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Exploding Dots

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Exploding Dots EXPLODING DOTS MIDDLE-SCHOOL VERSION JAMES TANTON www.jamestanton.com This material represents sample chapters from the MIDDLE-SCHOOL series: MATHEMATICAL THINKING ! available at the above website. CONTENTS: CHAPTER 3: DOTS AND BOXES ……………………………… 5 CHAPTER 4: DIVISION ……………………………… 25 CHAPTER 5: PILES AND HOLES ……………………………… 43 CHAPTER 14: DECIMALS (EXCERPTS) ……………………………… 65 SERIES CONTENT EXPLORATION 6: THE VINCULUM AND PARENTHESES (14 pages) A snazzy symbol from the 1500s makes EXPLORATION 1: THE POWER OF A order of operations easy and obvious. PICTURE (22 pages) Why don’t we use it today? How long do you think it would take to add TOPICS COVERED: Parentheses via the up all the numbers from one to a million vinculum. Basic order of operations via the and back down again? In this exploration vinculum. you will learn how to do it in under one second! EXPLORATION 7: TROUBLESOME TOPICS COVERED: The counting ZERO (16 pages) numbers; their addition and multiplication. Is zero easy to work with or tricky to Clever sums and figurate numbers. work with? Is it even a number? Let’s play with tricky zero and sort out all its EXPLORATION 2: FACTORS AND sneaky behaviors. PRIMES (12 pages) TOPICS COVERED: Is zero a number? Open doors and closed doors. Learn how Basic arithmetic with zero, and the locker doors reveal patterns in factors of danger of division. The use of zero in base numbers. 10 arithmetic. TOPICS COVERED: Factors of numbers. Composite numbers and prime numbers. EXPLORATION 8: MULTIPLICATION The great locker experiment! (26 pages) Slicing rectangles and slicing cheese is all EXPLORATION 3: DOTS AND BOXES one needs to multiply lengthy numbers (26 pages) (even if those rectangles and cheeses are Exploding dots reveal the secret to negatively long and wide – and high!) understanding how numbers are written TOPICS COVERED: Expanding brackets and added and multiplied and … and long-multiplication. Why the product TOPICS COVERED: Place-value and its of two negative numbers is positive. role in arithmetic algorithms. EXPLORATION 9: ACTING CLEVERLY EXPLORATION 4: DIVISION (22 (14 pages) pages) Clever ways to avoid ghastly work! And The mysterious long-division algorithm is clever ways to amaze your friends! finally explained. It makes sense! TOPICS COVERED: Grouping and TOPICS COVERED: Long division via dots factoring. Estimation. Mathematical party and boxes. Extension to polynomial tricks. division. EXPLORATION 10: FRACTIONS: EXPLORATION 5: PILES AND HOLES ADDING AND SUBTRACTING (24 (24 pages) pages) How a story that isn’t true can explain the Why do people find this topic so scary? mysteries of negative numbers! Pies and boys (and pies and girls) makes it TOPICS COVERED: Basic introduction to easy! negative numbers: addition and TOPICS COVERED: Fractions and their “subtraction,” and more polynomials. basic arithmetic. EXPLODING DOTS 3 EXPLORATION 11: FRACTIONS: EXPLORATION 15: EQUALITY AND MULTIPLYING AND DIVIDING (32 INEQUALITY (26 pages) pages) The mathematics of staying balanced – or Pies and Boys continue to save the day as staying unbalanced if you prefer! we explore further properties of TOPICS COVERED: Properties of equality. fractions and all the annoying jargon that Properties of inequality. Ranges of values. goes with them. TOPICS COVERED: More arithmetic of EXPLORATION 16: CONFUSING fractions. Mixed numbers. Extension to THINGS THAT ARE THE SAME (22 Egyptian fractions. pages) Let’s be honest: Many things in EXPLORATION 12: EXPONENTS (18 mathematics are confusing. Let’s see if we pages) can sort out a slew of confusing things Does folding a piece of paper in half once and for all. multiple times explain everything there is TOPICS COVERED: The word “more” and to know about the powers of two? the word “less.” Percentages. The word TOPICS COVERED: Exponents and their “of.” Ratios. The equals sign. properties EXPLORATION 17: NUMBERS ON EXPLORATION 13: RETURN OF THE THE LINE (22 pages) VINCULUM (28 pages) How much of the number line actually has The vinculum returns to save the day in anything to do with numbers? problems with division! TOPICS COVERED: Development of the TOPICS COVERED: The use of vinculum in number line. Rational and irrational fractions. Division in algebra. Division as numbers and the hierarchy of numbers. multiplication by fractions. Repeating decimals. Attempts to define the reals. EXPLORATION 14: DECIMALS (34 pages) EXPLORATION 18: SOLUTIONS Exploding and unexploding dots return to