# Common Misunderstandings - Levels 5–9 Understanding Scale Factor Tool

## Levels 5–9: Proportional Reasoning

### Instructions:

Place the two cards in front of the student and say, “What can you tell me about these two shapes? … Note student’s response, then say, “How would you tell a friend to draw the large shape if you could only show them the small shape?” Note student’s response.

Place the Dot Paper Worksheet in front of the student and say, “Could you make this shape half as big please?” Note and retain student’s response.

Place the Map Worksheet in front of student and say, “This is a map of a suburb in Perth. Can you find Nicholson Road?”….  Point to the scale and ask, “Can you tell me what this means? … Can you give me an example?”

If no response, say, “If you walked the full length of Arthur Street (indicate this, it is just to the right of the red star), about how far would you have walked?” … Note student’s response, then say, “Jo walks to Rosalie School which is here (indicate the red star). She lives on the corner of Nicholson and Rupert Street (indicate). About how far does she walk to school in the morning?” … Indicate that the bottom of the page can be used for any working required. Note student’s response and ask him/her to explain their reasoning.

If correct (ie, about 375 metres), say, “Thuan lives on the corner of Redfern Street and View Street (indicate upper left hand corner of the map). About how far does he have to ride to school?” …. Note student’s strategy and response in each case, retain any working.

Proportional reasoning is often more apparent in relation to visual images, eg, recognising shapes that have been enlarged or reduced, than it is in word problems that require interpretation relative to context. However, where students have had a limited exposure to the skills and strategies needed to enlarge or reduce shapes and/or to construct and interpret scale drawings there is a distinct possibility that misunderstandings will arise. One of these is the tendency to focus on area when attempting to identify ‘how many times larger’ one shape is of a smaller, similar shape.

This task examines the extent to which students are able to recognise and describe enlargements, and use a scale factor to reduce a shape and estimate distances on a scale map.

Observed response Interpretation/Suggested teaching response
Little/no response, possibly recognises shapes and/or notes one is bigger than the other, may be able to explain meaning of map scale May not understand the spatial task or have access to the skills and strategies needed to interpret maps
• Ensure that what is meant by statements such as, “3 times as big as” or “half the size of” are understood, ie, they can be modelled and interpreted relative to context
• Use peg boards, dot paper, cm grid paper etc to enlarge and reduce shapes by simple scalar amounts, discuss this in terms of what happens to corresponding sides (they are multiplied or divided by the same factor)
• Discuss which attribute is relevant and why for 2D shapes (ie, length not area, as object is to produce similar shapes)
• Provide opportunities to work with maps and scale diagrams, make thinking explicit, scaffold appropriate strategies for calculating or estimating distances
Recognises shapes are the same, may identify scale factor (3) but unable to halve the quadrilateral, although may make a start (eg, draw relevant diagonal), explains meaning of map scale but may not be able to use this to provide an example or reasonable estimates for both map questions (eg, may treat as 1 cm is 350 metres) Suggests a limited understanding of the scale factor idea for multiplication,
• Use cm grid/dot paper, peg boards etc to review the processes and language involved in enlarging and reducing 2D shapes by a range of different factors starting with simple shapes such as rectangles and moving to more complex shapes such as scalene triangles and irregular quadrilaterals
• Practice map reading skills and strategies, talk about the use of scales, construct scale drawings of the classroom, school grounds, students homes and/or backyards etc. discuss equivalent scales (eg, 2 cm to 150 metres is the same as 1 cm to 75 metres)
• Explicitly link the use of scales to multiplication using the term scale factor, explore the impact of different scale factors, including scale factors less than 1
Recognises scale factor for pentagons, able to halve the quadrilateral, may use diagonal from right angle vertex to opposite vertex to locate corresponding point, can explain meaning of map scale and provide reasonable estimates of distances Indicates a solid understanding scale factors and how it relates to multiplication in this context
• Provide opportunities for students to work with an extended range of scale factors, eg, very large whole numbers, decimals, mixed fractions, percentages, ratios, etc
• Extend scale drawing skills and strategies to include the idea of perspective and the use of a centre of enlargement (or dilation)
• Link solution strategies to proportional reasoning problems more generally, eg, finding for 1 and multiplying or finding for a common composite unit and multiplying
• Practice map reading skills and strategies, talk about the use of scales