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April 21, 2009 0011 0010 1010 1101 0001 0100 1011 1 2 “Energy and persistence conquer all things.” 4 ~Benjamin Franklin April 21, 2009 0011 0010 1010 1101 0001 0100 1011 • Section 5.4 – Decimals • Exploration 5.16 1 2 4 5.4 – Decimals 0011 0010 1010 1101 0001 0100 1011 • 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 1 2 4 5.4 (cont’d) 0011 0010 1010 1101 0001 0100 1011 • 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! 1 2 4 5.4 (cont’d) 0011 0010 1010 1101 0001 0100 1011 Another look at place value: 1 2 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 4 5.4 (cont’d) 0011 0010 1010 1101 0001 0100 1011 What does the number 1.234 mean? 1 2 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 4 5.4 (cont’d) 0011 0010 1010 1101 0001 0100 1011 From our work with fractions, you should recognize the difference between “tens” and “tenths”: 1 Each “ten” represents 10 ones, while each “tenth” is one of the 10 equal pieces that one whole was divided into. 2 4 5.4 (cont’d) 0011 0010 1010 1101 0001 0100 1011 Let =1 1 Then the shaded area below represents a “tenth”: 2 4 And the figures on the next slide make a “ten”: 5.4 (cont’d) 0011 0010 1010 1101 0001 0100 1011 10 = 1 2 4 5.4 (cont’d) 0011 0010 1010 1101 0001 0100 1011 In symbols, a “tenth” is 1/10. In decimal form, 1/10 = 0.1 1 2 Similarly, a “hundredth” is 1/100, and a “thousandth” is 1/1000. In decimal form, 1/100 = 0.01 and 1/1000 = 0.001 4 Can you see how base 10 blocks can be used to visualize these? 5.4 (cont’d) 0011 0010 1010 1101 0001 0100 1011 When decimals are equal: 3.56 = 3.56000000 But, 3.056 ≠ 3.560. 1 2 4 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 5.4 (cont’d) 0011 0010 1010 1101 0001 0100 1011 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. 1 2 4 Exploration 5.16 0011 0010 1010 1101 0001 0100 1011 Do #1, 2, 4, 7 and 8. For #8, draw a picture of blocks to represent each decimal. 1 2 4 5.4 (cont’d) 0011 0010 1010 1101 0001 0100 1011 Rounding 1 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 2 4 3.79 5.4 (cont’d) 0011 0010 1010 1101 0001 0100 1011 Rounding So, is 3.784 closer to 3.78 or 3.79? 3.78 3.785 1 2 4 3.79 5.4 (cont’d) 0011 0010 1010 1101 0001 0100 1011 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? 1 2 4 5.4 (cont’d) 0011 0010 1010 1101 0001 0100 1011 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! 1 2 4 5.4 (cont’d) 0011 0010 1010 1101 0001 0100 1011 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 1 2 4 5.4 (cont’d) 0011 0010 1010 1101 0001 0100 1011 Multiplying decimals: (Easiest to see with the area model) 2.1 × 1.3 1 + 1 Where is 2 × 1? 1 2 × 0.3? 1 × 0.1? 0.1 × 0.3? + .3 1 2 + .1 4 5.4 (cont’d) 0011 0010 1010 1101 0001 0100 1011 Dividing decimals: Find the quotient 7.8 ÷ 3.12 What steps did you take? Why do they work? 1 2 4 Homework 0011 0010 1010 1101 0001 0100 1011 Link to online homework list: http://math.arizona.edu/~varecka/302AhomeworkS09. htm 1 2 4 *Note: approximate grades from before test 3 are posted on D2L; I will update as soon as I can.