Graphing Fractions
Fractions
Fractions are numbers which are not whole; or, numbers that are more than zero, but less than one. Fractions are made up of a numerator and a denominator. The denominator describes the number of parts that we need to split a whole unit into in order to describe the value. The numerator describes the number of those parts needed to create the fraction. The ratio of the numerator over the denominator are what make our fraction (NUM⁄DEN).
Plotting Fractions
To plot a fraction on the real number line, first we split each unit of the number line up into number of parts equal to the denominator. Then, we move forward (right on the line if the fraction is positive, left if negative) by the number of those parts equal to the numerator.
To plot a mixed number (or a whole number and fraction put together), we first move forward the number of whole units equals to the whole number part of the value. Then, we divide the units of the number line up into the number of parts equals to the denominator. Finally, we move forward the number of those parts equal to that of the numerator to reach our final coordinate.
Example 1
- For example, say we ordered a mini pizza from a local restaurant. It was quite small, and we ate the entire pie; so, we order another. Each pizza is cut-up into 4 slices, and this time we're only able to eat 3 of the 4 slices of pie. Describe the amount of pie that we ate using a mixed number; then, graph the mixed number onto the real number line.
- We ate one whole (mini) pizza, so the whole number part is 1. Now, the second pizza is split-up into 4 parts, which represents the denominator of the fraction. We ate 3 of parts, which is the numerator. The ratio between the numerator and denominator is our fraction, that combined with the whole number gives us the amount of pizza we ate as a mixed number.
To plot this mixed number on the real number line, we equate each pizza that we've eaten with a unit on the line. Starting at zero, we begin by moving right the number of units equals to the whole number part of the value. We ate one entire ("mini") pizza, so we move right 1 unit. Now, we need to include the fractional part. We do this by dividing the next unit up into a number of parts equal to the denominator. The pizza is cut into 4 pieces, so we split the next unit up into 4 parts. Finally, we move forward the number of parts equal to that of the numerator. We ate 3 of the 4 slices of pizza, so we move forward 3 of the 4 parts (each step represents a slice of pizza). The final location is the coordinate of the mixed number.
Checkpoint 1
- Graph the number of pizza slices eaten by the family on the real number line.
Click for solutions...Checkpoint 2
- Plot the amount of cake left over after the party onto the number line.
Click for solutions...Graphing Decimals
Decimal Form
A decimal is a fraction which is in Base-10 Format. This means that the denominator is a power of 10.
Whole numbers and integers are also in Base-10 format; as we move left each digit, we multiply by another power of 10 to get the digits value (123 = 1*100 + 2*10 + 3*1). Having both the fractional and whole components of the value in the same base format allows us to represent both of the components together in the same number. We separate the two parts with a decimal point (.), with the number to the right representing the fractional component, and the number to the left, the whole component.
Plotting Decimals (Tenths)
Plotting a decimal is similar to plotting a fraction; however, instead of using the denominator to determine the number of parts that we need to split the next unit into, we use the position of the last digit to the right of the decimal point (.). Starting from the decimal point, each digit that we move to the right, we split each part of the unit up into 10 further parts (or we multiply the denominator by ten). For example, if we want to plot a digit that is in the tenths place (or one place to the right of the decimal point), we need to split the unit from 0 to 1 up into 10 parts.
A tenth (0.1) can also be shown in fractional form as 1⁄10. It is in this fashion that we can use fractions to divide the number line into tenths instead of decimals.
For example, to plot the number 0.2 on the number line, we start by splitting the unit from 0 to 1 up into tenths. Then, we move up the line 2 tenths to reach our final coordinate.
Plotting Decimals (Hundredths)
Now, to plot a number in the hundredths place (or a digit that is two places to the right of the decimal point), we need to split each unit up into 100 parts (or into 10 parts of 10).
We can see (and have the room to label) the individual tick marks if we zoom in on the line, and only graph from zero to the first tenth (0.1).
Same as before, we can use fractions to split the number line up into hundredths; however, as we move further to the right of the decimal point, it becomes increasingly difficult to understand and label the fractional representations of each value. This is when the benefits of Base-10 Format become increasingly clear, especially when combined with scientific notation which radically reduces the space needed to express extremely large and small values.
To plot the number 0.02 on the number line, we split the unit into 100 parts (10 parts of 10), then move forward 2 of those parts to reach our final coordinate.
Checkpoint 4
Graph the following decimals on the real number line:
a. 0.3
b. 0.9
c. 0.6
d. 2.0
e. 4.7
f. 3.8
Click for solutions...Checkpoint 5
A local car club recently had a 0-60 competition and wants your help making a display of the results. The times are exact down to the tenth (1⁄10 or 0.1) of a second. To do this, plot each time with the name of the car on the number line. Note: to make the data more meaningful to the team, the number line is rotated so that the faster times appear closer to the top.
Fatal Impact: 4.2 sec
Toxic Rain: 5.2 sec
Silver Bullet: 2.8 sec
Flash Bang: 1.9 sec
Catnip: 3.6 sec
Blade: 3.2 sec
Hellhound: 6.5 sec
Thunderbolt: 3.5 sec
Rusty John: 5.8 sec
Alexus: 4.0 sec
Checkpoint 6
A candy company wants a number line illustrating the prices of the closest competitors. Plot the following prices labeled with the name of the candy on the number line.
