Non-linear scales

Non-linear scales

Pencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
This task is about finding values on non-linear scales.
On some scales the markings are not placed at equal intervals.
 
Often, the gap between them gets smaller and smaller. The two scales below are both like this.
 
kilohertz-scale-tp.png and  non-linear-scale-tp.png

Question 1Change answer

a)  AM radio stations have frequencies measured in kilohertz and these are often shown on a scale like the one below.
i)  What is the frequency of radio station
     SportsAM?  kilohertz
     RadioRock?  kilohertz

Question Change answer

ii)  Draw an arrow labelled Kidz at 1350 kilohertz on the scale below.

Question 1Change answer

b) Exposure to radiation is measured in milliRoentgens per hour (mR/h)
i)  What radiation exposure level is indicated by
     arrow A?  mR/h
     arrow B?  mR/h

Question Change answer

ii) Draw an arrow labelled C at 35 mR/h on the scale below.
Task administration: 
This task can be completed with pencil and paper or online.
Copyright: 
Graphics: copyright NZCER.
Level:
5
Curriculum info: 
Maths, Geometry and Measurement, Measurement
Keywords: 
reading scales, plotting scales, non-linear scales
Description of task: 
Students read from, and mark points on, non-linear scales.
Answers/responses: 
  Y10 (05/2007)
a)

i)

ii)

570 [Accept 570 to 572]
910 [Accept 905 to 915]
Plotted between 1340 - 1360

very easy
easy
easy
b) i)

ii)

7 [Accept up to 7.1]
2.5 [Accept 2.4 to 2.7]
Plotted between 34 and 36

easy
moderate
easy
Based on a representative sample of 165 students. 
Teaching and learning: 
Scales
Not all scales are evenly spaced. Scales with unevenly spaced intervals are called "non-linear scales". Two examples of these are given in the questions, namely the markings on an AM radio dial, and the measure of radiation in mR/h. Other examples include the Richter scale for earthquake magnitude, or decibels which is the measure of noise intensity. Both of these are logarithmic scales, but neither are base-10 logarithms.
 
The non-linearity of the scale often reflects the physical properties of what is being measured. If a linear scale were used for earthquakes, the numbers would quickly get extremely large and cumbersome. An earthquake of 8 on the Richter scale is nearly a million times more powerful than one of magnitude 4. The aperture scale on a camera (f4, f5.6, f8, f11 etc.) is also non-linear, but in this case the larger the f-number, the less light is let into the camera. Older spring-operated bathroom scales are non-linear, as the spring compacts more slowly at higher weights, so the tick-marks get closer together.
Diagnostic and formative information: 
  Common error Likely misconception
a) i)

a) ii)
b) i)

b) ii)

 53.4, 534 etc.
800.6, 850.2, 860, 901 etc
1310
5.2, 5.4 etc.
24 etc.
3.5 etc.		 Place value error. Does not give appropriate place value magnitude to minor tick marks.
a) i)

a) ii)
b) i)

b) ii)

 530
900
1300
5
2
30		 Reads or plots point to the nearest major tick mark.
a) ii)
b) i)
b) ii)		 Plotted at 1330 (or 1340)
2.75 (or 2.7)
Plotted at 32, 33 (or 34)		 Insufficient accuracy. This may indicate that the student interpolates as if the scale is linear rather than non-linear.
Next steps: 

Place value errors:

  • Students should be encouraged to write the correct place value amount below the relevant tick marks. For example, the tick mark between 800 and 1000 is 900 rather than 810 or 850.
  • If they are still unclear, further investigations about their place value knowledge. For ARB resources to help assess this, click on the link or use the keyword place value.

Plotting to nearest tick mark:

  • Students should be encouraged to interpolate between given tick marks. They can pencil in smaller intermediate tick marks between the given marks and use these to estimate the actual reading.

Insufficient accuracy:

  • Students may not have put in intermediate tick marks between the given ones, or do not realise that these intermediate marks are not equally spaced, but get closer together on a non-linear scale.