Math 20

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Welcome to Math 20!
Normally class begins with a warm-up problem
here. Since today is the first day of class,
please take a moment until class starts to
introduce yourself to other students sitting near
you.
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Exploration Dollars, Cents, and Bigger Bills
Little Joey and Melinda save up change during the
week. Every Sunday afternoon they walk to the
pizza store to buy bread sticks with change. On
the menu the bread sticks cost 1.49 but Joey and
Melinda think about the price as 149 Switching
between dollar-format to cent-format is easy.
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Exploration Dollars, Cents, and Bigger Bills
Switch these three amounts to cent-format 5.29
1,234.5650 Switch these three amounts to
dollar-format 3,099 42 87,654,321 How do
you know you are right?
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Exploration Dollars, Cents, and Bigger Bills
We have seen the cost of the bread sticks written
in dollar-format and cent-format. 1.49 149
How about dime-format? ten-dollar-bill-forma
t? hundred-dollar-bill-format?
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Guided Activity Powers of Ten
To switch between dollar-format and cent-format
you scoot the decimal point two
places. 32.50 423 This is one case of a
more general rule. What we called dime-format,
and the others, were other examples of this rule.
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Guided Activity Powers of Ten
A number is a power of ten if it belongs to this
pattern 10 1 10
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Guided Activity Powers of Ten
A number is a power of ten if it belongs to this
pattern 10 1 1010 10 100
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Guided Activity Powers of Ten
A number is a power of ten if it belongs to this
pattern 10 1 1010 10 10010 10 10
1,000
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Guided Activity Powers of Ten
A number is a power of ten if it belongs to this
pattern 10 1 1010 10 10010 10 10
1,00010 10 10 10 10,000and so on Find
three more powers of ten using multiplication.
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Guided Activity Powers of Ten
Note that we could also write this pattern as 10
1 1010 10 10010 100 1,00010 1000
10,000and so on In this version of the
pattern we are multiplying ten by the previous
answer.
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Guided Activity Powers of Ten
A number is also a power of ten if it belongs to
this other pattern 10 1 10
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Guided Activity Powers of Ten
A number is also a power of ten if it belongs to
this other pattern 10 1 1010 10 1
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Guided Activity Powers of Ten
A number is also a power of ten if it belongs to
this other pattern 10 1 1010 10 110
100 0.1
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Guided Activity Powers of Ten
A number is also a power of ten if it belongs to
this other pattern 10 1 1010 10 110
100 0.110 1,000 0.01and so on Find
three more powers of ten using division.
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Guided Activity Powers of Ten
So here is what our powers of ten look like ,
0.0001, 0.001, 0.01, 0.1, 1, 10, 100, 1000,
They are easy to recognize. How do we know
these numbers are not powers of
ten? 100.1.0101110
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Guided Activity Powers of Ten
Let's take a second look at these
numbers 100.1.0101110Which are decimals?
How do you know?Do whole numbers also count as
decimals?
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Guided Activity Powers of Ten
Any time we divide or multiply by a power of ten
we can find the answer by scooting the decimal
point. How big the power of ten is determines how
many scoots. Whether we are multiplying or
dividing, and whether the power of ten is bigger
or less than one, determine which direction to
scoot the decimal point.
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Guided Activity Powers of Ten
You can discover the rules for yourself. Try to
work out these problems through common sense.
Don't use a calculator! We'll discuss the
answers. 512.99 100 512.99 100 1,234.56
0.1 1,234.56 0.1
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Guided Activity Powers of Ten
How about these? Again, don't use a calculator
and we will discuss the answers. 67.8 1,000
67.8 1,000 34.5 0.001 34.5 0.001
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Lecture Place Value
There is one more thing to discuss about powers
of ten. The powers of ten have names! These names
are the same names we use for place value (except
that for place value we make them plural).
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Hundred-Thousands Ten-Thousdands Thousands Hundred
s Tens Ones Tenths Hundredths Thousandths Ten-Thou
sandths Hundred-Thousandths
Lecture Place Value
1 2 3 , 4 5 6 . 7 8 9
1 2
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Lecture Place Value
(One) Hundred-Thousand(One) Ten-Thousand(One)
Thousand(One) HundredTenOne(One) Tenth(One)
Hundredth(One) Thousandth(One)
Ten-Thousandth(One) Hundred-Thousandth
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Lecture Place Value
Also, there is an abbreviation used to show that
digits form a repeating pattern. If I write this
I am being vague 0.234 I could mean any of
these three decimals 0.234 0.2342342340.234
0.23434343434 0.234 0.234444444 That's why
we use the bar and not ellipses.
