Year Six - Taking apart Prisms & Polygons
Home - Taking Apart Prisms & Polygons Map
About this Big Idea
In this Big Idea students explore the concepts of area and volume. Students apply their reasoning skills to find the area of complex two-dimensional shapes and investigate strategies that might be applied to non-traditional shapes. Students then consider how these strategies can be applied to determine the base area of prisms constructed on top of these irregualr shapes such as may be required to determine the volume of a building based upon the area of its footprint.
At the heart of this Big Idea is an opportunity for students to reason with their mathematical thinking and to communicate their thinking. Students are encouraged to approach a problem that they have solved using one method with fresh eyes and in doing so consider an alternative approach. By approaching a problem from multiple perspectives and explaining, comparing and justifying the reasoning behind each students become mathematical thinkers.
Understanding Goals:
Students will understand:
- the concept of area
- the concept of volume
Background:
Students have explored the concepts of area and volume previously and in particular when exploring multiplication in Year Three. In this Big Idea students explore through the application of their reasoning skills strategies to determine the area of irregular shapes. Students will explore the relationship between the area of the base shape in a prism and its volume and consider how in buildings this base shape is the buildings footprint i.e. the area that building occupies on the ground.
Core Content from the Syllabus:
Working Mathematically
Area
- solve a variety of problems involving the areas of rectangles (including squares) and triangles
- establish the relationship between the lengths, widths and areas of rectangles (including squares)
- calculate areas of rectangles (including squares) in square centimetres and square metres
- recognise that rectangles with the same area may have different dimensions (Reasoning)
- connect factors of a number with the whole-number dimensions of different rectangles with the same area (Reasoning)
- record calculations used to find the areas of rectangles (including squares)
- apply measurement skills to solve problems involving the areas of rectangles (including squares) in everyday situations, eg determine the area of a basketball court
Volume
- Calculate the volumes of rectangular prisms (ACMMG160)
- describe the 'length', 'width' and 'height' of a rectangular prism as the 'dimensions' of the prism
- construct rectangular prisms using cubic-centimetre blocks and count the blocks to determine the volumes of the prisms
- construct different rectangular prisms that have the same volume (Problem Solving)
- explain that objects with the same volume may be different shapes (Communicating, Reasoning)
- describe rectangular prisms in terms of layers, eg 'There are 3 layers of 8 cubic- centimetre blocks' (Communicating)
- establish the relationship between the number of cubes in one layer, the number of layers, and the volume of a rectangular prism
- calculate the volumes of rectangular prisms in cubic centimetres and cubic metres
- recognise that rectangular prisms with the same volume may have different dimensions (Reasoning)
Language:
- area, measure, square centimetre, square metre, square kilometre, hectare, dimensions, length, width,
Connected to:
Mindset Mathematics Learning Activities
Visualise
Students develop methods for finding the area of irregular polygons by exploring ways to decompose two-dimensional figues and reason about partial square units.
Questions for reflection:
- How did you decompose the shape to find its area?
- How else migh you have decomposed the shape?
- What alternative approaches might you explore? How might the results differ?
- How did you account for the partial squares?
- How do you know that you have accurately counted the area?
- How do the different methods prove each other? What do they have in common? What differences do you see?
- What methods make the most sense for finding the area of irregular polygons?
- How can we decide what method to use for a given shape?
- What do you think are the best methods for finding the area of polygons? What makes you say that?
Play
Students play with different ways to decompose a complex polygon to find its area.
Begin by studying the image.
- What do you notice?
- What shapes do you see?
- What shapes matter most to the problem of the area of the whole?
- What strategies could you use to decompose?
Questions for reflection after sharing gallery of decompositions:
- What do the different ways of decomposing have in common?
- How are some strategies different?
- Which ideas are most interesting? What makes you say that?
- What strategy for decomposing to find area did you find most useful? Why?
Investigate
Students explore finding the volume of solids that are not rectangular by constructing building on shapes explored in the Visualise activity. Students then apply the same methods to explore the volume of buildings in the real world.
Questions for reflection:
- What strategies did you use?
- How did you deal with the fractional height?
- How did you find the volume of prisms
Credit:
Boaler, Munson & Williams (2018) - Mindset Mathematics: Visualizing and investigating big ideas Grade 5
NESA - Mathematics K-10 - 2012
