Representations and moving between them

Science and society, grade 10 (3 units) Lesson 4 of 15Learning

Representations and moving between them

Four representations of one relation

The same relation between two quantities can be described in four ways: in words, in a table, in a graph and in a formula. Each representation shows the same information from another angle.

In words: a taxi charges 12 ₪ for starting the ride, and 4 ₪ more for every kilometre.

In a table:

km ( $\displaystyle x$ )	0	1	2	5	10
Price in ₪ ( $\displaystyle y$ )	12	16	20	32	52

In a formula: $\displaystyle {y}={12+4x}$

In a graph:

\displaystyle y=12+4x

\displaystyle (0,12)

\displaystyle (5,32)

12, the price at the start, is the point where the graph meets the $\displaystyle y$ -axis.
4, the extra for each kilometre, is the slope: for every step of 1 to the right, the graph rises by 4.

Which representation answers which question

Each representation suits other questions, so move from one to another by the question:

A table – exact values at a few places: a 5 km ride costs 32 ₪.
A graph – the trend at a glance: where the value rises, how fast, and where two lines meet.
A formula – any value, even one not in the table: a 23 km ride costs $\displaystyle {12+4\cdot23}={104}$ ₪.
Words – what the numbers mean: what is paid, and for what.

A scientific phenomenon as a formula

Phenomena in nature are sometimes described by a linear relation too, and then the formula gives values that were never measured.

Going up, the air gets colder: by about 6.5 degrees for every kilometre.
On a summer day the temperature at sea level is 25 degrees.

As a formula: $\displaystyle {T}={25-6.5h}$ , where $\displaystyle h$ is the height in kilometres and $\displaystyle T$ the temperature in degrees.
As a table:

Height in km ( $\displaystyle h$ )	0	1	2	3
Temperature ( $\displaystyle T$ )	25	18.5	12	5.5

At the top of Mount Hermon, about 2.2 km high: $\displaystyle {T}={25-6.5\cdot2.2}={25-14.3}={10.7}$ degrees.

From a table to a formula

When the value of $\displaystyle y$ in a table changes by the same amount for every equal step of $\displaystyle x$ , the relation is linear, and it can be written as the formula:

\displaystyle y=mx+b

$\displaystyle m$ – the change in $\displaystyle y$ when $\displaystyle x$ grows by 1; $\displaystyle b$ – the value of $\displaystyle y$ when $\displaystyle {x}={0}$ .

The price of a phone plan by the number of minutes of calls a month:

Minutes ( $\displaystyle x$ )	0	100	200	300
Price in ₪ ( $\displaystyle y$ )	30	35	40	45

Every 100 minutes the price rises by 5 ₪ – always by the same amount, so the relation is linear.
For one minute: $\displaystyle {\frac{5}{100}}={0.05}$ ₪. That is $\displaystyle m$ .
When $\displaystyle {x}={0}$ the price is 30 ₪. That is $\displaystyle b$ .
The formula: $\displaystyle {y}={30+0.05x}$ .

Exercise 1

A gym charges a joining fee of 150 ₪, and 120 ₪ more for each month.

Write a formula for the price $\displaystyle y$ , in shekels, by the number of months $\displaystyle x$ .

How much does a whole year’s membership cost?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

A candle was lit, and its height measured every hour:

Hours ( $\displaystyle x$ )	0	1	2	3
Height in cm ( $\displaystyle y$ )	24	21	18	15

Write a formula for the candle’s height $\displaystyle y$ by the number of hours $\displaystyle x$ .

After how many hours will the candle burn out? Write an equation and solve it.

Exercise 3

The graph: the amount of water in a tank, in litres, by the number of minutes since the tap at its bottom was opened.

Mark on the graph the point where the tank is empty.

\displaystyle y=500-25x

The graph’s formula is $\displaystyle {y}={500-25x}$ . What do the numbers 500 and 25 tell about the tank?

Exercise 4

Two phone plans:
Plan A: 40 ₪ a month, and 0.5 ₪ more for each minute of calls – in the graph, the line $\displaystyle {y}={40+0.5x}$ .
Plan B: 70 ₪ a month, and 0.2 ₪ more for each minute of calls – in the graph, the line $\displaystyle {y}={70+0.2x}$ .

Mark the point where the two plans cost the same.

\displaystyle y=40+0.5x

\displaystyle y=70+0.2x

Noa talks about 150 minutes a month. Which plan is better for her? Explain with the graph or by a calculation.

Exercise 5

In a thunderstorm the lightning is seen at once, and the thunder heard only a few seconds later: sound travels 0.34 km a second.
The distance to the lightning, in kilometres: $\displaystyle {d}={0.34t}$ , where $\displaystyle t$ is the number of seconds between the lightning and the thunder.

6 seconds passed between the lightning and the thunder. How far away is the lightning?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Lightning struck 5.1 km from you. How many seconds will pass before you hear the thunder? Solve the equation.

Formulas: science and society, grade 10 (3 units)

Open as a page

Information and representations

$\displaystyle p$ percent of $\displaystyle n$
$\displaystyle p\%\cdot n$ $\displaystyle =$ $\displaystyle \frac{p}{100}\cdot n$
A rate per 1,000 people – $\displaystyle k$ the number of cases, $\displaystyle N$ the number of people
$\displaystyle \frac{k}{N}\cdot1000$
The average rate of change between the points $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$
$\displaystyle \frac{y_2-y_1}{x_2-x_1}$
A linear relation – $\displaystyle m$ the change in $\displaystyle y$ when $\displaystyle x$ grows by 1, $\displaystyle b$ the value of $\displaystyle y$ when $\displaystyle {x}={0}$
$\displaystyle y=mx+b$

Measures of centre

The mean of $\displaystyle n$ data
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
A weighted mean – $\displaystyle w_1, \dots, w_k$ the weights of the values $\displaystyle x_1, \dots, x_k$
$\displaystyle \bar{x}=\frac{w_1x_1+w_2x_2+\dots+w_kx_k}{w_1+w_2+\dots+w_k}$
The mean of grouped data – $\displaystyle m_1, \dots, m_k$ the class middles, $\displaystyle f_1, \dots, f_k$ their frequencies
$\displaystyle \bar{x}\approx\frac{m_1f_1+m_2f_2+\dots+m_kf_k}{f_1+f_2+\dots+f_k}$
The place of the median in an ordered list of $\displaystyle n$ data
$\displaystyle \frac{n+1}{2}$

Probability

Equally likely outcomes – $\displaystyle k$ the outcomes in the event $\displaystyle A$ , $\displaystyle n$ all the outcomes
$\displaystyle P(A)=\frac{k}{n}$
The complement $\displaystyle \overline{A}$ – $\displaystyle A$ does not happen
$\displaystyle P(\overline{A})$ $\displaystyle =$ $\displaystyle 1-P(A)$
A path of a tree diagram – the product of the probabilities $\displaystyle p_1$ , $\displaystyle p_2$ on its branches
$\displaystyle P=p_1\cdot p_2$
An event of several paths – the sum of their probabilities
$\displaystyle P(A)=P_1+P_2+\ldots$
Estimating a probability from relative frequency – $\displaystyle A$ happened $\displaystyle f$ times in $\displaystyle n$ experiments
$\displaystyle P(A)\approx\frac{f}{n}$
How many times an event $\displaystyle A$ is expected to happen in $\displaystyle n$ experiments
$\displaystyle n\cdot P(A)$