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The prompt
The prompt
Mathematical inquiry processes: Verify; extend patterns; create and test examples; reason. Conceptual field of inquiry: Expansion of brackets; algebraic expressions.
The prompt was inspired by Jonathan Hall, a teacher of mathematics in Leeds (UK). When looking at four place mats (see picture), Jonathan observed that (2x + 4)2 = 4(x + 2)2.
The equation is intriguing because Jonathan has expressed the area of the four mats in two ways: firstly, (2x + 4)2 represents the area in terms of the length of one side of the large square (four mats combined); and, secondly, 4(x + 2)2 uses the area of each of the four small squares (or individual mats). In this way, students can relate the expressions to the diagram in the prompt and thereby explain the origins of the equation. Alternatively, the teacher might make the connection in order to encourage students to combine spatial and algebraic reasoning in the remainder of the inquiry.
In the initial phase of the inquiry, students might request instruction on expanding brackets (through their choice of a regulatory card) before going on to verify that the equation is true. Whether instruction is required or not, the teacher should address the common misconception in secondary school classrooms that (2x + 4)2 = 4x2 + 16.
Lines of inquiry
Lines of inquiry
Line of inquiry 1
Line of inquiry 1
Generalisation 1 (number of squares and stripes)
Are there any other equations of this type? In searching for other examples, students might start by analysing the structure of the equation. They notice that a 2 and a 4 are on both sides of the equation, although reversed in order. Could this observation lead to more examples? Testing of individual cases shows that reversing the integers does not always lead to an equality. For example,
(2x + 1)2 ≠ 1(x + 2)2 and (3x + 5)2 ≠ 5(x + 3)2
As they explore, students might also come up with partial conjectures about the relationship between the two integers, such as,
For the general form (ax + b)2 = b(x + a)2, b > a.
Such an observation could lead to the realisation that a2 = b as the coefficients of x2 must be equal. It follows that the equation in the prompt is the second of a series:
(1x + 1)2 = 1(x + 1)2
(2x + 4)2 = 4(x + 2)2
(3x + 9)2 = 9(x + 3)2
(4x + 16)2 = 16(x + 4)2 and so on.
Indeed, returning to the diagram, b must be a square number as b small squares form the large square. The teacher might reinforce the links between algebraic and spatial reasoning by drawing diagrams to illustrate the equations, with the diagrams for the second and third equations shown. (It is helpful to rearrange the squares in the prompt so the 'growth' of the patterns follows the same rule.)
The final step in this line of inquiry sees students expanding the brackets in the general form to prove that (ax + b)2 = b(x + a)2 is true for all (a,b) when a2 = b.
The second and third diagrams.
Line of inquiry 2a
Line of inquiry 2a
Generalisation 2 (number of stripes)
In the first pathway, the changes to the diagram occurred after students had noticed the reversal of the integers in the equation. This numerical (and more superficial) approach is far more likely to arise in the classroom than visualising the diagram as one of a series of diagrams. If, however, we consider changing just one property of the diagram at a time, we might start by reducing or increasing the number of stripes. In this series, the diagrams follow the 'rule' for the arrangement of the squares in the prompt. The second and third diagrams are shown.
The algebraic representations (including for the first and fourth diagrams) follow:
(2x + 2)2 = 4(x + 1)2
(2x + 4)2 = 4(x + 2)2
(2x + 6)2 = 4(x + 3)2
(2x + 8)2 = 4(x + 4)2 and so on,
leading to the general form (2x + 2n)2 = 4(x + n)2
where n is the pattern number.
The second and third diagrams.
Line of inquiry 2b
Line of inquiry 2b
Generalisation 3 (number of squares)
The second change to a property involves the number of squares. Using the original pattern on the place mat and assuming an infinite supply of mats, we could 'grow' the diagrams by creating a smaller and bigger squares. The second and third diagrams are shown (and once again the original diagram has been rearranged).
The algebraic representations (including for the first and fourth diagrams) follow:
(x + 2)2 = 1(x + 2)2
(2x + 4)2 = 4(x + 2)2
(3x + 6)2 = 9(x + 2)2
(4x + 8)2 = 16(x + 2)2 and so on,
leading to the general form (nx + 2n)2 = n2(x + 2)2
The second and third diagrams.
Line of inquiry 3
Line of inquiry 3
Considering dimensions and the exponent
If we see the original prompt as the face of a cube, then the following two expressions each gives the cube’s volume: (2x + 4)3, 8(x + 2)3. The cube would be made up of eight of the smaller cubes shown in the illustration.
Further inquiry might lead to a general form, such as (nx + n2)3 = n3(x + n)3, which is true for all values of n.
Line of inquiry 4
Line of inquiry 4
Changing the shape
Another pathway starts by changing the shape. When students consider a rectangle, for example, its dimensions could be expressed as 2y + 4 (length) and 2z + 4 (width). When the teacher sets a condition, such as the area of the rectangle and square are equal, the inquiry would develop into finding the relationship between the three variables in the following equation,
(2y + 4) (2z + 4) = (2x + 4)2
This pathway could be extended by changing the number of stripes in the diagrams, such as,
(2y + 2) (2z + 2) = (2x + 4)2 or (2y + 4) (2z + 4) = (2x + 2)2
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