Stage 7: Advanced Multiplicative Part-Whole

Stage Seven: Advanced Multiplicative Part-Whole
Students at the Advanced Multiplicative stage are learning to choose appropriately from a range of part-whole strategies to solve and estimate the answers to problems involving multiplication and division. These strategies require one or more of the numbers involved in a multiplication or division to be partitioned, manipulated, then recombined.


For example, to solve 27 x 6, 27 might be split into 20 + 7 and these parts multiplied then recombined, as in 20 x 6 = 120, 7 x 6 = 42, 120 + 42 = 162, or 2 x 27 = 54, 3 x 54 = 162. The firststrategy partitions 27 additively, the second strategy partitions 6 multiplicatively.


A critical development at this stage is the use of reversibility, in particular, solving division problems using multiplication. Advanced Multiplicative Part-Whole students are also able tosolve and estimate the answers to problems with fractions using multiplication and division.


For example, to solve 2/3 of 27, 1/3of 27 is 27÷ 3= 9 , 2 x 9 = 18 (using unit fractions). Students, at this stage, can also understand the multiplicative relationship between the numerators and denominators of equivalent fractions, e.g. 3/4  = 75/100.


Here are two strategies used by students at the Advanced Multiplicative stage to solve multiplication and division problems:
(i) Halving and doubling or dividing by three and trebling
Example: 16 x 4 as 8 x 8 = 64 (halving and doubling)  
Example: 72 ÷ 4 as 72 ÷ 2 = 36, 36 ÷2 = 18 (dividing by four is the same as dividing by two twice).  
(ii) Reversibility and place value partitioning
Example: 72 ÷ 4, as 10 x 4 = 40, 72 – 40 = 32, 8 x 4 = 32, 10 + 8 = 18.
Advanced Multiplicative students also apply their strengths in multiplication and division to problems involving fractions, proportions and ratios. Generally, their strategies involve usingequivalent fractions.


Here is an example of the strategies they use:
(i) Multiplying within
Example: Every packet has 8 lollies in it. Three of the 8 lollies are raspberry. There are enough packets in the jar to make 40 lollies. How many of the lollies are raspberry? 
(3:8 as □:40, 5 x 8 = 40 so 5 x 3 = □) 
8 x 5 0          8                                                                      40 lollies in the jar 
0          3                                                                      15 raspberry lollies3 x 5  

 
 
 
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