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April 21, 2009 “Energy and persistence conquer all things.” ~Benjamin Franklin

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April 21, 2009 Section 5.4 – Decimals Exploration 5.16

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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

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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!

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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

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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

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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.

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5.4 (cont’d) Let= 1 Then the shaded area below represents a “tenth”: And the figures on the next slide make a “ten”:

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5.4 (cont’d) 10 =

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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?

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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

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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.

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Exploration 5.16 Do #1, 2, 4, 7 and 8. For #8, draw a picture of blocks to represent each decimal.

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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

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5.4 (cont’d) Rounding So, is 3.784 closer to 3.78 or 3.79? 3.78 3.785 3.79

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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?

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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!

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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 like 3 + 2 ones + 4 + 0 tenths + 6 + 9 hundredths = 5.55

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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 +.1 1 +.3

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5.4 (cont’d) Dividing decimals: Find the quotient 7.8 ÷ 3.12 What steps did you take? Why do they work?

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Homework Link to online homework list: http://math.arizona.edu/~varecka/302AhomeworkS09. htm *Note: approximate grades from before test 3 are posted on D2L; I will update as soon as I can.

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