Mathematical inquiry processes: Make connections; generate examples; find patterns and rules. Conceptual field of inquiry: Multiplicative relationships; multiplier; reciprocal.
This inquiry has tended to be shorter than others on the website, perhaps lasting two hours of classroom time. Moreover, it has also required more guidance (see Levels of Inquiry Maths). Often comments about the prompt from younger students relate to the difference between the terms. These can generate valid questions for inquiry:
The difference between the two amounts you add horizontally is one, the same as for the vertical amounts.
Is it possible to find other numbers that have the same horizontal and vertical differences?
The differences create a sequence: 1.5, 2, 2.5, 3.
Is it possible to create a linear sequence from the differences of other terms?
What about other types of sequences?
However, the mathematical focus of the inquiry should be on the multiplicative relations if students are to understand the concept of ratio. It is at this point that the teacher might need to guide the inquiry with pertinent questions or by stipulating that there can only be one horizontal 'rule' and one vertical 'rule'. 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). At this point in the inquiry, students will often decide to create their own examples (on template 1 in the 'Resources' section).
Three interesting changes to the prompt that have been suggested by students are presented below:
(1) Add a diagonal line
Compare the horizontal, vertical and diagonal multipliers. This can lead on to the multiplication of a fraction by a whole number or of two fractions. (See template 2 in the 'Resources' section.)
(2) Use algebra
As students have moved onto using algebra, they have invariably asked (through the cards) for a teacher's explanation and for the teacher to set the questions.
(3) Reverse the arrows
In finding the inverse multipliers, students can begin to develop an understanding of reciprocals.
Change 1: add a diagonal line; Change 2: use algebra; Change 3: Reverse the arrows.
Adapting the prompt
The prompt was adapted for a year 8 mixed attainment class to make it more accessible, yet still challenging at different levels. The questions and observations (below) show some students attempting to understand the meaning of the prompt, while others are questioning its structure.
After the initial phase, half of the class used the regulatory cards to choose a worksheet or make up more examples. Other students wanted to change the prompt, but the teacher decided to delay this until the class had become familiar with the structure of the original prompt. A few students created their own cards, with one writing that she wanted to "explore the prompt with another student."
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, 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 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, the teacher introduced the formal terminology by explaining that the reciprocal represented a multiplicative inverse.
Questioning and noticing
These are the questions and comments of Caitriona Martin's year 10 class on the ratio prompt. Notice the 'what if ...' section that encourages students to change the features of the prompt. Caitriona observed that students' personalities as mathematicians come out during inquiries. Some who are quick to finish classroom exercises find they are 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 maths department at St. Andrew's School, Leatherhead (UK).
These are the questions and observations of a year 8 mixed attainment class. The students' own example can be compared to the prompt to emphasise multiplicative relationships above additive ones. The horizontal multiplier changes from 1.5 to 1.2, even though the differences remain one (top line) and four (bottom line). The picture was posted on twitter by Helen Hindle.
The first inquiry
Andrew Blair reports on the first inquiry with a new class:
The first inquiry with a new class can be daunting. How will the students react? How open can the teacher be? I decided to plunge straight in with my year 9 class. So in the first lesson, I asked for questions and observations about the ratio prompt. In case the class found this too challenging, I had also planned for three structured continuations. The students’ responses 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 line show a high degree of confidence in transforming mathematical situations.
The predominant use of addition shows a lack of awareness of multiplicative relations, although one pair of students suggested multiplying 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 (top left), but could not explain why the two sides of their equations are equal.
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 brief period of practising 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. In the second lesson, we explored the relationship between the vertical and horizontal multipliers and the one along the diagonal. Other students reversed the arrows and inquired into the new multipliers (and divisors). We then had some presentations of work in progress.
In the third and final lesson, I suggested that the students try to find multipliers in the context of the area of shapes with a pre-prepared worksheet. A few wanted to continue their explorations from the lesson before. The inquiry ended with the pair of students who had suggested the use of fractions and ratios in the initial phase explaining their ideas to the class and answering questions.
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 the first inquiry of the year. However, the students showed they have the potential to become resilient and independent inquirers as the year goes on.
Direct and inverse proportion
The prompt could be used with students after the ratio prompt. The pairs of numbers in the left-hand diagram are directly proportional. Considering the horizontal pairs (and where the first term is a and the second is b), 2.4a = b and the multiplier can be expressed as 12/5. The relationship for the vertical pairs is 0.8a = b and the multiplier is 4/5. In the right-hand diagram of the prompt, the situation is different. The horizontal pairs of numbers are inversely proportional; the relationship is 60/a = b. The vertical pairs, however, are not proportional. Would that always be the case if the horizontal pairs are inversely proportional?
Properties of ratio and proportion
If a : b = c : d, then
b : a = d : c invertendo
a : c = b : d alternendo
(a + b) : b = (c + d) : d componendo
(a – b) : b = (c – d) : d dividendo
(a + b) : (a – b) = (c + d) : (c – d) componendo and dividendo
a : (a – b) = c : (c – d) convertendo
a : b = (a + c) : (b + d) addendo
a : b = (a – c) : (b – d) subtrahendo