Number pairs

Number pairs

Pencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about writing different number sentences with the same answer.

Question 1Change answer

a)  Write four different pairs of numbers that add to 8.
      +    = 8
 
      +    = 8
 
      +    = 8
 
      +    = 8

Question 1Change answer

b)  Write four different pairs of numbers that add to 17. 
  17 =   +  
 
  17 =   +  
 
  17 =   +  
 
  17 =   +  

Question 1Change answer

c)  Write four pairs of numbers to make these number sentences equal 7. 
    –    = 7
 
    –    = 7
 
    –    = 7
 
    –    = 7
Task administration: 
This task can be completed with pen and paper or online (with NO auto marking).
Level:
2
Curriculum info: 
Maths, Number and Algebra, Equations and expressions
Keywords: 
relationships, addition, subtraction, equality, linear equations
Description of task: 
Students complete addition and subtraction equations that explore part-whole relationships.
Curriculum Links: 
This resource can help to identify students' ability to apply basic addition facts to partition whole numbers.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Additive thinking, set 5
Using symbols and expressions to think mathematically, set 3
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y4 (11/07)
a)

Any unique number pairs that add to 8, i.e., (0, 8) or (8, 0); (7, 1) or (1, 7); (6, 2) or (2, 6); (5, 3) or (3, 5); (4, 4).

4 correct – easy
at least 3 correct – very easy
at least 2 correct – very easy
at least 1 correct – very easy
b) Any unique number pairs that add to 17, i.e., (17, 0) or (0, 17); (16, 1) or (1, 16); (15, 2) or (2, 15); (14, 3) or (3, 14); (13, 4) or (4, 13); (12, 5) or (5, 12); (11, 6) or (6, 11); (10, 7) or (7, 10); (9, 8) or (8, 9).

4 correct – moderate
at least 3 correct – easy
at least 2 correct – easy
at least 1 correct – very easy
c) Any two whole numbers that have a difference of 7.

4 correct – moderate
at least 3 correct – easy
at least 2 correct – easy
at least 1 correct – easy
Based on a representative sample of 266 students.

NOTE: While students generally used whole numbers for their answers, accept other numbers (e.g., fractions, decimals, etc.) that add or subtract to the given total.

Teaching and learning: 
This resource is about students generating a set of unique number pairs to satisfy a given criteria.  It explores students' part-whole understanding of numbers.  While students may come up with commutative number pairs (e.g., 1 + 7, 7 + 1), the emphasis is on generating a range of different responses to demonstrate part-whole understanding rather than exploring commutativity. This resource can also be used to explore the relationships between the numbers in a more algebraic way.

Links to the Number Framework
This resource fits with the Grouping/Place Value aspect of the Advanced Counting stage (Stage 4) of the Number Framework and with Basic Facts in the Early-Additive stage (Stage 5).

Diagnostic and formative information: 
  Common response Likely misconception
a)
b)		 8 + 0 and 0 + 8 (for example)
10 + 7 and 7 + 10 (for example)		 Students repeat one or more of the number pairs, believing that having the numbers in a different order means it is a different number pair.
c) 3 and 4, 5 and 2, etc. The numbers add to seven - the subtraction operator is ignored.
c) 0 – 7, 1 – 6, etc. The smaller number is given before the larger number – students may believe that subtraction is commutative.

 

Next steps: 
If students give a number pair and its reverse, show how the numbers are the same even though the order is different and get them to try and think of another number pair they could use instead.  This is an opportunity, however, to explore the concept of commutativity.  Refer to the Algebraic Thinking Concept Map: Commutativity and Associativity for more information about this idea.

If students have added, rather than subtracted, two numbers in part c), have them generate numbers that, when subtracted, give the answer of 7.

If students have reversed the order of the numbers in the subtraction equations, use materials to reinforce the need to record the larger number first and to emphasise that subtraction is not commutative.  Alternatively, have students generate word problems to match their number sentences.  See Algebraic Thinking Concept Map: Commutativity and Associativity for more information about commutativity and subtraction.

While many students were able to identify pairs of whole numbers successfully, using 0 as one of the numbers in their pair was less frequent than other numbers.  There is an opportunity here to explore the additive identity with students and generate a rule about what happens to a number when it is added to zero or subtracted from zero.  For more ideas and information about this concept, see the Algebraic Thinking Concept Map: Additive Identity.

If students have been able to generate four number pairs for each question, have them try to write as many number pairs for each as they think there are.  Students could share their results and try to come up with as many answers as they can.  (NOTE: There are an infinite number of combinations for part c) - students may need to be restricted to numbers below 50 or 100).  Alternatively, give students another number to find totals for (e.g., 9, 12, and 20) and have them generate all possible number pairs.

The patterns of the number pairs can also be explored in an algebraic way too.  Students can explore the relationship between the numbers by looking at what happens to each of the numbers in the number pair to turn it into another number pair.  For example, use the number pairs (0, 8) and (1, 7).  Encourage students to notice that when one is added to one number in the pair, i.e., 0 + 1 = 1, then the other number has one subtracted from it, i.e., 8 – 1 = 7.  Students can use this relationship to help generate all the possible combinations of number pairs that add to a given number.

For more information about concepts related to algebraic thinking, see the Algebraic Thinking Concept Map.