During Ms Huang’s lessons students have been exploring a variety of calculations including the distributive law of multiplication, that a x (b + c) = a x b + a x c. Through procedural variation students got to grips with the concept, after which it was time to utilise it to efficientlycalculate the answer to the following questions.
There is more than one way to calculate the answer to these questions, but students were specifically asked to use the concept to make their calculations more efficient – can you see how?
The firs lesson by Ms Huang on calculations introduced students to the concept of the commutative law of addition (and multiplication) and how it can be utilised to perform mental calculations in the most efficient manor.
The aim of the lesson was for students to understand that changing the order of the addends did not change their sum.
Taking the concrete to the abstract
The story of the monkey king
One day the monkey king received many peaches and so he said to the little monkey:
‘I will give you 3 peaches in the morning and then I will give you 4 peaches in the afternoon.’
The little monkey did not feel good about this, as he wanted to receive more peaches! So the monkey king said:
‘Ok, no problem! I will give you 4 peaches in the morning and then I will give you 3 peaches in the afternoon.’
With that, the little monkey became very happy.
The students listened to this story, and then were asked for their thoughts. They were quick to identify that the little monkey was mistaken, as in both cases he received 7 peaches.
Students were asked to state the number sentences:
3 + 4 = 7
4 + 3 = 7
From which, they were asked ‘What is the same? What is different?’
This lead to identify the key terms of addend and sum, as well as concluding that:
3 + 4 = 4 + 3
Students then considered another example involving 2 students with different amounts of juice. From these concrete ideas of amount, students were quickly lead to the abstract generalisation of the commutative law of addition.
In applying the commutative law during exercises, students were challenged to look at applying it in instances using numbers, algebra and shapes. In question 4 of activity 1 there are infinite solutions as long as students identified that the number had to be the same in both blank spaces.
Students were then asked to identify where the commutative law had or could be applied through a true or false activity. Each option here has been carefully selected to identify a misconception and in each case students were selected to justify their solution.
In addition to this, students were also asked if the commutative law applied to the other operations, and were given the following examples to consider:
In exercises, students were the looking to calculate the sum of addends in an efficient manor by making use of the commutative law of addition or multiplication. The questions are included below, with the solutions in red – can you see why they have been chosen? What is the same in each question? What is different?
Through these questions, students were starting to develop their ‘intelligent practice’, that is to find the most efficient method of solving the problem as opposed to simply solving it. The last 2 questions given to the students were aimed to get students to actively seek the most efficient method, and continue to build on their intelligent approach to mathematical problems.
For the first problem the solution has been provided and for the second….well I shall leave the second one for you to consider!
Key considerations in the design of this lesson
Students were introduced to a concrete example before developing the abstract generalisation of the commutative law of addition (and multiplication).
Students experienced ‘what is’ and ‘what isn’t’ the commutative law, strengthening their understanding of the concept but also realising and justifying the non-concept.
Procedural variation started to build students approach to problems, where they began to look not only at solving the problem but also at what was the most efficient method (i.e. the most time efficient)
Intelligent practice – what was started in earlier procedural questions was then developed further through questions specifically chosen for their complexity. Whilst the solutions could be found through different approaches, using the commutative laws lead to the most efficient solution.