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
Overview
Build 16 rows of the triangle and discover at least six patterns hidden in it.
Skills you will use
Materials and tools
- Large paper
- Coloured pencils
Step-by-step roadmap
Start
You will finish with: a coloured triangle and a list of patterns with a reason for each.
- 1
Prerequisites
Nothing special to know first. Then collect the 2 items in the materials list.
- 2
Learn
Read up on combinatorics: binomial, Pascal's triangle, Sierpinski. Your textbook chapter and one short video are enough.
- 3
Plan
Write your question, your hypothesis and the one thing you will change. Draw the table you will fill in before you start.
- 4
Build
- Write out 16 rows.
- Sum each row and describe the pattern.
- Find the triangular and Fibonacci numbers.
- Shade the odd numbers and see the fractal.
- Link each row to a binomial expansion.
- 5
Test
Repeat every reading at least three times and check the result against what the science predicts.
- 6
Document
Record every reading with its unit, photograph the setup and draw a graph of the result. Report structure
- 7
Present
Explain the question, the method and the graph in two minutes; keep the setup ready to demonstrate. Presentation structure
- 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).
Materials and procedure are in Materials and the Roadmap.
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?
- Research question
- You write thisOne question you can answer with data, narrow enough to finish in the time you have.
- Hypothesis
- You write thisWhat you expect to find, and why.
- Variables
- You write thisWhat you change or compare, what you measure, and what you hold constant.
- Methodology
- You write thisDescribe the method in enough detail that someone else could repeat it: sample, tools, procedure.
- Data collection
- You write thisSay what you will record, how many times, and how you will keep the records safe.
- Analysis
- You write thisChoose the table, chart or test that answers the question; report averages with their spread.
- Limitations
- You write thisList what could have affected the result: small sample, instrument limits, things you could not control.
Literature review
Find five to eight sources (your textbook, review articles, reports) and note what each says about your question. Group them by idea, not one after another, and end with what is still not known.
References
List every source you used in one style throughout (author, year, title, where it was published, link and the date you opened it).
Report and presentation
Report structure
- Title and question
- Background: the science behind it
- Hypothesis
- Materials
- Method (steps, with a diagram or photo)
- Observations (table)
- Results (graph and calculation)
- Sources of error
- Conclusion
- What you would do next
- References
Presentation structure
- The question
- Why it matters
- The idea behind it
- Setup (photo)
- Method in three steps
- Results (one clear graph)
- What the result means
- Errors and limits
- 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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