International Baccalaureate · PYP 4

Position, Grids and Scale

⌚ ~130 min ▢ 14 blocks ✓ 61 exercises ➣ Prerequisites : Numbers to 999 999, Multiplication, Turns and Angles, Division, Grids, Maps and Compass Directions

I Coding a Route in a Neighbourhood

Definition — Route Instructions

On a grid, a path is coded with arrows, one for each step. In a neighbourhood, we no longer move square by square: we follow the streets and we only ever choose something at a crossing. So a route is coded with instructions, in order:

go straight on to the 1st, 2nd, 3rd… crossing;
turn left or turn right at that crossing;
turn around to go back the way you came.

Left and right depend on the way you are facing, never on the sheet: the same crossing is a left turn one way and a right turn the other way. That is why a route always starts by saying where you start and which way you face.

Method — Writing a Route

Say where you start and which way you face. Without that, no instruction means anything.
Follow the route with your finger, and write one instruction each time you arrive at a crossing where you change direction: count the crossings from where you last turned — the crossing you are leaving does not count, the next one you reach is the 1st, and you count up to the crossing where you turn — then say which way you turn.
End with the last straight run, the one that brings you to your destination, and name the place you arrive at. Without it, the reader stays at the last crossing and never reaches the destination.
Read your route again from the start, finger on the plan, without looking at the path you had in mind. If you land somewhere else, the route is wrong, not the reader.

Example

Aya starts at the School, facing the way the arrow shows, and follows this route: go straight on to the 2nd crossing, turn left, go straight on to the 1st crossing. Where does she arrive?

Answer

At the Pool. Facing that way, the 2nd crossing is the one at the Bakery: she goes straight past the first one. There she turns left — she is now facing the top of the plan — and the next crossing is the Pool. Notice that the Bakery is not written in the route: it is the place at the 2nd crossing, and a route only counts the crossings, it does not name the places you pass.

Example

Now write the route from the School to the Park. You start at the School, facing the way the arrow shows.

Answer

Go straight on to the 1st crossing, turn left, go straight on to the 2nd crossing — and you arrive at the Park. And now the check: finger on the School, facing the arrow. First crossing — turn left, facing the top of the plan. One crossing straight past, a second one: the Park. It works. Had we written “1st crossing” at the end, the finger would have stopped one street short, and the route would have been wrong even though the path we had in mind was correct. Other routes reach the Park too; this one is one of the shortest, and any route that really arrives at the Park when read back with the finger is correct.

Skills to practice

Writing and Checking a Route

II Long Programs with Quarter Turns

Definition — The Robot Code

On squared paper, a path can be coded with the four arrows $\uparrow\ \rightarrow\ \downarrow\ \leftarrow$, one per step, each one pointing the way to go on the sheet. A robot does not know the sheet: it only knows where it is looking. Its program uses three instructions:

$\rightarrow$	move forward one square
$\curvearrowright$	quarter turn to the right (the robot stays on its square)
$\curvearrowleft$	quarter turn to the left (the robot stays on its square)

A program is the list of these instructions, read in order. A half turn is two quarter turns the same way: $\curvearrowright\ \curvearrowright$. Careful: the arrow $\rightarrow$ does not mean “towards the right of the sheet” — it means “forward, the way the robot is looking”. The same program takes the robot to different places depending on which way it looks at the start. At a crossing of the neighbourhood we said “turn left”; the robot does the same, square by square, and we write $\curvearrowleft$.

Method — Executing a long program

The robot starts at the School, square A1, looking to the right, and receives this program of 10 instructions: $\rightarrow\ \rightarrow\ \rightarrow\ \rightarrow\ \rightarrow\ \curvearrowleft\ \rightarrow\ \rightarrow\ \rightarrow\ \rightarrow$ Where does it arrive? On the map, the blue squares are a lake and the grey squares are buildings: the robot never goes through them. A long program is executed one group of instructions at a time, never all at once.

Note the start. Write the starting square and draw a small arrow showing where the robot looks.
Read by groups. Several $\rightarrow$ in a row are one straight run: count them. A $\curvearrowright$ or $\curvearrowleft$ changes only the direction: after it, redraw the arrow before going on.
Keep a table with, for each group, the new direction and the square reached. With a long program, the table is what saves you from getting lost.
Check at the end that you have used every instruction: count them.

instructions	what the robot does	now looking	square
$\rightarrow\ \rightarrow\ \rightarrow\ \rightarrow\ \rightarrow$	5 forward	to the right	F1
$\curvearrowleft$	quarter turn	upwards	F1
$\rightarrow\ \rightarrow\ \rightarrow\ \rightarrow$	4 forward	upwards	F5

The robot arrives at the Library, square F5, looking upwards. Its path is drawn below.

