Math 103 Lecture 6 Notes page 1
Math 103 Lecture 6 Notes:
A Tale of Two Rain Totals:
In Urbana, fussy weatherman Felix Fractions noted this week’s precipitation as:
5/12”, 1/6”, 3/8”, 45/9”, 0”, and 6/11”.
What was the total precipitation in Urbana for the week?
In Champaign, weatherman Dag Decimal, a devil for detail, noted the week’s precipitation as:
0.42” 0.17”, 1.29”, 0.38”, 0.44”, 0”, 0.55”.
What was the total precipitation in Champaign for the week”?
Which got more rain for the week, Urbana or Champaign?
In 1584 Simon Stevin, a quartermaster in the Dutch army, wrote La Thiende (The Tenth), a work that
gave rules for computing with decimals. He not only stated rules for decimal computations, but also
suggested practical applications for decimals and recommended that his government adopt a system
similar to the metric system. To show place value, Stevin used circled numerals between digits. For
example, he wrote 0.4789 as 4 1 7 2 8 3 9 4.
Even today, there is no universally accepted form of writing a decimal.
-
International form of decimals –
The NCTM Standards for grades 3-5 state:
"All students should develop understanding of fractions as parts of unit wholes, as parts of a collection,
as locations on number lines, and as divisions of whole numbers; use models, benchmarks, and equivalent
forms to judge the size of fractions."
"All students should recognize and generate equivalent forms of commonly used fractions, decimals and
percents."
"As students solve problems in context, students also can consider the advantages and disadvantages of
various representations of quantities. For example, students should understand not only that 15/100,
3/20, 0.15, and 15% are all representations of the same number but also that these representations may
not be equally suitable to use in a particular context. For example, it is typical to represent a sales
discount as 15%, the probability of winning a game as 3/20, a fraction of a dollar in writing a check as
15/100, and the amount of the 5% sales tax added to a purchase of $2.98 as $0.15."
From the ISAT Math Materials, grade 3
1. Joe bought a drink for $.69 and gave the clerk $1.00. How much change should he receive?
From the ISAT Math Materials, grade 5
2. A waiter receives a tip that equals 15% of a customer's bill. If the meal costs $6.50, about how
much is the tip?
Math 103 Lecture 6 Notes page 2
Because decimals look similar to the familiar whole numbers, it seems reasonable to predict that
children might understand them without much difficulty. However, appearance is deceiving. The
research on learning decimals agrees on one point: there is a lack of conceptual understanding.
NAEP found that only about half of the eighth graders tested were able to correctly identify the
fraction closest to 0.52.
Similarly, the results to the following item were disappointing: George buys two calculators that cost
$3.29 each. If there is no tax, how much change will he receive from a $10 bill? Only 21% of the fourth
graders were able to correctly answer this question.
The following two exercises help children to develop meaning of decimal numbers by connecting them to
whole numbers.
1. Begin by asking children to respond to the following:
How many hundreds in 66 600?
How many tens in 6660?
How many ones in 666?
How many tenths in 66.6?
How many hundredths in 6.66?
How many thousandths in 0.666?
2. Have children read and write numbers in the following way:
444 000
44 400
4 440
444
44.4
4.44
0.444
Reading numbers as describes helps to reinforce the fact that no matter where the decimal point is,
each digit is 10 times as much as the digit on its right and one-tenth of the digit on its left.
The common practice of reading decimals by naming the digits rather than expressing them properly as
decimal fractions thwarts the ability to view decimals as fractional numbers.
- examples –
Children who understand fractions prior to studying decimals should be able to make connections
between the two systems of representing rational numbers.
ex:
8.45 –> 8 and 0.4 + 0.05 –> 8 and 0.45
8 and 4/10 +5/100 –> 8 and 45/100
8 45/100
What is the center of the decimal system, the decimal point or the ones place?
List the ways the decimal 2.86 can be read:
Math 103 Lecture 6 Notes page 3
See table in Billstein showing expanded fraction form of decimals:
Another factor that can contribute to the difficulty in determining the value of a decimal number is the
lack of concrete experiences that children have when working with decimal numbers.
