4-by-3 rectangle inquiry

The prompt

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.

Student-driven inquiry

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 picture (top left) shows the students' initial responses to the prompt. The other pictures show a planning sheet and examples of the students' inquiries.

November 2019

Building resilience, developing creativity

Michelle Cole, Leader of Learning 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.”

Students' questions

The class posed questions on a wide range of topics: perimeter and area; symmetry, angles, and other properties of the shapes; coordinates; volume; and enlargement by a scale factor of a half.

Other questions suggested novel lines of inquiry:

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?

Students recorded their questions and ideas on A3 paper (see pictures below.)

Michelle describes how she approached the inquiry:

I have been experimenting with prompts, which in the past I have structured a little more. This was the first time I simply gave them the diagram 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 many and varied ideas.

When we discussed their ideas we also talked about which questions we could answer (What is the perimeter? What is the area) compared to those we could not (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 by putting 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.

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!

December 2019

Question-driven inquiry

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 the pupils' questions show the power of the generative process to drive inquiry.

December 2016

Questioning, wondering and speculating

The picture shows the questions and observations from Aine Carroll's‏ 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 (below) show individuals' contributions to the inquiry.

October 2017

Notice, think, wonder

Amanda Klahn, a PYP and inquiry-based maths teacher, used the 4-by-3 prompt with her grade 4 class at the Western Academy of Beijing. She divided the pupils' responses into three categories. The 'I wonder ...' statements could initiate novel lines of inquiry:

How many rectangles (and other shapes) fit into the grid.
If the perimeters are all the same.
If there are any other rectangles with an area of 12 squares.
If there are rectangles with the same perimeter but different areas.
If there are perimeters where there is only one possible area.
If there are areas where there is only one possible perimeter.

A new line of inquiry

Samia Henaine posted the diagram on twitter with the caption, "Eliminate three unit squares from a 4-by-3 rectangle and observe what happens."

The prompt gives rise to new lines of inquiry with the 4-by-3 rectangle:

What happens to the area of the rectangle?
What eliminations give the longest and shortest perimeters?
How many different perimeters are possible when eliminating three squares?
What happens if you eliminate more or less squares?
Is it possible to create the same perimeter by eliminating two, three or four squares?
What happens if you eliminate a square in the middle of the rectangle?

What do you notice if we start with a bigger or smaller rectangle?
What eliminations always give the longest and shortest perimeters?

...................

Samia is a math teacher, and PYP and ICT Coordinator at Houssam Eddine Hariri High School in Saida (Lebanon). For more inquiry prompts, see her website Math Teachers as Bridge Builders.

May 2022

Genesis of the prompt

Mark Greenaway, an Advanced Skills Teacher in the UK, contacted Inquiry Maths to suggest using a 4-by-3 rectangle as a prompt. He was developing prompts for students with lower prior attainment.

There are two risks to using the rectangle on its own as an inquiry prompt. Paradoxically, it might elicit too many responses from students or too few.

Too many

While students might link the rectangle to concepts of area and perimeter, they might also associate it with, amongst others, angles, properties of shapes, symmetry, and transformations. The ensuing inquiry, if left open, could encompass many different lines of inquiry.

A teacher under pressure to 'cover' the curriculum would have to limit the inquiry to the target concept. However, teacher intervention reduces the opportunity for students to work on their own questions, which is key to increasing levels of motivation during inquiry.

It is better, therefore, for the prompt to 'suggest' the specific concept that appears in the curriculum. Students' initial responses would then generate a narrower inquiry but one in which they were addressing their own questions.

Too few

The converse risk of using the rectangle as a prompt, and the one more likely to arise in the classroom, is that students respond with too few questions. A rectangle is such a familiar object that it might fail to spark curiosity (see Creating a prompt).

Beyond noticing the rectangle's obvious features, the class might struggle to wonder, conjecture, or make general statements. For example, the observation that the area is 12 square units is unlikely to lead students to wonder about other rectangles with the same area. Such a step requires the use of well-developed inquiry skills and high levels of independence and creativity.

Three rectangles

For these reasons, the Inquiry Maths prompt contains three rectangles with the same area. A comparison draws attention to what they have in common. This simultaneously reduces the number of concepts that students link to the prompt and increases the questions they ask about one concept.

Now even a seemingly obvious statement can lead to speculation about other cases. For example, when a student says that all the shapes are rectangles, it is a short step to wondering whether there are more with the same area.

Guided inquiry

If the aim of the inquiry is to draw out the connections between mathematical concepts, then a guided inquiry might be appropriate. The teacher guides the inquiry into other lines by offering prompts in which the 4-by-3 rectangle is an integral part.

The two additional prompts (right) link the 4-by-3 rectangle to sequences and reflection symmetry.

Prompt 1: Sequences

The prompt invites students to pose questions and make observations about sequences of rectangles: 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 rectangle n? What are the term-to-term or position-to-term rules for the sequences? How many sequences are there that contain the 4-by-3 rectangle?

Prompt 2: 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?

Resources

Prompt sheet

PowerPoint

Page updated

Google Sites

Report abuse