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Right-angled triangles inquiry

  
This prompt can spark an open, exciting and multi-faceted inquiry that combines Pythagoras' Theorem and trigonometry with algebraic expressions for sequences and area. Sometimes it does this too well, and the teacher has to guide the inquiry to ensure students have access to necessary mathematical concepts. The prompt can develop along different pathways based on the students' questions or observations:
  • The sides of the triangles form ascending linear sequences. What is happening to the hypotenuse? Is its length also increasing in a linear sequence? Can the length be made to increase linearly?
  • The angles in each triangle are (not) the same. Is the angle in the bottom right-hand (or top) corner getting bigger or smaller? If bigger (smaller), can you make it go smaller (bigger) with ascending sequences? What happens with descending sequences? Can you find two sequences that keep the angles the same? What happens when you use other types of sequences?
  • The area increases in a quadratic sequence. How would you find an expression for the nth term of the sequence? Is there another set of triangles that has the same expression for the nth term of the area sequence?
The inquiry can be used to develop a conceptual understanding of the tangent ratio and, by calculating the length of the hypotenuse, the sine and cosine ratios as well. Students come to appreciate that the size of an angle is determined by the ratio of the lengths of two sides of the triangle. This explains how both sides can get longer, yet the angle gets smaller. Indeed, it is possible in the initial stages to use a unit ratio before introducing trigonometry in a formal way. The bottom right-hand angle in the prompt, for example, is increasing because the opposite side increases at a faster rate than the adjacent (as shown in the table below).
Triangle in prompt 1
 Ratio
(opposite:adjacent)
 4:37:510:713:9
 Unit ratio 1.33:11.4:1 1.43:1  1.44:1
  
Additionally, finding an expression for the area can be accomplished by considering the general form of the triangles in the prompt (right), which leads to ½(2n+1)(3n+1). If students derive the expression 3n2 + 2.5n + 0.5 by taking differences between the terms in the area sequence (6, 17.5, 35, 58.5), then reconciling the two expressions has proved rewarding. The inquiry often ends with groups of students presenting conjectures, methods, and examples from their pathways to their peers.
   
NEW
 
GeoGebra worksheet
This GeoGebra interactive worksheet enables students to see the triangles drawn to scale. The dynamic constructions are created by filling in the two-step functions and setting the input on the slider. The worksheet was produced by Professor Smudge (@ProfSmudge on twitter) and appears on the Maths Medicine website.
   
Do the lengths of the hypotenuses form a linear sequence?
Often students will claim that the lengths of the hypotenuses for the four triangles in the prompt form a linear sequence. This is certainly the case when the hypotenuses are given accurate to one decimal place. But is the sequence linear? Paul Foss, a maths teacher in the UK, has established that the sequence is not linear, although it tends to being linear. Below are links to a spreadsheet he devised to check the students' contention and a description of his lines of inquiry. This pathway of the inquiry teaches students about the importance of not rounding too early in mathematical exploration.
Spreadsheet
Lines of inquiry
   

Resources
Inquiry sheet (right-angled triangles for exploration)
Promethean flipchart    download
Smartboard notebook   download
  
T
he notebook and flipchart below were created by Colm Sweet (a UK maths teacher) for a structured inquiry he ran as part of a collaborative exploration of inquiry teaching with teachers of six other subjects. Colm restricted students' choice to nine regulatory cards.
Promethean flipchart (structured)    download

Teacher training
Meera Raghavan (a teacher trainer from Bengaluru, India) used the right-angled triangles prompt when she ran workshops with teachers in Aurangabad, Mumbai and Hassan. Previously, Meera had described Inquiry Maths as "what math teaching should be like!" She reports that the prompt was "a big hit with teachers" and, in particular, they were very interested in the way it "connects so many concepts." The picture (above) shows one of the teachers in the Aurangabad workshop jotting down questions about the prompt.

You can follow Meera on twitter @chikarleo.
  
Classroom inquiry 
These are the observations and questions of Helen Hindle’s year 10 (grade 9) class about the prompt:
The class decided to form groups that would follow different inquiry pathways. Each group recorded its findings on A3 paper (see examples below), before making a presentation to the class at the end of the second one-hour lesson. The class concluded that:
  • Using Pythagoras' Theorem, the lengths of the sides of the 100th triangle are 201, 301, and 361.94 (accurate to 2 decimal places).
  • Using the tangent ratio, the angles are not the same in the triangles; they are not similar triangles.
  • The areas of the triangles form a quadratic sequence.
  • The hypotenuses do not increase in a linear sequence as conjectured at first.
  • The bottom unknown angle is increasing, but at a decreasing rate.

Helen Hindle
is an Advanced Skills Teacher for maths and teaches at Longhill High School, Brighton and Hove (UK). She has an excellent website on developing a growth mindset in maths. You can follow Helen on twitter @HelenHindle1.





  
Evaluating conjectures  
Students' initial questions and statements about the prompt
The inquiry started with students in a year 10 class calculating the length of the hypotenuse and extrapolating from the results (accurate to one decimal place) a linear sequence. Is the sequence really linear? (See box above opposite.Two or three students had come across trigonometry in a former class and recalled the words 'opposite' and 'adjacent.' The teacher was then requested to explain how to work out the size of the angles, which she did using the 'top' angle in the first triangle as an example.
  
Students' conjectures
After some practice at using the tangent ratio, the class gave the sizes of the four 'top' angles, noted they were decreasing in size and discussed if they would always decrease if the lengths of the sides are terms in a linear sequence. Four conjectures arose in class discussion (see below). At this point, individuals, pairs or groups chose other sequences to explore.
   
Presentations of work in progress
Students came to the board at the end of the first 75 minutes and presented the following findings:
  • If you swap the lengths of the sides (that is, the adjacent becomes the opposite and vice versa), the top angle gets bigger. For the first four triangles in this case, the students gave 52.43o54.46o, 55.03o and 55.30as the top angles.
  • The top angle is the same (36.9o) if the ratio of the adjacent to the opposite is 4:3. The students claimed it 'worked' in the following cases:
 adjacent opposite
 4
 8
 12
 1612 
  • The top angle is always 45o if the adjacent and opposite sides are equal lengths - that is, the ratio of adjacent to opposite is 1:1.
  • The top angle gets bigger when the difference between the adjacent and opposite sides decreases (see below).
  • If the first triangle has the same dimensions as in the prompt, then adding 357 each time to the adjacent and opposite sides leads to a 'jump' in the top angle from the first to the second triangle before they become approximately the same (see below).
  
Preliminary conclusions about the conjectures
  

This is the sheet of a year 10 student who decided to work independently on the prompt over 50 minutes. He has developed the inquiry through the expressions for the nth terms of two sequences of his own.