Number line addition and subtraction II

Number line addition and subtraction II

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about using a number line to solve addition and subtraction problems.
For the questions below show how to solve the equations on the number line. Read through the example in the box before you start.
 
Example: 
Sarah had to solve this problem: 96 – empty square =  64.
She knew that 96 – 30 = 66, and that 66 – 2 = 64. 
So she showed this on the number line and then wrote the answer in the box:

           96 – =  64

 

Use the number lines below to show how to solve the equations.  

 

 

 

a)   128 +  = 519     

 

 

 

b)   345 +  = 603     

 

 

 

c)   530 –  = 327     

 

 

 

d)   906 –  = 625    

Task administration: 
This task is completed with pencil and paper only.
Level:
4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Keywords: 
number lines, addition, subtraction
Description of task: 
Students use a number line to show how to solve whole number addition and subtraction problems.
Curriculum Links: 
This resource can help to identify students' ability to apply additive strategies flexibly to whole numbers to solve addition and subtraction problems.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Additive thinking, sets 7-8
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
    Y8 (06/2006)
a)

391
The number line is used to show jumps added/subtracted that use the sequential nature of the number line, e.g.,

or other acceptable jumps that represent the equation.

moderate
easy
b)

258
The number line is used to show jumps added/subtracted that use the sequential nature of the number line, e.g.,

or other acceptable jumps that represent the equation.

 moderate
easy
c) 203
The number line is used to show jumps added/subtracted that use the sequential nature of the number line, e.g.,

or other acceptable jumps that represent the equation.
NOTE: Also accept a "reversed" number line as a suitable representation of the equation, but encourage students to use the convention of a number line incrementing left to right.]

Complementary addition
 moderate
easy
d) 281
The number line is used to show jumps added/subtracted that use the sequential nature of the number line, e.g.,

Complementary addition

or other acceptable jumps that represent the equation.
NOTE: Also accept a "reversed" number line as a suitable representation of the equation, but encourage students to use the convention of a number line incrementing left to right.]
 difficult
moderate
 
Based on a representative sample of 189 students.
NOTE:  Accept even if arrows are missing, as long as the intent is clear.
Teaching and learning: 
This assessment resource is about whether students can show what strategies they use to solve a subtraction equation on a number line.  Students who have not used a number line to solve problems will need to learn how to show addition and subtraction jumps on a number line.
Diagnostic and formative information: 
A number of students represented the subtraction problems as addition problems on the number line (known as complementary addition).

For the subtraction problems a number of students set up the number line beginning with the higher number on the left and jumping to the lower number on the right.  While this is not how number lines are conventionally set up, this is still “showing how to subtract on a number line”.  This common strategy could be a step toward recognising the complementary nature of addition and subtraction, and students could be further supported by being asked how they could show these jumps if the number line was increasing (i.e., either subtracting by jumping back or adding by jumping forward from the smaller number).

Over half of students solved the equations by either using place value partitioning (grouped by place value, i.e., + 30 not + 10 + 10 + 10), jumping to and from tidy numbers or recognising compatible numbers and jumping to numbers that related to the target number on the number line.

Next steps: 
Some students solved the problems using another method (e.g. vertical algorithm) and then attempted to represent the solution on the number line rather than showing how the solution could be reached by using the number line.  In this case the lines drawn on the number line do not necessarily represent additions or subtractions.  Students may write the answer on the number line and try to draw appropriate visual lines.  For example, 128 + 391 = 519

            
Without the size of the jumps indicated there is no evidence that the number line is actually being used to represent or solve the equation (although they have solved the equation using some other strategy).
As this resource is about representing operations on a number line, students who responded like this may need to look at how or why it can be useful (e.g., being able to identify and show a range of strategies, and compare for efficiency) to show addition and subtraction as jumps on a number line.  If required, get students to use basic addition problems such as those found in Number line addition.  Using the examples given, ask questions like "What happens if I start here and add 10?", or "How do we show the addition on the number line?"  Encourage students to represent an approximate jump – not count out and draw in marks, but just draw.  Mark the size and direction of the jump, and the new location on the number line.

For students who can show basic place value jumps on the number line, encourage them to use fewer jumps to solve the equation more efficiently and reflect on which strategies best suit the particular equation.  This may involve using tidy numbers or compatible numbers rather than jumps made strictly by place value partitioning, and may also involve jumping past the target number, then back to it.

For students who solved the equation using the vertical algorithm and then represented the answer on the number line, encourage them to use jumps on the number line as a tool to solve and reflect on the jumps that could make this most efficient (as above).

Links to the Numeracy project
Showing how to solve addition problems on a number line can require jumping to tidy numbers, or using tidy numbers to make the jumps, and then making an adjustment to complete the equation.  Students can also use elements from place value partitioning, or compatible numbers, to jump along the number line.  Number lines can provide a way for students to image addition or subtraction as forward or backward jumps.
A simple place value partitioning strategy with unitised jumps – not grouped (i.e.,  + 10 + 10 + 10 + 10 instead of + 40) – may indicate an early additive part-whole understanding.  To indicate advanced additive part-whole thinking, students would use fewer jumps (more efficient jumps) to solve equations and be able to use or identify different ways to show how to solve the equation (multiple strategies).
 
Figure it out:
  • Maths Detective (Number sense and Algebraic thinking, L3, book 2, pages 2-4), and
  • Tidying up (Number sense and Algebraic thinking, L3, book 1, pages 2-4).
 
Numeracy:
Using tidy numbers refer to: Jumping the number line, Problems like 23 +  = 71, and for using place-value partitioning refer to: Problems like 37 +  = 79  (both Book 5: Teaching addition, subtraction and place value, pages 33-36).