We usually measure the characteristics of objects if we want to compare them. One of the characteristics of things that we can measure is their lengths. There are specific tools for measuring lengths which use units like centimetres, inches, feet, meters, etc.

Let us say, for instance, that we can measure the lengths of school supplies like erasers, notebooks, and pencils using a ruler. A meter stick is a tool to measure the length of the classroom or the distance of the school library from the school canteen.

The data collected using length measurements can be represented using a graph called line plots which we can make interpretations.

**Length**

**Definition**

If we want to identify how long an object is or want to know the distance from one point to another, we are referring to length. Length also measures the height, width, and how apart two objects are.

**Measuring Lengths**

There are two classifications of measuring lengths that use different units: the non-standard and standard units of measure. The non-standard units of measuring lengths have handspan, foot span, cubit, etc. Standard units of measuring lengths or distances include millimetres, centimetres, meters, kilometres, etc.

**Non-Standard Units of Measuring Lengths **

Non-standard units of measuring lengths do not use scales; instead, it uses real-world objects to identify measurements or compare them with other things, including handspan, foot span, cubit, etc. The measurement results using non-standard units vary from person to person or object to object since one object may be smaller or larger than the other.

For example, a 5-year-old child measures a table as ten handspans while his mother measures the same table as six handspans. Using handspans, a 5-year-old child and his mother gave different measurements since one handspan of the mother is longer than of her child.

**Standard Units of Measuring Lengths**

The standard unit does not change from person to person, unlike the non-standard units, since this uses systems with specific measurements. The standard units can be divided into two groups: the metric system units and the imperial system units.

Meter, usually denoted with the letter *m,* is the standard unit of length based on the metric system. Although there are other standard units for measuring lengths, conversion of units may be applied depending on the most suitable for the object.

Here is the list of some standard metric units of measuring lengths :

*Millimetre (mm)*

A millimetre is a small unit of length, which equals 0.001 meters.

*Centimetre (cm)*

A centimetre is larger than a millimetre since one centimetre equals 10 millimetres.

*Meters (m)*

A meter is larger than a centimetre. A meter has 100 centimetres.

*Kilometre (km)*

A kilometre is a larger unit of measurement of length, and this is equal to 1000 meters.

The ** imperial system** of measurement is a system that originated in Britain that came to formal use in the early 19th century. The most used units in the imperial system are the inch, ton, gallon, etc.

Below is the list of some standard imperial units of measuring lengths :

**Inch ( in. )**

An inch is equal to 1/12 of a foot.

**Foot ( ft. )**

A foot is larger than an inch since one-foot equals 12 inches.

**Yard ( yd. )**

A yard is larger than a foot since this is equal to 3 feet.

**Mile ( mi. )**

A mile is a large unit of measurement of length, and this is equal to 1760 yards.

**Length Conversion**

The information below shows the conversion between the different units of length measurement. The knowledge of length conversion allows us to solve word problems involving lengths of various measures.

*Metric System Units *

Conversion of millimetres to other units of length measurement

1 millimetre = 0.1 centimetre

1 millimetre = 0.001 metre

1 millimetre =0.000001 kilometre

Conversion of centimeters to other units of length measurement

1 centimetre = 10 millimetres

1 centimetre = 0.01 metre

1 centimetre = 0.00001 kilometre

Conversion of meters to other units of length measurement

1 metre= 1000 millimetres

1 metre = 100 centimetres

1 metre = 0.001 kilometre

Conversion of kilometers to other units of length measurement

1 kilometre = 1000000 millimetres

1 kilometre = 100000 centimetres

1 kilometre = 1000 metres

*Imperial System Units *

**Conversion of imperial units to other units of length measurement**

- 1 inch = 1/12 foot
- 1 foot = 12 inches
- 1 yard = 3 feet
- 1 mile = 1760 yd.

**Steps for Measuring an Object with the Use of a Ruler**

**Step 1:** Identify the type of ruler and its unit that you are using.**Step 2:** Take one end of the object you are measuring and line it up at the ruler’s beginning or the ruler’s zero mark.**Step 3: **Figure out where the other end of the object lines up with the ruler.**Step 4: **Count the number of lines and determine the fraction to use when needed.**Step 5: **Record the measurement.

Let us follow the steps in measuring the length of the ribbon and leaf below using a ruler.

