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# Pascal's triangle Maths Investigation

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This is lesson where students can guide themselves through exploring and investigating patterns in Pascal's triangle.

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• Symmetry
• Exponents – power of 2 If x is the previous number the answer to the following row is x to the power of 2.
• 0 = 1s all the way 1 = counting 2= triangular numbers but adjacent numbers make square numbers 3 = tetrahedral numbers
• Odd + odd = even Even + odd = odd Even + even = even
• ### Pascal's triangle Maths Investigation

1. 1. Pascal’s Triangle WALT: investigate and describe patterns
2. 2. What is Pascal’s triangle? <ul><li>Named after the French mathematician Blaise Pascal (1623-62) who brought the triangle to the attention of Western mathematicians </li></ul><ul><li>It was known as early as 1300 in China, where it was known as the &quot;Chinese Triangle“ </li></ul><ul><li>It is used to solve problems </li></ul><ul><li>of probability </li></ul>
3. 3. FUNCTION – How does it work? <ul><li>What is the rule? </li></ul><ul><li>Use the rule to complete a triangle. </li></ul>
4. 4. What can you see? Here is a hint to help you finish the triangle. You may use a calculator.
5. 5. Finding patterns <ul><li>Find the total of each </li></ul><ul><li>row and record this. </li></ul><ul><li>What do you notice? </li></ul><ul><li>Can you use exponents to record this number sequence? </li></ul><ul><li>Can you write a general statement for this number sequence? </li></ul>
6. 6. Explore diagonal patterns within the triangle. <ul><li>Look at the diagonals: </li></ul><ul><li>Is there a pattern along each diagonal? </li></ul><ul><li>Describe the pattern and its rule. </li></ul>
7. 7. More Diagonal Patterns <ul><li>2 nd diagonal = triangular numbers AND the adjacent numbers make square numbers </li></ul><ul><li>3 rd diagonal = tetrahedral numbers (add the layers) AND the adjacent numbers make pyramid numbers (add the layers.) </li></ul>
8. 8. Investigate Pascal’s triangle – ODDS and EVENS <ul><li>Shade in all the even numbers in Pascal’s triangle. What do you notice? </li></ul><ul><li>This called - The Sierpinski Triangle </li></ul>
9. 9. Are there more odd or even numbers? <ul><li>Can you remember the addition properties of odd and even numbers? </li></ul><ul><li>ODD + ODD = </li></ul><ul><li>EVEN + ODD = </li></ul><ul><li>EVEN +EVEN = </li></ul><ul><li>How can you relate this to your prediction? </li></ul>
10. 10. Are there more odd or even numbers? <ul><li>Design a table or graph to record your data in two ways: </li></ul><ul><li>By row </li></ul><ul><li>Accumulative </li></ul><ul><li>Challenge! </li></ul><ul><li>What is the ratio of even to odd numbers after 3, 7,15 rows? </li></ul>
11. 11. Find your own patterns! <ul><li>Colour multiples of nine </li></ul><ul><li>What do you see? </li></ul><ul><li>Try multiples of other numbers are there repeating patterns? </li></ul>
12. 12. More information <ul><li>http://www.mathsisfun.com/pascals-triangle.html </li></ul>