This booklet contains 4 challenges, around the topic of “sums of n consecutive positive numbers”:
1. Determine all positive integers n
such that there are n consecutive positive integers that sum to a square.
2. Can the sum of the first n consecutive positive integers be equal to the square of a prime number?
3. Find all possible ways to write 2014 as a sum of n consecutive positive integers.
4. Show that the sum of 12 consecutive positive integers can never be a perfect square.
The booklet has the following outline:
• A one pager containing the challenges – grouped together as one assignment.
• Three visual hints – diagrams explaining a key mechanism or concept, needed to solve the challenges These diagrams can be printed on cards and handed to the students as extra hints.
• For each challenge: detailed step-by-step explanations and workings, assuming no or very limited prior algebra or calculus knowledge.
I use these math booklets with 15 to 18-year-old students, preparing for math Olympiads or Challenging Tests, such as IB Maths Higher Level. The challenges in the booklets can also be used as extension homework, practice during term breaks, or as part of an in-class differentiation approach.
I use these booklets to complement my one-on-one and group tutoring sessions .
First, I give my students the assignments, and a due date for their solutions. At any given time, they can request a hint, if they feel stuck. They then get the hint cards, which point them in the right direction.
The booklets themselves I give to the students when they hand in their solutions to the challenges.
Where the booklets refer to algebra, calculus or trigonometry concepts, I always include a brief explanation of the concept such that the booklets can be used independently of any curriculum or syllabus the students are using.
When talented students start working on math problems that go beyond their syllabus, or problems that are presented “out-of-context”, they often think that the solution is some magical formula. They become frustrated when they do not immediately “see” the answer. By grouping challenges around a specific topic, and taking them step-by-step through the problem, the students will experience that by the time they reach the last challenge in the booklet, they will breeze through the explanations and usually do not even need them anymore.
This booklet uses several ways to explain concepts: graphs, diagrams, mathematical notation, plain-language explanations, and examples.