*Step 1:** Identify the type of ruler and its unit that you are using.*

Let us use the inch side of this ruler to measure the lengths of the ribbon and leaf.

*Step 2:** Take one end of the object you are measuring and line it up at the ruler’s beginning or the ruler’s zero mark.*

*Measuring a ribbon using a ruler*

*Measuring a leaf using a ruler*

*Step 3: **Figure out where the other end of the object lines up with the ruler.*

The ribbon measures 5 inches while the leaf measures 4 inches.

*Step 4: **Count the number of lines and determine the fraction to use when needed.*

Both the lengths of the leaf and the ribbon are whole numbers. The marking did not fall in the fractional part of the ruler.

*Step 5: **Record the measurement.*

Length of the ribbon: **5 inches**

Length of the leaf: **4 inches**

**Line Plot**

**Definition**

A line plot is a graph showing the number of times a value in a data set occurred. Apart from frequency, a line plot gives different information about the data collected, like the differences among values, transitions among the data collected, etc. Thus, a line plot allows people to evaluate and interpret the data set collected.

These are some examples of how we can construct a line plot.

- The number of minutes the students spend doing their homework.
- The lengths of pencil that a person has in his collection.
- The travel time it takes for the students to reach school.
- The number of candies sold in a grocery store.
- The number of hours spent watching television.

**Constructing a Line Plot**

The following are the steps to follow in constructing a line plot:

**Step 1: **Gather and organize the data by arranging the values from least to greatest.**Step 2: **Draw a number line.**Step 3: **Write the title of the line plot.**Step 4:** Use “X” to mark the data on the line plot.**Step 5: **Interpret the data.

**Interpreting a Line Plot**

Once you have entirely constructed your line plot, there are components of the data that you may observe, evaluate, and interpret. The common things to look for when looking at line plots are the most frequent data, outliers, gaps, and clusters. These characteristics influence the analysis and interpretation of the data collected.

*Gaps* are the missing areas in the given data set. For example, in the recorded temperatures of classrooms in XYZ Elementary School, it shows that the range is between 60 degrees and 80 degrees Fahrenheit. If the line plot shows nothing between 74 and 78 degrees, then there is a gap in that data set.

*Outliers* are the values that lie in extreme areas. For example, if a value in a line pot is located at some distance away from other values, it is an outlier.

*Clusters* are the isolated groups based on the data set. The values in clusters are bunched together, or these values stick together away from other values in the data set. For example, in the record of ages of employees in ABC restaurant, there are five employees with age 18, 4 employees with age 19, 3 employees with age 20, 0 employees with age 21 to 24 and 2 employees with age 25. So from age 18 to 20, there is a cluster of employees at the restaurant.

The *most frequent* data are the values that appear the most in the data set.

**Constructing a Line Plot with Whole Numbers**

Line plots with whole numbers have data without fractions. Let us answer the problem below and follow the steps in measuring lengths and constructing data.

*Marivic has ribbon ten pieces of ribbon. Find the measure of each ribbon and create a line plot to display the data.*

Let us use a ruler to measure each ribbon and use inches as the unit of measurement. Remember to take one ribbon end and line it up at the ruler’s zero mark.

Find out where the other end of the ribbon lines up with the ruler and record the measurement.

** ****The table below shows the recorded lengths of each ribbon Marivic has.**

Ribbon | Length |

1 | 7 inches |

2 | 6 inches |

3 | 3 inches |

4 | 8 inches |

5 | 4 inches |

6 | 5 inches |

7 | 6 inches |

8 | 5 inches |

9 | 1 inch |

10 | 5 inches |

Now, let us follow the steps in constructing a line plot:

*Step 1: **Gather and organize the data by arranging the values from smallest to largest.*

Based on the gathered data, the lengths of the pieces of ribbon Marivic have are, 1 inch, 3 inches, 4 inches, 5 inches, 5 inches, 5 inches, 6 inches, 6 inches, 7 inches, and 8 inches. The shortest length is 1, while the longest is 8.

**1 , 3 , 4, 5 , 5, 5, 6, 6, 7, 8**

*Step 2: **Draw a number line.*

The number line must have a 1-interval unit, including the numbers 1 to 8. * *

*Step 3: **Write the title of the line plot.*

The title of the line plot is Lengths of the Ribbons Marivic Has.