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Lecture Rounding Decimals
40 7 5.742557 40 7 5.7 40 7
5.74 40 7 5.743 Next we want to talk
about how to write decimal answers. But before we
do this we need to review the rules for rounding.
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Lecture Rounding Decimals
40 7 5.742557 Round to the
nearest Tenth 5 . 7 4 2 5 5 7 Hundredth
5 . 7 4 2 5 5 7 Thousandth 5 . 7 4 2 5 5
7 Ten-Thousandth 5 . 7 4 2 5 5 7
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Lecture Rounding Decimal Answers
When we do math (especially with a calculator) we
often get answers that have long decimals. 40 7
5.742557 With how many decimal places should
we write our answers? This question is actually
more about philosophy than mathematics. Let us
consider a few cases. In all these cases we will
show that we changed the answer by using the
about equal sign, which is written like .
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Lecture Rounding Decimal Answers
40 7 5.742557 Usefulness of the
Answer Perhaps I have a 40 inch board, and I need
to cut it into seven equal lengths. When doing
carpentry with a tape measure I can only measure
to tenths or hundredths of an inch. It does not
make sense in this situation to use more decimal
places. 40 7 5.74
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Lecture Rounding Decimal Answers
40 7 5.742557 Indivisible Units Perhaps I
have a six children coming to my kid's birthday
party, and I need to put 40 pieces of candy in
the 7 bags of party favors. I do not want to cut
pieces of candy into pieces. To avoid having
kids argue I should just drop the decimal
entirely. I am not rounding! 40 7 5
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Lecture Rounding Decimal Answers
40.2 7 5.742857 Communicating Accuracy of
Measurement Imagine that a pharmacist has about
40.2 milligrams of a medicine, and divides it
into 7 doses. This measured value was not
notably precise 40.2, not 40.268 or 40.0035.
Our answer should not add more than one decimal
place. We do not want to pretend we have more
accuracy than we do. 40.2 7 5.7 or
40.2 7 5.74
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Lecture Rounding Decimal Answers
So those are our guidelines If the
situation requires a certain number of decimal
places to be useful, round appropriately. If
the amount cannot be divided into meaningful
parts, drop or round the decimal digits
appropriately. Otherwise, communicate how
precisely values were measured by adding no more
than one decimal place.
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Guided Activity More on Rounding
Normally we do not round in the middle of a
problem. By waiting until the end, we avoid
introducing error. Let's consider a few
examples. One of the examples involves percents.
Don't worry about having to do percent math yet
we'll discuss that soon.
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Guided Activity More on Rounding
Example 1 In Eugene it rains, on average, about
1.15 inches per week. 1.15 inches 52 weeks
59.8 inches annually If we rounded, the
difference is noticeable! 1 inch 50 weeks 50
inches annually That difference of 9.8 inches
would matter a lot if you were a farmer, or
planning the sewer system.
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Guided Activity More on Rounding
Example 2 You do assistant contractor work that
earns a little over 1,000 per month after
taxes. You work for 3 months and 1 week, then
have time between jobs, then work for 5 months
and 1 week. (3.25 5.25) 1000.20 8,501.70
8,502 If we rounded first, the difference is
significant! (3 5) 1000 8,000
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Guided Activity More on Rounding
Example 3 You are a realtor who earns a 6
commission. You sell a house for 210,000. 6
210,000 12,600 If we rounded only the
commission 5 210,000 10,500 If we rounded
only the house value 6 200,000 12,000
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Guided Activity More on Rounding
Your turn. Work with a partner to invent a word
problem. Solve it accurately, doing any rounding
at the end. Then round first, and see how much
the answer changes.
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Guided Activity More on Rounding
Rounding first is called estimating. This is
sometimes a helpful thing to do, but as we have
seen will often change the answer a lot! During
future lectures, as we discuss different math
topics, you will see some cases where estimating
is useful even though it changes the answer. In
particular, it can often be a quick thing to do
at the start of a problem to get a rough idea how
big the answer will be.
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Homework
Tonight's homework is to start a handout about
place value, powers of ten, place value,
rounding, and estimating. We will have a total of
six handouts to cover the material of chapters 1
through 4. Once we get to chapter 5 we will use
the textbook homework. This system keeps students
who just went through chapters 1 through 4 in
Math 10 from being too bored.