Method — Writing a long program

Write the program that takes the robot from the Town Hall (square D5, looking upwards) to the Station.

Draw the path on the map, square by square, avoiding the lake and the buildings.
Follow it with your finger from the start, looking where the robot looks. First turn the robot if it is not looking along the first straight line; write as many $\rightarrow$ as squares; then, at each change of direction, write $\curvearrowright$ or $\curvearrowleft$ — right or left for the robot, not for the sheet. Then count the squares of the straight run and write as many $\rightarrow$.
Execute your program again from the start, with the table of the previous method. If the robot does not arrive where you wanted, the program is wrong, not the robot.

Here: $\rightarrow\ \curvearrowright\ \rightarrow\ \rightarrow\ \rightarrow\ \rightarrow$, that is 6 instructions.

Repeating. Long programs often repeat the same little group. Instead of writing $\rightarrow\ \rightarrow\ \rightarrow$, we can write $3 \times \rightarrow$; instead of writing the group $\rightarrow\ \rightarrow\ \curvearrowright$ four times, we write $4 \times (\rightarrow\ \rightarrow\ \curvearrowright)$: the robot goes forward two squares and turns right, four times, and comes back to its starting square after drawing a square. This is how programs on a screen are written, with a “repeat” block.

Report an error

Example

Léo's robot starts at the Library, looking to the right, and receives this program: $\curvearrowright\ \curvearrowright\ \rightarrow\ \rightarrow\ \rightarrow\ \curvearrowleft\ \rightarrow\ \rightarrow$ Where does it arrive, and which way is it looking at the end?

Answer

instructions	what the robot does	now looking	square
$\curvearrowright$	quarter turn	downwards	F5
$\curvearrowright$	quarter turn	to the left	F5
$\rightarrow\ \rightarrow\ \rightarrow$	3 forward	to the left	C5
$\curvearrowleft$	quarter turn	downwards	C5
$\rightarrow\ \rightarrow$	2 forward	downwards	C3

The robot arrives at the Stadium, square C3, looking downwards.

Example

Same program, but this time the robot starts at the Library looking upwards. Where does it arrive?

Answer

Not at the Stadium at all: the same instructions, followed from a different direction, draw the same path turned a quarter turn. The robot arrives at the Pool, square H2, looking to the right. A program means nothing without the starting square and the starting direction.

Skills to practice

Executing a Program
Finding the Arrival Square and Direction
Finding the Wrong Instruction
Writing a Program

III Coordinate Plane

The nodes of a grid, named by two numbers, become the points of a coordinate plane: the two lines numbered $0$ become the axes, and the node $(0,0)$ is the origin.

Report an error

Definition — Coordinate Plane

A coordinate plane is a grid formed by two number lines that cross at zero. One number line goes left and right (called the x-axis), and the other goes up and down (called the y-axis). The point where they cross is called the origin and has coordinates $(0,0)$.
The coordinates of a point are a pair of numbers, like $A(3,2)$. The first number is the $x$-coordinate and tells you how far to move along the $x$-axis. The second number is the $y$-coordinate and tells you how far to move along the $y$-axis.

Example

Plot these points on a coordinate plane: $A(2,3)$, $B(4,2)$, and $C(0,1)$.

Answer

For $A(2,3)$: Start at the origin $(0,0)$, move 2 units along the $x$-axis, then 3 units up.
For $B(4,2)$: Start at $(0,0)$, move 4 units along the $x$-axis, then 2 units up.
For $C(0,1)$: Stay at $0$ on the $x$-axis, then move 1 unit up along the $y$-axis.

Skills to practice

Finding the Coordinates of a Point
Plotting Points
Targeting Ships with Coordinates

IV Reading a Plan

Discover

Here is the plan of a park, drawn on a grid of $1$ cm squares.

Tom measures the pond on the plan: $3$ cm. He says: "The real pond is $3$ cm long."
Say why Tom is wrong, then find the real length of the pond.

