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1. Place Value

1. Place Value. Powers of 10. Can help us represent decimals as fractions: 0.2, 0.45, 0.20, 4.6, etc. Decimals. Most decimal numbers are rational numbers: but some are not.

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1. Place Value

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  1. 1. Place Value • Powers of 10. • Can help us represent decimals as fractions: 0.2, 0.45, 0.20, 4.6, etc.

  2. Decimals • Most 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. • Repeating decimals: 7.6666…; 0.727272… • Terminating decimals: 4.8; 9.00001; 0.75

  3. A decimal like 3.5655655565555655556… 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. It is rational.

  4. 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 • 3.560 = 3 + 5 • .1 + 6 • .01 + 0 • .001 • Think of units, rods, flats, and cubes.

  5. 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.

  6. 3.78 3.785 3.79 Rounding • 3.784: round this to the nearest hundredth. • 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.

  7. 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

  8. 1 + 1 + .1 1 + .3 Multiplying Decimals • Easiest to see with the area model. • 2.1 • 1.3

  9. 3. 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 • 3.560 = 3 + 5 • .1 + 6 • .01 + 0 • .001 • Think of units, rods, flats, and cubes-how could we use them here?

  10. 4, 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.

  11. 3.78 3.785 3.79 5. Rounding • 3.784: round this to the nearest hundredth. • 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.

  12. 6. 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

  13. 1 + 1 + .1 1 + .3 7. Multiplying Decimals • Easiest to see with the area model. • 2.1 • 1.3

  14. 4, 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.

  15. 3.78 3.785 3.79 5. Rounding • 3.784: round this to the nearest hundredth. • 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.

  16. 6. 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

  17. 1 + 1 + .1 1 + .3 7. Multiplying Decimals • Easiest to see with the area model. • 2.1 • 1.3

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