measuring fractions of an inchvoyager.dvc.edu/~lmonth/prealg/lesson7student.pdfmeasuring fractions...

Free Pre-Algebra Lesson 7 ! page 1

© 2010 Cheryl Wilcox

Lesson 7

Measuring Fractions of an Inch

Measuring length to the nearest inch or centimeter sometimes isn’t accurate enough. To be more precise in measuring, most rulers include marks measuring fractions of a unit.

Hands-On Fractions The easiest way to get a feel for the fractions on a ruler is to make them yourself in a larger format. Take an ordinary sheet of paper, and cut or tear it into strips. You’ll need four or five strips.

Let’s call the length of the paper “one strip.” It will represent a single inch on the ruler.

Step 1 If you fold a strip in half the long way, then unfold it, you’ll have a line at the halfway point. Fold carefully to make the two lengths identical.

Each part is one-half strip, usually written ! or 1/2 or

1

2. The 2 in the fraction

notation refers to the fact that two of the halves makes one whole strip, and the 1 is because we’re talking about 1 of the two parts .

The length of the black line is 1/2 strip.

Step 2 Now take a new strip, fold it in half, and then in half again. Unfold and see

There are now four equal parts, and each is called " (spoken “one-fourth” or “one quarter”) of the strip. The 4 in the fraction notation refers to the fact that four parts make one whole strip.

Fraction Notation

In fraction notation the first or top number is called the numerator – it numbers the parts. The number written after the fraction bar, or at the bottom, is called the denominator. It names the type of fraction by telling how many parts make up one whole.

numerator

denominator



When we’re measuring with fourths, we count how many we’ve used, and that number goes at the top of the fraction.

The line is 1/4 strip long.


Notice that this is the same as the line 1/2 strip long.



This is the same as a line 1 strip long.

Since the lines 2/4 strip and 1/2 strip long are the same length, we call 1/2 and 2/4 equivalent fractions, and write

2

4=

1

2.

The name 1/2 for the fraction is usually preferrable, since it uses smaller numbers in the numerator and denominator. Similarly, the fraction 4/4 and the number 1 are equivalent, but the name 1 is often preferred.

Step 3 Save the previous strips and take a new one. Fold this one in half, in half again, and then in half again. You may prefer an accordian pleat here as the folds get thicker.

You now have three strips, one with halves, one with fourths, and a new strip which has eight parts. Each part of the new strip is 1/8 of the strip. If you number the lengths on the strip using eighths, it will look like this:

0

1

8

2

8

3

8

4

8

5

8

6

8

7

8

8

8

Notice that the denominator is always 8, because all the fractions are eighths. The numerators count the number of eighths 1 2 3 4 5 6 7 8. When we get to 8 eighths, that is one whole strip, so 8/8 is another name for 1.



A line 3/8 strip long would begin at the left edge and end at the fold labeled 3/8.

0

1

8

2

8

3

8

4

8

5

8

6

8

7

8

8

8

Since we started this strip of eighths by folding in half, the first fold is the same as the one we did for 1/2.

4

8=

1

2

And since the second fold was in half again, all the fourths are there as well.

2

8=

1

4

Put all your strips together to see where the matching lengths are.

The red fraction names are equivalent to fractions with smaller denominators. If there is an equivalent fraction with a smaller denominator, that name is preferred when measuring. The preferred fraction names are in black.

Observe that any fraction with an even numbered numerator is red. That’s because all our denominators are even numbers. If you’re measuring with fourths or eighths and the numerator of the fraction is even, you know that there’s a better name for that fraction. Also note that if the numerator and denominator of the fraction are equal, that fraction length is equal to 1. So 2/2, 4/4, and 8/8 are equivalent fraction names for 1, because those lengths are the same as one whole strip. Real Rulers On a ruler, each inch is divided into fractions. Each of the three inches on this ruler is marked in eighths, fourths, and halves.

The longest lines are the whole inches, and are marked 1, 2, 3. The next longest lines mark the halves, the next longest the fourths, and finally the eighths are the shortest lines on this ruler. The labels in red are for clarification and would not be marked on a real ruler.