See figure 12-7 Cathcart (adding mixed decimals with base-ten blocks)
Algorithm for addition of terminating decimals:
2.16 + 1.73 = (2 + 1/10 + 6/100) = (1 + 7/10 + 3/100)
= (2 + 1) + (1/10 + 7/10) + (6/100 + 3/100)
= (3 + 8/10 + 9/100
= 3.89
Algorithm for multiplying terminating decimals:
(4.62) (2.4)
= 462/100 x 24/10 = 462/102 x 24/10 = 11088/103 = 11.088
If there are n digits to the right of the decimals point in one number and m digits to the right of the
decimals point in a second number, multiply the two numbers, ignoring the decimals, and then place the
decimal point so that there are n + m digits to the right of the decimal point in the product.
n
m
n+m
Note: there are n + m digits to the right of the decimal point in the product because 10 • 10 = 10
See Figure 12-10 and 12-11 Cathcart (area model for multiplying decimals)
Mental computation with decimals:
1. "Breaking and bridging"
1.5 + 3.7 + 4.48
4.5 + 0.7 + 4.48
5.2 + 4.48
9.2 + 0.48
9.68
2. "Using compatible numbers"
7.91
3.85
4.09
0.15
3. "Making compatible numbers"
9.27
= 9.25 + 0.02
3.79
= 3.75 + 0.04
13.00 + 0.06 = 13.06
Math 103 Lecture 6 Notes page 4
4. "Balancing with decimals in subtraction"
4.63
– 1.97
=
4.63 + 0.03 = 4.66
= – (1.97 + 0.03) = –2.00
2.66
5. "Balancing with decimals in division"
0.25√8 -≥ 1√32
Rounding Decimals:
7.456 to the nearest hundredth
7.456 to the nearest tenth
7.456 to the nearest one
7456 to the nearest one
7456 to the nearest ten
7456 to the nearest hundred
7456 to the nearest thousand
Scientific notation: a positive number is written as the product of a number greater than or equal to 1
and less than 10 and an integer power of 10.
8
10
–6
Ex :
8.3•10
1.2•10
7.84•10
5
4
Not in scientific notation:
0.2•10
823.6•10
*Calculators use EE key to represent numbers in scientific notation.
Write in scientific notation: 413,600,000
0.0000234
–5
Convert to standard notation: 6.84•10
3.12•10
7
Rational numbers
Divide to change to a decimal: 1/5
3/4
3/8
7/16
What do you notice about the remainder in each case? What happens to the decimal as a result?
Divide to change to a decimal: 7/9
4/15 1/22 5/27
What do you notice about the remainder in each case? What happens to the decimal as a result?
Rational numbers are either terminating or repeating decimals.
Change these decimals to fractions: 0.95
0.666…
0.818181…
0.8333…
How can you tell if a fraction will terminate, repeat forever, or never repeat without going thought all of
the work of diving it out?
Math 103 Lecture 6 Notes page 5
Irrational numbers are non-terminating, non-repeating decimals.
The discovery of irrational numbers by members of the Pythagorean Society is one of the greatest
events in the history of mathematics. This discovery was very disturbing to the Pythagoreans, who
believed that everything depended on whole numbers, so they decided to keep the matter secret. One
legend has it that Hippasus, a society member, was drowned because he relayed the secret to persons
outside the society. In 1525 Christoff Rudolff, a German mathematician, became the first to use the
symbol √, for a radical or a root.
Tree diagram for Real-number hierarchy
What kind of number? knowing that there are 3 types of decimals and that they are either rational or
irrational, classify each as rational or irrational:
• terminating decimal
• repeating decimal
• infinitely non-repeating non-terminating decimal
• √2/3 and where would you place this number on the number line?
Unlike rational numbers, all irrational numbers can be expressed as infinitely non-repeating decimals.
Ex: √2 can be represented by the non-terminating, non-repeating decimal 1.414213562…
Ex: π = 3.141592653…
Decimals provide a convenient way of expressing irrational numbers as well as fractions. Without
decimals, it would be difficult to compute with irrational numbers. Discussing square roots of nonperfect squares is a natural introduction to the topic of irrational numbers. How do we know irrationals
won’t terminate or repeat? Mathematicians can construct a proof that numbers like √2 and √3 are not
rational. The proof for √2 is sometimes examined in middle-school mathematics and goes like this.:
2
2 2
2
Suppose √2 is rational. Then it can be expressed in the form a/b. Thus, (a/b) = 2, a /b = 2, and a =
2
2
2
2b . According the fundamental theorem of arithmetic, the equivalent natural numbers a and 2b
2
2
must have the same prime factorization. A square such as a and b must have an even number of prime
2
factors (4=2x2, 9 =3x3, 25=5x5). The number 2b has an additional prime factor 2 and thus, an odd
2
2
2
number of prime factors. Therefore, a can’t be equal to 2b because a has an even number of prime
2
factors while 2b has an odd number of prime factors.
