Bigger or smaller?
0
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about division with fractions.
Task administration:
This task can be performed with pencil and paper or online (with SOME automarking).
Levels:
4, 5
Curriculum info:
Key Competencies:
Keywords:
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.
For more information see https://nzcurriculum.tki.org.nz/Keycompetencies
Learning Progression Frameworks
This resource can provide evidence of learning associated with 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
Descibes the effects of division by fractions and whole numbers
Partial explanation given
Partial explanation with some misconceptions included

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

Gets the correct answer but gives no explanation 
Evaluates 20 ÷ ^{1}/_{4} incorrectly — confuses 20 ÷ ^{1}/_{4} with 20 ÷ 4 = 5

Believes ^{1}/_{4} ÷ 20 does not exist, or is negative

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 crossmultiplication approach,
i.e 20 ÷ ^{1}/_{4} = ^{20}/_{1} ÷ ^{1}/_{4} = ^{20}/_{1} × ^{4}/_{1}* = 20 × 4 = 80
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 crossmultiplication 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.