Motion Math Blog

You Asked… Motion Math Answers


Recently, Motion Math participated in Maker Faire, an amazing all-ages celebration of creativity. We hosted a booth in the DIY Learning Pavilion sponsored by EdSurge and had a great time connecting with parents, teachers and kids (some of whom even coded their own game level – see tomorrow’s post). We were curious about what other people wondered about math, so we offered a board for parents, teachers and other makers to stop and ask their big math questions. Some of our favorites:

What is the golden ratio? Followed by – what is the goldfish ratio?

Golden Ratio

The golden ratio: the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one or (a+b) / a = a / b. In decimal form, the golden ratio is 1.618. Rectangle ABCD above is a golden rectangle, meaning that (BC + AB) / BC = BC / AB.

If we then divide ABCD with line segment EF so that both BC and AD are divided so that (BF + FC / BF) = BF / FC and (AE + ED) / AE = AE / ED, we have a new golden rectangle, FCDE.

If we keep creating golden rectangles, they will converge at the point where the diagonals of rectangles ABCD and EFCD intersect. Logarithmic curves from point to point create a pretty spiral that’s often seen in nature.

The goldfish ratio: the number of Hungry Fish you create on your iPad, iPhone or iPod touch to the number of learners you have playing Motion Math: Hungry Fish on the same device.

Discussion questions: Can you give examples from nature of where you’ve seen the golden ratio? What are other examples of the golden ratio, from art or architecture?

What is the last digit of π?

π is the ratio of the circumference of a circle to its diameter. π is also an irrational number so we can’t know the last digit of π. My answer begs the question – what is an irrational number?

An irrational number is a number that cannot be expressed as the ratio A / B, where A and B are integers. For example, 3.14 can be represented as 314 / 100, a ratio of two integers. But pi isn’t 3.14! It keeps going and going, and you could never represent it accurately as the ratio of two integers.

Is π + e rational?

A rational number is one that can be expressed as a ratio of two integers. We know that π is irrational, as is e. We know e is irrational because it is the sum of an infinite series of a simple continued fraction.

e = 1 + \frac{1}{1} + \frac{1}{1\times 2} + \frac{1}{1\times 2\times 3} + \frac{1}{1\times 2\times 3\times 4}+\cdots

However, since adding two irrational numbers doesn’t always get you another irrational number, we can’t say for certain π + e is irrational. For example, 4 – π = 0.858407346… (the sum is irrational) and if you take the irrational sum, 0.858407346… and add it to π, another irrational number, you get 4, which is rational. The rationality of π + e is an unsolved question in number theory (see #22 in Old and New Unsolved Problems in Plane Geometry and Number Theory).

If 1/1 = 1; 2/2 = 1; -1/-1 = 1, is 0/0 = 1?

Let’s break this question down to find an answer. We know that division is the inverse of multiplication, so 1/1 =1; 1*1 = 1.

and

2/2 = 1; 1*2 = 2

and

-1/-1 = 1; 1*-1 = -1

and if we multiply 1*0, we get 0. However, we could multiply any number by 0 and get zero, so 0/0 is indeterminate.

When will you have a game for quadratics?

Good question! Right now, we’re focusing on games that develop number sense (estimation, mental arithmetic, place value) for our players. Sign up for our newsletter and we’ll keep you updated as new games come out.

√(-1)   2^3   Σ   π

Can you decipher the cute, clever mathematical sentence above? I think it would be fun for you to translate this!

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