A percent is a part-to-whole ratio that has 100 as its second term.
Ex:
7 percent
7 parts per hundred
Math 103 Lecture 6 Notes page 6
7 per hundred
7/100
7 : 100
7%
A 7% tax on a $100 item is $7.
The $100 is the referred to as the base.
The 7% is referred to as the rate.
The $7 is referred to as the percentage.
Consider these:
Joe’s batting average = 0.300
Mel’s grade on her test was 30/100
The sale was 30% off of suits
What do these mean? Are they good or bad?
Applications with percent:
1. Finding a percent of a number
2. Finding what percent one number is of another
3. Finding a number when a percent of the number is known
Mental math with percent:
1. Using fraction equivalents
ex. Find 20% of 80.
2. Using a known percent
ex. Find 55% of 62.
Estimation with percent:
ex. Figure the 15% tip on a dinner costing $6.99.
- Zorkaian T.V. Programming example Recall, a proportion = two ratios which are equal, i.e., a comparison of 2 ratios
Property of proportions:
If a, b, c, and d are real numbers and b ≠0 and
d ≠ 0, then a/b = c/d if and only if ad = bc. (product of means = product of extremes)
-
Application of proportions to attendance grade in this course example
Thinking About Percent – Conceptual Basis
1. Unfair Grade: Kassandra was puzzled about the grade on her spelling test. “I only got 1 wrong of
the 20 words tested, so shouldn’t my grade be 99% rather than 95%?” she aked her teacher. What is
the basis of Kassandra’s argument? Is she correct?
Math 103 Lecture 6 Notes page 7
2. Costly Price War: Mr Allman was fed up with Mr. Bandoxz’s prices. Now his competitor across the
street was having a clearance sale for 80% off. Mr Allman decided he was going to oudo Mr. Bandosz
this time by havinga sale for 100% off. Evaluate Mr. Allman’s marketing strategy.
3. Pay Raise: Mr Bogle wasa salesman for the James Game Company. His commission was 12.75%. He
thought it was a good time to ask for a higher commission because he had just had an outstanding year
selling games. Besides, he was about to marry the daughter of the owner of the company. Mr. James,
the owner, loved playing games and taking risks, and agreed to consider changing Mr. Bogle’s slary if he
was willing to play a game. Mr. James agreed to insert a 0 into Mr. Bogles’ percent of commission. If Mr.
Bogle chose a heart from a deck of playing cards, then the 0 would be placed in position A. If he chose
a diamond, the 0 would be in position B. If he chose a club, the 0 would be in position C. If he chose a
spade, the 0 would be in position D. What are the financial implications for Mr. Bogle of each
possibility?
A
B
C
D
1
2
. 7
5
Representing Percentages Less Than One:
Consider how you could represent 1/2% on a 10x10 grid.
Representing Percentages Less Than One:
How could 125% and 350% be represented with 10x10 grids of base-ten blocks?
Activities for Developing Intuitive Feel for Percent:
Estimating Percentages:
1. A light green Cuisenaire rod covers approximately
what % of a black rod?
2.
A triangular pattern block covers approximately what percent of a hexagonal pattern block?
3.
Try to estimate the answer in no more than 15 seconds: The County Boy Scout Council has a
total membership of 417 boy scouts. If 77% attended the national camporee, how many
Scouts went?
4.
Isaiah mixed 10 qts. of 10% salt solution with 5 qts. of a 20% salt solution. Which of the
following is the best estimate of % concentration of salt solution?
720% 20% 15-19% 15% 11-14% 10% <10%
Problems to help develop percent concepts:
1. Prison Overcrowding: According to a newspaper account, the Illinois Department of Corrections was
holding more than 30,000 prisoners – nearly 50% over capacity. What approximately was the capacity
of the Illinois prison system at the time?
