4-by-3 rectangle inquiry
Mathematical inquiry processes: Interpret; define parameters; explore. Conceptual field of inquiry: Area and perimeter; formulae.
Students have posed some of the following questions in their first response to the prompt:
Are there other rectangles with an area of 12 square units?
What other shapes could have an area of 12 square units?
What is different and the same about the rectangles?
How many rectangles are possible with the same area?
Which rectangle has the longest perimeter? ... the shortest?
Is there a rectangle with an area equal to the length of its perimeter?
A discussion can ensue at this point about whether a 4 by 3 rectangle is the 'same' as one with dimensions of 3 by 4. Accepting that they are not, we might speculate that the number of rectangles with the same area is half the area (assuming the dimensions are whole numbers). So there are six rectangles with an area of 12 square units and four with an area of eight. However, if the area is a prime number, there will only be two. The inquiry might develop into a consideration of the factors of prime and composite numbers.
Perhaps, the teacher prefers to hold the length of the perimeter constant for the initial prompt (see below), when similar questions might arise:
What is different and the same about the rectangles?
How many rectangles are possible with the same perimeter?
Which has the greatest area? ... the smallest?
The conjectures that develop from the particular cases shown in the two prompts above might be combined. So, considering the 4 by 3 rectangle, there are six rectangles with an area of 12 square units and six with a perimeter of 14 units. Is this always the case with every rectangle? This pathway to the inquiry has the potential to reinforce the distinction between the concept of area and that of perimeter.
Matthew Bernstein, a teacher of a grade 5/6 class at the Fred Varley Public School (Markham, Ontario), reports on the inquiry his students carried out into the prompt:
Even having done only a little bit of initial work on area and perimeter, I felt my Grade 5s would do well with this inquiry as the Grade 4 curriculum in Ontario asks students to inquire into the formula for the area of a rectangle. There was lots of great thinking when I introduced the prompt. Students’ curiosity was aroused and they immediately wanted to know if there were other rectangles with the same area. Then they began to investigate other shapes with areas of 12 square units, including triangles. This eventually led one group to use pattern blocks to inquire independently into the formula for the area of a trapezoid. During the inquiry, which lasted over two days, the students had some great learning. This has made for an easy transition to inquiring into the formulas for parallelograms and trapezoids!
The pictures show the students' initial responses to the prompt and examples of the their inquiries and planning sheet.
Building resilience and developing creativity
Michelle Cole, Leader of Learning in the Mathematics at Ormiston Bushfield Academy, Peterborough (UK), gave the prompt to her year 7 class as an introduction to the concepts of perimeter and area. She reports that the students’ responses were “inspiring, amazing, and truly beyond any of my expectations.”
The students posed questions on a wide range of mathematical topics: perimeter and area; symmetry, angles, and other properties of the shapes; coordinates; volume; and enlargement by a scale factor of ½. Other questions could have led to novel inquiry pathways:
How many rectangles (or squares) can you see in each shape?
How many triangles from one point can you find?
What shapes can be made out of each shape?
What fraction of the grid do the shapes take up?
Michelle describes how she approached the inquiry:
I have been experimenting with prompts which in the past I have given a little more structure to but this was the first time I simply gave them the diagram on A3 paper and said “what questions could we ask?”. Students were a little reluctant to put things on paper but once they realised that they had free range to think about questions that we would then discuss they came up with some many and varied ideas. When we discussed their ideas we also talked about which questions we could answer (for example, What is the perimeter? What is the area?) compared to the questions we could not answer (for example, ' Where is the origin? Is there a reason why they are different colours?). We then centred the activity back on perimeter and area with the students investigating the perimeter when they put more than one of their desks together.
I use the ‘what is the same? what is different?’ prompt fairly regularly as a starter (you can see some students have used this as questions on their sheets). It is this that has helped build up their resilience and has got them thinking in a wider context than the most obvious.
Grade 6 pupils at the Luanda International School (Angola) began their inquiry into measurement by considering the rectangles prompt. As they attempted to make sense of the prompt, the pupils' questions connected their existing knowledge of relevant mathematical concepts to the prompt. The class went on to conduct personal inquiries, during which the generation of even more questions opened up new pathways for exploration. The quality and depth of this generative questioning attests to the sophistication of inquiry processes developed in the class.
