B. Number · Number: Number Sets, Powers and Rational Numbers in Action

Unit Fractions: Building Fractions and Measuring with Tools

B3.2 A1 A2About 65 minutes

Learning goals

I can explain how every fraction is built from unit fractions, such as 3/8 = 3 × 1/8.
I can compare fractions by thinking about unit fractions and the gap to a whole.
I can read imperial rulers and tape measures to the nearest 1/16 inch, and use measuring cups and spoons.
I can write fractions as sums of different unit fractions, the way ancient Egyptian scribes did.

Words to know

unit fraction
A fraction with numerator 1, such as 1/2, 1/8 or 1/100. It is one equal part of a whole.

non-unit fraction
A fraction with a numerator greater than 1, like 3/8. It is a count of unit fractions: 3/8 = 1/8 + 1/8 + 1/8.

imperial units
Units such as inches, feet and cups. They are still used in Canada for some building materials, tape measures and recipes.

Egyptian fraction
A sum of different unit fractions, like 3/4 = 1/2 + 1/4.

tablespoon (tbsp) and teaspoon (tsp)
Kitchen measures: 1 tbsp = 3 tsp, and 1 cup = 16 tbsp.

Remember from earlier grades

In earlier grades you added, subtracted, multiplied and divided fractions. This lesson looks at the building blocks of every fraction, the unit fractions, and at how they make measuring tools work.

Unit fractions are the building blocks

A unit fraction is one equal part of a whole: 1/2, 1/3, 1/8, 1/16. Every other fraction is a count of unit fractions:

3/8 means three pieces of size 1/8: 3/8 = 3 × 1/8.
7/4 means seven quarters: 7/4 = 7 × 1/4 = 1 3/4.

The denominator names the size of the piece and the numerator counts the pieces. The bigger the denominator, the smaller the unit fraction: 1/16 < 1/8 < 1/4 < 1/2.

A fraction wall, like a set of measuring cups. More parts means smaller parts: 1/8 cup is half of 1/4 cup, and two 1/6 cups make 1/3 cup.

Measuring tools are made of unit fractions

Imperial ruler or tape measure: each inch is split into halves, quarters, eighths and sixteenths. The longest marks are the halves; the shortest are the sixteenths. To read a length, count the sixteenths and simplify: 6/16 in = 3/8 in.
Measuring cups and spoons: typical sets have 1, 1/2, 1/3 and 1/4 cup. To measure 3/4 cup, fill the 1/4 cup three times, or use 1/2 + 1/4.
Metric tools are unit fractions too: 1 mm = 1/10 cm = 1/1000 m, and 1 mL = 1/1000 L. Metric works in tenths, so its unit fractions are written as decimals.

An imperial ruler, drawn to scale between its marks (not actual size). Each inch is divided into 16 equal parts. The pencil stub ends 5 marks past the 1 inch mark, so it is 1 5/16 in long.

Worked example

Example 1: Reading a tape measure

A board ends at the 11th sixteenth mark after 4 in. A second board ends at the 6th sixteenth mark after 2 in. Write both lengths in simplest form, and find the difference.

First board: 4 11/16 in. (11 and 16 have no common factor.)
Second board: 2 6/16 in = 2 3/8 in.
Difference in sixteenths: 4 11/16 − 2 6/16 = 2 5/16 in.

4 11/16 in and 2 3/8 in; the difference is 2 5/16 in.

Worked example

Example 2: Measuring cups

A bannock recipe needs 2 1/4 cups of flour, but you can only find the 1/4 cup measure. How many scoops? Then the recipe is cut in half. How much flour is that, using tablespoons?

2 1/4 cups = 9/4 cups = 9 × 1/4. So 9 scoops.
Half of 9/4 is 9/8 cup = 1 1/8 cups.
1/8 cup = 16 tbsp ÷ 8 = 2 tbsp. So 1 1/8 cups = 1 cup + 2 tbsp.

9 scoops; half the recipe needs 1 cup + 2 tbsp of flour.

Worked example

Example 3: Compare with unit fractions

Which is greater: 7/8 or 9/10? Do it without common denominators.

7/8 is 1/8 short of a whole. 9/10 is 1/10 short of a whole.
1/10 < 1/8, so 9/10 is closer to 1.

9/10 is greater.

