unication

Example 1Solve the equation 1

4 (x – 2) = 12 (3x + 4). Show the equivalence

transformation used at each step. Remember to check your solution.

14 (x – 2) = 1

2 (3x + 4)

(1)x – 2 = 2(3x + 4)

x – 2 = 6x + 8

(2)x – 10 = 6x

(3)–10 = 5x

–2 = x

(4)Check: LHS: 14 (–2 – 2) = – 4

4 = –1

RHS: 12 (3×(–2) + 4) = 1

2 (–2) = –1

LHS = RHS

Practice 1Solve the following equations. Show the equivalence transformation used at each step. Remember to check your solutions.

1 2x + 3 = x – 7

2 5x – 4 = 2x + 6

3 –(x + 2) – 3x = 2(x + 1)

4 1 – 3(x + 2) = 12

(2 x – 8) + 3

Crea

tive

thin

king

Multiply both sides by 4.

Add –x to both sides or subtract x from both sides. Multiply both sides by 1

5or divide both sides by 5.

Add -8 to both sides, or subtract 8 from both sides.

Equivalence Transformations

è To solve an equation you can use these mathematical principles:

Principle Example

Addition Principle: Add the same value or variable to both sides of an equation.

Add 3 to both sides of 2x – 3 = 5 to get the equivalent equation 2x = 8

Multiplication Principle: Multiply by the same non-zero value or variable on both sides of an equation.

Multiply both sides of 11 = 5x by 15 to

get the equivalent equation 115

= x

An equivalence transformation uses these principles to transform an equation to an equivalent equation.

54 8 Equivalence 8.1 Equivalent transformations

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DRAFT

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