April 21, 2009 “Energy and persistence conquer all things.” ~Benjamin Franklin.
-
date post
21-Dec-2015 -
Category
Documents
-
view
214 -
download
0
Transcript of April 21, 2009 “Energy and persistence conquer all things.” ~Benjamin Franklin.
42510011 0010 1010 1101 0001 0100 1011
April 21, 2009
“Energy and persistence conquer all things.”
~Benjamin Franklin
4251
0011 0010 1010 1101 0001 0100 1011
5.4 – Decimals
• Many decimal numbers are rational numbers, but some are not.
• A decimal is a rational number if it can be written as a fraction. So, those are decimals that either terminate (end) or repeat are rational numbers.
• Repeating decimals: 7.6666…; 0.727272…
• Terminating decimals: 4.8; 9.00001; 0.75
4251
0011 0010 1010 1101 0001 0100 1011
5.4 (cont’d)
• A decimal like 3.56556555655556555556… is not rational because although there is a pattern, it does not repeat. It is irrational.
• Compare this to 3.556556556556556556…It is rational because 556 repeats.
• All rational numbers can be represented by terminating or repeating decimals!
4251
0011 0010 1010 1101 0001 0100 1011
5.4 (cont’d)
Another look at place value:
We know that the number 123,456 represents one 100,000 plus two 10,000’s plus three 1,000’s plus four 100’s plus five 10’s plus six 1’s or
1×100,000 + 2×10,000 + 3×1,000 + 4×100 + 5×10 + 6×1
4251
0011 0010 1010 1101 0001 0100 1011
5.4 (cont’d)
What does the number 1.234 mean?
1 is in the ones place
2 is in the “tenths” place (not the tens place!)
3 is in the “hundredths” place
4 is in the “thousandths” place
4251
0011 0010 1010 1101 0001 0100 1011
5.4 (cont’d)
From our work with fractions, you should recognize the difference between “tens” and “tenths”:
Each “ten” represents 10 ones, while each “tenth” is one of the 10 equal pieces that one whole was divided into.
4251
0011 0010 1010 1101 0001 0100 1011
5.4 (cont’d)
Let = 1
Then the shaded area below represents a “tenth”:
And the figures on the next slide make a “ten”:
4251
0011 0010 1010 1101 0001 0100 1011
5.4 (cont’d)
In symbols, a “tenth” is 1/10.
In decimal form, 1/10 = 0.1
Similarly, a “hundredth” is 1/100, and a “thousandth” is 1/1000.
In decimal form, 1/100 = 0.01 and 1/1000 = 0.001
Can you see how base 10 blocks can be used to visualize these?
4251
0011 0010 1010 1101 0001 0100 1011
5.4 (cont’d)
When decimals are equal:3.56 = 3.56000000
But, 3.056 ≠ 3.560.
To see why, examine the place values.
3.056 = 3 + 0 × .1 + 5 × .01 + 6 × .001
whole tenths hundredths thousandths
3.560 = 3 + 5 × .1 + 6 × .01 + 0 × .001
4251
0011 0010 1010 1101 0001 0100 1011
5.4 (cont’d)
Ways to compare decimals:• Write them as fractions and compare the fractions
as we did in the last section.• Use base-10 blocks.• Write them on a number line.• Line up the place values.
4251
0011 0010 1010 1101 0001 0100 1011
Exploration 5.16Do #1, 2, 4, 7 and 8.
For #8, draw a picture of blocks to represent each decimal.
4251
0011 0010 1010 1101 0001 0100 1011
5.4 (cont’d)
Rounding
3.784: round this to the nearest hundredth.• Look at the hundredths.• Well, 3.784 is between 3.78 and 3.79. On the
number line, which one is 3.784 closer to?• 3.785 is half way in between.
3.78 3.785 3.79
4251
0011 0010 1010 1101 0001 0100 1011
5.4 (cont’d)
Rounding
So, is 3.784 closer to 3.78 or 3.79?
3.78 3.785 3.79
4251
0011 0010 1010 1101 0001 0100 1011
5.4 (cont’d)
Practice Rounding:• Round to the nearest tenth: 5.249
– Closer to 5.2 or 5.3?• Round to the nearest hundredth: 5.249
– Closer to 5.24 or 5.25?• Round to the nearest whole: 357.82
– Closer to 357 or 358?• Round to the nearest hundred: 357.82
– Closer to 300 or 400?
4251
0011 0010 1010 1101 0001 0100 1011
5.4 (cont’d)
Practice Rounding:• Round to the nearest thousandth: 5.0099
– 5.010 Must have the last 0 for the thousandths place!
• Round to the nearest hundredth: 64.284– 64.28
• Round to the nearest tenth: 10.957– 11.0 Must have the last 0 for the tenths place!
4251
0011 0010 1010 1101 0001 0100 1011
5.4 (cont’d)
Adding and subtracting decimals:
• Same idea as with fractions: the denominator (place values) must be common.
• So, 3.46 + 2.09 is really like3 + 2 ones + 4 + 0 tenths + 6 + 9 hundredths = 5.55
4251
0011 0010 1010 1101 0001 0100 1011
5.4 (cont’d)
Multiplying decimals:
(Easiest to see with the area model)
2.1 × 1.3
Where is 2 × 1?2 × 0.3?1 × 0.1?0.1 × 0.3?
1 + 1 + .11
+
.3
4251
0011 0010 1010 1101 0001 0100 1011
5.4 (cont’d)
Dividing decimals:
Find the quotient
7.8 ÷ 3.12
What steps did you take?
Why do they work?