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Count the Small Triangles with a Power of Three

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Predicting What Comes Next: Exploring Sequences and Progressions · Fractals and geometric patterns · 3:41

You'll learn: Explain why each retained triangle produces three at the next stage.

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Count the Small Triangles with a Power of Three

3:41

Playing

Count the Small Triangles with a Power of Three

You'll learn: Explain why each retained triangle produces three at the next stage.

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Same topic

Fractals and geometric patterns

5

Build the Sierpiński Triangle One Removal at a Time
4:00
Build the Sierpiński Triangle One Removal at a Time
You'll learn: Join side midpoints and remove each central small triangle repeatedly.
More Triangles Can Occupy Less Total Area
3:41
More Triangles Can Occupy Less Total Area
You'll learn: Use the retained fraction to model the shrinking shaded area.
Plot AP and GP Behaviour Side by Side
3:55
Plot AP and GP Behaviour Side by Side
You'll learn: Compare constant additive change with multiplicative growth and decay.
Build and Count a Sierpiński Square Carpet
3:41
Build and Count a Sierpiński Square Carpet
You'll learn: Remove the central ninth in every retained square and count the remaining pieces.
Find the Shrinking Area of the Square Carpet
3:42
Find the Shrinking Area of the Square Carpet
You'll learn: Apply the retained fraction8/9 at every stage.

Next topic

Bouncing-ball applications

2

A Bouncing Ball’s Heights Form a Shrinking GP
3:50
A Bouncing Ball’s Heights Form a Shrinking GP
You'll learn: Generate bounce maxima and find the first bounce below a specified threshold.
Count Both Upward and Downward Travel Between Bounces
3:31
Count Both Upward and Downward Travel Between Bounces
You'll learn: Distinguish rebound height from total vertical distance travelled.

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Chapter challenge problems

8

A Shifted Recursive Sequence Can Reveal a GP
3:28
A Shifted Recursive Sequence Can Reveal a GP
You'll learn: Transform the recurrence into a geometric rule and identify a target term.
Count Multiples Within a Fixed Number Range
3:46
Count Multiples Within a Fixed Number Range
You'll learn: Use the smallest and largest allowable multiples and the AP term formula.
Recover an AP from Sums of Two Terms
3:41
Recover an AP from Sums of Two Terms
You'll learn: Convert term-sum information into two linear conditions on a and d.
When Does a Natural-Number Sum First Exceed One Thousand?
3:51
When Does a Natural-Number Sum First Exceed One Thousand?
You'll learn: Use the triangular-sum formula and test neighbouring integer positions.
Write One Hundred as Consecutive Natural Numbers
3:37
Write One Hundred as Consecutive Natural Numbers
You'll learn: Use a starting value and number ofterms to find every valid representation.
A Doubling Bacterial Culture Uses Stage Zero
3:43
A Doubling Bacterial Culture Uses Stage Zero
You'll learn: Model the initial count and subsequent hourly doublings.
Find a GP from an Initial Sum and Later-Term Relation
3:49
Find a GP from an Initial Sum and Later-Term Relation
You'll learn: Express the two conditions using a and r and find possible progressions.
Find Three GP Terms from Their Sum and Product
3:51
Find Three GP Terms from Their Sum and Product
You'll learn: Use the middle term and equal ratios to constrain the first and third terms.