B. Number · Operations: Adding and Subtracting to 1000, Multiplying and DividingGroups of Halves, Thirds and Fourths
B2.7 B2.8About 40 minutes
Learning goals- I can read and write fractions like 1/2, 1/3 and 3/4.
- I can add the same unit fraction again and again, like 1/4 + 1/4 + 1/4 = 3/4.
- I can explain that the top number counts how many unit fractions there are.
- I can solve problems with groups of one half, one third and one fourth.
Words to know
unit fraction
One equal part of a whole, like 1/2, 1/3 or 1/4. The top number is 1.
numerator
The top number of a fraction. It counts how many equal parts we have.
denominator
The bottom number of a fraction. It tells how many equal parts make one whole.
whole
One complete thing, like one whole pizza or one whole sandwich.
repeated addition
Adding the same thing again and again, like 1/3 + 1/3 + 1/3.
Remember from Grade 2
In Grade 2 you cut things into equal parts: halves, thirds, fourths and sixths. You learned that 2 halves, 3 thirds or 4 fourths make 1 whole. You also put equal groups of halves and fourths together. Now you will learn to write fractions with numbers, and to count groups of fractions.
Writing fractions with numbers
We write one fourth like this: 1/4. (In books it is often written with the 1 on top of the 4 and a line between them.)
- The bottom number (the denominator) tells how many equal parts make one whole. Here, 4 parts.
- The top number (the numerator) tells how many of those parts we have. Here, 1 part.
A fraction with a 1 on top, like 1/2, 1/3 or 1/4, is called a unit fraction. It is one single equal part.
Counting fourths
A granola bar is cut into 4 equal pieces. Each piece is 1/4. You eat one piece, then another, then another. How much did you eat?
Count the fourths: one fourth, two fourths, three fourths.
1/4 + 1/4 + 1/4 = 3/4
The 3 on top of 3/4 tells us we added three unit fractions of 1/4.
The top number counts
The numerator counts how many unit fractions there are.
- 2/3 = 1/3 + 1/3 (two thirds)
- 3/4 = 1/4 + 1/4 + 1/4 (three fourths)
- 5/6 = 1/6 + 1/6 + 1/6 + 1/6 + 1/6 (five sixths)
When you add the same unit fraction, the bottom number stays the same. The size of the pieces does not change; you just have more of them!
Worked exampleExample 1: Count thirds
Write 1/3 + 1/3 as one fraction.
- Each piece is one third.
- Count them: one third, two thirds.
- The pieces are still thirds, so the bottom number stays 3.
Groups of halves
Multiplication can be equal groups of fractions too!
6 kids each eat 1/2 of a sandwich. How many whole sandwiches do they eat?
That is 6 groups of one half: 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 = 6/2.
Every 2 halves make 1 whole. 6 halves make 3 wholes.
Worked exampleExample 2: Groups of fourths
8 kids each eat 1/4 of a pizza. How many whole pizzas do they eat?
- 8 groups of one fourth = 8 fourths.
- Every 4 fourths make 1 whole pizza.
- 8 fourths = 4 fourths + 4 fourths = 2 wholes.
Worked exampleExample 3: Groups of thirds
A baker uses 1/3 cup of oats in each muffin batch. The baker makes 5 batches. How many cups of oats are used?
- 5 groups of one third = 1/3 + 1/3 + 1/3 + 1/3 + 1/3 = 5/3.
- 3 thirds make 1 whole cup. That leaves 2 more thirds.
- So 5 thirds is 1 whole cup and 2/3 of a cup.
The baker uses 1 and 2/3 cups of oats (five thirds).
5 thirds: 3 thirds fill the first cup, and 2 thirds are in the second cup.| Groups | As a fraction | In wholes |
|---|
| 4 groups of 1/2 | 4/2 | 2 wholes |
| 3 groups of 1/3 | 3/3 | 1 whole |
| 6 groups of 1/3 | 6/3 | 2 wholes |
| 8 groups of 1/4 | 8/4 | 2 wholes |
| 5 groups of 1/4 | 5/4 | 1 whole and 1/4 |
Just for funWhy did one half of a sandwich invite the other half to the party?
Because together they make a whole lot of fun!
