In this Learning Module, students learn about Pythagoras’ Theorem, and how to apply it in real life contexts. They apply their conceptual mathematical understanding by designing a multimedia presentation about how to teach Pythagoras' theorem to future year 8 students.
Mathematics, Pythagoras, Addition, Subtraction, Sums, Products, Squares, Squared Numbers, Square Root.
As a result of completing this Learning Module, students will be able to:
These outcomes come under the following Essential Learning Achievements of the ACT Curriculum Framework: Every Chance to Learn.
16.EA.17 – Choose and use a range of strategies to solve problems including sensible choices about mental, written and electronic methods for calculation
17.LA.8 – … Apply Pythagoras Theorem . . . in appropriate situations to work out measures
18.EA.19 – Solve … simple algebraic equations using a variety of approaches and explain their reasoning
Create your own Critical Thinking Poster (CTP) on an A3 sized piece of paper. Alternatively, you can create an Update in Scholar, and keep adding to it throughout the Learning Module.
At the end of each lesson write about what you have learnt.
Comment: Why do you think it is important to reflect on what you have learnt? Is it best to reflect regularly or just at the end. Give reasons for your comments. You can also comment on other students' comments, explaining why you agree or disagree with the. Start with @Name and be respectful if you are critical in any way.
Critical Thinking Posters [CTP]
This is a reflection and recording strategy.
Use it at the end of each lesson in this Learning Module so students can reflect on what they have learnt or found interesting and document it in a cumulative way.
This has the advantage of providing evidence of learning over the wholemodule which can underpin meaningful summative reflections.
On one section of the CTP students will write about their prior knowledge of right angles and Pythagoras. Some focus questions that could be written up on the white board for students to consider might be:
With the placemat provided and working in small groups fill out what you know about right-angled triangles.
In a group of 3 or 4 discuss what each of you knows about triangles. See if you can:
Activity: right or not?
Fold a piece of paper to get a straight edge.
Fold it again along the folded straight edge to create a template for a right angle.
Measure it with a protractor. Predict things within the classroom that are right angles and then test them with your paper template.
What did you notice?
Comment: Share at least one main idea from your brainstorming session. Try not to repeat what other students state. Then comment on at least 1 other comment by starting with @Name.
Brainstorm
Group discussion using a blank placemat as a question reference guide. Teacher records student responses on poster paper to display in the classroom for student reference during the unit.
Activity: right angles in the student environment
Paper folding task: direct comparison with benchmark template
In all measurement activities it is critical that students understand and correctly identify the attribute being measured.
Here they are measuring angles. It is important that students have experience with direct comparison tasks before they attempt to measure angles with a protractor.
Using a simple template for a right angle [made by folding a piece of paper twice] familiarises students with procedural aspects of finer-grained measurements of angles using a protractor.
These aspects include:
Watch The Life of Pythagoras. Find and view other videoclips about Pythagoras.
Activity: Just in Time
In small groups, choose three major events in human history that you believe have had a great impact on how we live today.
Find out in which year these events happened and place them on a timeline. Position each event so that gaps between events are close to scale.
Include Pythagoras and current time on your timeline.
Comment: Share one link to a useful resource you found about Pythagoras. Use any useful links that you other studnets suggest. Make sure you thank them!
Who is Pythagoras?
Stimulate student interest in the background to this topic: Who is Pythagoras? What did he do?
Encourage students to research information and prompt them to look for pictures of Pythagoras himself, triangles, Greek islands, history, numbers and sun dials.
Activity: Close to Scale Timeline
This activity provides a bracket from approximately 600 BCE to the present. Students can be asked to explain how they know that they have correctly located their three selected events on their number lines. Explanations should include:
Nigel added 48 and 25 together in his head. He got the correct answer of 73.
How might he have done this? Describe as many ways that work as you can think of.
Are some ways better than others? Why?
Watch the YouTube videos to revise some good ways of adding numbers:
Choose two two-digit numbers and explain how you could use each of these methods to add the numbers correctly. Share these with your shoulder partner.
What about calculating 84 – 67 mentally? Which [if any] of the efficient addition strategies work well for subtraction?
Complete the worksheet on adding and subtracting two-digit numbers. Remember to explain the strategy you used to work out each answer.
Comment: Which strategy do you find most useful? Explain why.What is still hard for you to understand? Perhaps a student can help you.
Mental computation strategies for addition and subtraction
The purpose of revising efficient mental strategies for addition and subtraction is to scaffold students’ capacity to make reasonable estimates of partial calculations when solving tasks involving Pythagoras theorem.
