Equal sharing in halves

1.

Place two trees from Fantails trees on the mat or table.
Here are two kahikatea trees.
Eight fantails want to rest in the trees so that there are. equal numbers on each tree.

• How many fantails go on each tree?

2.

Invite students to work out the number of fantails in each tree. Discuss their strategies, which might include:

One by one: Dealing the fantails between the trees one by one. The process might be imaged or enacted by drawing a diagram or by using fingers. Students may want to recount to check that the numbers in each tree are the same.

Doubles facts: Using doubles facts, i.e., knowing 4 + 4 = 8.

3.

Use the fantail cards and trees to physically model each strategy before modelling how a calculator can be used to confirm the answer. Record the equation and discuss the meaning of the symbols.

• What do 8 and 2 mean in our problem?
• What does ÷ mean? (Shared equally)
• What does = mean?

You might also show 2 x 4 = 8, which is the multiplication equivalent.

4.

Discuss the efficiency of the strategies used.

• Which strategy is most efficient? Efficient means it takes the smallest amount of work.

5.

Pose similar problems involving two trees and an even number of fantails. Try not to make the numbers sequential, such as 10 fantails then 12 fantails, as students are likely to focus on the number pattern rather than the process of equal sharing. Encourage students to act out the strategies they use, using the tree and fantail cards. A good progression is:

• 12 fantails are allocated to two trees
• 6 + 6 = 12, 12 ÷ 2 = 6, 2 x 6 = 12
• 20 fantails are allocated to two trees
• 10 + 10 = 20, 20 ÷ 2 = 10, 2 x 10 = 20
• 14 fantails are allocated to two trees
• 7 + 7 = 14, 14 ÷ 2 = 7, 2 x 7 = 14
• 14 fantails are allocated to two trees
• 7 + 7 = 14, 14 ÷ 2 = 7, 2 x 7 = 14

You might allow students to work in groups comprised of students with a range of mathematical understandings, encouraging tuakana-teina and productive learning conversations. You might also encourage students to use a range of other means of action and expression (e.g., written equations, verbal explanations, and digital tools) to solve these problems.

6.

Gather together and have students share the strategies used. Record equations to represent each strategy as they are shared, and confirm the calculations with a calculator. Discuss the efficiency of the strategies. Strategies are likely to include doubles addition, division by two, and multiplication by two. If appropriate, you could introduce the te reo Māori kupu whakarea (times, of, multiply) and whakawehe (divide, division).

7.

Progress to problems in which the number of fantails is odd, such as 9 fantails, then 15 fantails. Encourage students to anticipate the result of equal sharing.

For example:

• You shared 15 fantails between two kahikatea trees. What happened?

Students should note that it was impossible to put equal numbers of fantails in each tree. There was one fantail left over. 
Use a calculator to demonstrate that 15 ÷ 2 = 7.5.

• What does 7.5 mean?

Students might suggest that 7.5 means “seven and one half.” 
We could put half a fantail in each tree, but that would not be kind.

• What else could we do?

Students might suggest putting five fantails in one tree and four fantails in the other. Others might say that the remaining fantail needs to find a different tree.
Record 15 ÷ 2 = 7 r 1. Discuss the meaning of r as the remainder.

8.

Provide students with some practice problems to solve independently.

For example:

• There are 18 fantails and two kahikatea trees. For there to be equal numbers, how many fantails should go on each tree?
• There are 13 fantails and two kahikatea trees. For there to be equal numbers, how many fantails should go on each tree?

9.

Roam and observe the strategies used by students, their understanding of the remainder, and their use of materials and mathematical language. You might gather together to address some key misconceptions, share some great strategy work, or you might work more closely with different individuals or groups of students. Consider which grouping and scaffolding approach is most appropriate for your learners. 

1.

Develop students’ fluent understanding of the connection between doubles facts and halving numbers. You might ask students to complete practice tasks, such as completing a table like this:
 

Alternatively, you might play Halving Bingo using a set of cards labelled with even numbers 2-24. Students can choose which numbers from 1-12 go in their 3 x 3 board and can use the same number as many times as they like. Shuffle the 2-24 even number cards. Show them one at a time. Students cover the number that is half of the card shown (e.g., if 6 was shown, they would cover 3). The first person to cover all their numbers wins.

2.

Encourage the transfer of halving strategies to division by four. Pose problems that relate division by two situations to division by four.

For example:

• There are 12 fantails and two kahikatea trees. For there to be equal numbers, how many fantails should go on each tree?
• There are 12 fantails and four kahikatea trees. For there to be equal numbers, how many fantails should go on each tree?

Help students see that division into four equal parts can be achieved by halving and then halving again.