Paying the bills

Paying the bills

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about ordering fractions, and finding different fractions of an amount.
 

Pere works out his budget to make sure he can pay his bills and have some spending money left over from his pay. The fraction of his budget that he spends on different things each week is shown in the table below.

Pere's Weekly Budget

Category Car Electricity Food Rent Internet Mobile phone
Fraction of his budget \(1 \over 10\) \(1 \over 20\) \(3 \over 20\) \(1 \over 5\) \(1 \over 30\) \(1 \over 20\)

   

a)
 Use the information in the table above to complete and label the strip graph below. 
Use the category headings from the table to label the strip graph of where Pere's pay goes. One label has been done for you.

Pere's Weekly Budget

b)

 
If Pere earns $600 this week, how much money will he put aside for ...
 
i)
the car? $__________
 
ii)
electricity? $__________
 
iii)
food? $__________
 
iv) rent? $__________
Task administration: 
This task is completed with pencil and paper only.
Level:
4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Keywords: 
fractions, fractions as operators, ordering fractions
Description of task: 
Students order fractions from smallest to greatest, and work out fractions of an amount.
Curriculum Links: 
This resource can help to identify students' ability to apply additive or multiplicative strategies flexibly to find fractions of sets, shapes, and quantities.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Multiplicative thinking, sets 6-7
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
      Y8 (05/2006)
a)   1/30 , 1/20 , 1/20 , 1/10 , 3/20 , 1/5
i.e.,  internet, mobile phone, electricity, car, food, rent or
internet, electricity, mobile phone, car, food, rent		 very difficult
b) i)
ii)
iii)
iv)		 60
30 [Accept if b ii) = ½ of b) i)]
90 [Accept if b iii) = 3 x b) ii)]
120 [Accept if b iv) = 2 x b) i)]		 difficult
difficult
difficult
difficult

Based on a representative sample of 201 students.

NOTE:
Students should be able to order all six fractions to indicate understanding of ordering fractions.  Various incorrect strategies can be used to order the fractions almost correctly. By using some of the incorrect strategies identified below, students can order 5 of the 6 fractions correctly, but will put 3/20 in the incorrect position.

Diagnostic and formative information: 
  Common error Likely misconception
a) 1/30 , 1/20 , 1/20 , 3/20 , 1/10 ,1/5
or
1/30 , 3/20 , 1/20 , 1/20 , 1/10 , 1/5		 Students apply the rule that the larger the denominator the smaller the fraction and then order the numerators when the denominators are equal. (16% ordered the numerators smallest to greatest, while 2% ordered them greatest to smallest.)
a) 1/5 , 1/10 , 1/20 , 1/20 , 3/20 , 1/30 Students apply the rule that the smaller the denominator the larger the fraction and then order the numerators from smallest to largest when the denominators are equal (6%).
a) 1/30 , 1/20 , 1/20 , 1/10 , 1/5 , 3/20 Students order the unit fraction from smallest to greatest, and place the non-unit fraction last (1%).
a) 1/5 , 1/10 , 1/20, 1/20 , 1/30 , 3/20
or
3/20 , 1/30 , 1/20 , 1/20 , 1/10 , 1/5		 Students order the unit fraction from greatest to smallest, and place the non-unit fraction last (13%), or order them from smallest to greatest and put the unit fraction first (1%). These students may order according to the product (or sum) of the numerator and denominator.
a) 1/10 , 1/20 , 3/20 , 1/5 , 1/30 , 1/20 Students order the fractions in the same way as presented in the table (5%).
b) i) 100 Students calculate 1/6 of 600.
b) i)-iv) 100, 100, 100, 100 Students share the 600 into 6 equal shares (2%).
b) i)-iv) 10, 20, 20, 5 Students use the denominator as the amount paid for each component (2%).
b) i)-iv) 10, 20, 60, 5 Students multiply the denominator and numerator to get the amount paid (1%).
b) i)-iv) 60, 120, 40, 30 or
100, 200, 67, 50 etc.
i.e. uses the rules:
     b) ii) = 2 x b) i)
     b) iii) = 1/3 b i)
     b) iv) = ½ b) i)		 Students attempt to use relationships between common fractions, but in a reciprocal way, seeing 1/20 > 1/10 > 1/5

Students apply the rule that 1/20 is twice as big as 1/10 (17%).
Students apply the rule that 1/20 is three times as big as 3/20 (1%).
Students apply the rule that 1/10 is twice as big as 1/5 (10%).
[some  do this consistently, especially for b) ii) and b)iv)]

Based on a representative sample of 201 students.

Next steps: 
Developing a part-whole understanding of fractions
Students who have incorrectly ordered the fractions need to develop a part-whole understanding of fractions before trying to devise a system to order or compare fractions.  A part-whole understanding refers to an understanding about the relationship between the top and bottom the numbers, not their absolute size. Students should be aware that the bottom number represents how many parts make up the whole, and the top number represents how many of these parts are of interest.  For further information about part-whole fractions see the Fractional Thinking concept map: part-whole fractions.  Following this, students can begin to look at comparing the size of simple fractions (below).

Comparing the size of fractions
For students who misplace one fraction such as 1/5 or especially 3/20 , ask them to explain how they know that the misplaced fraction is larger or smaller and to show this using a diagram – they can even begin by partitioning a rectangle, showing each fraction, and then comparing.  For a resource comparing fractions see NM0138: Larger fractions.

Understanding that fractions can be operators
For students who are having difficulty finding fractions of amounts, get them to explore similar situations that involve finding fractions of a number of physical objects. Click on the link practical fraction resources (or search with the Resource type as Practical and using the keyword fractions) to find these. Students need to be aware and talk about the distinction between fractions as operators that can be applied to an amount and the part-whole nature representation of fractions.

Flowers, books and sheep and Making necklaces involve calculating fractions of an amount.
For more information about fractions click on the Fractional thinking concept map.