How many are there?

How many are there?

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about working out how many of one number there are in a larger number.
 
illustration: selection of different numbers
 
a) Show how to work out how many lots of 3 there are in 24.
 
 
 
 
 
 
 
 
Answer: __________  
 
b)
 
Show how to work out how many lots of 8 there are in 40.
 
 
 
 
 
 
 
 
Answer: __________  
 
c)
 
Show how to work out how many lots of 5 there are in 75.
 
 
 
 
 
 
 
 
Answer: __________  
Task administration: 
This task is completed with pencil and paper only.
 
Level:
3
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
division, multiplication, work samples, key competencies
Description of task: 
Students show how to work out the number of 1-digit that there are in 2-digit numbers.
Curriculum Links: 
This resource can be used to help to identify students' understanding about multiplication. Students' strategies give a much better indication of curriculum level and progression than their answer. A possible progression for understanding could involve:
  • 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).
​Note: Students who use the vertical algorithm should be questioned on why the algorithm works. Their answer may indicate where their understanding fits against a progression of understanding of multiplication. For examples of the different strategies, click on the link, Student work samples for NM1327 [pdf]. Students who give the expressions 24 ÷ 8 or 8 × 3 should be asked how they got their answer especially for part c). 
Key competencies This resource involves recording the strategies they used to solve division problems. This relates to the Key Competency: Using language, symbols and text.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Multiplicative thinking, sets 4-5
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y7 (03/2010)
a)
8
Working showing any 1 of:
  • Using Basic Facts:

-  with doubling/halving (3×2=6, 6×2=12, 12×2=24);
-  in combination (5×3 = 15, 3×3=9, 3 + 5 = 8; or  5×10 + 5×5, 10+5=15);
-  with a single compensation (7×3 =21, 21 + 3 =24); or
-  followed by skip counting (3×5=15, 18, 21, 24)

  • Vertical division algorithm
  • Repeated addition (3+3=6, 6+3=9, 9+3=12, …)
  • Skip counting (3, 6, 9, 12, 15, 18, 21, 24)
  • Multiplication arrays
  • Grouping diagrams
  • Other correct strategies
 easy
moderate
b)
5
Working showing any 1 of the above strategies

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

moderate
moderate
Based on a representative sample of 142 students.

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).
Diagnostic and formative information: 
  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)
Next steps: 
Adds instead of multiplying or States the dividend
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 5-times 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 part-whole 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.
See Student work samples (NM1327) [pdf] for examples of the different student strategies.
​See the Analysis of student responses [pdf] for more detailed information about the strategies.