Name:

Theory Workbook Mathematics 2U

1a: Arithmetic

v2.1 2015enzuber 2015

Where do we want to go to in this topic?1. Be able to fluently perform the fundamental arithmetic operations

on fractions and surds.

2. Understand the difference between rational and irrational numbers.

3. Be able to work with different number representations fractions, decimals, surds, scientific notation.

You are free to share, copy, or modify this work for non-commercial purposes so long as you:

(i) Attribute the source : enzuber(ii) Share all derived works under a similar CC license.

enzuber@gmail.com exzuberant.blogspot.comFor my students: please use our class edmodo not email.

2U Topic 01a: Arithmetic v2015.1This resource is designed for teachers and senior high school students doing the NSW Board of Studies Mathematics 2 Unit course.

This work is licensed Creative Commons CC-BY-NC-SA. http://creativecommons.org/licenses/by-nc-sa/3.0/

If youre interested, explore beyond the syllabusBoxes with the rocket icon are not part of the syllabus however they will enrich your understanding of the topic.

code:

This code is only valid for a short time.

http://nsbhs.edmodo.com/

Topic 1a: Arithmetic

Fraction operations. Fractions and recurring decimals. Scientific notation. Significant figures. Rational and irrational numbers. Surd operations. Rationalising the denominator.

Fundamentals of Arithmetic

1. Simplify, add, subtract, multiply and divide fractions

The same number can be represented using different fraction expressions.

Add and subtract fractions by finding a common denominator.

2

Multiplying fractions.

Dividing fractions.

Cambridge Yr 11 2U Ex 2A3

2. Fractions and recurring decimals

The standard representation of a fraction is as a ratio of two integers.

All fractions can be represented as a recurring decimal.

Rational number

4

Integer

.Converting a recurring decimal to a fraction.

0. 2

0.42 Check your answer!

5

2. 53

4.253 Check your answer!

Cambridge Yr 11 2U Ex 2B6

3. Scientific Notation

Why do we care?

Scientific Notation

7

Very large numbers

Very small numbers

Veritasium:Scientific notation explained

8.Converting large numbers to scientific notation.

.Converting small numbers to scientific notation.

Common errors

Fitzpatrick 2U Ex 1.39

Multiplying and dividing numbers in scientific notation

Adding numbers in scientific notation

4. Rounding and significant figures

Does the number of decimal places mean anything?

10

This bag of chips contains 150gm of chips

The mass of an electron is 9.109382911031

My height is 1.8000m vs My height is 1.8m

Rounding rules

From a mathematical point of view, 1.34 is identical to 1.3400000.

From an engineer or scientists point of view there is a big difference!

Significant Figures

Watch out for .

Fitzpatrick 2U Ex 1.4 11

5. Rational and Irrational Numbers

12

Rational numbers

We can use counting to find the rational numbers.

We need some geometric thinking to help locate these irrational numbers.

Real numbers

How many irrational numbers are there compared to rational numbers?

Is there anything beyond the realnumber line?

Some infinite set are bigger than other infinite sets!

Numberphile: Counting to Infinity

Dimensions: Chapter 5 Complex Numbers13

6. Arithmetic with Surds

14

Definition of the symbol

Surds

.Basic operations with Surds

See Cambridge 2D

.Adding and subtracting surds

Convert all surds to the same simplest form Collect like terms

8 18 + 50

3 6 + 2 5 6

5 3 + 20 2 12 + 45

15

.Simplifying surds

12 3 28

Cambridge Yr 11 2U Ex 2D

.Multiplying and Dividing Surds

6 10805

24 4 2

16

Binomial Expansions of Surds

17

.Rationalising the Denominator

Cambridge Yr 11 2U Ex 2E

Single denominator

Denominator with two terms

12 661172 3

35 + 3

13 5 2

"Errors" tell you where your learning is today.

Errors are valuable signposts and guides where to go to next.

Don't run away or hide from them

Youre not the only one.

"Errors" are so valuable for learning we should welcome them.

If your classmates are generous enough to share an error with you, thank them - they just gave the class a special gift.

18

Common Arithmetic Errors

Division of surds722 = 2 6 62 Using a conjugate when not needed23 = 23 3 3 Correct but weird

Correct but slow

Not using the conjugate22 + 3 = 22 + 3 (2 + 3)(2 + 3) Not using the conjugate12 3 + 1 = 12 3 + 1 ( 3 1)( 3 1)

Valid but not helpful

Valid but not helpful

Weird fractions

Incorrect cancellations2 12 + 1 1+13 53 25 3 13 5

Incorrect attempt to remove a surd23 23 2 Just plain wrong12 3 + 1 + 33 1 43 3

19

20

21

Recommended Exercises

Foundation Development Challenge

2AIntegers and Rational Numbers

2BTerminating and Recurring Decimals

Fitzp. 1.3Scientific Notation

Fitzp. 1.4 Significant Figures

18

Recommended Exercises

Foundation Development Challenge

2DSurds and their Arithmetic

2EFurther Simplificationof Surds

2FRationalisingthe Denominator

2GChapter Review

19

01a: Arithmetic Cambridge Yr 11 2U

1 Can simplify, add, subtract, multiply and divide fractions. 2A

2 Can convert recurring decimals to fractions. 2B

3 Can write numbers in scientific notation. 2C

4 Can round to given number of significant figures. 2C

5 Can explain the difference between rational and irrational numbers. 2C

6 Can simplify, add, multiply and divide surds. 2D/2E

7 Can rationalise the denominator for a surd fraction. 2F

May I share your surds?

Little Miss MuffetSat on a tuffet,Eating her curds and whey;Along came a spider,Who sat down beside herAnd frightened Miss Muffet away

Peter Newell, 1907

From Wikipedia nth root page: The term surd traces back to al-Khwrizm (c. 825), who referred to rational and irrational numbers as audible and inaudible, respectively. This later led to the Arabic asamm (deaf, dumb) for irrational number being translated as surdus (deaf or mute) into Latin.

Back cover: The Spiral of Theodorus

Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28