MATHEMATICAL THINKING
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1 MATHEMATICAL THINKING Numbers and their Algebra James Tanton (with additional tidbits by Kit Norris) COMMENT: These notes are based on content from the THINKING MATHEMATICS! Volume 1: Arithmetic = Gateway to All available at James Tanton
2 2009 James Tanton TABLE OF CONTENTS
3 SERIES CONTENT EXPLORATION 1: THE POWER OF A PICTURE (18 How long do you think it would take to add up all the numbers from one to a million and back down again? In this exploration you will learn how to do it in under one second! TOPICS COVERED: The counting numbers; their addition and multiplication. Clever sums and figurate numbers. EXPLORATION 2: FACTORS AND PRIMES (10 Open doors and closed doors. Learn how locker doors reveal patterns in factors of numbers. TOPICS COVERED: Factors of numbers. Composite numbers and prime numbers. The great locker experiment! EXPLORATION 3: DOTS AND BOXES (24 Exploding dots reveal the secret to understanding how numbers are written and added and multiplied and TOPICS COVERED: Place-value and its role in arithmetic algorithms. EXPLORATION 4: DIVISION (20 The mysterious long-division algorithm is finally explained. It makes sense! TOPICS COVERED: Long division via dots and boxes. Extension to polynomial division. EXPLORATION 5: PILES AND HOLES (22 How a story that isn t true can explain the mysteries of negative numbers! TOPICS COVERED: Basic introduction to negative numbers: addition and subtraction, and more polynomials. EXPLORATION 6: THE VINCULUM AND PARENTHESES (10 A snazzy symbol from the 1500s makes order of operations easy and obvious. Why don t we use it today? TOPICS COVERED: Parentheses via the vinculum. Basic order of operations via the vinculum. EXPLORATION 7: TROUBLESOME ZERO (14 Is zero easy to work with or tricky to work with? Is it even a number? Let s play with tricky zero and sort out all its sneaky behaviors. TOPICS COVERED: Is zero a number? Basic arithmetic with zero, and the danger of division. The use of zero in base 10 arithmetic. EXPLORATION 8: MULTIPLICATION (24 Slicing rectangles and slicing cheese is all one needs to multiply lengthy numbers (even if those rectangles and cheeses are negatively long and wide and high!) TOPICS COVERED: Expanding brackets and long-multiplication. Why the product of two negative numbers is positive. EXPLORATION 9: ACTING CLEVERLY (10 Clever ways to avoid ghastly work! And clever ways to amaze your friends! TOPICS COVERED: Grouping and factoring. Estimation. Mathematical party tricks. EXPLORATION 10: FRACTIONS: ADDING AND SUBTRACTING (20 Why do people find this topic so scary? Pies and boys (and pies and girls) makes it easy! TOPICS COVERED: Fractions and their basic arithmetic James Tanton
4 TABLE OF CONTENTS EXPLORATION 11: FRACTIONS: MULTIPLYING AND DIVIDING (28 Pies and Boys continue to save the day as we explore further properties of fractions and all the annoying jargon that goes with them. TOPICS COVERED: More arithmetic of fractions. Mixed numbers. Extension to Egyptian fractions. EXPLORATION 12: EXPONENTS (14 Does folding a piece of paper in half multiple times explain everything there is to know about the powers of two? TOPICS COVERED: Exponents and their properties EXPLORATION 13: RETURN OF THE VINCULUM (26 The vinculum returns to save the day in problems with division! TOPICS COVERED: The use of vinculum in fractions. Division in algebra. Division as multiplication by fractions. EXPLORATION 14: DECIMALS (30 Exploding and unexploding dots return to explain decimals! TOPICS COVERED: Decimals and their arithmetic. EXPLORATION 15: EQUALITY AND INEQUALITY (24 The mathematics of staying balanced or staying unbalanced if you prefer! TOPICS COVERED: Properties of equality. Properties of inequality. Ranges of values. EXPLORATION 16: CONFUSING THINGS THAT ARE THE SAME (18 Let s be honest: Many things in mathematics are confusing. Let s see if we can sort out a slew of confusing things once and for all. TOPICS COVERED: The word more and the word less. Percentages. The word of. Ratios. The equals sign. EXPLORATION 17: NUMBERS ON THE LINE (20 How much of the number line actually has anything to do with numbers? TOPICS COVERED: Development of the number line. Rational and irrational numbers and the hierarchy of numbers. Repeating decimals. Attempts to define the reals. EXPLORATION 18: SOLUTIONS 2009 James Tanton
5 2009 James Tanton
6 2009 James Tanton TABLE OF CONTENTS
SUBTRACTION. My disbelief in subtraction comes from another story that isn t true. Briefly, it goes as follows.
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EXPONENTS EXPLORATION 12. Does folding a piece of paper in half multiple times explain everything there is to know about the powers of two?
EXPLORATION EXPONENTS Does folding a piece of paper in half multiple times explain everything there is to know about the powers of two? TOPICS COVERED: Exponents and their properties. James Tanton 9 A.
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