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The prompt
The prompt
Mathematical inquiry processes: Make connections; generate examples; find patterns and rules. Conceptual field of inquiry: Multiplicative relationships; multiplier; reciprocal.
The prompt is aimed at students in years 7 or 8 (grades 6 or 7) for whom ratio is a new concept.
The inquiry starts with a diagram. The arrows suggest there is a relationship between the numbers and, indeed, the horizontal pairs form equivalent ratios (3 : 4.5 = 5 : 7.5) and so do the vertical pairs (3 : 5 = 4.5 : 7.5). While 3 : 5 is the simplest form of the vertical ratio, the simplest form of the horizontal ratio is 2 : 3.
In the orientation phase of the inquiry, students often pose questions and make comments about the additive relationships in the prompt. These can generate valid questions for inquiry:
The difference between the two horizontal amounts you add (addends) is one, which is the same as the difference between the two vertical addends (horizontally: +1.5 and +2.5; vertically: +2 and +3).
Is it possible to design another diagram that has the same differences between the horizontal and vertical addends?
The differences between the addends create a sequence: 1.5, 2, 2.5, 3.
Is it possible to create another diagram in which the differences between the addends form a linear sequence?
However, the mathematical focus of the inquiry should be on the multiplicative relationships in the diagram if students are to understand the concept of ratio. It is at this point that the teacher might need to guide the inquiry by stipulating that there can only be one horizontal rule and one vertical rule.
The diagrams below show additive relationships (left) and multiplicative relationships (right).
The inquiry now focuses on multipliers. The teacher might explain how to deduce the multiplier by dividing the second number (of two linked by an arrow) by the first number.
The multiplier could be a fraction (before being simplified) in which the numerator is the second number and the denominator is the first number.
If fractional multipliers are likely to confuse the class, the teacher could start with a prompt that has whole number multipliers - without forgetting that prompts should be set above the class's current level of knowledge to encourage curiosity and speculation.
First published: October 2014
Revised: August 2025
Lines of inquiry
Lines of inquiry
The lines of inquiry that follow here have all been co-constructed by teachers and students during classroom inquiries. Resources for each line can be found in the Structured inquiry slides.
- Find missing numbers or terms
Students deduce the multipliers in the diagrams to find the missing integers, fractions, or algebraic terms.
2. Generate examples
2. Generate examples
Students use the templates to create their own examples. The teacher might encourage students to define parameters for their inquiry. For example, the top row of the diagrams could always be two and three.
3. Add a diagonal arrow
3. Add a diagonal arrow
Joining the starting number to the finishing number with an arrow invites students to find the connection between the three multipliers.
Using the diagram in the bottom left as an example, the horizontal multiplier is four and the vertical multiplier is two. The diagonal multiplier is eight - that is, 4 x 2.
4. Reverse the arrows
4. Reverse the arrows
By reversing the arrows, students consider inverse multipliers and begin to develop an understanding of reciprocals.
5. How many ways?
5. How many ways?
How many different diagrams can you create with the same start and finish numbers (not including reflections)?
Students can define parameters for their search: What if the missing numbers and the multipliers have to be integers? What if the multiplier can be a non-integer?
Questioning, noticing, and wondering
Questioning, noticing, and wondering
Generating an example
Generating an example
These are the questions and observations of Helen Hindle's year 8 mixed attainment class. Helen has adapted the prompt to ensure it is accessible while retaining its intrigue. The students have identified the multiplicative relationships in the prompt.
However, one pair of students has used additive relationships to generate an example of the same type (see the diagram in green). The difference between the numbers in the top row and the difference between the numbers in the bottom row are the same as in the prompt - one and four respectively.
While the horizontal multiplier changes from 1.5 in the prompt to 1.2 in the new example, the vertical multiplier stays the same.
The new example can be developed into lines of inquiry:
Can you generate other examples of the same type? Three and four in the top row, for example, give 12 and 16 in the bottom. Why will the vertical multiplier always be four?
Is it possible for the difference between the numbers in the bottom row to be, for example, three if the difference in the top row remains one? Why would 2, 3, 6, 9 work but not 2, 3, 5, 8?
What if the difference between the numbers in the top row is not one?