Place two two-digit numbers up on the board. Ask students to explain how they could add those numbers mentally. On the board record strategies used by students.
Efficient mental strategies include:
Use similar strategies with subtraction tasks. Use the empty number line to show both difference and take away interpretations of subtraction.
Explore how to adapt the break up/make up method using minus signs to indicate ‘more to take’. For example 72 – 37 could be done as:
For further teaching strategies and activities see Mental Computation: A Strategies Approach by Alistair Macintosh and Shelly Dole, Department of Education, Tasmania, 2004.
See Appendix: Mental Computation Overlay.
Now do some practice of the strategies. Make sure you explain how you calculated your answers.
Comment: Describe any challenges you met. Ask for help. Another studnet may be able to ofer you support.
These worksheets are quick and easy to create. They only take around five minutes for students to complete and can be collected as evidence of students’ level of strategy development. These tasks are particularly good for identifying any misconceptions students might have.
Insist that students explain how they calculated their answers. Promote efficient group-based arithmetic strategies and avoid count by ones approaches for all but the simplest of tasks [that is, plus or minus 1, 2 or 3].
On the white board there are some expressions. Use your counters to show what each expression means.
What do these expressions have in common? In what ways are they different? Share your thoughts on this with your shoulder partner.
Without using counters or drawings describe what is the same and different about:
What did you notice?
Establishing meaning for sums, products and squares
Some students do not understand the difference between 2 x 5 [two groups of five] and 5^{2} [five times itself]
Use arrays to model sums, products and squares. Discuss what they look like and describe each using pictures, words and symbols. Draw attention to the meaning of the mathematical symbols.
Write 5 + 5, 2 x 5 and 5 x 5 on the board. Give students counters and ask them to model each expression. Using an IWB or counters on an OHP ask students to show and describe what is the same and what is different about each expression.
Develop a refined class summary which students then record in their books. Important points include:
See Appendix: Mental Computation Overlay.
Draw a right-angled triangle with side lengths of 3, 4 and 5.
Put squares on each side [with areas 3², 4² and 5²]
Suppose these three squares were made of beaten gold, and you were offered either the one large square or the two small squares.
Which would you choose? How did you come up with your answer?
What if the triangle had side lengths of 5, 12 and 13? Explain your thinking.
This activity is designed for students to identify aspects of Pythagoras theorem before being introduced to the algebraic formula.
Have the students work in groups. One possibility for structured roles that could be adapted is in the appendix Structured roles for group work on mathematical problem solving tasks
Give students grid paper and Newman’s prompt table [appendix] to help them work out their choice. Then have each group share their responses with the class.
In groups students come up with a theory and test their theory of the relationship between the sides of a right-angled triangle. Students test their theory using more grid paper. Groups create a poster about their theory to share with the class.
What is a square number?
Where have we used squared numbers before?
Use your counters to show an example of a squared number. What do you notice?
We don’t always work with whole numbers. And sometimes we don’t need an exact answer, just an approximation. What is an approximation? Why are they sometimes useful?
How could you estimate what (5.7)^{2} is? How do I put this into a calculator? What answer comes up on the screen? Is the calculator answer exact or approximate?
What does 19^{2}mean? What picture would represent 19^{2}?
Can you calculate 19^{2} mentally? If not how can you mentally calculate an approximation that 19^{2} is close to? Is the exact answer more or less that your approximation? Explain how you know
Calculating squares: exact and approximate answers
Students are expected:
See Mental Computation: A Strategies Approach by Alistair Macintosh and Shelly Dole, Department of Education, Tasmania, 2004.
See Appendix: Mental Computation Overlay.
What makes squares special?
How are lengths of sides and the area of the square related?
If the side of a square was 9cm what would be its area?
A square has an area of 36cm^{2}. Barry said that each side must be 9cm long. What mistake is Barry making? How long must each side be, and how can know you are right?
What if the area of a square is 40cm^{2}? Is the length of each side 10cm? Explain your thinking.
Calculating square roots: exact and approximate answers
Find this button on your calculator [ √ ]
Use your calculator to find:
What do you notice? Without pushing any buttons what do you think √100 is?
Describe how to use the connection between squares and square roots of numbers. Give three examples to help explain your thinking.
√18 does not have a whole-number answer. What is a good estimate for √18? How do you know?
What about √23? What is the same and what is different about your estimates for √23 and √18?
Between which two whole numbers must these values be?
Use your calculator to find each value to one decimal place.
Demonstrate on the board how squares and square roots are related to the area and length of a square.
