Many problems involve three-dimensional objects or spaces. Pythagoras Theorem in 3D Shapes can be used as much with these problems as those in plane shapes.
The starter recaps applying Pythagoras Theorem as part of a larger problem involving the perimeter of a trapezium and square. The aim of this starter is for students to experience the range of problems that can be solved using the theorem. In this instance there are no obvious right-angled triangles yet the problem cannot be solved without Pythagoras.
When you work with 3D shapes it is important to look for and draw out right-angled triangles. Labelling the vertices of the triangle with the same label as those used in the 3D shape helps students to visualise it two dimensionally.
The base ABCG is in a horizontal plane and triangle ACD is in a vertical plane. First look at the right-angled triangle ABC and use Pythagoras’ Theorem to find length AC.
Next, look at triangle ACD. Label the length of AC that was found in triangle ABC, leave the answer in its most exact form as the root will be cancelled when squared. Use Pythagoras’ Theorem to work out length AD.
We work through the second problem as a class. I encourage students to draw the cuboid on one half of their mini-whiteboard and in the second half draw the right-angled triangles. Similar to the diagram above. It’s important to leave the working of the first example on the board to help students see the various stages.
The interactive Excel file can be used for additional practise before students work independently through the questions on the third slide and worksheet.
In the plenary students are challenged to find the total perpendicular height of a composite solid. In addition to this being a composite shape a new level of difficulty is added as students now have to use the hypotenuse to find a shorter side.
I like to set the problem shown below for homework after this lesson as it challenges students to link Pythagoras’ Theorem with volume of pyramids.
The diagram shows a pyramid.
ABCD is a square with lengths 8 cm.
The other faces of the pyramid are equilateral triangles with sides of length 8 cm.
Calculate the volume of the pyramid.
To find the area of compound shapes students need to understand what the word compound means. Therefore, I ask students to discuss in pairs a definition for the word compound and to extend it to include the shapes below. As a result of their learning in science students agree that a compound can be defined […]
At the start of the Spring Term these are three main priorities for me as the Head of Mathematics.
I teach mutually exclusive outcomes directly after students have encountered Venn diagrams. This is the fifth Year 8 Probability lesson.