Deriving from a known division or multiplication fact

1. Discuss situations in which making teams is important, such as sports days or quizzes.
   If there are ten teams of four students, how many students are there altogether?
   Ask students to predict the total. Some may try less efficient strategies like skip counting or repeated addition. Others may use place value knowledge in combination with the commutative property. This strategy can be represented as 4 x 10 = 40 so 10 x 4 = 40.
    
2. If needed, use connecting cubes or a drawn diagram to show students how 4 x 10 and 10 x 4 are related.
   
    
3. Confirm the result and record it using words and symbols.
    
4. Tell students: Now we know that 10 x 4 = 40, we can solve this problem. 48 students are sorted into teams of four. How many teams are there?
   Let students work in pairs to solve the problem. Roam as they work. Look for students to:
   • Attempt to connect 10 x 4 = 40 with the situation.
   • Record equations to help them remember and work through related facts. For example, 10 x 4 = 40 so 11 x 4 = 44 and 12 x 4 = 48. 48 ÷ 4 = 12.
   • Use schematic diagrams that display only the important information (as shown below) or pictorial (e.g. focused on drawing realistic people)?
     
   • If students rely on materials, encourage them, increasingly to use number knowledge and recording. This will ease memory load.
      
5. Gather the group after a suitable time and share strategies with a focus on the following:

• The connection between 4 x 10 = 40 and 4 x 12 = 48
• Equations that can be used to connection 4 x 10 and 4 x 12. Discuss the meaning of symbols in these equations, relating the meaning of multipliers, multiplicands, dividends, and divisors back to the contexts of teams of students.
• How schematic diagrams can be used to easily show important information about the problem.
   

6. Confirm the answers using a calculator. Emphasise the meaning of division as measurement and sharing.
    
7. Pose similar problems using the same scenario of forming teams. Examples might be:
   • There are six teams of five students at the maths quiz. How many students are at the quiz altogether?
     Use that answer to solve this problem:
     35 players turn up for the maths quiz. How many teams of five students can be made?
   • There are six crews of four students at the waka-ama competition. How many students are at the competition altogether?
     Use that answer to solve this problem:
     32 students turn up for the waka-ama race. How many teams of four students can be made?
   • Eight teams of ten students are made for the tug-o-war meet. How many students is that altogether?
     Use that answer to solve this problem:
     72 students turn up for the tug-o-war meet. How many teams of nine students can be made? (Note this is a division by sharing situation – not measurement)
   • Five teams of seven students are made for a basketball competition. How many students are in the competition altogether?
     Use that answer to solve this problem:
     49 students turn up for the basketball competition. How many teams of seven students can be made?

Allow students to work in groups comprised of students with a range of mathematical understandings, encouraging tuakana-teina and productive learning conversations. Ask students to work out the answers with their own strategies, and explain them using symbols, words, and diagrams, before using connecting cubes (if needed) and a calculator. Early finishers can be challenged to make up their own "teams" problems for classmates to solve.

8. As students work, roam and look for the following:

• Do students represent the first situation as a multiplication equation?
• Can students create matching division equations for the multiplication equation?
• Are students able to apply the distributive property of multiplication to the second division problem?
   

9. Gather the group and share the strategies used. 

Next steps

1. Support students to memorise some basic multiplication facts and the matching division facts, especially the multiples of two, five and ten. Play games and set memory targets to motivate students. Investigate patterns that the multiples of two, five and ten make on the hundreds board. Progress to anticipating the number of twos, fives or tens that are made from a given number. For example, “how many fives are in 30? How many tens are in 70?”
    
2. Explore measurement situations in which there is a remainder and discuss what the remainder means in context of the problem. For example, “each car takes five passengers. How many cars are needed to take 43 passengers?” 43 ÷ 5 = 8 remainder 3. In context this means that another car is needed to transport the remaining passengers.
    
3. Pose division word problems with larger numbers to improve students’ capacity for recognising the required operation. Allow access to calculators as students solve these problems. For example, “102 players enter the volleyball competition. Each team has six players. How many teams are made?” Look for students to enter the required calculation, 102 ÷ 6 =, and use the context to find the answer "17 teams". ​​​​​​​Calculators should be used to support students' solutions to equal sharing problems, and problems with remainders.