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

• From Class Notes: Describe/Use 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 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

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?)

• 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.

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 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)

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/related facts• Models for division:• Partition, Repeated subtraction,

missing factor/related facts

Chapter 3• Vocabulary:• Addition: addend + addend = sum• Subtraction: minuend - subtrahend =

difference• Multiplication: multiplier • factor = product• Division: dividend ÷ divisor = quotient• Dividend = quotient • divisor + remainder

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?)

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)

Chapter 5• Fractions models:

Part of a wholeRatioOperatorQuotient

• Make up a real-world problem for each model above for 6/10.

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)

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.

Chapter 5• If represents

12/25, show what 2 will look like.

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.

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.

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.

– Multiply by a form of 1.

Chapter 5• Compare these fractions without using

decimals or common denominators. • 37/81 and 51/90

691/4 and 791/7200/213 and 199/2147/19 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.• 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.

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.

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.

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%

Chapter 6• Which is faster?• 11 miles in 16 minutes or 24 miles in 39

minutes? Explain.• Use the rate miles/minutes. Then

11miles/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.

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.

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)

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%

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%

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

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• Iga Tahavit has 150 mg of fools’ gold.

Find the new amount if:• She loses 30%?• She increases her amount by 90%?• She decreases her amount by 40%?• 105 mg; 285 mg; 90 mg

Percent & Proportion Questions

• In Giant World, a giant tube of toothpaste holds one gallon. If a normal tube of toothpaste holds 4.6 ounces and costs $2.49, how much should the giant tube cost?

• One gallon is 128 ounces.Ounces = 4.6 = 128 Dollars $2.49 x4.6x = 128 • 2.49 About $69.29

Estimate• In Giant World, a giant tube of toothpaste

holds one gallon. If a normal tube of toothpaste holds 4.6 ounces and costs $2.49, how much should the giant tube cost?

• If we round, we can think: 4 ounces is about $2.50. Since we want to know how much 128 ounces is, think: 4 • 32 = 128, so $2.50 times 32 is $80. (or, $2.50 • 30 = $75)

Try this one• The admissions department currently

accepts students at a 7 : 3 male/female ratio. If they have about 1000 students in the class, how many more females would they need to reduce the ratio to 2 : 1?

• Currently: 7x + 3x = 1000, so x = 100; 700 males and 300 females. To keep 100 students in class, they want 2y + 1y = 1000, so y = 333; 666 males and 333 females. They need to accept 333 - 300 = 33 more females to achieve this ratio. OR, with 700 males, they need 350 females, or 50 more.

Try this one• Lee’s gross pay is $1840 per paycheck, but

$370 is deducted. Her take-home pay is what percent of her gross pay?

• Part = percent = 370 = x Whole 100 1840 100

• 370 • 100 = 1840x; About 20% is taken out, so about 80% for take-home pay.

• Could also do: 1840 - 370 = 1470: 1470 = x 1840

100

Last one• Estimate in your head:• 16% of 450• 10% of 450 = 45; 5% = 22.5, about 67.5 OR

10% of 450 = 45; 1% of 450 = 4.5, or about 5; 6 • 1% = 6 • 5 = 30; 30 + 45 = 75.

• 123 is approximately what percent of 185?• Approximate: 120 is approximately what

percent of 200; 120/200 = 60/100, so about 60%.

Good Luck!• Remember to bring pencil, colored pencils or

markers, and calculator to the exam. • Study hard!• Show up on time!!!

– 11:00 - 12:15 section: 16 Dec at 11:00 AM– 12:30 - 1:45 section: 18 Dec at 11:00 AM