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What is wrong with saying 372 base 6? What is a proper way of writing this number in base 6? Announcements • Last day to drop without a W is Friday, September 17th • First exam: Tuesday, September 28th • Sample questions will be posted on the website. Answers later. • Class Notes: purchase at the bookstore; please get your copy as soon as possible. Let’s add in Alphabitia BCD + DB BCD is DB is BCD + DB is You try: B0B +ABC Use the manipulatives or draw pictures to help explain your answer. Exploration 3.1 In your groups: Part 1 • #1 Make up one addition story problem in Alphabitia and solve it. • Read your problem to the other members of your group and discuss how you solved your problems. • #2a Operations with Whole Numbers Whole numbers: 0, 1, 2, 3, 4, 5, … Operations: + ─ × ÷ Addition of Whole Numbers • Ways to represent addition: • 1. Counters • 2. Number Line 0 • 3. Part-Part Whole 2 4 6 a c 8 10 b Vocabulary • • • • A+B=C A and B are called Addends C is called the Sum Since A + B = B + A every time, we give it a special name: the commutative property. • Since A + (B + C) = (A + B) + C every time, we give it a special name: the associative property. Addition and Place Value • If we have 154 + 69 Mental Arithmetic Add these numbers in your head—no pencil, paper, or calculator. 39 + 57 How did you do it? Mental Arithmetic 39 + 57 • • • • • Leading digit (30 + 50) + (9 + 7) Compensation (40 - 1) + 57 Compensation (40 + 57) - 1 Break and Bridge (39 + 50) + 7 Compatible Numbers (39 + 51) + 6 You try… • 23 + 96 • 489 + 723 Exploration 3.2 Mental Addition • Find the sums in the table mentally. • Write a description of your method for finding each sum. • Did you use the same technique for all of the sums? • Discuss your methods with your group members. • Did you all use the same technique for each sum? • How many different techniques were used? Estimation • Get a QUICK answer that is close but not necessarily exact! You have 5 seconds 382 + 147 Estimation • • • • • • • 382 + 147 Actual answer: 529 Ways to estimate: 380 + 150 = 530 400 + 150 - 20 = 530 375 + 150 = 525 390 + 140 = 530 You try: • 45 + 87 • 391 + 1472 Estimation vs. Exact Answer • Are the exact values of the addends given? • Is the exact value of the sum required? • In general: changing values (like population), measurement values (like distance), and predictions (like 35% chance of rain) are estimates Rounding • When you estimate, you are rounding to the nearest ten, hundred, million, etc. • If you round up, you are making an overestimation. • If you round down, you are making an underestimation. Explain the math 348 + 472 10 11 7 820 Explain the math 471 + 299 10 160 600 770 How is this different from the previous example? Explain why the child made the mistake • 341 + 287 is the same as 340 + 290 + 7 + 3 • 749 + 684 is the same as 700 + 600 + 49 + 1 + 84 + 16 • 322 + 468 is the same as 320 + 460 A problem: • I am thinking of 3 numbers whose sum is greater than 70. • Look at each statement below. Mark it as true or false. Be ready to explain why. • All three numbers are greater than 20. • If two of the number are less than 20, the other must be greater than 20.