Times as many

1. Pose problems that involve ‘times as many’ using facts that are readily accessible. Consider your students' multiplication facts knowledge. There might be opportunities in these problems for students to work in tuakana-teina partnerships and scaffold their peers' multiplication facts knowledge.
   Here is a stack of two cubes. 
   Can you build a stack that is five times higher?
    
2. Let the student solve the problem with materials and talk aloud as they do so. Look to see that students build a higher stack that contains 10 cubes. Pose questions focused on anticipating how many cubes are needed. 
   How many cubes should your stack have?
   How do you know?
   What fraction of the big stack is the small stack? (one fifth or 1/5)
   If the student/s cannot interpret the problem, model solving it yourself. Use colour breaks of two cubes to make the stack, as shown. Model iterating the unit of two cubes along the length of the 10-cube-stack to show it is “five times higher.” You might introduce the kupu hautau (fraction) as part of this modelling.
    
3. Show the student how the problem could be recorded using an equation: 5 x 2 = 10. Discuss what role the symbols play in the context and support the use of relevant mathematical language and te reo māori kupu (e.g, times, of, multiply - whakarea).

• What does 2 mean in the problem? (The height of the original stack)
• What does 10 mean in the problem? (The height of the bigger stack)
• What does 5 mean in the problem? (Five times as many cubes are needed. The big stack is five times as high as the small stack)
• What does the x symbol mean in the problem? (It means ‘of’ in the sense that the large stack is made of 5 lots of the small stack).
   

4. Model building the stack of 10 cubes, drawing attention back to the equation and language used (e.g. there are 5 times as many cubes in this stack, there are 5 lots of 2 in this stack). Provide time for students, either independently or in pairs, to build their own cube tower and then describe it with a partner. Support students to make comparisons being the two cube towers using relevant mathematical language. You might record key statements and vocabulary on a class chart.
    
5. Display your stacks of cubes. Ask students to change their stack of 10 cubes to be three times as high as the small stack.
   How many cubes will be in the big stack now?
   What equation should we write now?  3 x 2 = 6
   What fraction of the big stack is the small stack? (one third or 1/3)
    
6. Provide time for students to change their stacks, then model changing your own stack. Confirm the height of the stack and, as a class, describe it in comparison to the 2-cube stack.
    
7. Ask students to change their stack of 6 cubes to be seven times as high as the small stack.
   How many cubes will be in the big stack now?
   What equation should we write now?  7 x 2 = 14
   What fraction of the big stack is the small stack? (one seventh or 1/7)
    
8. Provide time for students to change their stacks and discuss them in comparison to the 2-cube stack. Ask a student or student pair to model this change for the class and describe the 14-cube stack for the class. As a class, confirm the height of the stack and share the comparative statements made,
    
9. Pose further “times as many" problems. Focus students’ attention on predicting how many cubes the big stack will need, before building. You might provide a graphic organiser or mini whiteboards for students to use for drawing or recording their estimations and working out. As variations in problems occur, ask what changes and what stays the same among those problems. Examples of problems include:
   Here is a stack of three cubes. Make a stack that is … two times higher…five times higher … three times higher. (2 x 3 = 6,  5 x 3 = 15, 3 x 3 = 9)
   Here is a stack of five cubes. Make a stack that is … two times higher…four times higher … six times higher. (2 x 5 = 10,  4 x 5 = 20, 6 x 5 = 30)
   Support students to make the connection that the multiplier tells how many times the unit stack must be copied (iterated) to make the target stack. The multiplier also gives the denominator of the fraction.

Next steps

1. Increase the level of abstraction by covering the materials, asking anticipatory questions, and working with more complex facts (e.g. 6 x 7 or 9 x 3).

A suggested sequence for extending the difficulty of the additions and subtractions is:

• Using unit stacks of two, five, and ten cubes with multipliers five or less.
• Using unit stacks of three, and four cubes with multipliers five or less.
• Using unit stacks of six, seven, eight and nine cubes with multipliers five or less.
• Using unit stacks of six, seven, eight and nine cubes with multipliers more than five.

More complex examples are likely to ‘sell’ the idea of multiplication to students.

2. Reverse the task. Give students a multiplication fact, such as 4 x 8, and ask them to draw or make the corresponding array with materials. Find ways to work out the product.