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; reverse percentages.

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.

After this phase, students are ready to follow new lines of inquiry. They might inquire into the other prompts about percentage increases and decreases below.

Building confidence

Alfie Benson, a trainee teacher at the Ormiston Academies Trust (UK) and a Maths Scholar, used the percentages prompt with his year 8 class to boost confidence, strengthen mathematical thinking, and develop oracy.

In his blog post, Alfie explains that explicit instruction and independent practice, on the one hand, do not reflect the nature of mathematics and take away the joy of learning.

The Inquiry Maths lesson, on the other hand, was "rich in student discussion, thinking, and questioning which are not only skills needed for a good mathematician but also builds oracy and collaborative skills."

The students were impressed by their ability to create a proof for the prompt, and, as Alfie says, they left the lesson "very happy with themselves."

August 2025

Prompts for other lines of inquiry

(1) 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?

(2) 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 after the increases, the increase in each case (28) is 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 slides).

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.

(3) 40 increased by 70% is the same as 70 decreased by 40%.

This second statement is also false, but it has led students to search for two numbers that would make such a statement correct. One example that arises regularly during this inquiry is 25 increased by 50% is the same as 50 decreased by 25%.

The relationship between the two numbers is shown below:

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.

(4) 40 increased by 20% is the same as 60 decreased by 20%.

The prompt is true as both calculations give 48. In this line of inquiry the percentage increase and the percentage decrease are the same, but the starting number is different.

Students are required to understand reverse percentages when they make up their own examples. For example,

40 increased by 10% is 44; ....... decreased by 10% is 44.

44 represents 90% of the original number and so: 44 ÷ 90 x 100 = 48.89 (rounded to two decimal places).

Students can explore sets of examples that have a starting number in common:

40 increased by 10% is 44; 48.89 decreased by 10% is 44.

40 increased by 20% is 48; 60 decreased by 20% is 48.

40 increased by 30% is 52; 74.29 decreased by 30% is 52.

What do you notice about the sequence 48.89, 60, 74.29 , ...? Would it help to plot the sequence on a graph (with 'percentage' on the x-axis and 'original number' on the y-axis)?

The illustration shows Ayub's attempt to find an example when a percentage increase and a percentage decrease lead to the same outcome, although he has not used the same percentage in this case.

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. 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.

"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 aren’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 4/10 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:

"This lesson was to introduce the new topic of percentages to my year 9 class. 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."

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.

October 2017

Learning through inquiry

Differentiated lines of inquiry

Year 8 students in a mixed attainment class at Haverstock School (London, UK) started the inquiry by writing questions on the prompt sheet. Their teacher, Nina Morris-Evans, was excited by the quality of the students' responses and their contributions to the class discussion that followed.

Nina structured the remainder of the inquiry by setting differentiated lines of inquiry. Some students found more examples and tested whether reversing the numbers always works. Others went on to verify other statements, such as a increased by b% equals b decreased by a%, or find counter-examples. The pictures show the questions and observations from Aysha, Mordecai, and Courtney.

Extending learning

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

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

Amazing progress

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

The department reports that the inquiry led to “fantastic learning” and students “made so much progress that we are planning to use one inquiry in each unit.” One student asked, “Can we do inquiries every lesson?”

Overall, the department summarises the students’ response to the prompt as “amazing”.

Structured inquiry

Percentages structured inquiry

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

Percentages inquiry - year 7

The slides show the development of a structured inquiry with a year 7 mixed attainment class. The inquiry lasted two lessons and involved three lines of inquiry.

Making conjectures and reasoning

During the first lesson of the inquiry, year 8 students in a mixed attainment class made conjectures and gave reasons to explain what they noticed. Andrew Blair, their teacher, recorded the students' ideas.

In the next lesson, Andrew presented the four statements (pictured) to the class. He chose them because they represented different types of statements: a conjecture, a generalisation, an assertion, and a reason. The students attempted to match the statements with the correct type.

Students then evaluated whether the statements were true or false, looking for examples and counter-examples to support their reasoning.

During a class discussion, the students came to the following conclusions:

The first is a conjecture that is true in all cases. 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 ≠ b.
The third is a reason for 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.

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?' For them, the prompt was not intriguing and the inquiry died before it had started. The students in set 4 did not know where to start. They could not formulate questions from which to launch the inquiry.

The Head of Maths contacted the Inquiry Maths website to ask for advice before using the prompt with the other half of the year group.

We suggested redesigning the prompt to contain appropriate levels of support and challenge for each class:

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 prompt for students with lower prior attainment is more accessible because they were more likely to be familiar with calculating 50% and 10% of an amount.

Students with higher prior attainment were more likely to be intrigued because they had not previously considered calculating the percentage of a percentage of an amount.

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.

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