**Lengths of Ribbons Marivic Has**

*Step 4:** Use “X” to mark the data on the line plot.*

Mark one *X *above 1 because it occurs once, mark one *X *above 3 since it occurs once, mark one *X *above 4 since it occurs once, mark three *Xs *above 5 since it occurs three times, mark two *Xs *above 6 since it occurs twice, mark one *X *above 7 since it occurs once and mark one *X *above 8 since it occurs once.

**Lengths of Ribbons Measured**

It helps to cross out the numbers list to ensure accurate marking of the line plot.

*Step 5: **Interpret the data.*

From the data collected, the most frequent length of ribbon from Marivic’s collection is 5 inches. The outlier is the 1-inch length since it is the measurement far away from other data. There is a gap between 1 inch and 3 inches of ribbon lengths. There are clusters of data between 5 and 6 inches since many pieces of ribbon fall in these measurements.

**Constructing a Line Plot with Fractions**

A line plot with fractions includes whole numbers and fractions on the data collected.

Let us try to answer the scenario below by measuring lengths and creating line plots.

*Trisha collected different types of leaves in the garden. Create a line plot for the length measures of the leaves she collected.*

The table below shows the recorded length measures of the leaves collected by Trisha.

Leaves | Length (cm) | Leaves | Length (cm.) |

1 | 11 ½ cm. | 8 | 12 ¾ cm. |

2 | 12 ½ cm. | 9 | 11 ¼ cm |

4 | 11 ¼ cm. | 10 | 12 ¾ cm. |

4 | 17 ½ cm. | 11 | 15 ½ cm. |

5 | 15 ¾ cm. | 12 | 12 ½ cm. |

6 | 12 ½ cm. | 13 | 15 ½ cm. |

7 | 15 ¾ cm | 14 | 15 ½ cm. |

Now, let us follow the steps in constructing a line plot:

*Step 1: **Gather and organize the data by arranging the values from least to greatest.*

Arranging the data collected from least to greatest gives us: 11 ¼ , 11 ¼ , 11 ½ , 12 ½ , 12 ½ , 12 ½ , 12 ¾, 12 ¾, 15 ½ , 15 ½ , 15 ½ , 15 ¾ , 15 ¾ , 17 ½

*Step 2: **Draw a number line.*

Since the data collected involves fractions, let us create a line plot with ½, ¼, and ¾ of an inch. Ensure that the line plot includes the number of all the data from the least to the greatest data collected. Notice also that the number line has the label “lengths ( cm )”, which describes the unit used in the measurement as centimetres.

Each 1-each interval has three vertical marks or four spaces between them, dividing them into four equal parts. The first vertical mark is ¼ inch away from the lower number, the second vertical mark is ½ inch away from the lower number, and the third vertical mark is ¾ away from the lower number.

**Step 3: **Write the title of the line plot.

The number line is titled Lengths of Leaves to describe that the line plot shows the data of lengths of leaves collected.

**Lengths of Leaves**

*Step 4:** Use “X” to mark the data on the line plot.*

Mark two *Xs *above 11 ¼ since it occurs twice, mark onw *X *above 11 ½ since it occurs once, mark three *Xs *above 12 ½ since it occurs three times, mark two *Xs *above 12 ¾ since it occurs twice, mark three *Xs *above 15 ½ since it occurs three times, mark two *Xs *above 15 ¾ since it occurs twice and mark one *X *above 17 ½ since it occurs once.

**Lengths of Leaves**

Here is the arrangement of data from least to greatest, and crossing out the numbers as you put *X *marks in the line plot helps ensure that you have all the data: ~~11 ¼~~ , ~~11 ¼~~ , ~~11 ½~~ , ~~12 ½~~ , ~~12 ½~~ , ~~12 ½~~, ~~12 ¾~~, ~~12 ¾~~, ~~15 ½~~, ~~15 ½~~, ~~15 ½~~, ~~15 ¾~~, ~~15 ¾~~, ~~17 ½~~

**Step 5: **Interpret the data.

The most frequent lengths of leaves Trisha collected in the garden are 12 ½ cm and 15 ½ cm. The outlier is 17 ½ cm in length since it is the measurement far away from other data, while there is a gap between 12 ¾ cm and 15 ½ cm of leaves length. There are clusters of data between 12 ½ cm and 12 ¾ cm and 15 ½ cm and 15 ¾ cm because there are a lot of leaves lengths that fall in these measurements.