Engaging younger learners in inquiry
Amelia O’Brien, a teacher at the United World College Thailand in Phuket, tried out the prompt with her grade 3 class. She reports on discovering that the Inquiry Maths approach is just as effective with younger learners as it is with secondary students:
Grade 3 tried out the prompt by first using Project Zero’s Visible Thinking Routine 'see, think, wonder' as a collaborative group, sharing and building on each other's ideas. At first, some students could 'see' a face (if another rectangular eye was added!) and others could 'see' a similarity to the Chinese character 'up'. "After being asked to think like a mathematician, students made immediate connections to arrays. It is worth noting that we are concurrently inquiring into multiplication and division.
After modelling expectations and discussing possible ways of unpacking and responding to the prompt, including sentence starters and suggestions of other ways we could represent the prompt using a multidimensional approach, we continued our thinking in small groups. Students identified patterns, experimented using symbols and numbers and were encouraged to ask questions. They then choose a question to explore that interested them. As the prompt and associated questions inherently differentiate themselves, most students chose an appropriately challenging question based on their own prior knowledge and understanding. We then planned how we would carry out our inquiries, selecting and connecting concepts that might help focus our thinking. Most students decided to experiment by modelling (using blocks, counters or grid paper) or research using maths dictionaries and discussion.
I had not used mathematical prompts with students this young before and was unsure of how it would play out. With some differentiated modelling, the prompt and subsequent inquiries proved to be as engaging and meaningful as using this approach with older students!
Questioning, wondering and speculating
The picture shows the questions and observations from a year 7 class. They initially focus on the area and perimeter of the rectangles in the prompt before students start to wonder whether it is possible to create other shapes with the same area. One student speculates about the number of rectangles with an area twice the size of those in the prompt. The other two pictures show individual students' contributions to the inquiry.
The pictures were posted on twitter by the students' teacher Aine Carroll.
Genesis of the prompt
Inquiries can begin with the simplest of prompts. Mark Greenaway (an Advanced Skills Teacher in the UK) contacted Inquiry Maths about developing prompts for students with lower prior attainment. He proposed using a 4 by 3 rectangle as a prompt. This could potentially lead to a very open inquiry encompassing a number of different directions. However, a prompt like the 4 by 3 rectangle is so familiar to students that it might fail to meet the first criterion for creating a prompt: "A prompt must promote curiosity and questioning in students of the sort 'that can't be right' or 'I've noticed ...'. Prompts should be engaging, and ripe for speculation or conjecture."
There is no guarantee that, beyond stating the obvious features of the rectangle, students will be able to isolate a key concept on which to build an inquiry. For example, a student might be able to identify the area as 12 square units, but not be able to extrapolate the concept of area as a foundation for inquiry. Such a step requires the use of well-developed inquiry skills, and particularly high levels of confidence and creativity.
If students lack those skills, then the teacher will need to define the inquiry further. Teacher intervention at this point reduces the possibility of students working on their own questions and statements, which is a key motivational aspect of inquiry. It is better for the prompt to 'suggest' the key concept so that students can generalise to other cases for themselves. When the rectangle is placed in the context of a series of rectangles sharing the same characteristic (for example, the area), then an initial inquiry can develop out of students' observations.
After the first phase, the teacher can highlight the constraint in the prompt, and invite students to change the prompt by holding another characteristic (for example, the perimeter) of the 4 by 3 rectangle constant. While suggesting changes to the prompt is empowering for students, it remains a highly developed skill. Not only do students have to learn how to be creative, they also have to learn how to make mathematically-valid suggestions.
The teacher can run a fully open inquiry by starting with the 4 by 3 rectangle only. The risk is that, with pressures to 'cover' a curriculum, the inquiry could go in one of many different directions. Alternatively, the teacher could guide the inquiry into those directions by offering prompts in which the 4 by 3 rectangle is an integral part. Indeed, a teacher who starts a series of inquiries with one key component thereby emphasises the inter-connected nature of mathematics.
For more on the differences between different types of inquiry and on the factors involved in deciding whether to choose an open or guided inquiry, see Levels of Inquiry Maths.
Prompt 1 Sequences
This prompt invites students to pose questions and make observations on sequences: How many sequences are there that contain the 4 by 3 rectangle? Is there another rectangle before the sequence in the prompt? Can you find an expression (in words or algebra) to describe the area and perimeter for shape n? What are the term-to-term or position-to-term rules for the sequences?
Prompt 2 Reflection symmetry
A second alternative prompt invites students to remove squares to create rectangle patterns with lines of symmetry. How many patterns can be created with one line or two lines of symmetry? What if you remove more than two squares? Why can you not make patterns with more than two lines of symmetry? What shape would you need if that was your aim?