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 (including for percentages greater than 100). After a period in which students ask questions and make observations about the prompt, the inquiry often involves a fast-paced period of exploration with students testing different types of numbers. Once students have explained why the prompt is true, they can extend their reasoning by comparing the original prompt to other statements.
Statement 1: 70 increased by 40% is the same as 40 increased by 70%.
Although the additional statement is false if "same as" is taken to refer to the outcomes, the increases (28 in the case of the statement) 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. A proof (above) that if the outcomes are equal, then the two starting numbers (a and b) must be the same is accessible to students as early as year 7.
Statement 2: 40 increased by 70% is the same as 70 decreased by 40%.
This second additional statement is also false, but it has in the past led to an inquiry to find two numbers that would make such a statement correct. The relationship between the two numbers is shown on the right. 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.
Statement 3: 51% of 640 = 34% of 960
This additional statement invites students to use proportional reasoning. What does the new statement have in common with the prompt that mean they are both true? The support sheet (below in 'Resources') invites students to find more pairs of the same type.
A proof of the general case
The inquiry has developed into finding fractions of amounts (and fractions of fractions) when students change the prompt. Why does 5/8 of 2/3 = 2/3 of 5/8? The prompt has, furthermore, led to students trying to express their observations algebraically.
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."
Questions 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." 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.)
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 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
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”.
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 of their attempts to generalise from the properties of the prompt.
Adapting the prompt
An important feature about a prompt is that it is set just above the understanding of the class to arouse curiosity and generate conjectures without intimidating students by being too 'difficult'. The prompt, then, must take into account the prior learning of the class. One secondary mathematics department decided to use the percentages prompt with all their setted year 8 classes. Teachers were concerned that the prompt would be too easy for the higher attaining classes, but too difficult for students with the lowest prior attainment. With these concerns in mind, the prompt was changed 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.