1

2

1

4

3

4

1

8

3

8

5

8

7

8

8

8

6

8

4

8

2

8

2

4

2

2

4

4

1

1 32

1

8

1

4

1

2



Example: Measure the line using the fractions on the ruler.

1 Place the ruler with the left edge at one end of the line and find

the nearest fraction marking.

2 The line is longer than 1 whole inch. (If we were measuring to

the nearest inch, it would be approximately 2 inches long.)

The length is 1 inch and a fraction more. To find the name of the fraction, look at the length of the mark it is nearest. The righthand end of this line is nearest a mark for 4ths.

3 Count the 4ths by counting the spaces after 1, but count only

spaces between marks at least as long as 4ths.

We don’t count the spaces between the shorter eighth marks.

The fraction part of the line is 3/4 inch.

4 Since the line measures 1 inch and 3/4 more, we write the length as

13

4 inches ,

(spoken “One and three-quarters inches”).

Back to Paper Folding Most inch rulers have more fraction markings than the halves, fourths, and eighths shown above. Let’s make another strip and then add sixteenths to our ruler.

To make a strip with sixteen parts, fold in half, (forming halves), in half again (fourths), in half

again (eighths), and then in half yet again. When you unfold, you should see sixteen equal parts.

The number of parts doubles each time you fold in half. If you were to fold your strip of sixteenths in half, you’d get 32 equal parts. Some inch rulers are marked in 32nds, and some in 16ths. We’ll stick to measuring in 16ths.

Each part is 1/16th, and we count the sixteenths just as we did the eighths and fourths. Just as before, if the fraction has a previous name with a smaller denominator we prefer to use that name. For example, 2/16ths is the same as 1/8th.

1 32

1 32

1 32

1 2 3

This is 1 inch

What fraction

is this?



When the ruler is marked in sixteenths, they are very short little marks, shorter than the eighths.

Example: Draw a line 23/16 inches long using the given ruler.

1 The line is 2 whole inches and another fraction, so start by

drawing the 2 inches. We know the line will extend a fraction further.

2 Each space between the shortest markings is 1/16th, and we

need 3/16ths so count 3 spaces between lines at least as long as 16ths.

3 Extend the line to the mark at 3/16 that comes after 2.

If you want more practice than is offered in the homework, you can use this online practice. Printable rulers – Use the first ruler, labeled One foot ruler, For US letter-size paper.

!

1 32

?

1 32

1 32

1 32

1

16

1

8

1

4

1

2



Lesson 7: Measuring Fractions of an Inch

Worksheet Name

1. What is the length of the line?

2. What is the length of the line?

3. Draw a line that is 7/16 inch long.

4. Draw a line that is 23/8 inches long.

5. Measure the line.

6. Measure the line.

7. Write the preferred fraction name for the mark at each arrow.

8. a. How many halves are in 1 inch?

b. How many 4ths are in 1 inch?

c. How many 8ths are in 1 inch?

d. How many 16ths are in 1 inch?

1 32 1 32

1 32 1 32

1 32 1 32

1 32



Using a real inch ruler, measure Raging Dino’s

a. Frill

b. Mouth

c. Tail

d. Foot claw

e. Hand claw

to the nearest sixteenth of an inch.

Draw lines to show the length and measure Raging Dino’s

a. Distance between knees.

b. Total height

c. Tongue length

to the nearest sixteenth of an inch.




Homework 7A Name

1. a. Round using your judgment: “The current U.S. population is 305,344,591 people, of whom 46,218,447 smoke cigarettes.”

b. Now round the numbers to the nearest ten thousand.

305,344,591

46,218,447

2. A rectangle has length 23 inches. The width is twice as much as the length.

a. What is the width?

b. What is the perimeter?

c. What is the area?

3. a. Write the multiplication in exponential form:

6 • 6 • 6 • 6 • 6

b. Write the exponential expression 74 as a multiplication.

4. A box has length 30 inches. The width is 2 more than the length. The height is 2 less than the length.


b. What is the height?

c. What is the volume of the box?