(At the time of the inquiry, Helen was Lead Practitioner for mathematics at Longhill School, Brighton, UK.)
Asking 'What if ... ?'
Asking 'What if ... ?'
The picture shows the questions and comments of Caitriona Martin's year 10 class. The students have noticed additive relationships and use fractions and decimals to explore multiplicative relationships. They have deduced the horizontal and vertical multipliers.
By asking 'What if ...?', Caitriona has encouraged the class to develop lines of inquiry, such as generating examples with other numbers and reversing the arrows.
Caitriona reported that the students' personalities as mathematicians came out through the inquiry. Moreover, some students who are quick to finish classroom exercises found themselves out of their comfort zone when required to pose questions and structure their own learning.
(At the time of the inquiry, Caitriona was Second-in-Charge of the mathematics department at St Andrew's School, Leatherhead, UK.)
Structured or guided inquiry?
Structured or guided inquiry?
Andrew Blair used the prompt with two of his classes. Even though the classes followed similar lines of inquiry and reached similar mathematical outcomes, the balance between teacher structure and student independence was different. Andrew explains the approach he took to each inquiry with reference to the Levels of Inquiry Maths.
Structured inquiry
Structured inquiry
The ratio inquiry was the first inquiry with my new year 9 class. I was not sure how the students would respond to the prompt or whether they were used to taking the initiative in lessons.
Their responses to the prompt (pictured) are revealing:
The question about what the arrows represent shows a desire to construct meaning.
The idea that the arrows could be reversed and the addition of a diagonal arrow show a high degree of confidence in transforming mathematical situations.
The predominant use of addition shows a lack of awareness of multiplicative relationships, although one pair of students suggested multiplying the numbers in the top row by 1.5 and another suggested a way of getting from 3 to 7.5 with two calculations.
One pair of students used fractions and ratios to describe the connections.
At this point, I decided to structure the next phase of the inquiry by discussing why the vertical multiplier is 12/3. After one student's comment that the multiplier could also be expressed as 5/3 was met with some confusion, we had a briefly practised the conversion of mixed numbers to improper fractions.
The remainder of the first lesson involved students in creating their own examples. They moved from whole number multipliers to mixed numbers or improper fractions as their confidence grew.
I structured the second lesson to follow two lines of inquiry suggested by the students' initial responses:
(1) We explored the relationship between the vertical and horizontal multipliers and the one along the diagonal; and
(2) We reversed the arrows, deduced the new multipliers, and compared them to the ones going in the original direction.
We finished the lesson with some presentations that included examples and explanations.
Overall, I structured the inquiry. I had the regulatory cards ready, but judged that the students would have found operating at a metacognitive level too demanding in their first inquiry.
However, the students showed they have the potential to become resilient and independent inquirers as the year goes on.
Guided inquiry
Guided inquiry
I adapted the prompt for a year 8 mixed attainment class to make it more accessible, yet still challenging at different levels. The questions and observations (pictured) show some students attempting to understand the meaning of the prompt, while others are questioning its structure.
As the students had participated in inquiries throughout year 7, they were used to taking more responsibility for directing the course of the lesson.
After the initial phase, therefore, they decided how to proceed by using regulatory cards. Half the class selected the card Use a worksheet (the worksheet required them to find a missing number once they had deduced the multipliers), while many of the rest chose to Make up more examples.
A few students selected the card Change the prompt, but I decided to delay this until the class had become more familiar with the structure of the original prompt. One girl created her own card - Explore the prompt with another student - and she created examples collaboratively with a partner.
Changing the prompt became the focus of the second lesson of the inquiry. One group introduced the diagonal line from the first to last numbers and went on to explain that, in the case of the prompt, that multiplying by 1.5 and then by four was equivalent to multiplying by six.
A group of students created pairs of diagrams with the same numbers, but with the arrows going "forwards and backwards" (see the picture below). They attempted to explain the relationship between the multipliers in the forwards diagram and their "reverse equivalents" in the backwards diagram.
This led on to a presentation at the end of the inquiry on reciprocals in which students explained why multiplying by 4 in the first diagram was "equivalent" to multiplying by 1/4 in the second.
At the end, I introduced the formal terminology by explaining that the reciprocal represented a multiplicative inverse.
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