Decimals

PLACE- VALUE INTERPRETATION Students consider
the place value of digits and what that means
when adding or subtracting numbers.

Zeke buys cider for $1.97 and donuts for $0.89. The clerk said
the bill was $10.87. What did the clerk do wrong?
The cider is and the donuts are
So the cost is In 1.97 + 0.89, we add
hundredths to hundredths tenths to tenths
and ones to ones

The clerk incorrectly added dollars and pennies (ones and
tenths, tenths and hundredths).

DECIMALS AS FRACTIONS Write decimals as
fractions, multiply, write the answer as a
decimal, and relate the number of decimal
places in the factors to the answer .

PLACE-VALUE INTERPRETATION Students see
why counting decimal points make sense and
use the short-cut algorithm: multiply the
decimals as whole numbers and adjust the
place of the decimal in the product.

We can look at a problem using equivalent fractions .

The product as a fraction is  as a decimal 0.69.

The 100 in the denominator shows that there should be two
decimal places (hundredths) in the answer. The denominator of
the fraction tells us the place value needed in the decimal.

Using the fact that students reason about a
related product: (2.5 is a tenth of 25, 0.31 is a
hundredth of 31, so the product is a thousandth of 775) 0.775.

DECIMALS AS FRACTIONS Write decimals as
fractions with common denominators and
divide the numerators .

PLACE-VALUE INTERPRETATION Write an
equivalent problem by multiplying both the
dividend and the divisor by the same power
of ten until both are whole numbers.

This makes a whole number problem with the same quotient as
the original decimal problem.

The fraction approach explains why moving decimal places
works.

FINITE ( OR TERMINATING ) DECIMALS are
decimals that “end.” The simplified fraction
has prime factors of only 2s or 5s in the
denominator.

FNFINITE REPEATING DECIMALS are decimals
that “go on forever” but show a repeating
pattern. These fractions have prime factors
other than 2 or 5 in the simplest
denominator form.

In simplified fraction form has only factors of five
in the denominator.

In simplified fraction form

PERCENT OF A PRICE “A CD costs $7.50. The
sales tax is 6%. How much is the tax?”

ON WHAT AMOUNT THE PERCENT WAS FIGURED
“Customers left Jill $2.50 as a tip. The tip was
20% of the total. How much was the bill?”

WHAT PERCENT ONE NUMBER IS OF ANOTHER
NUMBER “Sam got a $12 discount off a $48
purchase. What percent discount did he get?”

20% of some number equals $2.50
Find how many 20%s it takes to make 100%. In this case we
need five. So,5 × $2.50  gives us $12.50.

Find what % 12 is of 48. Students can solve this by computing
how many 12s in 48. It takes four, so the percent is 1/4 of 100%
or 25%.