Scale factor and dimensions

Scale factor and dimensions

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Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about how scale factor works for shapes of 1- 2- and 3- dimensions.

Question 1Change answer

Finding scale factors is different for 1-, 2-, and 3-dimensional shapes.
 
a)  A thin piece of rubber that is 50cm long is stretched until it is 300cm long.
 
stretch-rubber.png [Not drawn to scale]
     What is the scale factor for this enlargement? 

Question 1Change answer

enlarging a splat[Not drawn to scale]
 
b)  A splat has an area of 150 mm2. 
     The image of the splat has an area of 1350 mm2 when it is enlarged on an overhead projector.
     i)  How many times bigger is the area of the big splat than the area of the small splat?
     ii) What is the scale factor for this enlargement?

Question 1Change answer

enlarging balloon[Not drawn to scale]
c)  After a few breaths a balloon is inflated to a volume of 400cm3. 
     When it is fully inflated it has a volume of 3200 cm3.
     i)  How many times bigger is the volume of the big balloon than the volume of the small balloon?
          
     ii) What is the scale factor for this enlargement?
Task administration: 
This task can be completed with pencil and paper or online with auto-marking.
Level:
5
Curriculum info: 
Maths, Geometry and Measurement, Transformation
Keywords: 
enlargement, resizing, scale factor, length, area, volume
Description of task: 
Students calculate increases in length, area, and volume, and relate this to the scale factor for an enlargement.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Geometric thinking, set 6
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y10 (05/2007)
a)

 

 6 very difficult
b)

i)
ii)

 9
3

moderate
very difficult
c) i)
ii)		 8
2

moderate
very difficult
Based on a representative sample of 152 Year 10 students. 
Diagnostic and formative information: 
  Common error Likely misconception
a)
b) i) or b) ii)
c) i) or c) ii)		 250      [300 – 50]
1200    [1350 – 150]
2800    [3200 – 400]		 Uses an additive model  (may be unfamiliar with "scale factor")
Student gives the absolute amount that the length, area, or volume has increased, rather than using a multiplicative model.
b) ii)
c) ii)		 9
8		 Unfamiliar with "scale factor" and its effects
Confuses the increase of area or volume with the scale factor, which is a linear measure of increase.
a)
b) ii)
c) ii)		 ×6, 50×6, etc.
×9, 150×9, etc.
×8, 400×8, etc.		 Unfamiliar with "scale factor" and its effects
Student expresses the number of times the original shape has increased by.
a)
b) ii)
c) ii)		 1-6, 1/6 , 1:6, 50/300 , 6/1 etc.
1-9, 1/9 , 1:9, 150/1350 , 9/1 etc.
1-8, 1/8 , 1:8, 400/3200 , 8/1 etc.		 Unfamiliar with "scale factor" and its effects
Student expresses a movement (1-6 or 1→6), a fraction, or a ratio to describe the relative amount the original shape has increased by.
Next steps: 

Mathematical language – scale factor
Students need to know and use the correct mathematical language of enlargement, otherwise ambiguity arises. In particular, they need to know and use the term scale factor, and that it describes linear change. This is the clearest and most unambiguous way to describe the degree of an enlargement.

Uses an additive model
Students need to look carefully at the language clues in the question. The phrase "how many times bigger … compared with" in parts b)ii) and c)ii) needs to be interpreted as a multiplicative question. If students have misinterpreted the question this way, see if they can perform the question once this has been clarified.

Unfamiliar with the term "scale factor" or of the concept of enlargement
Work with students to understand that enlargement increases each dimension linearly by an amount known as the scale factor. Get the students working with enlargement on grid paper. Enlarging shapes describes the process without using the term scale factor. Use keywords enlargement AND scale factor for further resources that assess this. Also look at the Figure it out resource Enlargement Explosion  (Geometry, L4+, Book 2, page 12).

Unfamiliar with the effects of enlargement on area or volume
Students firstly need to know that "scale factor" is a linear measure. They then need to explore the relationship between scale factor and area.

  • The increase in area goes up by the square of the scale factor. This is because area is a square measure (m2, cm2 etc). Resources Enlargement, length and area, Enlarge the arrow, and Projecting with a scale factor are particularly useful for exploring invariant properties. Also look at the Figure it out resource Growing Changes (Geometry, L3, page 24).
  • The increase in volume goes up by the cube of the scale factor. This is because area is a cubic measure (m3, cm3 etc).