# Substitution inquiry

# The prompt

Mathematical inquiry processes: Generate examples and counter-examples; conjecture and reason; change conditions. Conceptual field of inquiry: Substitution of different types of numbers into expressions.

The prompt was devised by Mark Greenaway (an Advanced Skills Teacher in Suffolk, UK). In his original version, the prompt included values for a and b (2 and 3 respectively). In the prompt’s more open form, students might suggest their own values to substitute into the inequality in order to develop conjectures about the relationship between a and b.

In classroom inquiry, students’ questions and observations about the prompt have included:

What are they trying to find out?

What values of a and b satisfy the inequality?

What does the sign mean?

a could be 2 and b could be 4.

This can only work when a ≥ 2 and b ≥ 3 with b > a.

What if a and b are negative numbers? Would the inequalities be reversed?

What if a and b are fractions?

Is it possible to change the order of the terms and expressions and find the same values of a and b that satisfy all permutations?

What if we changed the operations?

Could we use square roots?

This one must work all the time: a/b < a - b < a + b < ab.

Is it possible for a2, a + b, ab, b2 to be equal?

Inquiries have often led students into changing the order of the expressions in the inequality. Other lines of inquiry include substituting decimal and negative numbers into the inequality, and using more complex expressions. (See the PowerPoint for more lines of inquiry.)

# Lines of inquiry

## Conjectures

Students have made many conjectures about the relationship between a and b. Here are five that have arisen in classroom inquiry:

If a and b are consecutive whole numbers, then the inequality will never work. By making a = n and b = n + 1, a year 9 student showed how the four parts of the inequality become n2 < 2n + 1 < n2 + n < n2 + 2n + 1.

The values of a and b must be in the same times table to make this correct. A counter-example (for example, a = 3 and b = 7) shows this conjecture to be false.

As long as a is smaller than b, the prompt will work. This can also be shown to be false with a counter-example, such as a = 3, b = 5.

If a = 1, then the inequality will never work. This is true because when a = 1, ab < a + b.

The inequality will not work if both numbers are negative. This is also true because if a and b are negative, a2 > a + b.

## Line of inquiry

A group of year 9 students developed a line of inquiry by deciding to find the lowest value of b when a is a given positive integer. The results from a = 2 to a = 7 led them to a generalisation for any value of a.

The students explained their results by focusing on a2 and a + b. In order for a + b > a2, b = a2 - a + 1.

# New resource for a structured inquiry

During an Inquiry Maths workshop at St Leonards School (St Andrews, Scotland) in August 2024, three teachers - Hilary Ballantine, Tom Gough, and Sean Pennycook - collaboratively planned the substitution inquiry for their year 9 MYP classes.

As the classes had not experienced inquiry prompts before, the teachers decided to run a structured inquiry.

Tom designed a completion table as the first step in the inquiry so that students could connect their prior knowledge of inequalities to substitution.

After choosing values for a and b, students work out the values of the expressions in each case, and then decide what signs (<, >, or =) go in the boxes.

# Generating excitement and commitment

The picture shows the questions and observations from Shawki Dayekh's year 9 class at Haverstock School in Camden (London, UK). The class had low prior attainment in mathematics.

Shawki reflects on how the inquiry developed:

"I didn’t expect the level of some of the questions from the class. Students were so excited to prove or disprove each other’s comments, observations and conjectures. One student, who was disengaged at the beginning of the inquiry, asked 'How long will it take to work out?' Well, I said it can take a lifetime if you want! Then the students dived into the problem.

"In the second lesson they went straight to making generalisations from their tables of values. One group wrote, 'If a and b are both positive and b > a, then the inequality will always work.' Another group, referring to their table of results, said, 'The inequality fails if a and b are both negative.' I have to say that I didn’t expect that level of commitment and mathematical language from the class. It was one of the best lessons I have had with them!"

Extending the inquiry

Shawki was so enthusiastic about the depth of learning during the inquiry that he decided to extend it into the third lesson. Describing the process as "zooming in on a mathematical property", Shawki gave the class the following conjectures to test:

(1) If a < b, then a2 < b2

(2) If a < b, then a - b < 0

(3) If a and b are integers, then a + b < ab

(4) If 0 < a < 1 and b > 1, then ab > b

Shawki reports that all the students tested and wrote about one conjecture and many tackled two or three.

Sofian, one of the students, even suggested revising the first conjecture to see if it was true for a3 and b3.

Tasnim, another student, tested the first three conjectures and her results are pictured below. She shows that conjecture (1) is not always true when a is negative; conjecture (2) is always true; and conjecture (3) is true when both a and b are negative or positive, but not if one is negative and the other positive.

# Question-driven inquiry

These are the questions of a year 9 class (a middle set) in a UK comprehensive school. The students are curious to see what happens when they change the prompt.

In order to help students decide on their own line of inquiry, the teacher produced this list based on the students' questions for the second lesson. In the pictures below, you can see students' inquiries into some of the questions.

