Equal sharing into more than two shares

1. Place four tree cards and 16 fantail cards from the Copymaster on the mat or table.
   There are four kahikatea trees.
   Sixteen 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 anticipate the result of the equal sharing without access to calculators. Ask them to record their thinking with diagrams or equations so they can explain their strategies to others. Ask students to share their strategies with a partner before you bring the group together.
    
3. Provide time for students to share their strategies with the wider class. Look for these strategies:
   • One by one dealing: This involves dealing fantail cards out, one by one, before checking that all the fantails are used up, and counting to confirm the equal size and number of the shares. This strategy is common with students whose preference is counting.
   • Successive trial and improvement: This might involve trying three fantails for each tree then adding one more fantail to each tree until all 16 fantails are allocated. Use questioning to draw attention to the fact knowledge used. For example: Asking how three fantails per tree was worked out might lead to an explanation relying on 3 + 3 = 6 and 6 + 6 = 12.
   • Halving and halving using doubles facts: This shows emerging multiplicative thinking. Half of 16 equals 8 and half of 8 equals 4. Use the physical model to show students how the repeated halving works. Half of 16 equals 8. So one quarter of 16 equals 4.
   • Using multiplication facts: For example, 4 x 4 = 16. Ask the student what the fours refer to in the multiplication equation and how knowing 4 x 4 = 16 solves the problem.
      
4. Let students use the calculators. Demonstrate how keying in 16 ÷ 4 = 4 solves the problem of sharing 16 fantails among 4 trees. Record the equation and discuss what the numbers and division symbol refer to. You might also record the matching multiplication equation to see if students notice the relationships.
   4 x 4 = 16
   16 ÷ 4 = 4
    
5. Pose a sequence of problems based on the same scenario of fantails occupying kahikatea trees. Physically model each problem with the cards from the copymaster, whilst encouraging students to use mental or diagrammatic strategies to anticipate the equal shares. 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, digital tools) to solve these problems. If appropriate, you could introduce the te reo Māori kupu whakarea (times, of, multiply) and whakawehe (divide, division). Good examples are:
   • There are ten kahikatea trees.
     Thirty fantails want to rest in the trees so that there are equal numbers on each tree.
     How many fantails go on each tree? (30 ÷ 10 = 3).
   • There are five kahikatea trees.
     Fifteen fantails want to rest in the trees so that there are equal numbers on each tree.
     How many fantails go on each tree? (15 ÷ 5 = 3).
   • There are four kahikatea trees.
     Forty fantails want to rest in the trees so that there are equal numbers on each tree.
     How many fantails go on each tree? (40 ÷ 4 = 10).
   • There are three kahikatea trees.
     Eighteen fantails want to rest in the trees so that there are equal numbers on each tree.
     How many fantails go on each tree? (18 ÷ 3 = 6).
      
6. 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. 
    
7. Gather together and have students share the strategies used. As students explain their strategies, act them out using the cards, record relevant division equations, and confirm the calculations with a calculator. 
    
8. Discuss the efficiency of the strategies. Be mindful that the use of any strategy is based on knowledge and practice.
    
9. Give students the opportunity to make up their own division story involving fantails and trees. Require them to provide a model answer that shows how the problem might be solved. They should include the division calculation performed on the calculator.

Next steps 

1. Explore the connection between multiplication facts and division. Begin by establishing a multiplication fact. For example:
   There are four kahikatea trees. In each tree there are three fantails. 
   How many fantails are there altogether?
   If needed, use a calculator to establish that 4 x 3 = 12.
   What division problem could we solve with 4 x 3 = 12? Use the fantails and trees.

• Look for students to frame the problem as “12 fantails shared equally among 4 trees” and record 12 ÷ 4 = 3.
   

2. Create a display that shows connected multiplication and division facts and relevant stories. These stories might include situations like sharing items among friends (marbles, toy cars, all black or silver fern collection cards), allocating farm animals to pens, sharing money from a job, or serving hāngi vegetable pieces among guests. Discuss what is the same about each situation to see if students can link equal sharing to division.