Answer

A real pond of $3$ cm would be smaller than a hand. The plan is a small drawing of the park: every length has been made much smaller.
The sentence under the plan says how much smaller: $1$ cm on the plan stands for $5$ m in the park. The pond is $3$ cm long on the plan, that is $3$ times $1$ cm, so it stands for $3$ times $5$ m: $$\begin{aligned} \text{real length} &= 3 \times 5~\mathrm{m} &&\text{(}1~\mathrm{cm} \text{ stands for } 5~\mathrm{m}\text{)}\\ &= \textcolor{colorprop}{15~\mathrm{m}} \end{aligned}$$ The real pond is $15$ m long.

Definition — Plan and Scale

A plan (or a map) is a drawing of a real place, made much smaller. Every length is made smaller in the same way, so the shapes stay the same.
The scale of the plan says what $1$ cm on the plan stands for in real life, for example "$1$ cm stands for $5$ m". It is often drawn as a scale bar: every piece of $1$ cm stands for the same real length, and the numbers under the bar count the metres from $0$. Here each piece stands for $5$ m:

A map often has a legend, which says what each symbol stands for.

Method — From the Plan to Real Life

To find a real length from a plan:

Measure the length on the plan in centimetres, with a ruler or by counting the squares of a $1$ cm grid.
Multiply this number by what $1$ cm stands for. With "$1$ cm stands for $5$ m", $4$ cm stand for $4 \times 5 = 20$ m.
Write the unit of real life (m or km), not the centimetre of the plan.

Example

Here is the plan of a school playground, drawn on a grid of $1$ cm squares.

Find the real length and the real width of the playground.

Answer

On the plan, the playground is $6$ squares long and $3$ squares wide: $6$ cm and $3$ cm. The scale bar shows that $1$ cm stands for $10$ m. $$\begin{aligned} \text{real length} &= 6 \times 10~\mathrm{m} &&\text{(}1~\mathrm{cm} \text{ stands for } 10~\mathrm{m}\text{)}\\ &= \textcolor{colorprop}{60~\mathrm{m}}\\ \text{real width} &= 3 \times 10~\mathrm{m}\\ &= \textcolor{colorprop}{30~\mathrm{m}} \end{aligned}$$ The playground is $60$ m long and $30$ m wide.

Example

On this map, a straight road goes from Léa's home to the school. A ruler is placed along the road.

Read the length of the road on the map, then find the real length of the road.

Answer

The ruler shows that the road measures $7$ cm on the map. The scale bar shows that $1$ cm stands for $100$ m. $$\begin{aligned} \text{real length} &= 7 \times 100~\mathrm{m} &&\text{(}1~\mathrm{cm} \text{ stands for } 100~\mathrm{m}\text{)}\\ &= \textcolor{colorprop}{700~\mathrm{m}} \end{aligned}$$ Léa walks $700$ m from her home to the school.

Skills to practice

Reading a Scale
Finding a Real Length

V Drawing a Plan

Method — From Real Life to the Plan

To draw a real length on a plan:

Find how many times the real length contains what $1$ cm stands for. With "$1$ cm stands for $5$ m", $30$ m is $6$ times $5$ m, because $6 \times 5 = 30$: this is the division $30 \div 5 = 6$.
Draw that many centimetres on the plan: $6$ cm.
Check by going back to real life: $6 \times 5$ m $= 30$ m.

Example

A vegetable garden is a rectangle $20$ m long and $15$ m wide. Tom wants to draw its plan on squared paper with $1$ cm squares, with the scale "$1$ cm stands for $5$ m". Find the length and the width of the garden on the plan, then draw the plan.

Answer

Each real length is divided by $5$, the number of metres that $1$ cm stands for: $$\begin{aligned} 20 \div 5 &= 4 &&\text{(}4 \times 5 = 20\text{)}\\ 15 \div 5 &= 3 &&\text{(}3 \times 5 = 15\text{)} \end{aligned}$$ So the garden is $4$ cm long and $3$ cm wide on the plan. Check: $4 \times 5$ m $= 20$ m and $3 \times 5$ m $= 15$ m. The plan is a rectangle of $4$ cm by $3$ cm:

Skills to practice

Finding a Length on the Plan
Drawing a Plan to Scale

VI Going further

Skills to practice

Solving Route and Program Problems
Reasoning with Coordinates
Quiz

\(\rightarrow\)	move forward one square
\(\curvearrowright\)	quarter turn to the right (the robot stays on its square)
\(\curvearrowleft\)	quarter turn to the left (the robot stays on its square)