# Adapting the prompt

The new prompt was devised for a year 9 class with high prior attainment. The class had carried out mathematical inquiries before and were beginning to inquire independently.

Students came up with a wide range of questions and observations that the teacher or, in the case of the example with a = 2 and b = 5, a student wrote on the board (see the picture). Another student explained that as b2/a is greater than a2/b, b must be greater than a.

Regulatory cards

The class was then given the opportunity to select a regulatory card to decide how the inquiry should proceed. Most students chose to Find more examples by which they meant to substitute values for a and b into the inequality. Their aim was to determine the conditions under which the inequality is true.

One table of four students wanted to Use a worksheet - a regulatory card they had selected on previous occasions - even though it was not included in the set. The teacher gave those students the structured exploration sheet (see the PowerPoint in Resources below).

Proof

As they explored, students began to appreciate that it was difficult to find values for a and b that satisfy the central inequality. One pair of girls proved that the middle two expressions should be reversed by creating two equivalent fractions with the denominator ab and using the inequality b > a to show that ab + b2 > a2 + b2.

Another student proved the prompt was false by showing the middle inequality contradicted the condition b > a. He presented his proof (see picture) to the class at the end of the lesson. If the middle inequality is correct, he argued, then b < a, which we know to be false.

# Introducing substitution through inquiry

Rachel Mahoney, a mathematics teacher at Carre's Grammar School in Sleaford (Lincolnshire, UK), posted this picture online. It shows the questions and observations from Rachel's year 7 class.

The students who were carrying out only their second inquiry have already begun to suggest changes to the prompt, such as changing the inequality to greater than, changing the indices and extending the 'sequence'.

Rachel reports that the prompt is "a great way to introduce substitution."

# Students' rich questions

Amanda Kirby a teacher of mathematics and the Numeracy Across the Curriculum Coordinator at St Clement Danes School, Hertfordshire (UK), used the substitution prompt for an observation lesson with her year 10 foundation GCSE class.

She reports that the students were "fabulous at asking questions and questioning each other" (see their initial comments and questions in the picture). The class quickly realised that the prompt is never true if a and b are negative numbers because, in that case, a2 > a + b.

As the inquiry progressed, students created their own inequality: a - b2 < a2 - b. They wondered if a has to be greater than b for the inequality to be true.

Overall, Amanda was pleased with the way the class was 'buzzing' during the inquiry and with students' willingness to challenge each other's reasoning.

# "Oh yes, we're doing inquiry"

Aine Carroll posted the picture online. It shows her year 9 students' questions and observations in response to the prompt.

The students wonder about the values of a and b that satisfy the inequality and consider if there is one pair of values or more. They have started to speculate about the relationship between a and b:

Does a have to be less than b? Will it work if a > b? Can a = b?

They also consider using different types of numbers, and wonder if it is possible to satisfy the inequality with non-integers and negative numbers.

One student's suggestion to swap the order of the expressions or change the direction of the inequality signs has the potential to open new lines of inquiry.

Lines of inquiry

The next two pictures show examples of the students' lines of inquiry. One student has reached conclusions about the possibility of using different types of numbers, while another has calculated the lowest value of b for each value of a and then devised the formula:

Lowest value of b for a given a = n(n - 1) + 1 (where n = a).

After the success of inquiry lessons Aine reports that, "Inquiry Maths is becoming a core part of my year 9 lessons." When she wrote an inquiry prompt on the board, one student remarked, "Oh yes, we're doing this today."

# Extending the inquiry

Permutations

The extension to the inquiry developed out of a student's question about the original prompt. She asked, "Are there values of a and b to make all the different orders of the inequality?"

As there are 24 permutations of the four parts of the inequality, it might be better to start with three expressions. In that case, there are six permutations. In the example (illustration above), we have:

b - a < b/a < ab b - a < ab < b/a

b/a < b - a < ab ab < b - a < b/a

b/a < ab < b - a ab < b/a < b - a

In this PowerPoint designed for online inquiry there are four challenges. Students can create their own by using more than two variables, by making up their own expressions or by using more than three expressions.

Permutations of the prompt

The final suggestion might lead students to consider the inequality in the original prompt and attempt to find values of a and b that satisfy each of the 24 permutations.

It is possible to find values of a and b that satisfy the permutations shaded in green below (1 is the lowest value and 4 the highest).

There are no values of a and b that satisfy the permutations in red. That is because ab can never be greater than both a2 and b2 as would be required in those permutations.

February 2021

# An alternative prompt

In Mathematics Teaching 223 (July 2011), Geoff Tennant describes using this prompt with a class on his subject knowledge enhancement course (pre-teacher training) at Reading University.

In order to keep the arithmetic simple, the students decided to substitute perfect squares into the inequality. This led to the following results in the table:

Geoff was taken aback. A quick check of a = 1 and b = 9 led to the conjecture: If a and b are consecutive perfect squares, then ½(a + b) - √(ab) = ½. The class went on to prove the result, using a = n2 and b = (n + 1)2.