2. Eight seconds is what percent of 4 hours?
Math 103 Lecture 6 Notes page 8
3. A Dismal Performance: On the Fourth NAEP 21% of the seventh graders did not respond to the test
item that required writing 0.9% as a decimal. Of those responding, only 25% answered the item
correctly. What percent of the seventh graders taking the test answered the item correctly?
4. Disappointing Improvement: Mr. Finster was handing back the monthly tests to his class. When he
got to Maurice, he said, “The good news is your grade improved 20% over last month’s grade – the
largest increase percentage-wise in the class. The bad news is that this represents an improvement of
only 5 points.” What was Maurice’s new test grade?
5. Crowded Class: Mr. Arning was a popular professor on campus. On the first day of class, 175 students
showed up for his course. This was 125% of the capacity of the classroom. What was the capacity of
the classroom?
6. Comparative Wealth: Morley is bringing 33 1/3% more money on the class trip than Leslie. Leslie has
what percent less money than Morley?
7. A Smaller Square: If each side of a square is decreased by 10%, by what percent is the area of the
square decreased?
8. Setting an Auction Price: Francine, a stamp collector brought a stamp at the bargain price of $120.
Soon afterward, she found the same stamp in much better condition and decided to sell her less
desirable bargain item. If she wanted to break even and a stamp auctioneer’s commission was 20%, what
minimum bid should Francine specify for the stamp? (In other words, what is the least amount the
stamp can be sold for if Francine is to get back her $120 and the auctioneer gets his 20% commission?)
9. Copying Complications: Dr. Ying needed to reduce a figure to half its size. Unfortunately, his copying
machine had the capacity of reducing figures by 1% increments down to 64% but not lower. Dr. Ying
immediately recognized that he would have to reduce his figure twice, but he did not know by how much
each time. The first copy of his figure should be reduced by how much? The copy of this copy should
be reduced by how much? Is it possible to get a copy that is exactly 50% smaller than the original
10. Depreciation Problem: In 1986, $1 = 160 pesates; in 1988, $1 = 115 pesates. By what percent has the
dollar depreciated in respect to the pesata?
11. CD Cash Dilemma: Five area stores ran ads this week offering special deals on CD’s.
Store 1: buy 3 and get 1 free
Store 2: 20% discount on all CD’s purchased
Store 3: buy 2 and get the third one free
Math 103 Lecture 6 Notes page 9
a)
b)
Store 4: buy 3 and get 2 free
Store 5: 25% discount on all CD’s purchased
Store 6: Buy 6 and get 3 free
From which store should you purchase CD’s to get
the best buy? Let’s have fun and assume you have enough money to purchase as many CD’s as
you want.
If you had enough money to buy only 2, which store offers the best buy?
1. Winning Season: The Physical Wrecks baseball Team won 40 games and lost 25. If it had 55 games
left to play, how many of the remaining games must the team win to give it a 60% winning record for the
season?
2. Discounted Twice: The ad said 30% off the price for every car on the lot. The car Darby liked had
the old price, $9840, painted on the windshield. The salesman showed Darby the sale price but Darby
thought it was too high. The salesman realized he had only deducted 10% so he then deducted 20% from
his first sale price. Darby wasn’t sure this was the same as deducting 30%. Is a single discount of 30%
the same as successive discounts of 10% and 20%?
Newspapers and magazines, radio and TV reports, entertainment, and sports are topics of interest to
children and can be used for concept and skill development activities related to ratio and percent.
ex:
3. collect sports cards to explain what statistics such as batting average of .317 means, or a players 3point shooting percent of 46%
4. write part-to-part or part-to-whole ratios based on the number of medals won by a country at the
Olympic Games
5. TV ratings
6. Surveys on soft-drink preference
- JOKE If we could shrink the Earth’s populations to a village of precisely 100 people, with all existing human
ratios remaining the same, it would look like this:
57 Asians
21 Europeans
14 from Western Hemisphere (North and South America)
8 Africans
70 would be non-white and 30 would be white
50% of the entire world wealth would be held by 6 people
70 would be unable to read
50 would suffer from malnutrition
80 would live in sub-standard housing
only 1 would have a college education . . .