Square and cubic numbers

Square and cubic numbers

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about recognising square and cubic numbers without using a calculator.

A square number is an integer that can be written as the square of some other integer. So 4 is a square number because 2 × 2 = 22 = 4.

A cubic number is an integer that can be written as the cube of some other integer. So 1000 is a cubic number because 10 × 10 × 10 = 103 = 1000.

 

(Second fold along this line)  ………………………..……………………………………………………..
  

a) Circle all the square numbers in this list:
 
 

2

8

9

12

16

20

25

30

33

39

44

49

50

60

64

84

100

111

121

125

200

 
 
 

 


(First fold along this line)  ………………………..……………………………………………………..
 

b) Circle all the cubic numbers in this list:
 
 

3

6

8

12

16

18

25

27

32

36

48

64

75

81

90

100

111

121

125

144

200
Task administration: 
This task is completed with pencil and paper, and other equipment.
 
Equipment:
Accrate watch to time 45 seconds
One pre-folded sheet per student
  1. Fold the paper along the lower dotted line, then along the upper dotted line.
  2. Hand out with just the picture and definitions face up. Students should not turn them over until they are asked to.
  3. Read out what it means to be a square or a cubic number.
  4. Get the students to turn the page over (without unfolding it) and give them 45 seconds to answer part a).
  5. Ask students to unfold the paper and give them 45 second to answer part b).
Level:
5
Curriculum info: 
Maths, Number and Algebra, Number Knowledge
Keywords: 
square numbers, cubic numbers, basic facts
Description of task: 
Students identify square and cubic numbers mentally in a timed test.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Multiplicative thinking, sets 8-9
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y10 (07/2007)
a)

9, 16, 25, 49, 64, 100, and 121
(with no others circled)

All 7 correct (with no others circled)
or
4 or more correct (with no others circled)
or
1 or more correct (with no others circled) or
4 or more correct (with 1-2  others circled)

very difficult

difficult
 
moderate
b)

8, 27, 64, and 125
(with no others circled)

All 4 correct (with no others circled)
or
2 or more correct (with no others circled)
or
1 or more correct (with no others circled) or
2 or more correct (with 1-2  others circled)

very difficult
 
very difficult
 
difficult
Based on a representative sample of 64 Year 10 students in three different classes. 

NOTE: The marking gives three stages of achievement. The top level is reserved for students who identify all squares (or cubes) with no falsely identified ones. The next tier down is for those who identify half or more or the correct answers with no false ones. The third tier is for those who have some limited success at identification. With a sample of only 64, the difficulty level estimates are less accurate than usual.

Teaching and learning: 
Instantly recognising numbers as being square or cubic can empower students by offering them possible hints or insights into the structure of problems, as well as freeing their minds to focus on deeper aspects of problems other than just the computational. Knowing the exact square roots of these numbers is also highly beneficial even though this resource does not assess this.

Students who had full (or near full) recognition of the cubic basic facts in the sample also recognised square numbers, so start with remembering the square numbers first.

Diagnostic and formative information: 
  Common errors Likely misconception
a) Circles all (or most of) the even numbers. Confuses square numbers with even numbers
Confuses 2x and x2 ( 31% of students made this error.)
a) Incorrectly circled:
2, 8, 30, 44, 50, or 200
(excluding those who circled just even numbers)		 Wrongly identifies a square number

  • These were the most common numbers to be incorrectly identified as square. Each of these was identified by 5 or more students as being square. 2 and 8 were identified by more than 10 students.
  • The most common wrongly identified number was 2 (13 students, or 30% of the 44 who did not circle just even numbers). The confusion is most probably seeing that:
    1 + 1 = 2 rather than 1 × 1 = 1.
b) Circles all (or most of) the square numbers. Confuses cubic numbers with square numbers
Confuses x3 and x2 ( 3% of students made this error.)
b) Incorrectly circles:
3, 6, 12, 16, 18, 36, 48, 90, 100, 200		 Wrongly identifies a cubic number
These were the most common number to be incorrectly identified as cubic. Each of these was identified 10 or more of the 62 students who appeared to identify just cubic numbers.
6, 12, 18, 48, and 100 were identified by more than 20 students.

 

Next steps: 
Students who don’t recognise square or cubic numbers

  1. Motivate students to learn square and cubic numbers by discussing that instantly recognising them at least up to 102 or 103  (or even 123 and even some higher ones), will assist them with problem solving as it may often suggest ways of approaching a problem.
  2. Discuss what it is to be a square number. It can be represented by an array with the same number of rows and columns. A square number must have a square root, a factor that can be multiplied by itself to equal that square number. Click on the links square numbers or square roots for resources on this.
  3. Ensure students understand that square numbers are of the form x × x or x2.
  4. Encourage students to learn the square numbers as basic facts.
  5. Discuss and explore cubic numbers as 3-dimensional arrays, and having the form x × x × x or x3.
  6. Move them on to memorising some cubic numbers.

Students confuse square numbers with even numbers

  1. Discuss what it is to be a square number. It is an array with the same number of rows and columns. Click on the link square numbers for resources on this.
  2. Contrast this with even numbers. This could be seen as an array of two columns only. Alternatively, the number must be divisible by 2. This means the last digit is 0, 2, 4, 6, or 8. Click on the link even numbers for further resources on this.

Students wrongly identify a square number

  1. Ask the student why they thought a number was square. Ask them to identify the factors of that number, and which of these would indicate that it is a square number (i.e. "What is the square root?" or "What factor can be squared to arrive at the number they thought was square?").
  2. Discuss what it is to be a square number. It is an array with the same number of rows and columns.
  3. Click on the link square numbers or on NM0051 or AL6147 part e) for resources on this.

Students confuse cubic numbers with square numbers or wrongly identify a cubic number
Contrast square numbers as 2-dimensional arrays and cubic numbers as 3-dimensional arrays.

Figure it Out:
Number, L4
Starting with stamps (pp. 22-23),
Building squares (p. 14),
Superior side lengths (p. 24),
Cubic capacity (p. 17).
Algebra, L4+
Square number differences (p.1)

Numeracy:
Book 6: Teaching multiplication and division, Powerful numbers (pp. 40-41)

Book 8: Teaching number sense and algebraic thinking, Squaring (p.28), Square roots (p.29), Cubes and cube roots (p. 30).