Fraction long jump

Fraction long jump

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about showing fractions of a distance along a number line.

Seven friends are practising for a long jump competition. Tana made the longest jump. The other 6 students did not jump as far as Tana.

Use the information below to show how far each of the other students jumped as a fraction of Tana's jump.

  1. Jake jumped \(1 \over 2\) as far as Tana.
  2. Maraea jumped \(3 \over 4\) as far as Tana.
  3. Sylvia jumped \(1 \over 4\) as far as Tana.
  4. Tipene jumped \(1 \over 3\) as far as Tana.
  5. John jumped \(1 \over 5\) as far as Tana.
  6. Amelia jumped \(2 \over 3\) as far as Tana.

Instructions:
Cut out the fractions below and place them on the number line to show how far each students jumped. When all fractions are placed, check them and then glue them down.

 

diagram of the where the friends jumped to on the long jump

 

Jake
\(1 \over 2\)		 Maraea
\(3 \over 4\)		 Sylvia
\(1 \over 4\)		 Tipene
\(1 \over 3\)		 John
\(1 \over 5\)		 Amelia
\(2 \over 3\)
Task administration: 
Equipment
Scissors; glue.
  • This resource looks at developing understanding of ordering proper fractions (both unit fractions and non-unit fractions).
  • Students could be asked to explain how they know that one fraction is larger/smaller than another when they are placing them.
  • Some students may be able to identify the order of the fractions without cutting out and comparing the fractions before ordering.
  • Encourage students to check the order before gluing the fraction down.
Copyright: 
Image: NZCER
Person: iStock
Level:
3
Curriculum info: 
Maths, Number and Algebra, Number Knowledge
Keywords: 
fractions, number lines, ordering numbers
Description of task: 
Students cut out and attach fractions onto a number line in order from smallest to largest.
Answers/responses: 
John 1/5 , Sylvia 1/4 , Tipene 1/3 , Jake 1/2 , Amelia 2/3 , Maraea 3/4
Teaching and learning: 
Ordering fractions involves a number of multiple comparisons and students should already explored partitioning, part-whole understanding of fractions, and had some experiences comparing fractions, and explaining their reasoning.
Diagnostic and formative information: 
Common error Likely misconception
1/2 , 1/3 , 2/3 , 1/4 , 3/4 , 1/5 Whole number (denominator first)
Ordering the fractions by the denominator and then the numerator – indicating a lack of understanding of the nature of a fraction as a rational number – instead it is some kind of system with the two numbers: bottom then top.
1/2 , 1/3 , 1/4 , 1/5 , 2/3 , 3/4 Whole number (larger numbers)
Ordering fractions by the larger the numbers (top and/or bottom) the larger the fraction.

Based on a trial set of 22 Y5 students.

NOTE:
Students should be able to order all six fractions to indicate understanding of ordering simple proper fractions.  Two important ideas in this resource are:

  • where they place 1/5 , 2/3 and 3/4 ; and
  • how they justify their choice.
Next steps: 
After cutting out and placing the fractions, students could be asked to mark down which fractions they started with to work out the order, i.e., did they start with 1/2 and then halve to get 1/4 then build up to 3/4 ? or did they start with 1/2 then 1/3 , 1/4 , 1/5 and then the non-unit fractions.  By sharing their strategies for ordering the strategies can be critiqued to find the easiest or most efficient.

Understanding partitioning and the part-whole relationship
Students who have either of the whole number misconceptions identified previously need to develop a part-whole understanding of fractions before trying to devise a system to compare or order fractions.
If required, students could go back to partitioning and explore constructing the parts (unit fractions), combining these parts to make non-unit fractions that are between 0 and 1 (also called proper fractions), and naming these new fractions (part-whole fractions).  Encourage students to explore a range of many new fractions such as 3/13 , 11/27 , etc, or even include some top heavy (or improper) fractions and discuss how large these fractions are.

Using diagrams to compare
After this, encourage students to compare just two fractions (including non-unit fractions) before trying to order a number of unit and non-unit fractions.  For example, students could explain (using materials, diagrams or reasoning) how they know the larger of 1/2 and 1/5 , and then the larger of 2/3 and 1/5 .  Appropriate drawing of fractions can promote understanding and help with comparing the size of fractions. For a resource comparing fractions see Larger fractions (level 2).

Simple fractions correctly placed
For students who correctly placed all simple fractions, it is important to ensure that they do not develop the misconception that fractions are always between 0 and 1.  Ask students if someone jumped where a fraction such as 5/4 would be.  Students could draw this on another number line and start to look at where other fractions such as  1/4 , 3/4 , 5/4 ; or   1/2 , 2/2 , 3/2 might go.  Scaffold the students to construct these top heavy fractions by starting with the unit fraction of each and asking for non unit fractions with the same denominator.  Try different fractions, 1/5 , 3/5 , 6/5 and even strange fractions such as 7/11 .

 

For similar ARB resources click on the link use the keywords, fractions AND number lines.
For ARB resources about comparing fractions click on the link or use the keywords fractions AND ordering numbers
Click on the link for further information about Fractions and number lines (Fractional thinking: conceptual map).
 
Numeracy resources
Book 7: Teaching Fractions, Decimals and Percentages
  • Fraction circles, (p.9), Advanced counting/early additive part-whole
  • Trains (p.19) Early additive/Advanced additive/Early multiplicative.