**Examples**

**Example 1:** Find the measure of each pencil in centimetres with the ruler shown.

a)

b)

c)

d)

** Solution:** To measure the pencils with the ruler shown, find the measurement where the other end of the pencil lines up. Check if the other end is lined up at the ruler’s zero mark. Record the measurements.

These are the lengths in centimetres of the pencils:

- 8 cm. b. 5 ½ cm. c. 11 cm. d. 7 ½ cm.

**Example 2:** Mario measures the lengths in inches of his friends’ pens and records the data in the table below. Create a line plot using Mario’s data and answer the questions below:

**Length of Pens**

Name | Length ( in inches ) |

Leni | 5 ¼ |

Rodrigo | 4 ½ |

Martha | 4 ¾ |

Ana | 5 |

Peter | 5 ¾ |

Joseph | 5 ½ |

Mark | 4 ¾ |

Anthony | 5 ½ |

- How many pencils did Mario measure?
- What is the measure of the shortest pencil?
- What is the measure of the longest pencil?
- What pencil lengths appear more than once?
- How many pencils have lengths greater than 5 centimetres?

*Solution:*** **Let us first arrange the length measurement of the pens from least to greatest

: 4 ½, 4 ¾, 4 ¾, 5, 5 ¼ , 5 ½ , 5 ½ , 5 ¾

The measurements show quarters and halves, so we have to set the number line with quarter intervals, as illustrated below. It should include all the values in the table from 4 ½ inches to 5 ¾ inches.

Mark 1 *X *above 4 ½ since it occurs once, mark 2 *Xs *above 4 ¾ since it occurs twice, mark 1 *X *above 5 since it occurs once, mark 1 *X *above 5 ¼ since it occurs once, mark 2 *Xs *above 5 ½ since it occurs twice, mark 1 *X *above 5 ¾ since it occurs once.

**Lengths of Pens**

Do not forget to label the line plot properly and put a title to it to describe what the collected data is all about.

- How many pencils did Mario measure?

Mario measured a total of 8 pencils.

- What is the measure of the shortest pencil?

The shortest pencil measures 4 ½ centimetres.

- What is the measure of the longest pencil?

The longest pencil measures 5 ¾ centimetres.

- What pencil lengths appear more than once?

The pencil lengths that appear more than once in the data set are 4 ¾ and 5 ½ .

- How many pencils have lengths greater than 5 centimetres?

There is a total of 4 pencils with lengths greater than 5 centimetres.

**Summary**

- Length measures how long an object is or how far away a point is from another.
**Steps for Measuring an Object with the Use of a Ruler****Step 1:**Identify the type of ruler and its unit that you are using.**Step 2:**Take one end of the object you are measuring and line it up at the ruler’s beginning or the ruler’s zero mark.**Step 3:**Figure out where the other end of the object lines up with the ruler.**Step 4:**Count the number of lines and determine the fraction to use when needed.**Step 5:**Record the measurement.*The image below is an example of measuring the length of a pencil with a ruler.*

- There are two classifications of measuring lengths that use different units: the non-standard and standard units of measure.
- Examples of non-standard units for measuring length are handspan, foot span, cubit, etc.
- Examples of standard units of measuring length are millimetres, centimetres, metres, kilometres, etc.
- Examples of imperial units measuring lengths are the inch, foot, yard, mile, etc.
- A line plot is a graph showing the number of times a value in a data set occurred.
- The following are the steps to follow in constructing a line plot:
**Step 1:**Gather and organize the data by arranging the values from least to greatest.**Step 2:**Draw a number line.**Step 3:**Write the title of the line plot.**Step 4:**Use “X” to mark the data on the line plot.**Step 5:**Interpret the data.

- Below is a line plot of the Lengths of Ribbons Measured.

- Aside from marking the line plot, the following must be set clearly in the line plot: the title, the label of the units used for measurement, and scales or the intervals in the number line.

The common things to look for when looking at line plots are the most frequent data, outliers, gaps, and clusters. The *most frequent* data are the values that appear most *often. Gaps* are the missing areas in the given data set*. Outliers* are the values that lie in extreme areas or the values that are far away from the data, and *clusters* are the isolated groups on the data set.

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