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Pythagoras’ Theorem Mathematics Stage 4
Summary of Substrands
Duration: 3 weeks
Substrand 4: Right-Angled Triangles
(Pythagoras)
Start Date:
Completion Date:
Teacher and Class:
Outcomes
›
MA41WM communicates and connects
mathematical ideas using appropriate
terminology, diagrams and symbols
›
MA42WM applies appropriate mathematical
techniques to solve problems
›
MA416MG applies Pythagoras’ theorem to
calculate side lengths in right-angled triangles,
and solves related problems
›
MALS1WM responds to and uses mathematical
language to demonstrate understanding
1
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›
MALS2WM applies mathematical strategies to
solve problems
›
MALS3WM uses reasoning to recognise
mathematical relationships
›
MALS30MG recognises, matches and sorts three-
dimensional objects and/or two-dimensional
shapes
›
MALS31MG identifies the features of three-
dimensional objects and/or two-dimensional
shapes and applies these in a range of contexts
Overview
Key Words
Throughout this unit of work student will
investigate Pythagoras’ theorem and its
application to solving simple problems involving
rightangle triangles.
Exact and approximate answers, surd, square
root, squared, converse, triad, irrational, length,
perimeter, area, hypotenuse, opposite,
adjacent,side, right-angle, accuracy, vertical,
horizontal, incline,
Rego
Suggested Assessment
●
Content
Teaching, learning and assessment
Stage 4 - Right-Angled Triangles
(Pythagoras)
Pre-Assessment
Students:
Find squares and square roots, using a calculator and noncalculator techniques. Begin with square numbers
Investigate Pythagoras' theorem and its
application to solving simple problems
involving right-angled triangles
(ACMMG222)
● identify the hypotenuse as the
longest side in any right-angled
triangle and also as the side
Resources
Before beginning this topic, students will need to be able to:
Round to a predetermined number of decimal places, with the
understanding that their solution is now an approximation of the full
solution.
Label and identify parts of a triangle using mathematical
Sides of a right-angle triangle
2
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opposite the right angle
conventions
● establish the relationship
between the lengths of the sides
of a right-angled triangle in
practical ways, including with ●
the use of digital technologies
●
● describe the relationship
between the sides of a rightangled triangle
(Communicating)
●
Basic properties of a right-angled triangle
http://www.wisconline.com/Objects/ViewObject.
aspx?ID=TMH401
Identify the hypotenuse as always opposite the right angle
using a number of triangles of different orientations
Physically measure the distances of all sides of a number of right
angles triangles, with the intention of the students “discovering” that
the hypotenuse is always the longest side and its relative position to
the right angle
Adjustment: For less able students use 2 coloured highlighters
to represent hypotenuse and right angle in one colour and
other sides in the other.
Water Activity
Complete activity where squares on sides are cut up to show that
the smaller two side have the same area as the larger side
(hypotenuse)
http://www.youtube.com/watch?
v=CAkMUdeB06o
Adjustment: lower ability students may benefit from a 1cm grid
division on the squares, so they can cut up individually and
move to hypotenuse.
Showing Pythagoras theorem
visually
More able students can approach this using tangrams
http://www.pbs.org/wgbh/nova/p
roof/puzzle/theorem.html
Discussion to follow this activity asking questions such as
What does this show you in relation to sides?
●
Watch the You Tube clip of Pythagoras being demonstrated with
water
●
Use computer animations to demonstrate the theorem
At this stage the teacher must formalize the theorem. It is important
that c2 = a2 + b2 is not the only format given to students
● use Pythagoras' theorem to find the
length of an unknown side in a
Finding the Hypotenuse
http://www.mathsisfun.com/pyth
agoras.html
http://www.brainingcamp.com/re
sources/math/pythagoreanformula/lesson.php
Changing side lengths
3
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right-angled triangle
➢ explain why the negative
solution of the relevant quadratic
equation is not feasible when
solving problems involving
Pythagoras' theorem
(Communicating, Reasoning)
Initially only find the length of the hypotenuse. It is important for
students to be very proficient at this before proceeding to finding a
shorter side.
