# Right-angled triangles inquiry

# The prompt

**Mathematical inquiry processes: **Generate more examples; conjecture; create and test particular cases. **Conceptual field of inquiry: **Pythagoras' Theorem; trigonometry; sequences; algebraic expressions for the *n*th term of a sequence; expanding double brackets; area.

The 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 n^{th}term of the sequence? Is there another set of triangles that has the same expression for the n^{th}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).

Additionally, finding an expression for the area can be accomplished by considering the general form of the triangles in the prompt. Students can derive expressions for the base (2*n* + 1) and height (3*n* + 1) in the prompt, which gives an area of ½(2*n *+ 1)(3*n *+ 1). If students also derive the expression 3*n*^{2} + 2.5*n* + 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.

# Creating examples to test conjectures

**Zane Latve**, an MYP and DP teacher of mathematics in Yokohama (Japan), posted this picture on twitter. Her grade 10 students focused on the size of the top angle in each triangle. They made conjectures about the conditions necessary for the size of the angle to increase. Zane reports that the students generated lots of ideas before creating examples to verify the truth of their conjectures.

The second conjecture in the picture shows the lengths of the opposite and adjacent sides increasing by three each time with the opposite always one less than the adjacent. As the lengths increase (and the ratio of opposite to adjacent gets closer to 1:1), the angle approaches the limit of 45 degrees.

April 2022

# Connecting concepts through inquiry

**Stephanie Asciak**, Head of Mathematics at St. Clare College (Malta), posted these pictures on **twitter**. They show the ideas of her year 9 students during an inquiry lesson. Stephanie reports: "I presented the students with the prompt with which they had to discuss in groups. I wanted them to use their mathematical knowledge to find anything they could come up with, such as the length of the hypotenuse, the area and perimeter of the triangles and so on. During the plenary the students discussed their work and we also discussed other topics that nobody came up with, such as sequences." Stephanie said that the prompt is her favourite on the Inquiry Maths site as it is "superb for making mathematical connections."

# Exploring angles

The questions and observations (see picture) come from a year 10 class at **Haverstock School** (Camden, UK). The students suggested a variety of pathways, which were to generate five one-hour lessons of inquiry under the teacher's direction. The first pathway involved the students in using Pythagoras' Theorem, which they had studied before, to find the lengths of the hypotenuse of each triangle.

The main inquiry pathway related to the angles. To determine if the angles stayed the same, the class needed the trigonometric ratios. The teacher introduced these one-by-one with the tangent ratio first. After calculating the size of the angles, students began to reason that the bottom right-hand angle in the prompt was increasing because the length of the opposite side was increasing at a faster rate than the adjacent.

The class then started to change the linear sequences to see what would happen to the angle. (Having calculated the length of the hypotenuse, the teacher introduced the sine and cosine ratios in subsequent lessons.) Students posed further questions as the inquiry developed:

Can we use descending sequences?

Does the angle always get bigger if the sequences ascend?

Does the angle always get smaller if the sequences descend?

Will we get different answers for the size of the angle if we use the sine, cosine or tangent?

What happens if I calculate the size of the top angle instead?

The teacher used the final question as the basis for a teaching phase on identifying the opposite and adjacent sides and the hypotenuse.

The pictures below show students' responses to the supplementary questions. They come from **Valbone, Agnesa, Nafisa **and **Catarina**.

### Sequences ascend, size of angle increases

### Sequences ascend, size of angle decreases

### Sequences ascend, size of angle stays the same

### One sequence descends, size of angle decreases

# Teacher training

These are the notes of **Michael Shkurka** and **Robert England** who are teachers of mathematics at the British International School of Bratislava. They were participating in a departmental training day about Inquiry Maths in February 2020. The training with Andrew Blair was organised by **Robert Euell** (Head of Mathematics and Computer Science). In the final session of the day, participants were asked to choose a prompt for one of their classes, map out possible inquiry pathways and select** ****regulatory cards** that students could use to direct the inquiry.

**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.

# Classroom inquiry

The picture shows the observations and questions of **Helen Hindle**’s year 10 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 100

^{th}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.

At the time of the inquiry,** Helen Hindle** was an Advanced Skills Teacher at Longhill High School, Brighton and Hove (UK).

# Engagement through inquiry

The pictures show the questions and approaches of year 9 students at Adeyfield School in Hemel Hempstead (UK). Their teacher, **David Robertson**, shared the pictures on **twitter**. He reported on the inquiry: “Year 9 had their first attempt at Inquiry Maths. It was a great lesson with loads of great questions from the students. I was blown away by their engagement and love of the concept. They started to develop their own rationale through some brilliant discussions on different topic areas. Inquiry is such an easy way to differentiate and show progress for all students.”

# Evaluating conjectures

## Students' initial questions and statements

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 52.43

^{o}, 54.46^{o}, 55.03^{o }and 55.30^{o }as the top angles.The top angle is the same (36.9

^{o}) if the ratio of the adjacent to the opposite is 4:3. The students claimed it worked in the following cases: 4:3, 8:6, 12:9 and 16:12.The top angle is always 45

^{o}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

# Resources

**GeoGebra worksheet**

**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**.

**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?**

**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 teacher of mathematics 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.

**Paul Foss**, a teacher of mathematics 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.