A Peek Beyond the Point · Lesson 1 of 11
The Need for Smaller Units and A Tenth Part
“Discover why smaller equal parts help us describe lengths more precisely.”
• Explain why a whole-unit ruler may not distinguish nearby lengths. • Interpret one-tenth as one of ten equal parts of a chosen unit. • Write a length as whole units and tenths or entirely as tenths. • Compare and order measurements expressed in tenths.
The Need for Smaller Units
Two screws may look almost identical, yet only one fits a toy correctly. A small difference in length can make a real difference. If we describe both screws only as being between 2 cm and 3 cm long, we hide the difference that matters. We need a way to describe positions between whole centimetres.
A ruler with only whole-centimetre divisions tells us which two whole numbers surround an endpoint. A ruler with smaller divisions can tell us more. Suppose the gap from 2 cm to 3 cm is divided into ten equal parts. Each small part is one-tenth of a centimetre, and an endpoint seven small parts beyond 2 cm has length 2 cm and seven-tenths of a centimetre.
A chosen quantity used as a reference for measurement. In a length measurement, the unit might be a centimetre, a metre, or a specified unit strip.
Before comparing measurements, check that they use the same unit. Four parts of a long strip can be longer than five parts of a short strip. The word “part” tells us little unless we know the whole that was divided. In this lesson, every fraction refers to the stated unit.
Problem
A screw begins at 0 cm and ends at the seventh small division after 2 cm. Ten small divisions make 1 cm. Find its length.
- 1.Identify the whole units first: the screw reaches beyond 2 cm.
- 2.Each small interval is 1/10 cm, so seven intervals give 7/10 cm.
- 3.Its length is 2 + 7/10 cm, read as two and seven-tenths centimetres. Another screw ending two intervals beyond 3 cm is 3 + 2/10 cm long.
A Tenth Part
Splitting a unit into ten equal parts gives a useful new building block. We can count these smaller blocks just as we count whole units. The important difference is their size: ten of the small blocks are needed to make one complete unit.
One of ten equal parts of one unit. It is written as 1/10 of that unit.
In the fraction 4/10, the bottom number 10 is the denominator: it tells us that the unit has ten equal parts. The top number 4 is the numerator: it counts how many of those parts are taken. This way of reading a fraction will also help when we use hundredths and thousandths.
A length of 3 units and 4 tenths contains three whole groups of ten tenths, followed by four more tenths. Counting entirely in tenths gives 30 + 4 = 34 tenths. These descriptions refer to the same length; one groups the parts into units and the other counts all the small parts.
Problem
A pencil is 3 units and 4 tenths long. Express its length entirely in tenths.
- 1.Each unit contains 10 tenths, so 3 units contain 30 tenths.
- 2.Add the remaining 4 tenths: 30 + 4 = 34 tenths.
- 3.The length is 34/10 units, which is the same quantity as 3 + 4/10 units.
Problem
A cable measures 48 tenths of a unit. Express this using whole units and tenths.
- 1.Group 48 tenths into groups of 10.
- 2.Four groups use 40 tenths and make 4 units. There are 8 tenths left.
- 3.The cable measures 4 + 8/10 units. Neither description changes the unit or the physical length.
Comparing Lengths in Tenths
To compare two lengths, express them using the same-sized parts. For example, 4 + 1/10 units is 41 tenths, while 4/10 unit is only 4 tenths. Similar-looking digits can represent very different lengths when the units attached to them differ.
Problem
Arrange 4 + 1/10, 4/10, 41/10, and 41 + 1/10 units in increasing order.
- 1.Convert each quantity into tenths: 41 tenths, 4 tenths, 41 tenths, and 411 tenths.
- 2.Compare the counts: 4 < 41 = 41 < 411.
- 3.The order is 4/10 < 4 + 1/10 = 41/10 < 41 + 1/10. The middle two quantities are equal.
Measure a pen, a sharpener, and an eraser with a ruler. Keep one end at zero. Record each length in whole centimetres and tenths of a centimetre, then convert it entirely into tenths. If an endpoint falls between the smallest divisions, describe the reading as approximate rather than pretending the ruler gives an exact value.
A tenth is an equal part, not simply any small piece. Also, if an object starts at a nonzero ruler position, its length is the difference between its endpoint readings. Reading only the right-hand endpoint would include the unused gap before the object.
Check Your Understanding
Use the ideas from this lesson to choose an answer. Explain your choice to yourself before opening the explanations below.
Quiz
A unit is divided into ten equal parts. What is the size of each part?
How many tenths are in 3 units and 4 tenths?
Which quantity equals 41/10 units?
Which length is smallest?
Why does a ruler with tenth-unit divisions help distinguish two nearby endpoints?
Ten equal parts together make one unit, so each is 1/10 unit.
Practice Problems
- Write 6 units and 7 tenths entirely in tenths.
- Write 130 tenths as whole units and tenths.
- Arrange 9/10, 1 + 7/10, 130/10, 13 + 1/10, 10 + 5/10, 7 + 6/10, 6 + 7/10, and 4/10 units in increasing order.
- An object starts at 1 cm and ends at 4 cm and 8 tenths. How long is it?
- A friend says five tenths of any unit always measure the same physical length. Explain the mistake.
- Measure an object using a ruler and explain how its smallest divisions affect your reading.
Six units contain 60 tenths. Add 7 tenths to get 67 tenths, or 67/10 units.
Key Takeaways
• A chosen unit gives meaning to a measurement. • Smaller equal divisions help describe positions between whole units. • One-tenth is one of ten equal parts of the same unit. • Ten tenths make one unit. • Whole units and tenths can be regrouped without changing the quantity. • Compare quantities using the same unit and equal-sized parts.
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Adding and Subtracting Tenths