Egyptian fractions

Scribes in ancient Egypt, more than 3500 years ago, wrote fractions as sums of different unit fractions (with a special symbol for 2/3). For example, they wrote 3/4 as 1/2 + 1/4, and 5/6 as 1/2 + 1/3. A papyrus scroll in the British Museum, the Rhind papyrus, has a whole table of these.

A method that always works (the greedy method): subtract the largest unit fraction that fits, then repeat with what is left.

Worked example

Example 4: Write 7/12 as an Egyptian fraction

Write 7/12 as a sum of different unit fractions.

The largest unit fraction that is not more than 7/12 is 1/2 (since 7/12 > 6/12 = 1/2).
7/12 − 1/2 = 7/12 − 6/12 = 1/12. That is already a unit fraction.
Check: 1/2 + 1/12 = 6/12 + 1/12 = 7/12 ✓

7/12 = 1/2 + 1/12

Worked example

Example 5: A unit fraction of an amount

A team has driven 3/5 of the 240 km trip to the hockey tournament. How far have they driven?

Find the unit fraction first: 1/5 of 240 = 240 ÷ 5 = 48 km.
3/5 is three of those: 3 × 48 = 144 km.

144 km

Hands-on activity

Measure and scale a recipe

You need: measuring cups and spoons, a tape measure or ruler with inch marks, a recipe card or a recipe from a website.

Use the tape measure to measure 3 things at home (a book, a phone, a spoon) to the nearest 1/16 inch. Write each in simplest form. Then measure them in centimetres and millimetres.
Choose a recipe. For each ingredient, decide which cup or spoon you would use and how many times, if you could only use one measure. (Example: 3/4 cup → the 1/4 cup, 3 times.)
Rewrite the recipe for half the amount. Where an amount like 3/8 cup appears, change it to tablespoons (1 cup = 16 tbsp, so 3/8 cup = 6 tbsp).

If you actually bake, do it with a grown-up, especially when using the oven.

Just for fun

Riddle: Why was 1/2 so calm when it met 1/16?

Because it knew it was eight times as big as the problem.

Your turn

Write 5/6 as a count of unit fractions.

Order from least to greatest: 1/5, 1/3, 1/9, 1/4.

A tape measure reads 3 inches and 12 sixteenths. Write the length in simplest form.

How many 1/3 cup scoops make 2 2/3 cups?

Which is greater: 11/12 or 13/14? Use the gap to 1.

Write 5/8 as an Egyptian fraction.

What is 2/7 of 63 L?

More practice: warm-up

Is 3/4 a unit fraction?

How many 1/8 in are in 1 inch?

How many 1/4 cups make 1 1/2 cups?

Which is smaller: 1/12 or 1/10?

Find 1/6 of 54.

More practice: on your own

A shelf is 23 5/8 in wide. How many sixteenths of an inch is the fractional part? Write the full length in sixteenths of an inch.

A recipe uses 3/4 cup of maple syrup. You want to make 2 1/2 times the recipe. How much syrup do you need?

Write 2/3 as a sum of two different unit fractions.

How many tablespoons are in 3/4 cup? (1 cup = 16 tbsp)

Order 5/6, 7/8, 4/5, 9/10 from least to greatest using unit fractions.

Challenge

Use the greedy method to write 3/7 as an Egyptian fraction.

Show that 1/n − 1/(n + 1) = 1/(n(n + 1)) for n = 2, 3 and 4. What pattern do you see?

A carpenter needs 6 pieces of trim, each 7 3/8 in long, from a board that is 48 in long. Each saw cut removes 1/8 in. Is the board long enough? Explain.

Check your understanding

Quiz

Which is a unit fraction?

Which unit fraction is largest?

A length is 14/16 in. What is it in simplest form?

How many 1/3 cup scoops make 1 2/3 cups?

Which shows 5/6 as an Egyptian fraction?

Which is greater: 6/7 or 8/9?

For parents and teachers

Big idea: Expectation B3.2 asks students to apply their understanding of unit fractions and how they relate to other fractions, in various contexts, including measuring tools. Seeing 5/8 as "five eighths" helps with comparing, estimating and every fraction operation.

Common difficulties:

Thinking a bigger denominator means a bigger fraction (1/8 > 1/4).
Misreading rulers: counting the marks instead of the spaces, or not knowing which marks are eighths and which are sixteenths.
Forgetting to simplify (8/16 in is 1/2 in).

How to help: Cook or build something together. Let your teen do the measuring and the scaling of the recipe. Canada uses metric officially, but many tape measures and recipes use inches and cups, so both are life skills.