# Percentages inquiry

# The prompt

Mathematical inquiry processes: Verify; test other cases; conjecture, generalise and prove. Conceptual field of inquiry: Percentages, including percentages greater than 100; percentage of a number.

On first inspection, the prompt seems rather trivial and easy to confirm. Yet, it has proved to hold a fascination for younger secondary school students who can verify its truth but are rarely certain that the relationship will hold for more complicated statements of the same type.

Explore and discuss

After a period in which students question, notice, and wonder in response to the prompt the inquiry often involves a fast-paced period of exploration. Students test different types of numbers, including decimal and three-digit numbers, and, as they do so, they become fluent in finding the percentage of a number.

The concept of percentages greater than 100 can lead to a rich class discussion. The use of negative numbers can also give rise to deep thinking. For example, what does (-40)% of 70 = 70% of (-40) mean?

The support sheet Percentage Match (in the Resources section) combines procedural practice in finding the percentage of a number with proportional reasoning. For example, why does 51% of 640 = 34% of 960?

Reason and prove

Once students have convinced themselves that reversing the numbers always works and counter-examples have been examined for calculation errors, the inquiry moves onto explaining and proving. Why does it always work? One approach is to use a chain of reasoning:

40% of 70 = 0.4 x 70 (equivalent decimal) = 70 x 0.4 (multiplication is commutative) = 0.7 x 40 (divide and multiply by 100) = 70% of 40

Another approach, which some students might adopt spontaneously and early in the inquiry, involves using algebra to generalise the equation.Â

After this phase, students are ready to follow new lines of inquiry by considering the other prompts below about percentage increases and decreases.

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Teachers of year 8 mixed attainment classes used the structured inquiry in Drayton Manor High School (Ealing, UK) in September 2024.

# Lines of inquiry

20% of 30% of 40 = 40% of 30% of 20

Students with high prior attainment might launch their inquiry with this prompt. There are six possible permutations involving 20, 30, and 40. Are they all equal? How can we prove any three numbers arranged in any permutation will give the same result?

70 increased by 40% is the same as 40 increased by 70%.

Although the statement is false if "same as" is taken to refer to the outcomes, the increases (28) are the same. It follows that the gaps between the two starting numbers (40 and 70) and the outcomes (68 and 98) are also the same.

The teacher's introduction of bar models and multipliers to carry out the percentage increase can enrich this line of inquiry (see the structured inquiry).

A proof that if the outcomes are equal, then the two starting numbers (a and b) must be the same is accessible to the youngest students in secondary school.

40 increased by 70% is the same as 70 decreased by 40%.Â

This second statement is also false, but it has in the past led to a search to find two numbers that would make such a statement correct. The relationship between the two numbers is shown below:

One example that arises regularly during this inquiry is: 25 increased by 50% is the same as 50 decreased by 25%. Once the general relationship is found, pairs of numbers can be generated. For example, 30 increased by 75% is the same as 75 decreased by 30%. Students can then plot values of a and b on a graph (or use graphing software) in order to explore the relationship further.

# Making connections

Devon Burger, a middle school math teacher at a charter school in Brooklyn, New York, used the percentages prompt to launch an inquiry with her grade 6 class. She described the experience as "an awesome inquiry-based lesson."

The picture shows the students' responses to the prompt.Â They range from different ways to verify its truth to thinking about the general case by asking 'Is it always true?' and 'If so, why?'

Devon reports on how the inquiry developed:

"This was the very first time we tried an inquiry lesson, and it was the students introduction to finding the percent of a number, so we mostly asked questions and explored the prompt in different ways.Â

"Some students worked simply on whether this case was true and then tried other whole numbers they felt comfortable with; other students worked on proving to me that this would be true in any case; others tried it with percents that included decimals, percents greater than 100%, percents that arenâ€™t multiples of 10, negative percents, etc.

"We came back together to talk about what we found. It was a lot of fun, and it was great practice for the students to look at something that appears really foreign at first but then be able to manipulate the numbers and words to create a statement that is familiar to them. Some students panicked when they first saw this, but after realizing that they could re-write it as 4/10Â x 70, they made all kinds of connections."