explain decimals! TOPICS COVERED: Decimals and their arithmetic. © 2009 James Tanton www.jamestanton.com EXPLODING DOTS 4 © 2009 James Tanton www.jamestanton.com EXPLODING DOTS 5 MATHEMATICAL THINKING! Exploration 3 DOTS AND BOXES © 2009 James Tanton www.jamestanton.com EXPLODING DOTS 6 A. GETTING STARTED Here are some dots (nine of them, I believe): and here are some boxes: In the game we are about to play boxes explode dots! In fact, these boxes like to follow the following rule: THE 1← 2 RULE: Whenever there are two dots in any one box they “explode,” disappear and become one dot in the next box to their left We start by placing our nine dots in the right-most box: There are certainly two dots somewhere in this box and they explode to become one dot one place to the left. It does not matter which two dots we circle. © 2009 James Tanton www.jamestanton.com EXPLODING DOTS 7 And it can happen again: And again! And it can happen again in right-most box, but it can now also happen in the second box. Let’s do it here now just for fun: Okay, now we have to go back to the right-most box: And another time: © 2009 James Tanton www.jamestanton.com EXPLODING DOTS 8 And one final time! After all this, reading from left to right we are left with one dot, followed by zero dots, zero dots, and one final dot. Let’s say: OUR CODE FOR THE NUMBER 9 IS: 1001 Here’s what happens with seven dots: EXERCISE: Circle the pair of dots that “exploded” at each turn in the above diagram. OUR CODE FOR THE NUMBER 7 IS: 0111 © 2009 James Tanton www.jamestanton.com EXPLODING DOTS 9 Your turn! Question 1: Draw 10 dots in the right -most box and perform the explosions. What is our code for the number ten? OUR CODE FOR TEN IS: _______________ Question 2: Drawing this on paper is hard. Maybe you could use buttons or pennies for dots and do this by hand. What could you use for the boxes? Use your chosen objects to find the code for the number 13. Also find the code for the number 6. OUR CODE FOR 13 IS: _______________ OUR CODE FOR 6 IS: _______________ Question 3: CHALLENGE: What number has code 0101? © 2009 James Tanton www.jamestanton.com EXPLODING DOTS 10 B. OTHER RULES Let’s play the dots and boxes game but this time with … THE 1← 3 RULE: Whenever there are three dots in any one box they “explode,” disappear and become one dot in the next box to their left Here’s what happens to fifteen dots: We have: THE 1← 3 CODE FOR FIFTEEN IS: 0120 Question 4: a) Show that the 1← 3 code for twenty is 0202. b) Show that the 1← 3 code for four is 0011. Question 5: What is the 1← 3 code for 13? For 25? Question 6: Is it possible for a number to have 1← 3 code 2031? Explain. Question 7: HARD CHALLENGE: What number has 1← 3 code 1022? © 2009 James Tanton www.jamestanton.com EXPLODING DOTS 11 Let’s keep going … Question 8: What do you think is the 1← 4 rule? What is the 1← 4 code for the number thirteen? Question 9: What is the 1← 5 code for the number thirteen? Question 10: What is the 1← 9 code for the number thirteen? Question 11: What is the 1← 5 code for the number twelve? Question 12: What is the 1← 9 code for the number thirty? AHA MOMENT! Qu estion 13: What is the 1← 10 code for the number thirteen? What is the 1← 10 code for the number thirty-seven? What is the 1← 10 code for the number 238? What is the 1← 10 code for the number 5834? © 2009 James Tanton www.jamestanton.com EXPLODING DOTS 12 C. WHAT’S REALLY GOING ON Let’s go back to the 1← 2 rule for a moment. THE 1← 2 RULE: Whenever there are two dots in any one box they “explode,” disappear and become one dot in the next box to their left Two dots in the right-most box is worth one dot in the next box to the left. If each of the original dots is worth “one,” then the single dot on the left must be worth two. But we also have two dots in the box of value 2 is worth 1 dot in the box just to the left … This next box must be worth two 2s. That’s four! © 2009 James Tanton www.jamestanton.com EXPLODING DOTS 13 And two of these fours makes 8. Question 14: If there was one more box to the left, what must it be worth? We said earlier that the 1← 2 code for 9 was 1001. Let’s check: Yep! Nine 1s does equal one 8 plus one 1. 9= 8 + 1 We also said that 13 has code 1101. This is correct. 13= 8 + 4 + 1 © 2009 James Tanton www.jamestanton.com EXPLODING DOTS 14 What number has code 10110? Easy: 16+ 4 + 2 = 22 Question 15: What number has 1← 2 code 100101 ? Question 16: What is the 1← 2 code for the number two hundred? FANCY LANGUAGE: People call numbers written in 1← 2 code binary numbers .
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