For example a 4 x 4 square has an area of 4² [16] and a side length of 4
So if we know the length of a square we can calculate its area [squares]
And if we know the area of a square we can calculate the length of each of its sides [square roots]
A likely misconception is confusing the area of a square with its perimeter [which is a length not an area]. This needs to be explored by students and unpacked as a class group.
Calculating square roots: exact and approximate answers
Show students how to use a calculator to find the square and square root.
Have students estimate the square root of any two-digit number by bracketing that number between perfect squares. For example the square root of 67 is between which two whole numbers? [8 and 9, but closer to 8]
Discuss this strategy and how it can be refined.
See Mental Computation: A Strategies Approach by Alistair Macintosh and Shelly Dole, Department of Education, Tasmania, 2004.
See Appendix: Mental Computation Overlay.
[Present students with mini task relating to mental computation activities from previous sessions. For example …]
This assessment task will help you to establish students’ levels of understanding around mental computation/estimation/approximation in using Pythagoras’s Theorem with simple decimals.
It is an ‘assess as you go’ task. Use it to determine if you need to revisit any number concepts to further develop student understanding.
This could be a 5 – 10 minute warm up activity or used as a break between activities.
See appendix :Mental Computation – Squared Numbers and Square Roots
Extension activity: Some students could represent other decimals such as 3.2² in a similar way.
The focus here is not so much the sketch in each case but the explanation of the value of each component area and how the total of these relates to the overall squared value.
Have a look at this triangle. What can you tell me about it? [it is right-angled, the three sides look like they have different lengths]
[point to the shortest side] How long is this side? [dunno, haven’t measured it …]
I’ll give that side a name so we can talk about it easily.
Let a = the length of the shortest side in this triangle. I know that a is that length, even if I don’t know how many millimetres that length is. [write a next to the shortest side; repeat the process for naming the other short side b and the hypotenuse c]
I reckon a and b must each be less than c. Am I right? How do you know? [discuss strategies for justifying this assertion; perhaps use a triangle on the board and a piece of string to hold a length constant to show a difference …]
Earlier we look at some right triangles. One triangle had sides 3, 4 and 5 cm long. The other had sides 5, 12 and 13 cm long. What did we see about the squares that were built on each side? [adding the two smaller areas gives the same area as the big square]
Pythagoras’ theorem is a powerful mathematical relationship. It says that this connection between the areas on the sides of right triangles is true for all right triangles … and only for right triangles … go figure …
Using the names of the sides of this triangle [a, b and c] what would be the areas of the squares on each side? [a², b² and c²]
What then would Pythagoras’ theorem be for this triangle? [a² + b² = c²]
Names [for the lengths of sides] can change but the Pythagoras relationship doesn’t change.
Here is a different right triangle. [draw a right triangle that has no side horizontal and the right angle in the top left corner]
How do I know it’s got a right angle? [the square in the corner]
Give me some algebra here. What will we call each side? [students suggest letters as labels, such as p, q and r] Let’s be really clear about what these letters mean. Tell me, what are p, q and r in this situation? [the lengths of the three sides] Do I need to know the length of each side (say in mm) to talk about the lengths? [no]
Given that this is a right triangle, how are p, q and r related? [if p is the hypotenuse say, then q² + r² = p²]
Now, draw two right triangles.
With each triangle name the sides, describe what the names represent, and describe the relationship between the three sides. [have each student do this on a half A4 sheet of paper and then collect and review these for indicators of conceptual understanding of algebraic representations of Pythagoras’ theorem]
Naming Pythagoras’ theorem [move this to after the next section: ‘square and square root examples’?]
Introduce students to the algebraic representation of Pythagoras theorem.
Point out that here the algebra is one way of expressing the special relationship between the lengths of the sides of right triangles.
The language here is critical. We need to say:
“Let a = the length of this side. Let b = the length of this side and then c = the length of the hypotenuse.”
Look to establish meaning for the algebraic symbols by getting students to describe what a, a² and a² + b² = c² represent. A good response would be something like:
a, b and c are lengths of sides; a², b² and c² and the areas of the squares we could put on each of those sides … a² + b² is the total area of those two squares put together, and that total s the same as c²’
It is important to give examples that challenge shallow conceptions, such as:
Remember, Pythagoras’ theorem only works for right-angled triangles.