Begin with examples that lead to whole number answers.
During the completion of this section it is worthwhile to stress that
the longest side is the hypotenuse and so your answer will be the
biggest.
Proceed to examples with answers that are not exact square roots
interactive
Scootle L6559
Worked examples
http://www.mathwarehouse.com/
geometry/triangles/how-to-usethe-pythagorean-theorem.php
Finding the shorter side
Revise solution of equations.
Ask: If the hypotenuse is 10m, and one of the other sides is 6m
long, how would you find the length of the unknown side?
Some students will use equation solving techniques whilst others
will develop the “subtraction” method.
This needs to be consolidated through extensive practice in a
variety of orientations. (triangles given)
● write answers to a specified or
sensible level of accuracy, using an
'approximately equals' sign,
ie or
Adjustment: Lower ability students may only be able to work
with exact answers, as errors may occur with rounding etc.
Discuss sensible answers and levels of accuracy. This will need to
be discussed again once problems have been solved and the
context of their answers. Review “rounding” rules
Investigate the concept of irrational
numbers (ACMNA186)
● use technology to explore decimal
approximations of surds
➢ recognise that surds can be
represented by decimals that are
neither terminating nor have
a repeating pattern
Practical application problems
http://www.mathsteacher.com.a
4
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(Communicating)
●
●
solve a variety of practical
problems involving Pythagoras'
theorem, approximating the
answer as a decimal
solve a variety of practical
problems involving Pythagoras'
theorem, giving exact answers
(ie as surds where appropriate),
eg
➢ apply Pythagoras' theorem to solve
problems involving the perimeters and
areas of plane shapes (Problem
Solving)
u/year10/ch14_measurement/01
_pythag/18pythag.htm
View SCOOTLE S4693 video on how the Egyptians built the
pyramids so accurately.
Follow with a discussion of where else Pythagoras can be used
practical applications such as building, orienteering, navigation
etc.
A variety of real life problems should be investigated.
Adjustment: Lower ability level students to stay with 2D problems
whilst more able may progress to 3D problems
Practical applications should be extended to find unknown sides for
perimeter and area of plane shapes
http://voyager.dvc.edu/~lmonth/
PreAlg/lesson51student.pdf
http://www.regentsprep.org/Reg
ents/math/ALGEBRA/AT1/PracP
yth.htm
Lessons, Interactive, Questions
and Application of Pythagoras
http://www.brainingcamp.com/re
sources/math/pythagoreanformula/
http://www.bbc.co.uk/schools/gc
sebitesize/maths/geometry/pyth
agorastheoremact.shtml
Applying Pythagoras’ theorem to
decode a puzzle
http://www.themathlab.com/Alge
bra/pythagorean%20theorem%2
0intro%20to%20trig/pythagtest.h
tm
● identify a Pythagorean triad as a
set of three numbers such that
the sum of the squares of the first
two equals the square of the third
● use the converse of Pythagoras'
theorem to establish whether a
Triads
Begin with 3,4,5 triad . Set out as follows
e.g. 3,4,5
LHS = 52
= 25
RHS = 32 + 42
= 9 + 16
= 25
5
Patterns in triads and creating
triads
http://www.maths.surrey.ac.uk/h
ostedsites/R.Knott/Pythag/pythag.html
#moretriples
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triangle has a right angle
LHS = RHS
Therefore this is a Pythagorean Triad
Ensure examples used include non triads.
e.g. 3,4,6
LHS = 62
RHS = 32 + 42
= 36
= 9 + 16
= 25
LHS ≠ RHS
Therefore this is not a Pythagorean Triad
Adjustment: More able students can investigate patterns in
triads and how to create a triad
Use the format as set up above to illustrate the converse
Evaluation
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