Deriving by doubling

Note: te reo Māori kupu for numbers (e.g. ono - six) can be used throughout this lesson.

1. Show the students the cards made from the animals copymaster.
   Can you name the animals?
   Tuatara, kea, octopus (wheke), weta
   How many legs does each animal have?
   Students might use the pictures to count legs if they do not know. A weta is an insect, so it has six legs. “Octo” is ancient Greek for eight so an octopus has 8 legs. If needed, you could adapt this context to reflect characters from recent shared texts.
    
2. Arrange three kea cards in a line.
   
   How many kea are there?
   Each kea has two legs. What multiplication tells me how many legs there are in total?

• Students might suggest that 4 x 2 = 8 gives the number of legs.
• Discuss the role of 3, the multiplier, as ‘how many kea” there are. 
• Discuss the role of 2, the multiplicand, as “number of legs per kea.” Note that per means “for every.” 
• Use calculators to check the multiplication agrees with observations from the picture, 3 x 2 = 6.
   

3. Imagine we double the number of kea. How many kea would we have then?
   How many legs would that be altogether?
   Let students predict the outcome of doubling first before building the model.
   
   What multiplication can we use to find the total number of legs for six kea?
   Students should suggest 6 x 2.
   If 3 x 2 = 6 what does 6 x 2 equal?
   Students might suggest that double 6 is the product. Check with the calculator though the product is easy to see in the picture.
    
4. Record the two multiplication equations to see if students can see patterns.

• 3 x 2 = 6 and 6 x 2 = 12
• You might allow students to use drawings, counting, written expressions, and materials to explore this equation. Look for students to see that as the multiplier doubles, so too does the product.
   

5. Pose similar problems using the animal cards, in which the multiplier is doubled. Good examples might be:
   • Five kea have a total of 10 legs. (5 x 2 = 10). Ten kea have a total of 20 legs. (10 x 2 = 20)
   • Three tuatara have a total of 12 legs. (3 x 4 = 12). Six tuatara have a total of 24 legs. (6 x 4 = 24)
   • Four weta have a total of 24 legs. (4 x 6 = 24). Eight weta have a total of 48 legs. (8 x 6 = 48)
   • Three octopuses (wheke) have a total of 24 legs. (3 x 8 = 24). Six octopuses (wheke) have a total of 48 legs (6 x 8 = 48)
      
6. Broaden the doubling to include doubling the multiplicand. 

• Here are five tuatara. How many legs are there altogether?
  What is the multiplication equation for this story? (5 x 4 = 20).
  
   
• If I change the tuatara for octopuses, how many legs will there be in total?
  What multiplication equation matches the story now? (5 x 8 = 40)

  

• Look for students to recognise that the total number of legs doubles.
• Record the equations and invite students to look for patterns: 5 x 4 = 20 and 5 x 8 = 40.
• You might allow students to use drawings, counting, written expressions, and materials to explore this equation.
   

7. Pose similar problems in which a known fact is used, and the multiplicand is doubled. Allow students to work on these in groupings that support the scaffolding and extension of students' thinking. Encourage students to demonstrate their thinking using appropriate means of action and expression. 
   Examples of questions might include:
   • Here are six kea. How many legs are there altogether?
     If I change the kea for tuatara, how many legs will there be in total?
   • Here are eight tuatara. How many legs are there altogether?
     If I change the tuatara for octopuses, how many legs will there be in total?
      
8. Provide time for students to share their working with other students. Display the pairs of equations and, as a class, identify any patterns. Use calculators where appropriate to check the derived answer is correct.

Next steps

1. Challenge the students by posing problems in which the multiplier or multiplicand is trebled or quadrupled. For example:
   • I have three tuatara. How many legs is that in total? (3 x 4 = 12)
     If I triple the number of tuatara to nine, how many legs will there be? (9 x 4 = 36)
   • I have five kea. How many legs is that? (5 x 2 = 10)
     If I change the kea into weta, how many legs will there be? (5 x 6 = 30)
      
2. Expect students to memorise some basic multiplication 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.