Let’s apply this is a similar way to what carpenters and bricklayers do when they build a house. We’ll start just with pen, paper and a ruler but the process is similar …
[example on OHP or white board] Here is an angle that looks like a right angle … but I’m not sure …
I’ll make a triangle by drawing a line across the two arms of the angle …
Can you see the triangle I’ll be working with? [trace out the triangle with the tip of a finger]
Let’s name the sides … who’s got some algebra we can use here? [label each side and get students to describe the meaning of the terms]
How can I use Pythagoras’ theorem to help me work out if the corner is a right angle or not? [there is more than one solution here … one option is to measure each side, calculate the sum of the squares, find the square root of this and compare that to the length of the longest side]
Because my triangle is pretty big I’ll measure each side to the nearest centimetre … [construct a table with a, b and c, and a² and b² and a² + b²; fill in the agreed lengths for a, b and c]
Notice not all the lengths line up right on the cm markings. What will this mean for our calculations? [open the discussion about measurement error and how close is close enough; calculate a² and b² and a² + b² and the square root of a² + b² and compare this last value to c]
Try one for yourselves. [prepare a half A4 sheet with two angles, one on each side of the paper, with space to record measurements and calculations]
Because your triangles will be smaller you can measure to the nearest millimetre. Why might this be better? [discuss why the test works, issues in the process and interpreting results of calculations, and how this process can be used in construction]
Proving and disproving: practical measurement
In this practical task students:
This task involves the application of a range of understandings and skills. These include:
This last point is an important and integral part of the exercise. It is unlikely that the calculations will tally exactly due to the need to round all measurements of lengths to the nearest millimetre. Discussion around ‘how close is close enough’ is essential for making sense of this activity.
Also there are several ways of approaching this task. These include:
An alternative to placing the transverse line somewhat randomly is to mark off convenient values for the lengths of the two arms of the angle [say 30 mm and 40 mm where the arms are long enough]. This is more like what carpenters and bricklayers are likely to do in practice. However do not jump to this too early. Allow students time to engage with the measurement process and perhaps suggest a similar refinement in the process. If they do suggest a refinement [this or another] push them to explain why their suggestion is an improvement.
Who is Pythagoras’s theorem for and how do you use it?
T Charts
Activity
Describe to students how builders use Pythagoras when constructing a house to ensure that all corners are 90 degrees.
Provide groups with measurements of the short sides of a right angles triangle. Have them determine the hypotenuse and show all working out.
Using their measurements they then need to construct the outline of a bedroom as if they were the builders on a building site. Outside there will be pegs and a string line. The task is to measure the short sides and place pegs at a point as both short sides of their triangle meet here. Then use the hypotenuse to align the right angle. Students then need to re measure the remaining two sides to form a rectangle.
Peers groups then check other group’s bedrooms to see if their corners are square. A test for a quadrilateral to be a rectangle is if the diagonals are equal, therefore the two hypotenuses ensure the room is square.
T-Charts are a type of graphic organiser in which a student lists and examines two facets of a topic, like possible pros and cons, advantages and disadvantages, facts vs. opinions and so on.
Students work in groups using three different T charts.
Each group starts with one T chart and then the charts are rotated to the next group to add to. On the last rotation that group shares the results.
Activity
Prior to this activity the teacher will need to set up 5 sets of string lines attached to metal pegs that are placed into the ground at about 8 metres in distances so that students can use the string line as a guide line to set up their room outline.
Take photos as part of a portfolio task and stick on as part of a work sample and reflection.
This could be an item to place in student’s portfolios with the self assessment sheet.
Students reflect on this activity with a self-assessment sheet.
Students’ answer the question “Who gains by using Pythagoras Theory? Who loses by not using Pythagoras Theory?”
This Update is important to understand the relevance of learning about Pythagoras' Theorem.
Look at these two right triangles. The triangles aren’t drawn to scale but there is information there as part of each diagram.
How are the sides in each right triangle related?
In pairs write down as many things as you can … what did you find?
[discuss ways of using the Pythagoras relationship to calculate the missing length in a right triangle]
What’s a good way of recording our thinking as we use Pythagoras’ theorem work out the length of the third side of a right triangle?
Where does it make sense to use a calculator? How do we use a calculator effectively for this task? How much do we need to write down so that our thinking is recorded [for future reference, for communication or for assessment]?
Finding the length of a side by calculation [without measuring]
So far the applications of Pythagoras’ theorem have involved practical measurement.
But if we know two sides of a right triangle Pythagoras’ theorem allows us to calculate the length of the third side without physically measuring it.
There are two options:
A strategy for starting with these tasks is to describe Pythagoras’ theorem for the triangle you are working with. In these examples:
Discuss the similarities and differences between these expressions.
Consider using a blank box to represent the missing or unknown value [which can be calculated from arithmetic necessity]
For example if 53 = + 17 what number must go in the ?
