Bigger or smaller?

Bigger or smaller?

Pencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about division with fractions.

Question

bigger or smaller expressions
 
Which symbol should go into the box to make the statement above true?

    • >

    • =

    • <

    • You cannot tell.

Explain your answer
Task administration: 
This task can be performed with pencil and paper or online (with SOME auto-marking).
Levels:
4, 5
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
multiplication, fractions, fractions as operators, division
Description of task: 
Students demonstrate their understanding of the effect of division and multiplication involving fractions.
Curriculum Links: 
Key competencies
This resource involves explaining why one division involving fractions is bigger than another, which relates to the Key Competencies: Thinking, and Using language, symbols and text.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Multiplicative thinking, sets 6-7
Multiplicative thinking, sets 8-9
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y10 (04/2016)
 1/4 ÷ 20 < 20 ÷ 1/4 * [Option 3]
 
Full explanation given 
Evaluates both sides  
  • Because 20 divided by 1/4 = 80, and 1/4 divided by 20 = 0.0125
  • Because 1/4 divided by 20 is 1/80 and 20 divided by 1/4 is 80
Descibes the effects of division by fractions and whole numbers  
  • 20 divided by 1/4 is basically multiplying the numbers, while 1/4 divided by 20 is dividing them both.
  • Because 20 divided by 1/4 is saying how many quarters are in 20 whereas 1/4 divided by 20 is saying how many twenties are in a quarter.
Partial explanation given
  • Because when you divide a number with a fraction the number becomes bigger. (doesn't mention division by a whole number)
  • Because 20 is bigger than 1/4 and it is saying how many times does 1/4 go into 20.
  • If you divide 1/4 by 20 it would be a very small number. (doesn't mention 20 ÷ 1/4)
Partial explanation with some misconceptions included
  • 20 divided by 1/4 is 5, 1/4 divided by 20 is a number less than one. (first part incorrect)
  • 1/4 divided by 20 = 5 and 20 divided by 1/4 = 80. (first part incorrect)
  • 1/4 divided by 20 =  1/80 and 20 divided by 1/4 = 16. (second part incorrect)
easy
 
very difficult
(Full explanation)
 
very difficult
(Partial explanation)
 
difficult
(Partial explanation with misconceptions)
Based on a representative sample of 114 year 10 students.
* Many students who got the correct response gave no explanation.
Teaching and learning: 
This resource requires dividing a number by a fraction. This is is better achieved by students who use the quotitive approach to division (i.e., say to themselves "How many quarters are there in twenty?".
For more on quotitive division, see Division with fractions.
Diagnostic and formative information: 
Common incorrect explanations
Gets the correct answer but restates the answer that they have selected
  • I think it's because 1/4 ÷ 20 is smaller than 20 ÷ 1/4
  • Because the one to the left is less.
 
Gets the correct answer but gives no explanation
Evaluates 20 ÷ 1/4 incorrectly — confuses 20 ÷ 1/4 with 20 ÷ 4 = 5
  • Because 20 ÷ 1/4 is 5
Believes 1/4 ÷ 20 does not exist, or is negative
  • Cause 1/4 ÷ 20 will be a negative.
  • You can't do 1/4 ÷ 20
  • Because 1/4 ÷ 20 has no number specific
Believes division is commutative, i.e., 1/4 ÷ 20 = 20 ÷ 1/4
 
Next steps: 
Gets the correct answer but restates the answer that they have selected or Gets the correct answer but gives no explanation gives no explanation
Ask the student to verbally articulate how they selected their answer. If they can, get them to write it down using words, symbols or pictures/diagrams.
 
Evaluates 20 ÷ 1/4 incorrectly — confuses 20 ÷ 1/4 with 20 ÷ 4 = 5
Students could perform  20 ÷ 4. Then ask them if 20 ÷ 1/4 is bigger or smaller than this.
If a student does not follow this reasoning, get them to say what 20 ÷ 4 means. If they respond "It's 20 divided by 4", ask them to phrase it another way. If their response is  "How many 4's are there in 20?" then ask them to phrase 20 ÷ 1/4 as "How many quarters are there in 20?".
If they are still stuggling, encourage them to use drawings or materials such as cuisenaire rods (see, for example Cuisenaires and fractions, especially the Working with students section).
Once students have mastered this, they can be exposed to the cross-multiplication approach,
i.e  20 ÷ 1/4 = 20/1 ÷ 1/4 = 20/1 × 4/1* =  20 × 4 = 80
* The step before this is left out. It requires multiplying both 20/1 and 1/4 by 4/1. Because this is equivalent to multiply by 4/4 = 1, it is still equivalent to the original expression, as multiplying any number by 1 leaves it the same. The number 1 can therefore be called the multiplicative identity.

Believes 1/4 ÷ 20 does not exist, or is negative
Give a problem such as: "Cut a cake into 20 equal pieces" and get students to write this as an arithmetic expression. If they write 1/20  ask them if they can write this as a division (1/20 = 1 ÷ 20).
Next ask them to write an expression for "Cut a quarter of a cake into 20 equal pieces" and get them to write this as an arithmetic expression.

Believes division is commutative, i.e., 1/4 ÷ 20 = 20 ÷ 1/4
Students could explore commutativity with multiplication (for example What's the same as ...   ). Then get them to do some division problems, such as 8 ÷ 2 then compare this with 2 ÷ 8. The answers are 4 and 1/4 respectively. The  multiplicative inverse (or reciprocal) of 4 is 1/4 and vice versa, because 4 × 1/4 = 1.