Review for the FinalReview for the Final

302A, Fall, 2007302A, Fall, 2007

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, 1.4, 1.7; 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, 5.14, 5.15, 5.16, 6.3, 6.4, 6.5, 6.7

• From Class Notes: Describe the strategies used by the students--don’t need to know the names.

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

Chapter 1• Fred Flintstone always says

“YABBADABBADO.” If he writes this phrase over and over, what will the 246th letter be?

• D

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.

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.

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.

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.

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.

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.

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.

Chapter 3• Multiply 39 • 12 using at least 5 different

strategies. • Lattice Multiplication• Rectangular Area• Egyptian Duplation• 39 • 10 + 39 • 2• 40 • 12 - 1 • 12• 30 • 10 + 9 • 10 + 30 • 2 + 9 • 2 =

(30 + 9)(10 + 2)

Chapter 3• Divide 259 ÷ 15 using at least 5

different strategies.• Scaffold• Repeated subtraction• Repeated addition• Use a benchmark• Partition (Thomas’ strategy)

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• Models for division:• Partition, Repeated subtraction, missing factor

Chapter 4• An odd number:

• An even number:

• • •

• • •

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

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

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.

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.

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?)

Chapter 5• Fractions models:

Part of a wholeRatioOperatorQuotient

• Make up a situation for 6/10 for each of the models above.

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, but I lost 6 of them. What

fraction describes what is left?• In a room of students, 50 wore glasses and

10 did not wear glasses.

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 above pictorial models.

Chapter 5• Errors in comparing fractions: 2/6 > 1/2

• Look at the numerators: 2 > 1

• Look at the denominators: 6 > 2

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

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.

Chapter 5• Compare these fractions without using

decimals or common denominators.

• 37/81 and 51/90

• 691/4 and 791/7

• 200/213 and 199/214

• 7/19 and 14/39

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

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

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

Chapter 5• Explain the error:

• 3/5 ÷ 4/5 = 4/3

• 12 1/4 ÷ 6 1/2 = 2 1/2

Chapter 5• Decimals:

• Name a fraction and a decimal that is closer to 4/9 than 5/11.

• Explain what is wrong:

• 3.45 ÷ .05 = 0.69

Chapter 5• True or false? Explain.

• 3.69/47 = 369/470

• 5.02/30.04 = 502/3004

Chapter 5• Order these decimals:

• 3.95, 4.977, 3.957, 4.697, 3.097

• Round 4.976 to the nearest tenth. Explain in words, or use a picture.

Chapter 6• An employee making $24,000 was

given a bonus of $1000. What percent of his salary was his bonus?

• 1000/24,000 = x/100

• 100,000 = 24,000x x ≈ 4.17%

Chapter 6• Which is faster?

• 11 miles in 16 minutes or 24 miles in 39 minutes? Explain.

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.

Chapter 6• Which ratio is not equivalent to the

others?

• (a) 42 : 49

• (b) 12 : 21

• (c) 50.4 : 58.8

• (d) 294 : 357

Chapter 6• Write each rational number as a

decimal and a percent.• 3• 4/5• 1/11• 2 1/3

Chapter 6• Write each decimal as a fraction in

simplest form and a percent.

• 4.9

• 3.005

• 0.073

Chapter 6• Write each percent as a fraction and a

decimal.

• 48%

• 39.8%

• 2 1/2%

• 0.841%

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.

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.

Chapter 6• DO NOT set up a proportion and solve: use

estimation instead.• (e) What percent of 80 is 39?

(f) What percent of 120 is 31?(g) 27 is what percent of 36?(h) 87 is 20% of what number?

• Now, go back and set up proportions to find the exact values of (a) - (h). Were you close?

Chapter 6• David has 150 mg of fools’ gold. Find

the new amount if:

• He loses 30%?

• He increases her amount by 90%?

• He decreases her amount by 40%?

• Study Hard, and show up on time!