Download Day 30

302A final exam review
Final Exam
• Tuesday, May 12
• 11am – 1pm
• In our usual classroom
• Review session:
Thurs, 5/7 @ 12:30pm here.
Some Answers to 6.1
homework
• 28) We need a constant ratio of men to
women.
9 men
=
x men
4 women
360 women
Solve: 9 • 360 = 4x; x = 810 men now.
• If we want a ratio of 2 : 1, then we want 405
women. We have 360 women, so we need
45 more women to make the ratio 2 : 1.
• 31a)
Let’s draw a picture.
Crew of 4 men
In 5 days: 1 building
In 10 days: 2 buildings
In 15 days: 3 buildings
In 20 days: 4 buildings
• So, a crew of 4 can build 4 buildings in 20
days. Two crews of 4 can build 8 buildings in
20 days. 10 crews of 4 can build 40
buildings in 20 days, so 40 people altogether.
Some Answers to 6.2
homework
• 1a: 4% of 450. Think 1% of 450 is 4.5,
so 4% of 450 is 4 • 4.5, or 18.
• Or, 4% of 450 is close to 5% of 450.
We know that 10%of 450 is 45, and so
5% is half of 10%, so half of 45 is 22.5.
• 1h: 30 is what percent of 35?
• Think: part : whole = percent : 100.
• So, 30/35 = x/100? Well, 30/35 is close
to 30/36, or 5/6, which is about 83%.
• Or, 30/35 is close to 35/40, or 7/8,
which is 87.5%.
• 1p: What is 100% more than 35?
• 100% of 35 is 35, so 100% more is 35
+ 35 = 70.
#8. Two dresses
• $119 with 40% off
• Pay 60%
• 119 • 0.60 = $71.40
• $79.99 with 20% off
• Pay 80%
• 79.99 • 0.80 =
$63.99
• Second dress is
cheaper.
• 30. To find 0.5%, we know that this
means 0.5/100. If we multiply
numerator and denominator by 10, we
get (0.5 • 10)/(100 • 10) = 5/1000, or
1/200.
Exploration 6.7
Do Part 1 #1 with your group.
What is on the final exam?
• 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; 2.8, 2.9; 3.1,
3.3, 3.6, 3.13, 3.15, 3.18, 3.19; 4.2; 5.8, 5.9,
5.10, 5.12, 5.13, 5.14, 5.15; 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
A test 3 problem
• If
= 3/5, carefully draw 1/6.
Two approaches:
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 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 + 60
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 + 90  20  7.
So, 300 + 70  7 = 370 7 = 363.”
• Yes, this student is correct.
Chapter 3
• Multiply 39 × 12 using at least 5 different nontraditional 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 + remainder/divisor
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
• The number 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
a factor of every number in 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, 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); partwhole (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-whole
discrete,
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
791/7
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.
• 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 new 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%:
39.8%:
2 1/2%:
0.841%:
48/100 = 12/25;
39.8/100 = 398/1000 = 199/500;
2.5/100 = 25/1000 = 1/40;
841/100000;
0.48
0.398
0.025
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 original amount by
90%?
• She decreases her originalamount 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 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.
• To keep 1000 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, eraser and
calculator to the exam.
• Study hard!
• Show up on time!
11:00 am – 1:00 pm Tuesday, May 12 (here)