How many are there?


a)  Show how to work out how many lots of 3 there are in 24.  


b)

Show how to work out how many lots of 8 there are in 40.




c)

Show how to work out how many lots of 5 there are in 75.



 Using grouping diagrams
 Using only array diagrams (Number Framework Stage 3).
 Using skip counting (Number Framework Stage 4). For this resource, skip counters were of similar mean ability as repeated adders.

Using only repeated addition (Number Framework Stage 5).
 Using partitioning followed by repeated addition or skip counting (Number Framework Stage 6).
 Using fully multiplicative doubling and halving, or a mix of multiplicative and additive strategies (Number Framework Stage 6).

Fully multiplicative partitioning strategies (Number Framework Stage 7).
Y7 (03/2010)  
a) 
8
Working showing any 1 of:
 with doubling/halving (3×2=6, 6×2=12, 12×2=24);

easy moderate 
b) 
5
Working showing any 1 of the above strategies

moderate 
c) 
15
Working showing any 1 of the above strategies.

moderate 
NOTE:
 Students often just stated the result or the relevant division expression (e.g., 24 ÷ 8) or multiplication expression (8 × 3 or 3 × 8). Some students mentioned both the multiplication and the division expression in the same part of a question. These were of higher mean ability than those who mentioned just one expression.
 If students just state the answer (or use the vertical algorithm), ask them how they got their answer or how the algorithm works, especially for part c).
Common response  Likely misconception  
a) b) c) 
27 48 80 
Adds instead of multiplying 
a) b) c) 
24 40 75 
States the dividend 
a) b) 
7 (or 9) 4 (or 6) 
One off the correct answer (quotient) 
This may be a reading issue. Get the students to read the question out loud. Read it to them if they are struggling to do this.
If it is not a reading problem, the student is probably still at the counting or early additive stage. They need to explore the situation using physical objects. This will lead to grouping diagrams or multiplication arrays. If the student finds these easy, move them from their diagrams onto skip counting or repeated addition.
One off the correct answer (quotient)
This may be caused by faulty basic facts. If so get the student to further work on their 3 and 5times tables.
It may be a problem with skip counting, where they get their tally one short of the correct answer. Encourage these students to record their tallies in a consistent manner. Click on Students' work samples for NM1327 [pdf]: skip counting for some examples of ways students do this
Student strategies
 The mean ability of students relates to the location of their strategy on the Number Framework, with more sophisticated strategies being used by students with higher mean ability. The possible exception is that repeated addition and skip counting were being used by students with roughly equal mean ability. This indicates that skip counting is of equal sophistication as repeated addition. Only a few students used partitioning or partwhole strategies higher up the Number Framework than repeated addition.
 The success rates for different strategies did not vary in accordance with the level of sophistication of the strategy. The highly successful strategies (over 90% success rate) were the vertical division algorithm and skip counting where a clear tally of the number of skips was recorded or even just showing tally marks (see Students' work samples for NM1327 [pdf]: skip counting (only tally marks recorded). Other strategies tended to have success rates of about 70–80% regardless of their sophistication.
 Students who gave an expression (e.g., 24 ÷ 8, 8 × 3 or both) had high success rates (over 90%). This may well indicate that these students are using basic facts rather than strategies, especially for parts a) and b). Students who state both forms of the expression have higher mean abilities than those who state just one. This indicates that understanding the reversibility of division shows deeper understanding.
 Stacking wood
 Buying books
 Counting sheep
 Numbers at an art show
 Cough medicine
 Cans of fruit drink
 Dividing with remainder
 No remainder
 Giveaways
 Packing cherries
 Sharing fruit and vegetables
 Division wheel
 Calculating with time
 Writing word problems
 Sharing Jelly beans
 Town hall concert
 The price of flour
 Shopping for vegetables
 Estimating cards, money and pinecones
 Equal sharing III
 How many groups? II
 Powerful twenty five
 Saving money
 Buying a phone
 Anahera's sweets