technifyed

Pure Mathematics · Combinatorics

Find patterns in Pascal's triangle

Build 16 rows of the triangle and discover at least six patterns hidden in it.

  • DifficultyEasy
  • Estimated time1–3 hours
  • Budget₹0
  • ClassClass 8–11
  • TeamSolo
  • You needHousehold materials

Prerequisites: None beyond your class work

Original · by TechnifyedInvestigationVisualizationBeginner friendlyFree to doResearch oriented

My projects

Overview

Build 16 rows of the triangle and discover at least six patterns hidden in it.

Skills you will use

Conjecture and checkingPattern finding

Materials and tools

  • Large paper
  • Coloured pencils
Expected cost₹0Nothing to buy
Estimated duration1–3 hours5 build steps, plus the report

Step-by-step roadmap

  1. Start

    You will finish with: a coloured triangle and a list of patterns with a reason for each.

  2. 1

    Prerequisites

    Nothing special to know first. Then collect the 2 items in the materials list.

  3. 2

    Learn

    Read up on combinatorics: binomial, Pascal's triangle, Sierpinski. Your textbook chapter and one short video are enough.

  4. 3

    Plan

    Write your question, your hypothesis and the one thing you will change. Draw the table you will fill in before you start.

  5. 4

    Build

    1. Write out 16 rows.
    2. Sum each row and describe the pattern.
    3. Find the triangular and Fibonacci numbers.
    4. Shade the odd numbers and see the fractal.
    5. Link each row to a binomial expansion.
  6. 5

    Test

    Repeat every reading at least three times and check the result against what the science predicts.

  7. 6

    Document

    Record every reading with its unit, photograph the setup and draw a graph of the result. Report structure

  8. 7

    Present

    Explain the question, the method and the graph in two minutes; keep the setup ready to demonstrate. Presentation structure

  9. 8

    Publish

    Share the report and photos with your class, a science fair or your school magazine.

Expected outcome

A coloured triangle and a list of patterns with a reason for each.

Other versions of this project

Advanced version

Prove the row-sum pattern and the hockey-stick identity.

Ways to do this project

Text marked "You write this" is a prompt for your own work; everything else is specific to this project.

Problem
You write thisWrite your question as: How does [what you change] affect [what you measure]?
Hypothesis
You write thisWrite it as: If [I change this], then [this will happen], because [reason].
Variables
You write thisName the one thing you change (independent), the thing you measure (dependent) and what you keep the same (controlled).

Observation table

You write thisDraw a table with one column for what you change, one for each repeat reading and one for the average.

Graph
You write thisPlot what you changed along the bottom and what you measured up the side; use a bar chart for categories and a line graph for numbers.
Results
You write thisState what the graph shows in one or two sentences, with numbers.
Conclusion
You write thisSay whether the result supports your hypothesis and why you think so.

Display board

  • Title and your name
  • Question
  • Hypothesis
  • Materials
  • Procedure (with photos)
  • Data table and graph
  • Conclusion
  • What you would do next

Questions judges ask

  • What would you change if you did this again?
  • What was your biggest source of error, and how did you reduce it?
  • Where is this idea used in real life?

Report and presentation

Report structure

  1. Title and question
  2. Background: the science behind it
  3. Hypothesis
  4. Materials
  5. Method (steps, with a diagram or photo)
  6. Observations (table)
  7. Results (graph and calculation)
  8. Sources of error
  9. Conclusion
  10. What you would do next
  11. References

Presentation structure

  1. The question
  2. Why it matters
  3. The idea behind it
  4. Setup (photo)
  5. Method in three steps
  6. Results (one clear graph)
  7. What the result means
  8. Errors and limits
  9. Conclusion and next step

Viva questions

Prepare your own answers to these before you present.

  • What is the aim of your project, in one sentence?
  • What is the principle behind it?
  • Why did you choose these materials or tools?
  • What result did you get, and was it what you expected?
  • What went wrong and how did you fix it?
  • How could this be improved or used in real life?

Resources

Source and attribution

Original

Written by Technifyed. Free to use for your own school or college project; write the report in your own words. Added 1 Oct 2026.

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