Let’s Experiment: Discovering Area and Volume Formulas

# 2. Quadrilaterals: Investigating areas of Kites and Rhombuses

Suggested Learning Intentions

- To investigate the area formulas of rhombuses and kites

Sample Success Criteria

- I can describe the similarities and differences between rectangles, parallelograms, kites and rhombuses
- I can develop the formula for calculating the area of kites and rhombuses
- I can use a range of manipulatives to model and explain my thinking

- Comparing Kites and Rectangles template: docx PDF
- Venn Diagram graphic organiser template: pptx PDF
- Geoboard App or Geoboard with rubber bands
- Computer, laptop or tablet access

This stage of the sequence aims to support students to observe and make connections between the area formula of rectangles, parallelograms, kites and rhombuses.

Ask students to draw a kite and a rhombus or show students a drawing of a kite and a rhombus. Encourage students to draw kites and rhombuses of different sizes and orientations. Ask students, “How can you tell that the shape is a kite or a rhombus?”

Encourage students to work collaboratively to record as many properties as they can of the two shapes. Prompt students by asking questions such as:

- How many lines of symmetry does it have?
- Which angles must be equal?
- What does it mean for its diagonals?

Facilitate a class discussion about the similarities and differences between the two shapes. A Venn Diagram can be used as an organisational tool (see Materials and texts section above for a downloadable template). Encourage students to consider the properties of the shapes such as their lines and angles. Create an anchor chart for later referral.

**Enable** students by encouraging them to cut out and fold the shapes to find the lines of symmetry. Encourage them to draw the diagonals on the kites and rhombuses. Explicitly teach the vocabulary used to identify and describe the shapes and create posters or a word wall of the shapes and refer to them regularly.

**Extend** students by encouraging them to write a definition of kites and rhombuses, labelling the properties and presenting this as a class display.

This activity is adapted from ‘Establishing the area of a kite’ lesson published by the Government of South Australia (2017). It aims to challenge students to experiment with methods for establishing the area formula of kites.

Provide students with drawings of kites and rectangles on grid paper, which can be found as a template in the Materials and texts section above.

Students compare the area of each pair of shapes – the purple kite with the purple rectangle, the green kite with the green rectangle and so on.

Ask students, “Which is bigger – the area of the kite or the area of the rectangle in each pair?”

Pair students to work collaboratively with their peers and encourage them to find more than one way of solving the problem.

As students are working, encourage them to convince you or their peers that their solutions are reasonable.

Students may find the area of a kite in various ways, such as:

- Noticing that there are two congruent triangles within a kite.
- Cutting along the diagonals of the kite and rearranging the pieces to form a rectangle, for example:

Ask students to label the base and the height of the rectangle they have formed to calculate the area, then reconstruct the original kite.

Model how to find the corresponding ‘base’ and ‘height’ on the original kite.

Despite the fact they may have cut the kite in different ways, they always use the dimensions of the diagonals, as shown:

Students can also apply known formulas for rectangles and triangles.

Encourage students to form generalisations by asking questions such as:

- Is there a rule that you could use to describe a way to work out the area of a kite?
- What if you change the size of your kite? Does your rule work for any size?
- What if you change the angles of your kite? Does your rule still work?
- What if all the sides of the kites are the same length?
*This will help students establish the area of a rhombus in a different way to the method they used when they connected it to a parallelogram.*

Students can research the formula for the area of kites and rhombuses and relate this to their own thinking.

**Enable** students by suggesting that they cut straight across the diagonals of the kite and asking, “Can you find a way to rearrange pieces to form a rectangle?”

**Extend** students by asking them to find another way of finding the formula of kites.

Formative assessment of student learning in this stage of the sequence could include observation of students’ approaches as they discover how kites are related to rectangles as well as noting how students proceed to develop their formulas for finding the area of kites.

Students create a presentation of how area formulas of kites and rhombuses are derived and to prove that their formulas are true for any kites and rhombuses, regardless of their size or angles.

To assess students’ ability to apply these formulas to solve problems, ask students to draw a kite and a rhombus, and show steps to find the areas of these shapes.

**Enable** students by suggesting that they could begin by drawing two rectangles with a given area, for example 36 square centimetres. Then ask students to cut one of the rectangles out to transform it into a kite. The other rectangle can be transformed into a rhombus.

**Extend** students by asking them to predict if kites can look different even if they have the same area. Ask: Can two rhombuses look different but have the same area? Encourage students to test their theory by attempting to make different kites and different rhombuses with the same area.

Black Douglas Professional Education Services, n.d. *Mathematics Task Centre: Sphynx. *[Online]

Available at: http://mathematicscentre.com/taskcentre/166sfinx.htm

[Accessed 15 March 2022].

Government of South Australia, Department for Education and Child Development, 2017. *Using units of measurement: Year 8. *[Online]

Available at: https://acleadersresource.sa.edu.au/wp-content/uploads/2018/05/Using_units_of_measurement_Year_8.pdf

[Accessed 15 March 2022].

Reys, R. E. et al., 2020. *Helping Children Learn Mathematics. *Milton: John Wiley & Sons Australia..

Siemon, D. et al., 2015. *Teaching Mathematics: Foundations to Middle Years. *Melbourne: Oxford University Press.

University of Cambridge, n.d. *NRICH: Kite in a Square. *[Online]

Available at: https://nrich.maths.org/kiteinasquare

[Accessed 15 March 2022].

Van de Walle, J., Karp, K., M, B.-W. J. & Brass, A., 2019. *Primary and Middle Years Mathematics: Teaching Developmentally. *Australia: Pearson.

Other stages

1. Quadrilaterals: Investigating Areas of Trapeziums

EXPLORESuggested Learning Intentions

- To investigate the area formulas of trapeziums

Sample Success Criteria

- I can describe the similarities and differences between a parallelogram and trapezium
- I can develop the formula for calculating the area of trapeziums
- I can demonstrate and justify my thinking using a range of manipulatives

3. Circles: Establishing Pi

EXPLORESuggested Learning Intentions

- To investigate the ratio of circumference to diameter

Sample Success Criteria

- I can describe how the radius, diameter and circumference of a circle are related
- I can demonstrate and justify my thinking using a range of manipulatives
- I can interpret the formula for the circumference of circles

4. Circles: Establishing Area Formula

EXPLORESuggested Learning Intentions

- To visualise and evaluate how the area of a circle is determined

Sample Success Criteria

- I can explain the strategies used to develop the area formula for circles
- I can demonstrate my understanding of how the formula for area of circles is derived using a range of manipulatives

5. Prisms: Establishing Volume Formulas

EXPLORESuggested Learning Intentions

- To build our understanding of the relationship between surface area and volume
- To develop the formula to find the volume of prisms

Sample Success Criteria

- I can identify the surface area of a prism
- I can describe the connection between surface area and volume
- I can describe the formula for calculating volume of prisms and explain why this is true for all prisms
- I can make connections between the area formula and volume formula
- I can demonstrate and justify my thinking using a range of manipulatives