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302A final exam review

302A final exam review. What is on the test?. From book : 1.2, 1.3, 1.4, 1.7; 2.3; 3.1, 3.2, 3.3, 3.4; 4.2, 4.3; 5.2, 5.3, 5.4; 6.1, 6.2 From Explorations : 1.1#2, 1.4#1; 2.8, 2.9; 3.1; 3.13, 3.15, 3.19, 3.20, 4.2, 4.3, 5.8, 5.9, 5.10, 5.12, 5.13, 5.15, 6.3, 6.5, 6.7

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302A final exam review

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  1. 302A final exam review

  2. What is on the test? • From book: 1.2, 1.3, 1.4, 1.7; 2.3; 3.1, 3.2, 3.3, 3.4; 4.2, 4.3; 5.2, 5.3, 5.4; 6.1, 6.2 • From Explorations: 1.1#2, 1.4#1; 2.8, 2.9; 3.1; 3.13, 3.15, 3.19, 3.20, 4.2, 4.3, 5.8, 5.9, 5.10, 5.12, 5.13, 5.15, 6.3, 6.5, 6.7 • From Class Notes: Describe/Use the strategies used by the students--don’t need to know the names.

  3. Chapter 1 • A factory makes 3-legged stools and 4-legged tables. This month, the factory used 100 legs and built 3 more stools than tables. How many stools did the factory make? • 16 stools, 13 tables

  4. Chapter 1 • Fred Flintstone always says“YABBADABBADO.” If he writes this phrase over and over, what will the 246th letter be? • D

  5. Chapter 2 • Explain why 32 in base 5 is not the same as 32 in base 6. • 32 in base 5 means 3 fives and 2 ones, which is 17 in base 10. • 32 in base 6 means 3 sixes and 2 ones, which is 20 in base 10. So, 32 in base 5 is smaller than 32 in base 6.

  6. Chapter 2 • Why is it wrong to say 37 in base 5? • In base 5, there are only the digits 0, 1, 2, 3, and 4. 7 in base 5 is written 12.

  7. Chapter 2 • What error is the student making? “Three hundred fifty seven is written 300507.” • The student does not understand that the value of the digit is found in the place: 300507 is actually 3 hundred-thousands plus 5 hundreds and 7 ones. Three hundred fifty seven is written 357--3 hundreds plus 5 tens plus 7 ones.

  8. Chapter 2 • True or false? • 578 + 318 = 1008 • 24015 – 4325 > 20005 • 139 < 228 < 417 • False: 578 + 318 = 1108; False: 24015 – 4325 = 14145; True: 12 < 18 < 29

  9. Chapter 2 • In what base is 44 + 65 + 26 = 201? • Decompose: 44 + 65 + 26 = 40 + 60 + 20 + 4 + 5 + 6 = 200 + 00 + 1. • Need 4 + 5 + 6 = ?1 (why?) • so 4 + 5 + 6 = 1510= 1410 + 110 • The base = 7 or 14 (why?)

  10. In what base is 44 + 65 + 26 = 201? Continued: • Check base 7: • 44 + 65 + 26 = 40 + 60 + 20 + 4 + 5 + 6 = 40 + 30 + 30 + 20 + 4 + 3 + 2 + 5 + 1 = 100 + 50 + 10 + 10 + 1 = 100 + 100 + 1 = 201 in base 7.

  11. Chapter 3 • List some common mistakes that children make in addition. • Do not line up place values. • Do not regroup properly. • Do not account for 0s as place holders.

  12. Chapter 3 • Is this student correct? Explain. • “347 + 59: add one to each number and get 348 + 60 = 408.” • No: 347 + 59 is the same as 346 + 59 because 346 + 1 + 60 - 1 = 346 + 60 + 1 - 1, and 1 - 1 = 0. The answer is 406.

  13. Chapter 3 • Is this student correct? • “497 - 39 = 497 - 40 - 1 = 457 - 1 = 456.” • No, the student is not correct because 497 - 39 = (497) - (40 - 1) = (497) - 40 + 1 = 458. An easier way to think about this is 499 - 39 = 460, and then subtract the 2 from 499, to get 458.

  14. Chapter 3 • Is this student correct? • “390 - 27 is the same as 300 - 0 + 90 - 20 + 0 - 7. So, 300 + 70 + -7 = 370 + -7 = 363.” • Yes, this student is correct. This is analogous to 390 = 380 + 10 = 27; 300 - 0 + 80 - 20 + 10 - 7 = 300 + 60 + 3. Note: to avoid this negative situation, we regroup.

