Intersections

Task 206 ... Years 4 - 10

Summary

It doesn't take a very big pile of sticks before you might begin to think that counting the number of intersections is too much of a challenge, let alone counting the maximum number. The pile begins to look like a jumble. The task begins with a simple case and is structured so the students explore even simpler cases before tackling more complex ones. It eventually asks for the maximum number of intersections for 100 sticks in a pile. Counting is not going to be a feasible option, so students need to look for information within the problem that allows the required number to be calculated. Mathematics eventually predicts what must be true even if the eye doesn't see it.
 

Materials

  • About 30 sticks

Content

  • algebra, concept of a variable / function
  • algebra, generalisation in words & symbols
  • algebra, quadratic
  • counting
  • equations, substitution & solution
  • graphical representation
  • language of space, position and order
  • numbers, triangle
  • patterns, number
  • shapes, properties (of lines)
Intersections

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
   

The first part of the problem is designed to help students realise that three sticks could cross to make 1, 2 or 3 intersections. There is one intersection if all sticks cross at the same point (coincident); two intersections if two of the sticks are parallel and the third crosses both (there is another arrangement for two intersections) and three intersections if the three sticks are non-parallel and non-coincident. Three must be the maximum number of intersections for three sticks.

So is two the maximum number of intersections for two sticks? Nice thought, but no. Two sticks either don't intersect (they would be parallel) or can only intersect once. That's true for any two sticks no matter how many others are around. This is one of the key realisations involved in the problem.

Question 3 encourages further exploration in the hope that students will also realise that:

  • To achieve the maximum intersections each new stick must intersect with all the current sticks without passing through any current intersections.
  • These new intersections are added to the number already present.
It's also worth noting that stick can't intersect itself.

So, using the strategies of breaking the problem into smaller parts and making a table, a pattern emerges:

Number of Sticks 1 2 3 4 5
Number of New Intersections 1 2 3 4
Total Intersections 0 1 3 6 10

The total number of intersections turns out to be a triangle number! Why? Because to make the maximum, each new stick must cross all of the current sticks and that maximum number is added to the previous number of intersections which itself is the sum of natural numbers that started with one intersection between two sticks.

For 100 sticks the total will be as follows, beginning with the intersections for 1 stick:

Total100 = 0 + 1 + 2 + 3 + 4 + ... + 97 + 98 + 99
This series has an even number of terms (no middle number) so the first and last pair to equal 99, so to the second and second last and so on. That's 50 terms that sum to 99, so the total number of intersections for 100 sticks is 4,950.

Extensions

  • The columns of the yellow rows of the table above make ordered pairs (1, 0), (2, 1), (3, 3), (4, 6) ... If we graphed these pairs what shape might the dots make?
  • If I tell you any number of sticks, can you tell me how to calculate the maximum number of intersections?
  • If I tell you the maximum number of intersections I made, can you tell me how many sticks I used?

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
   

All you need to convert this task to a whole class investigation is a packet of drinking straws, or similar. Begin with 4 straws for each pair and ask them to investigate and record ways of using all four to make intersections with each other. What's the smallest number of intersections that can be made? The largest? Is it possible to make intersections for the numbers between the smallest and largest? ... For this part you might make columns on the board for 0, 1, 2, 3, 4, 5, 6 intersections and ask students to sketch their ways of for each (four straws only remember).

  • Discuss what has been learnt about making intersections and record key points on the board.
  • Declare an interest in the maximum number and ask students to find the maximum number for 1, 2 and 3 straws.
  • Move forward with the problem as described above.

At this stage, Intersections does not have a matching lesson on Maths300.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
   

The Intersections task is an integral part of:

  • MWA Pattern & Algebra Years 9 & 10

Green Line
Follow this link to Task Centre Home page.