STEM-Plus Content offers K-12 math and other STEM content in both English and Japanese.

The content aids in vocabulary acquisition for both English and Japanese learners.

How does it achieve this?

Through the content, learners grasp words and phrases more intuitively because they already know their meanings in their native language.

It is like learning the word “apple” by looking at a picture of one.

STEM-Plus Content facilitates easy learning of abstract yet useful words.

To ensure effective and efficient learning, the content is designed to be concise and illustrative.

Learners can easily understand the material through images and examples.

In addition, the content provides useful tips on math and other STEM subjects, supporting

academic success in these important fields.

How to use:

Carefully selected English words and phrases related to “Proportionality” are highlighted in yellow in the images below. The words are also summarized at the bottom of this page.

We recommend reviewing them regularly during your daily math studies to enhance your vocabulary.

# TERMS, DEFINITIONS, AND ILLUSTRATIONS

## Direct Proportionality

- Two quantities x and y are
**directly proportional**when y=kx, where k is a nonzero constant.

y=kx でkがゼロでない定数のとき、ふたつの量xとyは**比例する**という。 - The number k is called the
**constant****of proportionality**.

kを**比例****定数**とよぶ。 - The ratio y/x is constant and equal to k.

比 y/x は一定で比例定数kに等しい。 - The graph of y=kx is a line that passes through the origin.

y=kx のグラフは原点を通る直線である。

## Inverse Proportionality

- Two quantities x and y are
**inversely proportional**when y=k/x, where k is a nonzero constant.

y=k/x でkがゼロでない定数のとき、ふたつの量xとyは**反比例する**という。 - The number k is called the
**constant of proportionality**.

kを**比例定数**とよぶ。 - The product xy is constant and equal to k.

積 xy は一定で比例定数kに等しい。 - The graph of y=k/x is a
**hyperbola**.

y=k/x のグラフは**双曲線**である。

# WORDS AND PHRASES

✅**quantity** 量 ⇔**quality** 質

✅**constant** 一定の

✅**increase** 増加する；を増やす／増加

✅**decrease** 減少する；を減らす／減少

# TIPS

## Increasing Functions and Positive Slopes

**When ‘x increases’** → How ‘y behaves’

↑ ↑

Developing the habit of fixing your attention here (**‘x increases’**) will bring various benefits in the future.

Specifically,

- terms like ‘increasing’ and ‘decreasing’ functions,
- expressions like a function is ‘increasing’ or ‘decreasing’ in a certain interval,
- and determining whether the slope is ‘positive’ or ‘negative’ as another way to say the above,

all refer to the ‘increase’ or ‘decrease’ of y **when ‘x increases’**.

It’s good to get accustomed to this early on.

## About Inverse Proportionality

Inverse proportionality is indeed difficult.

As we all know, x cannot be zero.

But it’s hard to intuitively understand the behavior of y before and after this point.

As x approaches zero, like at -0.1, -0.01, y becomes a large negative number, like -10, -100,

and as x gets even closer to zero, y also becomes a larger negative number (eventually approaching negative infinity).

Then, the moment x exceeds zero by even a little, y jumps to positive infinity.

It’s baffling.

And not to mention, drawing its graph requires skill!

In some English-speaking countries, inverse proportionality is often introduced much later as a rational function.

Specifically, after studying quadratic, cubic, n-th degree functions, and even exponential and logarithmic functions.

However, it’s a relationship we often encounter in daily life.

So Don’t be too averse to it. Instead, let’s just recognize that it’s like a character shrouded in mystery.