Comparing Apples with Apples

Note that the students will need to be familiar with the structure of the fraction strips,particularly ideas about equivalence, e.g., 1/12 = 1/4.

 Using Materials

Ask the students to model the following action with the fraction strips:

“Make four-sixths and five-sixths separately. Now join the two fractions together, end  on end. How many sixths is that altogether? How should we record that operation  using an equation?”

Students may suggest that the equation is 4/6 + 5/6 = 9/6. If not, record it for them.Ask them how else nine-sixths might be written (1 3/6 or 1 1/2).

Pose other similar problems for the students to model with the fraction strips, such as:
3/4 + 5/4 1/5 + 4/5 7/8 + 4/8

Record the answers to each problem using both fractions (e.g., 11/8 ) and mixed numbers(e.g., 1 3/11). Challenge the students to explain the link between the model of  the operation and what occurs with the symbols.
Ask why the numerators are being added but not the denominators. Link this to addition with equivalent units that the students already know, such as six tens plus five tens, seven hundreds plus eight hundreds. Develop a generalised rule for adding fractions that have the same denominator:

Ask the students to use the materials to confirm that this holds for subtraction as well.  Get them to model problems like: 12/10 – 3/10 = 9/10

Pose other related problems, such as: 7/5 – 3/5,  13/8 – 7/8,  7/2-4/2.
Tell the students that subtracting fractions with common denominators is similar to adding fractions with the same denominators. Investigate adding and subtracting fractions with different denominators, using problems such as: 3/4 + ½

Students may use the equivalence of one-half and two-quarters to realise that theanswer must be 5/4 = 1 1/4. The key idea in these examples is that one or more of the fractions must be renamed so that the denominators are the same. Provide otherproblems, such as: 1/2 + 2/3 , 7/6 – 2/3,  7/8 – 3/4,  3/4 + 3/8,  4/5 – 4/10

Record the answers using fractions and mixed numbers. Discuss the links between the numerators and denominators and the answers. Point out that the choice of common  denominator is influenced by the denominators of the fractions that are being added or  multiple of both denominators.  
both halves and thirds can be renamed as sixths:
1/2 + 2/3 = 3/6 + 4/6 = 7/6 = 1 1/6

In the example of 1/2 + 2/3, 

Using Imaging

Encourage the students to image or draw to show what fraction strips would be used to solve the following problems:

3/5 + 1/2,  3/4 + 2/3,  7/8 + 3/4,  5/6 + 2/3

2/3 – 1/2,  6/5 – 1/2,  3/4 – 1/3,  11/8 – 3/4

If necessary, fold back to using the materials to check the students’ ideas.

Using Number Properties

Use the following examples to generalise how fractions are added or subtracted:

3/4 + 2/5 2/3 + 4/5 1/6 + 3/8 5/7 + 2/3 5/6 + 3/4

7/8 – 3/4 9/10 – 4/5 8/6 – 3/8 11/12 – 1/3 5/4 – 5/10

Note that in these examples, the common denominators will require students to go outside the pieces available in the fraction strips. This requires students to generalise the number properties rather than rely on images of the materials.

Independent Activities

Students will enjoy playing the game Create 3 (See Material Master 7–9) to consolidate the addition of fractions. The game can be adapted to subtracting fractions if each student’s score starts on three and the students aim to subtract the fractions they land on until they reach zero.