Hands-on activityPaper strip fractions
You need: 4 strips of paper the same length (about as long as a ruler), a pencil, and crayons or markers.
- Fold two strips in half. Unfold them and draw a line on each fold. Label each part 1/2.
- Fold another strip into fourths (fold in half, then in half again). Label each part 1/4.
- Colour the fourths one at a time and say: "1/4, 2/4, 3/4, 4/4." What is 4/4 the same as?
- Try folding a strip into thirds (this is tricky; fold it like a letter). Label each part 1/3.
- Now cut the parts apart. A grown-up should help with scissors. Use the pieces to show: 3 groups of 1/2. How many wholes is that? (1 whole and 1/2.)
Math talk: "Why does the bottom number stay the same when I add more fourths?"
Your turn
14 halves make how many wholes?
2 wholes. 2 halves make 1 whole, so 4 halves make 2.
31/4 + 1/4 + 1/4 + 1/4 = ?
4How many 1/4 pieces are in 3/4?
3. The numerator 3 counts the fourths.
510 kids each eat 1/2 an apple. How many whole apples?
5 apples. 10 halves = 5 wholes, because every 2 halves make 1 whole.
6/3, which is 2 wholes. 3 thirds + 3 thirds = 2 wholes.
7Write 5/6 as a repeated addition of unit fractions.
1/6 + 1/6 + 1/6 + 1/6 + 1/6
More practice: warm-up
1How many halves make 1 whole?
2How many thirds make 1 whole?
3What is the numerator (top number) in 2/3?
5A pie is cut into 4 equal pieces. What fraction is one piece?
More practice: on your own
112 kids each eat 1/4 of an orange. How many whole oranges?
3 oranges. 12 fourths: 4 fourths make 1 whole, and 4 + 4 + 4 = 12, so 3 wholes.
27 halves is how many wholes?
3 wholes and 1/2. 6 halves make 3 wholes, and there is 1 half left.
3Write 4/6 as a repeated addition.
1/6 + 1/6 + 1/6 + 1/6. The numerator 4 tells us there are 4 sixths.
4A recipe needs 1/3 cup of milk. You make it 3 times. How much milk do you need?
1 cup. 1/3 + 1/3 + 1/3 = 3/3 = 1 whole cup.
5What is 5 groups of 1/4? Say it in wholes.
5/4, which is 1 whole and 1/4. 4 fourths make 1 whole, and 1 fourth is left.
Challenge
1A soccer team has 9 players. At half-time each player eats 1/2 of an orange. How many oranges does the coach need to cut?
9 halves = 4 and 1/2 oranges. The coach has to cut 5 oranges (there will be one half left over).
2How many 1/3 cups of flour do you need to make 2 whole cups?
6. 3 thirds make 1 cup, so 3 + 3 = 6 thirds make 2 cups.
3Lia says 1/4 + 1/4 = 2/8. Explain her mistake.
When you add fourths, the pieces stay fourths, so the bottom number stays 4. 1/4 + 1/4 = 2/4. (2/8 would mean 2 pieces from a whole cut into 8, which is smaller.)
Quiz
1. What does the bottom number in 3/4 tell you?
3. 6 kids each eat 1/2 of a sandwich. How many whole sandwiches?
4. Which is the same as 4/6?
Big idea: Grade 3 students solve problems involving groups of one half, one fourth and one third (B2.7), and see that the numerator counts repeated unit fractions: 3/4 = 1/4 + 1/4 + 1/4 (B2.8). This is also when children start using standard fraction notation like 3/4.
Common difficulties:
- Adding the bottom numbers (1/4 + 1/4 = 2/8). Use paper strips: the pieces are still fourths.
- Not seeing 4/4 or 3/3 as 1 whole. Fold and colour strips until the whole strip is coloured.
- Thinking bigger denominators mean bigger pieces. A third is bigger than a fourth: fewer pieces means each piece is bigger.
- Counting groups of halves. Pair up the halves: each pair is a whole.
How to help: Use food! Cut sandwiches, oranges or granola bars into halves, thirds or fourths, and count the pieces aloud: "one fourth, two fourths, three fourths, four fourths, one whole." Fraction words in the kitchen (1/2 cup, 1/3 cup) are great practice. Supervise any cutting with knives.