  15. Chapter 3 • Multiply 39 • 12 using at least 5 different non-traditional strategies. • Lattice Multiplication • Rectangular Array/Area Model • Egyptian Duplation • Repeated Addition, Use a Benchmark • 39 • 10 + 39 • 2 • 40 • 12 - 1 • 12 • 30 • 10 + 9 • 10 + 30 • 2 + 9 • 2 = (30 + 9)(10 + 2)

  16. Chapter 3 • Divide 259 ÷ 15 using at least 5 different strategies. • Scaffold • Repeated subtraction • Repeated addition • Use a benchmark • Partition (Thomas’ strategy)

  17. Chapter 3 • Models for addition: • Put together, increase by, missing addend • Models for subtraction: • Take away, compare, missing addend • Models for multiplication: • Area, Cartesian Product, Repeated addition, measurement, missing factor/related facts • Models for division: • Partition, Repeated subtraction, missing factor/related facts

  18. Chapter 3 • Vocabulary: • Addition: addend + addend = sum • Subtraction: minuend - subtrahend = difference • Multiplication: multiplier • factor = product • Division: dividend ÷ divisor = quotient • Dividend = quotient • divisor + remainder

  19. • • • • • • Chapter 4 • An odd number: • An even number:

  20. Chapter 4 • Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, … 2 factors • ONE IS NOT PRIME. • Composite numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, … at least 3 factors • Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, … an odd number of factors

  21. Chapter 4 • Prime factorization: many ways to get the factorization, but only one prime factorization for any number. • Find the prime factorization of 84. • 2 • 2 • 3 • 7, or 22 • 3 • 7

  22. Chapter 4 • Greatest Common Factor: The greatest number that can divide evenly into a set of numbers. • The GCF of 50 and 75 is 25. • You try: Find the GCF of 60, 80, and 200. • 20: 60 = 20 • 3, 80 = 20 • 4, 200 = 20 • 10.

  23. Chapter 4 • The Least Common Multiple is the smallest number that is divisible by a set of numbers. • The LCM of 50 and 75 is 150. • You try: Find the LCM of 60, 80, and 200. • 1200: 60 • 20 = 1200, 80 • 15 = 1200, 200 • 6 = 1200.

  24. Chapter 4 • What is the largest square that can be used to fill a 6 x 10 rectangle? • 2 x 2: You can draw it to see why. (Which is involved here, GCF or LCM?)

  25. Language Ch. 3-4 • Explain mathematically what we mean when talking about operations using an example: • Carry (+) • Borrow (–) • (Does or does not) go into (÷) • Bring down (÷) • Move over (x)

  26. Chapter 5 • Fractions models:Part of a wholeRatioOperatorQuotient • Make up a real-world problem for each model above for 6/10.

  27. Chapter 5 • Name the model for each situation of 5/6. • I have 5 sodas for 6 people--how much does each person get? • Out of 6 grades, 5 were As. • I had 36 gumballs, and I lost 5/6 of them. How many are left? • In a room of students, 50 wore glasses and 10 did not wear glasses. • Answers: quotient (5/6 soda per person); part-whole (5/6 As); operator (5/6 of 36 gumballs, or 30 are gone--6 remain); ratio (5:1)

  28. Chapter 5 • There are three ways to represent a fraction using a part of a whole model:part-wholediscrete,number line (measurement) • Represent 5/8 and 11/8 using each of the pictorial models above.

  29. Chapter 5 • If represents 12/25, show what 2 will look like.

  30. Chapter 5 • If these rectangles represent 12 of something, then each rectangle represents 3 of something. One third of a rectangle represents the unit fraction. • So, we need to show 50/25, which is 16 full rectangles, and 2/3 of another rectangle.

  31. Chapter 5 • Errors in comparing fractions: 2/6 > 1/2 • Look at the numerators: 2 > 1 • Two pieces is more than one piece. • Look at the denominators: 6 > 2 • We need 6 to make a whole rather than 2. • There are more pieces not shaded than shaded. • If we look at what is not shaded, then there are more unshaded pieces. The pieces are smaller in sixths than in halves.

  32. Chapter 5 • Appropriate ways to compare fractions: • Rewrite decimal equivalents. • Rewrite fractions with common denominators. • Place fractions on the number line. • Sketch parts of a whole, with the same size whole

  33. Chapter 5 • More ways to compare fractions: • Compare to a benchmark, like 1/2 or 3/4. • Same numerators: a/b > a/(b + 1) 2/3 > 2/4 • Same denominators: (a + 1)/b > a/b 5/7 > 4/7 • Look at the part that is not shaded: 5/9 < 8/12 because 4 out of 9 parts are not shaded compared with 4 out of 12 parts not shaded. • Multiply by a form of 1.