5. Evaluate

a. (3 + 7)2 ÷ 20

b. 35 – 53

c. 18 + 4 • 6 – 2

d. (18 + 4) • (6 – 2)

6. Find the distance a train travels in 17 hours at 44 miles per hour.

Find the distance a snail travels in 17 minutes at 44 inches per minute.

Find the number of words typed in 17 minutes at 44 words per minute.



7. “The high today will be 40ºC in Madrid, 20ºC in London, and 30ºC in Paris.”

Find the equivalent Fahrenheit temperatures using the formula F = 9C/5 + 32.

Madrid

London

Paris

What season do you think it is?

8. The cost formula for a manufacturing business (per month) is

Total Cost equals Fixed Costs plus Cost per Item times Number of Items Produced.

Write the formula in algebraic notation.

Find the cost (per month) for a business that has fixed costs of $2,485, and manufactured 9,244 items at a cost of $2 per item.

9. Measure each line.

a.

b.

c.

10. Draw lines of the given length.

a. A line 5/16 inch long:

b. A line 1 7/8 inches long:

c. A line 2 1/4 inches long:

1 32

1 32

1 32

1 32

1 32

1 32




Homework 7A Answers

1. a. Round using your judgment: “The current U.S. population is 305,344,591 people, of whom 46,218,447 smoke cigarettes.”

over 300 million people

more than 46 million

b. Now round the numbers to the nearest ten thousand.

305,344,591

305,340,000

46,218,447

46,220,000

2. A rectangle has length 23 inches. The width is twice as much as the length.


W = 2(23 inches) = 46 inches


P = 2(23 + 46) = 138 inches


A = (23)(46) = 1058 square inches


6 • 6 • 6 • 6 • 6

65


7 • 7 • 7 • 7



W = 30 + 2 = 32 inches


H = 30 – 2 = 28 inches


V = (30)(32)(28) = 26,880 cubic inches

5. Evaluate

a. (3 + 7)2 ÷ 20

= 102 ÷ 20 = 100 ÷ 20 = 5

b. 35 – 53

= 243 – 125 = 118

c. 18 + 4 • 6 – 2

= 18 + 24 – 2 = 42 – 2 = 40

d. (18 + 4) • (6 – 2)

= (22) • (4) = 88

6. Find the distance a train travels in 17 hours at 44 miles per hour.

d = rt = (44)(17) = 748

748 miles

Find the distance a snail travels in 17 minutes at 44 inches per minute.

d = (44)(17) = 748

748 inches


d = (44)(17) = 748

748 words





Madrid

F = 9(40)/5 + 32 = 104, Temp is 104ºF

London

F = 9(20)/5 + 32 = 68, Temp is 68ºF

Paris

F = 9(30)/5 + 32 = 86, Temp is 86ºF

What season do you think it is? Summer.




T = F + CN

Find the cost (per month) for a business that has fixed costs of $2,485, and manufactured 9,244 items at a cost of $2 per item.

T = (2485) + (2)(9244) = 20973

The total cost is $20,973


a. 9/16 inch

b. 1 1/8 inches

c. 2 1/2 inches





1 32

1 32

1 32

1 32

1 32

1 32




Homework 7B Name

1. There are an estimated 21,539,559 people in the U.S.

between the ages of 20 and 24, of whom 8,400,428 are smokers.

a. Write the numbers in words.

b. Round using your judgment.

2. A rectangle has length 4 meters. The width is three times the length.





6 • 6 • 6 • 6 • 6 • 6 • 6 • 6 • 6 • 6






5. Evaluate

a. (15 – 8)2 + (23 – 16)2

b. 152 – 82 + 232 – 162

c. 2(15 – 8) + 2(23 – 16)

d. 2 • 15 – 8 + 2 * 23 – 16

6. Find the distance a bus travels in 7 hours at 58 miles per hour.

Find the distance a motorcycle traveled in 7 seconds at 58 feet per second.






Madrid

London

Paris

What season do you think it is?




Find the total cost (per month) for a business that has fixed costs of $5,850, and manufactured 12,333 items at a cost of $1 per item.


a.

b.

c.





1 32

1 32

1 32

1 32

1 32

1 32

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