Embedded Files
The prompt
The prompt
Mathematical inquiry processes: Generate examples; analyse structure; determine necessary conditions. Conceptual field of inquiry: Expansion of brackets; factorisation of quadratic expressions.
The prompt, in presenting two equations, requires students to consider whether it is always possible to go both ways.
If they start with two binomial expressions in the form shown in the prompt, then it is possible to expand them. The result will be a quadratic expression. However, if students start with the quadratic expression, they realise that it is not always possible to factorise.
During the initial question, notice, and wonder phase of the inquiry, the teacher can assess the students' existing knowledge and what they can infer from the prompt. It might be appropriate to orchestrate a class discussion, particularly if the students request an explanation through the regulatory cards.
The discussion would focus a particular case to draw out the connections between the brackets and the expression. For example, multiplying (x + 2)(x + 3) with the grid method shows that, in the general expression ax2 + bx + c, b is the sum of 2 and 3 and c is the product of 2 and 3.
As students explore the second part of the prompt when a = 1 they begin to speculate that the quadratic expression can be factorised if there are two numbers whose sum equals b and whose product equals c.
Once the class has confirmed this with different examples, the students can move on to the case of a = 2, 3, or any other prime number (p). They return to the top part of the prompt, analyse the structure with grid multiplication, and then aim to generalise for those expressions that can be factorised. The general factorised form will be (px + m)(x + n).
The general case
The general case
The teacher can co-construct or call on students to explain the expansion of the two binomial expressions of the general form (in which a, b, c, d > 1).
The reverse process is more difficult. In the previous cases, the coefficient of x2 - that is, ab - was one or a prime number. That meant either both a and b were one or they were the prime number and one respectively. Now the coefficient of x2 is a composite number and the factorisation requires us to find all four variables.
From the quadratic expression, we know that the coefficient of x2 and the constant are ab and cd respectively. To factorise an expression we would list the possible values of a, b, c, and d and then determine which of the values give the correct value of bc + ad. (See two examples of the process.)
Once students have developed an understanding of mathematical structure, the teacher can show them the formal method that involves finding two factors of the product of ab and cd that sum to bc + ad.
Structured inquiry
Structured inquiry
The slides contain questions as part of a structured inquiry. The emphasis of the inquiry is on explaining patterns through variation in the questions and solutions.
The teacher could use two other Inquiry Maths prompts to develop the inquiry further:
Solving quadratic equations
Classroom inquiry
Classroom inquiry
Helen Hindle, head of mathematics at Park View School (London, UK) used the prompt with her year 10 class. The picture shows the students' initial thoughts.
Helen reports that the students followed the lead of two groups by placing numbers in the boxes. The class had met the expansion of two binomial expressions before and Helen used the upper half of the prompt to consolidate their use of the grid method.
As she says, the grid method facilitates an analysis of the lower half of the prompt, which can be more problematic for students. Through using the method 'in reverse,' the class began to develop an understanding of factorisation.
October 2019
Extending the inquiry
Extending the inquiry
The slides contain two challenge prompts that lead students into completing the square and cubic expressions.
Completing the square
Students find it easier to explore the prompt 'backwards' by putting numbers into the bottom row, squaring the binomial expression in the bracket, and simplifying the expression.
When they are confident with the process, they attempt to complete the square by placing even numbers in the top row before moving on to an odd coefficient of x.
Cubic expressions
Students start by expanding triple brackets. The aim is to understand how the terms in the cubic expression are formed (when the coefficient of x3 is one):
The coefficient of x2 is the sum of the numbers in the brackets;
The coefficient of x is the sum of the products of the three pairs of numbers; and
The constant is the product of the numbers.
Using this knowledge of mathematical structure, students now attempt to factorise cubic expressions.
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