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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, 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, 5.15, 5.16, 6.3, 6.5,
• 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 4legged 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 Array
• Egyptian Duplation
• Lightning-Cross
• 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 whole
Ratio
Operator
Quotient
• 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, 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-whole
discrete,
number line (measurement)
• Represent 5/8 and 11/8 using each of
the pictorial models above.
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.
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
• 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%?
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 x
4.6x = 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. 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.
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!!!