Angles, direction, position and movement
🎯 In this topic you will
- Compare and describe angles.
- Identify the directions north, south, east, and west.
- Use words to describe position, direction, and movement.
- Create simple grid maps.
🧠 Key Words
- cardinal points
- compass
- right angle
Show Definitions
- cardinal points: The four main directions—north, south, east, and west—used for describing where something is or where it is going.
- compass: A tool that shows direction, usually with a needle that points to north, to help you navigate or read a map.
- right angle: An angle of $90^\circ$, like the corner of a square or rectangle.
🔄 Measuring a Turn
An angle is a measurement of turn. It tells us the amount of turn from one position to another.
🟦 Right Angle Turn
A quarter turn in any direction is called a right angle.
➡️ A Straight Line Turn
Two right angles together make a straight line. This is also a half turn.
❓ EXERCISES
$1$. We are going to make a right angle measure.
Take a piece of paper.
Fold it in half.
Fold it in half again.
You have made a right angle.
a. Find right angles in your classroom using your right angle measure. Look at your table, your book or a window.
b. Draw what you find. Mark the angle.
👀 Show answer
a. Examples of right angles you might find are the corner of a table, the corner of a book, and the corner of a window frame (each is a quarter turn, a right angle).
b. Draw the objects you found and mark the right angle at the corner (for example, draw the table corner and draw a small square at the corner to show the right angle).
$2$. Use your right angle measure to find in your classroom four right angles, four angles that are greater than a right angle and four angles that are less than a right angle.
Write and draw the angles that you found.
| Right angle | Greater than | Less than |
|---|---|---|
👀 Show answer
Your answers will depend on your classroom. Here is one example set you could write/draw:
- Right angle: table corner; book corner; whiteboard corner; window frame corner.
- Greater than: an open door wider than a right angle; a pair of scissors opened wide; the angle between a chair back and the seat if it leans far back; a cupboard door opened wide.
- Less than: a slightly open door; scissors opened a little; the angle in a letter “V”; the corner of a set square that is smaller than a right angle.
$3$. You are at the campsite.
You are at the campsite.
a. What can you see if you look west?
b. What can you see if you look south?
c. In which direction is the bridge?
d. In which direction is the forest?
Now face west.
Which direction are you facing if you turn
e. two right angles clockwise?
f. one right angle anticlockwise?
g. three right angles clockwise?
h. Using the map, write two questions and answers of your own.
$1$ ________________________________
$2$ ________________________________
Will your partner be able to answer them?
👀 Show answer
a. The school.
b. The pond.
c. North.
d. East.
e. East (west → turn clockwise twice → north → east).
f. South (west → turn anticlockwise once → south).
g. South (west → turn clockwise three times → north → east → south).
h. Example: “What is east of the campsite?” Answer: “The forest.” / “What is north of the campsite?” Answer: “The bridge.”
$4$. a. Write in the missing cardinal points.
Imagine that you are facing south.
Write the instructions for the route from start to finish.
Use the words left, right and forward.
Start with: Move forward south one square, turn right ...
b. Find a different route to go from start to finish.
Write instructions for someone to be able to follow your route.
👀 Show answer
a. Missing cardinal points: E is to the right, S is down, and W is to the left (with N already shown).
Route instructions (facing south): Move forward south $1$ square, turn right, move forward $4$ squares, turn right, move forward $3$ squares, turn left, move forward $1$ square, turn right, move forward $4$ squares to finish.
b. One possible different route: Move forward south $1$ square, turn right, move forward $4$ squares, turn right, move forward $7$ squares to finish. (Any correct set of left/right/forward instructions that gets from start to finish is acceptable.)
🧠 Think like a Mathematician
Task: Use squared paper (or draw a grid) to investigate different routes.
The hippo wants to get to the river.
Each block is one step.
Rule: The hippo always walks along the paths and it always walks towards the river.
Method:
- Copy the grid onto squared paper (or use squared paper directly).
- Mark the hippo’s starting point and the river’s position.
- Trace possible routes using only steps that move closer to the river (east or south).
- Record each route using letters: E for one step east and S for one step south.
Follow-up Questions:
👀 show answer
- a) The hippo can walk $3$ steps east before it must turn (that takes it as far east as it can go while still staying on the paths).
- b) The total number of south steps is $3$.
- c) The total number of east steps is $3$.
- d) Every route is just a different order of $3$ E’s and $3$ S’s, so the number of routes is $\binom{6}{3}=20$.
- e)No. Walking north would move away from the river, so it breaks the “always towards the river” rule.
- f)No. Walking west would also move away from the river.
- g) Record routes using strings like $EESSES$, or use a table/tree diagram. A neat method is: write every route as a 6-letter code made of E and S (exactly three of each).
- h) If the grid changes size, the totals change. For a grid with $m$ steps east and $n$ steps south: $\text{routes}=\binom{m+n}{m}=\binom{m+n}{n}$. Smaller blocks usually mean more blocks (bigger $m,n$) so you get many more possible routes.
Note: These answers assume the grid is $3$ blocks east by $3$ blocks south (as shown).
❓ EXERCISES
$5$. Write the instructions to get from START HERE to the school.
👀 Show answer
From START HERE, go west$2$ squares, then go north$2$ squares to reach the school.