Mastering Decimals in Mathematics Study: Understanding Rational Numbers and Decimal Operations

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

2. April 21, 2009 • Section 5.4 – Decimals • Exploration 5.16

3. 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

4. 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!

5. 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

6. 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

7. 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.

8. 5.4 (cont’d) Let = 1 Then the shaded area below represents a “tenth”: And the figures on the next slide make a “ten”:

9. 5.4 (cont’d) 10 =

10. 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?

11. 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

12. 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.

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

14. 3.78 3.785 3.79 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.

15. 3.78 3.785 3.79 5.4 (cont’d) Rounding So, is 3.784 closer to 3.78 or 3.79?

16. 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?

17. 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!

18. 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

19. 1 + 1 + .1 1 + .3 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?

20. 5.4 (cont’d) Dividing decimals: Find the quotient 7.8 ÷ 3.12 What steps did you take? Why do they work?

21. 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.