Transcript Decimals and place value

Decimals and place value
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
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.
Comparing Decimals
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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.
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.
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.
Homework for Wednesday
• Read pp. 308-323 in the textbook
• Exploration 5.16
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.
3.78
3.785
3.79
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
Multiplying Decimals
• As with whole numbers and fractions,
multiplication of decimals is best illustrated with
the area model.
• 2.1 • 1.3
1
1
+
.3
+
1
+ .1
Dividing decimals
• Standard algorithm—why do we do what
we do?
Exploration 5.18
• Work on this exploration in class and finish
for homework.
• Part 1: 1-4
• Part 2: 1, 2