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Decimals and place value

Decimals and place value. Decimals as rational numbers. Some decimal numbers are rational numbers: but some are not.

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Decimals and place value

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  1. Decimals and place value

  2. Decimals as rational numbers • Some decimal numbers are rational numbers: but some are not. • A decimal is a rational number if it can be written as a fraction with integer numerator and denominator. Those are decimals that either terminate (end) or have a repeating block of digits. • Repeating decimals: 7.6666…; 0.727272… • Terminating decimals: 4.8; 9.00001; 0.75

  3. Irrational numbers • A number that is not rational is called irrational. • A decimal like 3.5655655565555655556… is not rational because although there is a pattern, it does not repeat. It is an irrational number. • Compare this to 3.556556556556556556…It is rational because 556 repeats. It is a rational number.

  4. Comparing Decimals • When are decimals 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. • Use a number line. • Line up the place values.

  6. Exploration 5.16 • Use the base 10 blocks to represent decimal numbers and justify your answers. • Work on this together and turn in on Wednesday.

  7. Homework for Wednesday • Read pp. 308-323 in the textbook • Exploration 5.16

  8. 3.78 3.785 3.79 Rounding • 3.784: round this to the nearest hundredth. • 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.

  9. 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

  10. 1 + 1 + .1 1 + .3 Multiplying Decimals • As with whole numbers and fractions, multiplication of decimals is best illustrated with the area model. • 2.1 • 1.3

  11. Dividing decimals • Standard algorithm—why do we do what we do?

  12. Exploration 5.18 • Work on this exploration in class and finish for homework. • Part 1: 1-4 • Part 2: 1, 2

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