If necessary explore all the equivalences that arise from a simple addition fact such as 10 = 7 + 3 [3 + 7 = 10, 10 – 7 = 3, 10 – 3 = 7, 3 = 10 – 7 and so on …]
Apply this to describing the relationship between the sides of a right triangle and the process of calculating the remaining side. For example:
to calculate the length of the hypotenuse
h^{2} = 7^{2} + 10^{2}
h^{2} = 149
so h = √149 ≈ 12.2
to calculate the length of a short side
12^{2} = a^{2} + 3^{2}
144 = a^{2} + 9 [what do we add to 9 to get 144?]
a^{2} = 144 – 9 = 135
a = √135 ≈ 11.6
[note: ≈ represents ‘approximately equal to’]
Several contexts could be used where students have to measure the length of sides a and b in order to work out the length of the hypotenuse. Some options are a square room, a basketball court or even a football field.
Possible questions are:
How far is it diagonally across a basketball court? Does Pythagoras help here? How do you know?
How did you measure the lengths of sides “a” and “b”?
Which length is “a” and which is “b”? Why? Does this matter?
‘On paper’ with Pythagoras
A powerful thing about mathematical relationships like Pythagoras’ theorem is that we can work out some things [like a length] without having to measure it.
A simple sketch or diagram can hold all the information we might need. The picture itself is not the main thing; it’s the information on the picture that matters. Things like the lengths of sides, if the side is the hypotenuse, where the right angle is and so on.
Here are some sketches of right triangles. What information is given? What are we asked to find? How might we find it?
Practical tasks outside the classroom
Students work through a number of practical problem solving tasks that require them to
Set up a number of stations in an open area for students to use and check Pythagoras theorem.
Theoretical tasks from texts
Students need experience in solving standard textbook tasks that involve Pythagoras’ theorem. This requires a capacity to:
Participate in a Rally Robin using questions and problems on the activity sheet [see Problem solving: Pythagoras’ theorem]
Find a work partner. One of you will be student A, the other student B.
Each of you individually work out the answers to your problems then take turns in talking your partner through your solutions.
Rally Robin
This is a Kagan cooperative learning strategy.
Here students are given a problem with multiple solutions, or multiple problems.
Each student individually works out a solution.
Partner A explains to B how they worked out a problem. Then partner B explains a different solution to B.
A and B take turns giving solutions until all solutions or tasks are done.
Use the table to help plan and document your solutions to the problem solving tasks involving Pythagoras theorem.
Draw a sketch of the main features of the shape/ object/ thing [not to scale] | 1: What is the question asking you to do? | 2: Describe how you might do this | 3: Work through your steps to answer the question |
Problem-Solving
This activity promotes metacognition as well as providing practice of using Pythagoras' Theorem.
Assessment
Complete the assessment activities to demonstrate your understanding of the Pythagoras' Theorem.
Problem-Solving
This activity promotes metacognition as well as providing practice of using Pythagoaras' Theorem.
Project Name: Pythagoras' Theorem
Description: Design a multimedia presentation as a tutorial that could be used for Year 8 classes to learn about Pythagoras’ Theorem next year.
Check the Work Request in your Notifications. Click on this link to open the “Untitled Work” in Creator. Then, change the title, and begin a first draft. Go to About This Work => Project => Description for further project information.
For what you need to do in order to create an effective multimedia presentation, go to Feedback => Reviews => Rubric. Keep the Rubric open and refer to it as you write.
When you are ready to submit, click “Submit Draft” below the work. This is the version of your work that will be sent to others for feedback.
Feedback Phase
Check your Notifications for a "Feedback Request". For more information, see Reviewing a Work and Submitting a Review and Annotations.
Revision Phase
The next stage of the writing process is to revise your own work. For more information, see The Revision Phase. You can also write a self-review, explaining how you have taken on board the feedback you received.
For more information, see The Revision Phase.
Comment: Do you have any questions about how Scholar works? Make a comment in this update. If you think you have an answer to another student's question, please answer it - be sure to name the student you are replying to in your comment by starting with @Name.
This task is one way to consolidate and draw out student understanding of Pythagoras' Theorem. Students could work in pairs to complete the assignment.
Title: (Source); Fig. 1: (Source);Fig. 2: What is a Right Angle Placemat by Lanyon High School teachers; Fig. 3: "Sanzio 01 Pythagoras" by Raphael - Web Gallery of Art: Image Info about artwork. Licensed under Public Domain via Wikimedia Commons (Source); Fig. 4: Compensation Strategy (Source);Fig. 5: (Source);