Loopies: $1.57
PawPaws: $1.62
BingBongs: $1.04
Fiddle Sticks: $1.91
Chestnutter: $1.18
Appies: $1.81
Click for solutions...Exercises
Exercise Set 1
- Write the fraction plotted on each number line:
1-1.)
1-2.)
1-3.)
1-4.)
1-5.)
1-6.)
1-7.)
1-8.)
1-9.)
1-10.)
1-11.)
1-12.)
Exercise Set 2
- Write the mixed number plotted on each number line:
2-1.)
2-2.)
2-3.)
2-4.)
2-5.)
2-6.)
2-7.)
2-8.)
2-9.)
2-10.)
Exercise Set 3
- Write the decimal plotted on each number line:
3-1.)
3-2.)
3-3.)
3-4.)
3-5.)
3-6.)
3-7.)
3-8.)
3-9.)
3-10.)
Exercise Set 4
- Plot each fraction illustrated below on the real number line:
4-1.)
4-2.)
4-3.)
4-4.)
4-5.)
4-6.)
4-7.)
4-8.)
4-9.)
4-10.)
4-11.)
4-12.)
Exercise Set 5
- Plot each fraction illustrated below on the real number line:
5-1.)
5-2.)
5-3.)
5-4.)
5-5.)
5-6.)
Exercise Set 6
- Plot the following numbers on the real number line:
6-1.) 9⁄10
6-2.) 3⁄5
6-3.) 1⁄4
6-4.) 4⁄9
6-5.) 5⁄6
6-6.) 3⁄10
6-7.) 1⁄7
6-8.) 2⁄8
6-9.) 6⁄9
6-10.) 8⁄8
6-11.) 6⁄10
6-12.) 8⁄10
Exercise Set 7
- Plot the following numbers on the real number line:
7-1.) 29⁄10
7-2.) 36⁄9
7-3.) 11⁄6
7-4.) 35⁄9
7-5.) 1⁄3
7-6.) 13⁄4
7-7.) 26⁄11
7-8.) 31⁄10
7-9.) 05⁄9
7-10.) 16⁄7
7-11.) 37⁄11
7-12.) 31⁄12
Exercise Set 8
- Plot each decimal on the real number line:
8-1.) 0.2
8-2.) 0.5
8-3.) 0.7
8-4.) 0.6
8-5.) 0.8
8-6.) 0.3
8-7.) 0.9
8-8.) 0.4
8-9.) 0.1
8-10.) 0.0
Exercise Set 9
- Louis, the coach of a local basketball team, wants help evaluating the endurance of his athletes. To do this he had each of them run as far as they could in 15 minutes and recorded the results. Now he wants your help visualizing the results below. Plot each distance and label it with the name of the athlete on the real number line below:
![Exercise Set 9: Real Number Line [0,10] (Increments of 0.1)](/assets/images/number_line/plotting-fractions/exercises/basketball_team_run_distance/line.svg)
Sonia: 1.7 mi
Sera: 3.4 mi
Zoe: 2.4 mi
Rose: 2.1 mi
Lea: 2.9 mi
Pili: 3.2 mi
Dalia: 1.4 mi
Anika: 3.0 mi
Exercise Set 10
Adrian, the local track and field coach, would like your help. The team recently had a meet, and Adrian wants help visualizing the results of the 1 mile-run event. Plot each time below on the real number line labeled with the runners name:
Duane: 6.5 min
Carlos: 4.5 min
Colby: 5.8 min
Archer: 6.3 min
Eli: 5.5 min
Xavier: 4.1 min
Exercise Set 11
- While playing a video game, your friend Olivia encounters an extremely difficult boss monster. She wants your help picking the best weapon in her inventory to use against the boss monster. To help her, create two separate number lines illustrating the damage and speed of each weapon.
Slapper - DMG: 3.3, SPEED: 4.8
Stretcher - DMG: 5.7, SPEED: 3.5
Angler - DMG: 2.7, SPEED: 5.8
Cage - DMG: 4.3, SPEED: 4.3
Boom Boom - DMG: 5.5, SPEED: 2.5
Pimple Remover - DMG: 4.9, SPEED: 1.7
Bear Claw - DMG: 5.3, SPEED: 5.6
Duck Foot - DMG: 1.5, SPEED: 3.6
Click for solutions...Exercise Set 12
- Read each number line below and write the decimal that's plotted:
12-1.)
12-2.)
12-3.)
12-4.)
12-5.)
12-6.)
12-7.)
12-8.)
12-9.)
12-10.)
12-11.)
12-12.)
12-13.)
12-14.)
12-15.)
12-16.)
12-17.)
12-18.)
12-19.)
12-20.)
12-21.)
12-22.)
12-23.)
12-24.)
12-25.)
12-26.)
12-27.)
12-28.)
12-29.)
12-30.)
12-31.)
12-32.)
12-33.)
12-34.)
12-35.)
12-36.)
