You have been working with various ways to
represent portions of a whole. These multiple
representations are shown in the diagram at
right. It is called a web. For some problems,
you might prefer to work with a percent,
while at other times, it might make sense to
use a fraction or a decimal. Today you will represent portions in multiple ways and
consider which of them allows you to work most efficiently.
As you work on today’s lesson, use these questions with your team to focus your
discussions:
How else can I represent the same portion?
How do I know the portions are equivalent (the same)?
3-55. BUILD IT, WRITE IT, DRAW IT
Sometimes it is easier to compare portions when they are written in a particular form.
For example, 0.25 and 0.8 can be compared easily when both are drawn on a 100%
block or are both in percent form. In this problem, your team will work with all of the
representations on the web and find ways to change one representation into another.
You may want to explore using either Base Ten Block (html5) or Base Ten
Blocks (GeoGebra: html5). For each portion of a whole described below:
Build the portion on top of a 100% block.
Draw the representation on your Lesson 3.1.3B Resource Page .
Write the portion as a percent, fraction, decimal, and as a description in words.
d. 3 tenths and 1 hundredth
e. 0.56
f.
g. 3%
h. Place each of the portions described in parts (a) through (d) on a number line.
The number line should range from 0 and 1 be marked in tenths.
i. Which representation (fraction, decimal, or percent) is most convenient to use
when placing values on a 0-to-1 number line? Explain your choice.
3-56. LOCATION, LOCATION, LOCATION
Sally was helping her younger sister Susie, who had
been absent from school, to understand decimals.
When Susie came to the problem 0.37 + 0.7, she got
very excited. “I know, I know!” Susie shouted, “37
and 7 make 44, so the answer is 0.44!”
“Well,” Sally said, “You’re right that 37 and 7 make 44, but 0.37 is not 37 and 0.7 is
not 7. The value of numbers depends on where they are located,” Sally
explained, “That is why you have to line up the place values, by lining up the decimal
point, when you add or subtract.”
What does Sally mean? What explanation can you give for lining up decimals when
adding or subtracting? Write a note
to Susie explaining why 0.37 + 0.7 is
not 0.44. Include the correct answer and an explanation of what each number in the
problem represents. Hint: It might help to rewrite each number as a sum of fractions
or draw them with hundred blocks.
3-57. Susie’s next question impressed Sally. “So does that mean that if I want to add
1.003 and 0.47, instead of adding 47 and 3, I need to add 470 and 3? Is this similar
to adding fractions, so I have to write them in equivalent forms?”
. What do you think? Is Susie right? Use what you know about representing
these numbers with fractions, percents, and 100% blocks to justify your
answer.
a. Find the answer to 1.003 + 0.47.
3-58. Complete each of the following computations using your understanding of
decimal place value and representations of portions.
. 0.375 − 0.2
a. 18.6 + 0.04
b. 2.008 − 0.46
3-59. Maya and Logan are playing “Guess My Decimal” again. You will play along
with them in parts (a) and (b) below.
. Maya challenges Logan by saying, “The decimal I’m thinking of is what you
would get if you subtract 0.01 from 0.3.” Visualize what her number might
look like on a 100% block. What decimal is Maya thinking of? Explainhow
you know.
a. Logan continues the game with this challenge: “The decimal I am thinking
about is halfway between 18 hundredths and 3 tenths.” Help Logan solve this
puzzle and show any work that might help explain your thinking.
3-60. Additional Challenge: Work with your team to rewrite
percent. Be prepared to explain your strategies to the class.
as a decimal and as a
3-61. LEARNING LOG
In your Learning Log, copy the web below. Since it is a web that
shows the connections between all the different ways a portion can
be represented, its technical name is “Representations of a Portion
web.” After copying the web, show as many representations of the
number 15% as you can. Title this entry “Representations of a Portion” and include
today’s date.
Percents
A percent is one way to write a portion of 100. It can always be written as a fraction
with a denominator of 100 and/or as a decimal.
3-62. Add or subtract the following fractions. Help (Html5)⇔Help (Java)
a.
b.
3-63. Copy the incomplete axes and fill in the missing numbers to make the scaling
consistent. Help (Html5)⇔Help (Java)
.
a.
3-64. For each of the representations below, write the portion in each of the forms
listed. Remember that a hundred block now represents 100%. Help (Html5)⇔Help
(Java)
Percent
Decimal
Fraction
Description in words
d.
e.
f.
g.
3-65. Estimate the amount of gas left in this car’s gas tank with a fraction. Help
(Html5)⇔Help (Java)
3
-66. Randall and Stephano work in a restaurant. Randall
earned $27.50 one day, $25.00 the next day, and $32.50 on the third day. Stephano
works fewer hours, but more days. He earned $17.50 one day, $22.50 the next day,
$12.50 the third day, $15.00 the fourth day, and $17.00 the fifth day. Who earned the
most money? How much more? Help (Html5)⇔Help (Java)
Lesson 3.1.4
3-55. See below:
a. 0.31, 31%,
b. 5 tenths and 6-hundredths (or 56 hundredths), 56%,
c. 0.84, 84%, 8 tenths and 3-hundredths (or 83-hundredths)
d. 0.03, 3 hundredths
e. See number line below.
f. Answers vary.
3-56. LOCATION, LOCATION, LOCATION
Correct solution is 1.07. Explanations will vary, see Suggested Lesson Activity
for suggestions.
3-57. See below:
. Yes, Susie is correct. To add the decimals the place value must be the same. Since 47hundreths is equivalent to 470-thousandths you can then add it to the 3-thousandths.
a. 1.473
3-58. See below:
. 0.175
a. 18.64
b. 1.548
3-59. See below:
. 0.3 – 0.01 = 0.29
a. 0.24
3-60. See below:
. 62.5%, 0.625
3.62. See below:
.
a.
3-63. Only the missing numbers are listed. a. 4, 8, 12, 20; b. 25, 75, 100, 125, 150
3-64. See below:
. 35%,
or
, 0.35, thirty five hundredths or three tenths and five hundredths or
thirty five out of one hundred
a. 80%,
or
, 0.8, eighty hundredths or eight tenths or eighty out of one
hundred or eight out of ten
b. 8.5%, I
or
, 0.085, eight and one half hundredths or eighty five
thousandths or eight and one half out of one hundred
c. 195%,
3-65.
or
, 1.95, one whole and ninety five hundredths
full
3-66. See below:
Randall earned $85 and Stephano earned $84.50, so Randall earned more. $0.50.