Triangular numbers inquiry
Mathematical inquiry processes: Extend patterns; find connections; reason using structure; prove. Conceptual field of inquiry: Triangular and square numbers; algebraic expressions for position-to-term rules.
Kirsten McGarrie, a teacher of mathematics, designed the prompt as the stimulus to structured inquiry for her year 7 mixed attainment classes. It shows that two consecutive triangular numbers equal a square number.
Kirsten wanted the prompt to give "every student the potential for growth" by suggesting multiple lines of inquiry. These include:
Extending the sequences by drawing more patterns;
Identifying and analysing the pattern of triangular and square numbers on a multiplication grid;
Developing rules, both term-to-term and position-to-term;
Exploring other patterns that involve triangular and square numbers; and
Kirsten used the prompt in a unit on sequences. Students had already learnt about term-to-term and position-to-term rules.
They had experience of posing questions about a prompt, noticing its properties and wondering where it could lead. The rich collection of their responses are shown in the picture below.
Kirsten incorporated the students' questions and observations into the inquiry. Her structure offered various levels of challenge to students with different prior attainment in mathematics lessons.
For example, the student's question about the tenth pattern can be approached in at least four ways:
Concrete: use tiles to create the design;
Pictorial: draw the pattern;
Numerical: follow a number pattern (for example, 1 + 0 = 1 x 1 = 1, 1 + 3 = 2 x 2 = 4, 3 + 6 = 3 x 3 = 9, ..., 45 + 55 = 10 x 10 = 100); or
Algebraic: Use the position-to-term rule - [(10)(10 + 1)]/2 and [(9)( 9 +1)]/2 to give the numbers of black and grey tiles respectively.
Kirsten reports that all the students grew mathematically during the inquiry. In particular, a student who had very low prior attainment was able to identify the position-to-term rule for square numbers. When the student informed the class of the rule, Kirsten says that "it was a splendid moment, joyful ... joyful for the whole class."
Lines of inquiry
1. Patterns on a multiplication grid
Students shade in the square and triangular numbers on a multiplication grid. (You can find a template in the resources section.)
They then describe the patterns:
The diagonal line of square numbers represents a line of reflection - a fact that students can use to check they have shaded the grid correctly.
The triangular numbers are arranged in pairs and each pair appears twice.
There is a line between each pair of triangular numbers.
The final stage is to attempt to explain the patterns. The triangular numbers are arranged in twos because each pair is a multiple of the same number.
There is a line between them because consecutive pairs are multiples of odd numbers. For example, 3 and 6 are multiples of three and 10 and 15 are multiples of five.
2. Derive a formula for triangular numbers
Students combine two sets of tiles to form a rectangle and then derive the formula for triangular numbers under the teacher's guidance. A further step would be to prove algebraically that the sum of two consecutive triangular numbers is a square number.
3. Explore more patterns
In a structured inquiry, the teacher poses a number of questions that promote structural reasoning:
Can you draw the next two patterns?
How many tiles of each colour are there?
What connections do these patterns have with square numbers?
What is the position-to-term rule for each sequence of patterns?
How could you rearrange the tiles to make squares and justify your rule?
4. Explore three-dimensional patterns
Explore three-dimensional designs with cube. Is the highest degree of the position-to-term rule still two or is it three because there is an extra dimension?