  34. Chapter 5 • Compare these fractions without using decimals or common denominators. • 37/81 and 51/90691/4 and 791/7200/213 and 199/2147/19 and 14/39 • <; >; >; >

  35. Chapter 5 • Remember how to compute with fractions. Explain the error: • 2/5 + 5/8 = 7/13 • 3 4/7 + 9/14 = 3 13/14 • 2 7/8 + 5 4/8 = 7 11/8 = 8 1/8 • 5 4/6 + 5/6 = 5 9/6 = 5 1/2

  36. Chapter 5 • Explain the error: • 3 - 4/5 = 2 4/5 • 5 - 2 1/7 = 3 6/7 • 3 7/8 - 2 1/4 = 1 6/4 = 2 1/2 • 9 1/8 - 7 3/4 = 9 2/8 - 7 6/8 = 8 12/8 - 7 6/8 = 1 4/8 = 1 1/2

  37. Chapter 5 • Explain the error: • 3/7 • 4/9 = 7/16 • 2 1/4 • 3 1/2 = 6 1/8 • 7/12 • 4/5 = 35/48 • 4/7 • 3/5 = 20/35 • 21/35 = 420/1225 = 84/245 = 12/35

  38. Chapter 5 • Explain the error: • 3/5 ÷ 4/5 = 4/3 • 12 1/4 ÷ 6 1/2 = 2 1/2

  39. Chapter 5 • Decimals: • Name a fraction and a decimal that is closer to 4/9 than 5/11. • 4/9 = 0.44…; 5/11 = 0.4545… ex: 0.44445 is closer to 4/9 than 5/11 • Explain what is wrong: • 3.45 ÷ .05 = 0.0145928… • This is 0.5 ÷ 3.45. If we divide by a number less than one, than our quotient is bigger than the dividend.

  40. Chapter 5 • True or false? Explain. • 3.69/47 = 369/470 • 5.02/30.04 = 502/3004 • Multiply by 1, or n/n.3.69/47 = 36.9/470, not 369/470;5.02/30.04 = 502/3004.

  41. Chapter 5 • Order these decimals: • 3.95, 4.977, 3.957, 4.697, 3.097 • 3.097, 3.95, 3.957, 4.697, 4.977 • Round 4.976 to the nearest tenth. Explain in words, or use a picture. • Is 4.976 closer to 4.9 or 5.0? Put on a number line, and see it is closer to 5.0.

  42. Chapter 6 • An employee making $24,000 was given a bonus of $1000. What percent of his take home pay was his bonus? • 1000/25,000 = x/100 • 100,000 = 25,000x x = 4%

  43. Chapter 6 • Which is faster? • 11 miles in 16 minutes or 24 miles in 39 minutes? Explain. • Use the rate miles/minutes. Then11miles/16 minutes compared to 24/39.0.6875 miles per minute > 0.6153… miles per minute. So the first rate is faster, or more miles per minute.

  44. Chapter 6 • Ryan bought 45 cups for $3.15. “0.07! That’s a great rate!”What rate does 0.07 represent?Describe this situation with a different rate--and state what this different rate represents. • $3.15/45 cups = $0.07 per cup. Another rate would be 45/$3.15 = 14.28 cups per dollar.

  45. Chapter 6 • Which ratio is not equivalent to the others?(a) 42 : 49 (b) 12 : 21(c) 50.4 : 58.8 (d) 294 : 343 • (b)

  46. Chapter 6 • Write each rational number as a decimal and a percent.3 4/5 1/11 2 1/3 • 3: 3.0 or 3, 300%4/5: 0.8, 80%1/11: 0.09…, 9.09%2 1/3: 2.3, 233.3%

  47. Chapter 6 • Write each decimal as a fraction in simplest form and a percent.4.9 3.005 0.073 • 4.9: 4 9/10; 490%3.005: 3 5/1000 = 3 1/200; 300.5%0.073: 73/1000; 7.3%

  48. Chapter 6 • Write each percent as a fraction and a decimal.48% 39.8% 2 1/2% 0.841% • 48%: 48/100 = 12/25; 0.4839.8%: 39.8/100 = 398/1000 = 199/500; 0.3982 1/2% = 2.5%: 2.5/100 = 25/1000 = 1/40; 0.0250.841%: 841/100000; 0.00841

  49. Chapter 6 • A car travels 60 mph, and a plane travels 15 miles per minute. How far does the car travel while the plane travels 600 miles? • (Hint: you can set up one proportion, two proportions, or skip the proportions entirely!) • Answer is the car travels 40 miles--the car travels 1 miles for each 15 miles the plane travels. 1/15 = x/600.

  50. Chapter 6 • DO NOT set up a proportion and solve: use estimation instead. • (a) Find 9% of 360. • (b) Find 5% of 297. • (c) Find 400% of 35. • (d) Find 45% of 784.

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