April 2022

# Questioning and discussion

These questions and observations come from Emma Rouse's year 9 mixed attainment class. Emma, a Lead Practitioner at Brittons Academy (Rainham, UK), explains how her students respond to inquiry lessons:Â

"This lesson was to introduce the new topic of percentages to my year 9 class.Â Last year I started to teach the students through inquiry and they love making up questions. The inquiry was full of conjecturing and learning and the students loved discussing other peoples' questions and comments."Â

On twitter, Emma declared that "If I could teach inquiry everyday I would." Below are examples of students' responses to the prompt and a display that Emma has created in her classroom.

# Making conjectures and reasoning

The conjectures and reasons above arose during a lesson involving a year 8 mixed attainment class. The teacher recorded the students' ideas and assertions and, in the next lesson, required the class to find examples and counter-examples and to explain if the conjectures were true or false. It was also an opportunity to decide if the class was dealing with a conjecture, generalisation, assertion or reason.

The first is a conjecture that is true in all cases. (The class decided that if 'in all cases' had been included in the statement, then it would have constituted a generalisation.)Â

Students found other examples and some went on to argue that it is always true by using algebra and presenting the percentage as a fraction.

The second is an assertion that turns out to be false. a% of a = b% of b can never be true when a â‰ b.

The third is an attempt to explain why the prompt is true by using an analogy involving the commutative law for multiplication of positive integers.

The fourth is a generalisation - that is, the two numbers must sum to 110. The 'in all cases' is inferred from the statement. One counter example (for example, 20% of 40 = 40% of 20) shows this to be false.

# Learning through inquiry

## Amazing progress

These pictures were posted on twitter by the Mathematics Department of Wellfield High School (Leyland, UK). They come from Miss Jacksonâ€™s Year 8 class.Â

The department reports that the inquiry led to â€śfantastic learningâ€ť and students â€śmade so much progress that we are planning to use one inquiry in each unit.â€ť One student asked, â€śCan we do inquiries every lesson?â€ť Overall, the department summarises the studentsâ€™ response to the prompt as â€śamazingâ€ť.

## Extending learning

After attending the Inquiry Maths workshop at the Mixed Attainment Maths conference in January 2018, Laura Katan used the percentages prompt with her girls' maths club. The girls were delighted with the results (see picture) of their attempts to generalise from the properties of the prompt.

Laura is a teacher on the Teach First programme at Park View School in Haringey (London, UK). Her head of department, Olly McGregor Hamann, reports that the department will be running its first inquiry with the whole of year 8 in February 2018.

# Adapting the prompt

The teacher should aim to set the prompt just above the understanding of the class to arouse curiosity and generate conjectures. The prompt should be accessible while not being too difficult.

In designing or choosing a prompt, therefore, the teacher must take into account students' prior learning to decide what concepts and procedures are 'just above' the level of the class.

One secondary mathematics department decided to use the percentages prompt with all their year 8 classes. The year group was split into two sides with classes being set from 1 (highest prior attainment) to 4 (lowest prior attainment) on both sides.

The prompt proved successful in generating inquiry only in the middle sets. The students in set 1 verified the prompt was true in seconds and asked, 'So what?' The inquiry died before it had started. The students in set 4 did not know where to start.

The Head of Maths contacted the Inquiry Maths website to ask for advice for the other half of the year group.Â

We suggested changing the prompt in the following ways to provide enough intrigue at each level:

### 10% of 50 = 50% of 10

### 40% of 70 = 70% of 40

### 47% of 74 = 74% of 47

20% of 30% of 40 = 40% of 30% of 20

The lower attaining students explored examples involving multiples of 10 guided heavily by the teacher; the higher sets regulated their activity with the regulatory cards and finished with students presenting their proofs for the conjecture that "the order does not matter."

# Questions and observations

Year 8 students in a mixed attainment class at Haverstock School (London, UK) started the inquiry by writing questions and observations on the prompt sheet. Their teacher, Nina Morris-Evans, was excited by the students' responses and the class discussion that followed. She structured the remainder of the inquiry by providing differentiated tasks to extend students' learning.

Some found more examples (including with decimals) and tested whether reversing the numbers always works. Others went on to verify or find counter-examples for other statements on the website, such as a increased by b% equals b decreased by a%. The sheets show the questions and observations from Aysha, Mordecai, Courtney and Rehan.

# Question, notice, and wonder

The pictures show the questions and observations from year 8 students at the start of the inquiry. Year 7 students in two mixed attainment classes contributed these questions and observations to their inquiries.

# Resources

Andy Gillen created this sheet with structured phases for inquiry. Andy is head of mathematics at The Hathershaw College, Oldham (UK).