Midpoints inquiry
The prompt
Mathematical inquiry processes: Test and classify cases; counter-example; draw conclusions; reason. Conceptual field of inquiry: Types of quadrilaterals; midpoints; coordinates and vectors.
The prompt makes a claim that students of all ages find difficult to believe. Even if they agree that it is true for some regular quadrilaterals, which, in itself requires an acceptance that squares, rectangles and rhombuses are types of parallelograms, they still find the claim difficult to believe for irregular quadrilaterals.
At the start of the inquiry, students pose questions and make observations in order to understand the prompt. They might recognise some of the terms, but cannot comprehend the overall meaning.
What does 'adjacent' mean? What does 'inscribed' mean?
A quadrilateral is a four-sided shape and could be a square, rectangle or trapezium.
'Adjacent' means next to.
Does the quadrilateral have to be regular?
The teacher might start the inquiry by encouraging students to list all the quadrilaterals they know and discuss their properties. The class could also attempt to visualise a square and the inscribed shape formed by joining the midpoints of adjacent sides.
Once the class is clear about the process described in the prompt, the exploratory phase of the inquiry begins. Using the regulatory cards, students often decide to draw their own examples. The inscribed shapes of a parallelogram and isosceles trapezium are shown below.
Results
As the class collects results for different types of quadrilaterals, the teacher writes a list on the board (see the table below) and leads a discussion on any contradictory results. To show that the contention in the prompt is true for all cases except one, the teacher should be prepared to convince students that the definition of a parallelogram - a four-sided plane rectilinear figure with opposite sides parallel - includes a square, a rectangle and a rhombus.
Joining the midpoints of adjacent sides in an arrowhead (a concave quadrilateral) also forms a parallelogram. However, the parallelogram overlaps the edges of the arrowhead and is, therefore, not inscribed within the original shape. This one counter-example means the prompt is not strictly true.
Irregular quadrilaterals
Perhaps the most surprising result is that the prompt is also correct for irregular quadrilaterals (see examples below). The teacher might use a dynamic geometry package (such as Cabri Express) to move a vertex of the irregular shape in order to show how that changes the position of two vertices of the parallelogram.
Using column vectors to find the midpoint of a side
When students draw their own diagrams, they might find it difficult to locate the midpoints of the sides. They could measure the length of each side. However, a more accurate method involves the use of column vectors. Using the diagram of a parallelogram above as an example, the diagonal sides can be expressed (from top to bottom) as 4 right and 8 down and, therefore, the midpoint is found at 2 right and 4 down.
Column vectors are also useful for checking that the new shape formed by joining the midpoints is indeed a parallelogram. Using the same diagram, the column vectors for each pair of sides of the new shape are 4 right, 4 up and 8 right, 4 down, which shows that each pair is parallel.
May 2022
Lines of inquiry
1. Similar triangles
What shape do you create by joining the midpoints of adjacent sides of triangles?
What happens if you start with equilateral or isosceles triangles? What is the scale factor of enlargement of the similar triangles? Why is it a half? Why is the area of the new triangle a quarter of the area of the original triangle?
What happens if you start with a scalene triangle? Do you also create similar triangles? Why?
2. Midpoints on Cartesian grids
Students might carry out the inquiry using coordinates. This could lead to a line of inquiry in which students generalise about the four coordinates that give particular shapes. For example, the shape whose vertices have the coordinates (2,3) (2,5) (6,3) and (6,5) is a rectangle. However, the following coordinates (1,2) (3,6) (5,5) and (3,1) also give a rectangle. Do the two sets of coordinates have anything in common?
An advantage of using coordinates is that students can use the formula to find the midpoints of the sides (M) and, thereby, locate them accurately.
3. Hexagons and ratio
What shape do you create by joining the midpoints of adjacent sides of a regular hexagon?
After carrying put the process twice, what is the ratio of the length of the sides of the three hexagons? Students could use the cosine rule or Pythagoras' Theorem to find the length of a side of the second hexagon (AC in the diagram below).
Pythagoras' Theorem
For triangle BCD, CD = √(22 - 12) = √(3) and AC = 2√(3)
Cosine rule
For triangle ABC, angle ABC = 120o
AC2 = 22 + 22 - 2(2)(2)Cos 120o = 12 and AC = √12 = 2√(3)
Thus, the ratio is 4 : 2√(3) : 3
4. Proof
