The purpose of this activity is for students to apply equivalent fractions and fractions as operators. The context is about maintaining proportionality of features on a face with reference to the total length of the face.
The background knowledge presumed for this task is outlined in the diagram below:
This activity should be used in a ‘free exploration’ way with an expectation that students will justify the solutions that they find.
The procedural approach (show more)
- The student uses visual perception or fractions as operators to see if the lengths match the desired measures.
Students who operate procedurally are likely to rely on their perception of each image and describe whether, or not, the proportions of the face ‘look right’. They may use the fractions between features of the face as operators on the total length of each face. They will compare the length found by calculation with the lengths on the portrait. Note that the total length of the face will depend on scaling of the image.Click on the image to enlarge it. Click again to close.
The conceptual approach (show more)
- The student uses knowledge of equivalent fractions or ratios to check whether or not the measures are in proportion.
Students who work conceptually will create fractions of measures between features (numerators) and the total measure of face length (denominator). They will use their knowledge of equivalence combined with qualitative adjustment to decide how close the measures on the portraits match the desired proportions. Qualitative adjustments are likely to be small changes to numerator or denominators to make simplification easier, with awareness of the effect of the adjustment on the fraction, e.g. reducing the numerator reduces the size of the fraction.Click on the image to enlarge it. Click again to close.
Some students may organise the desired proportions into ratio form. They may create a standard ratio and enlarge the picture, so the total length is a convenient multiple of the number of parts in the ratio.Click on the image to enlarge it. Click again to close.
Expect conceptually orientated students to realise that real faces vary a bit from the desired proportions, but these students have a sense of when the variation is unreasonable. With the Einstein caricature they should note that Einstein’s nose is almost one third of the face length, rather than one quarter. As a result, his chin to mouth length is less than one quarter. His eyes are set half way down the face which is surprising. Intelligent people are often depicted to have disproportionately large foreheads.