Coordinates & translations
🎯 In this topic you will
- Read and plot coordinates.
- Use knowledge of 2D shapes and coordinates to plot points to form lines and shapes.
- Translate 2D shapes on coordinate grids.
🧠 Key Words
- axes
- axis
- corresponds
- translate
Show Definitions
- axes: The two reference lines on a coordinate grid (horizontal and vertical) used to locate points.
- axis: A single reference line on a coordinate grid, such as the x-axis or the y-axis.
- corresponds: Means that one value or point matches or relates directly to another value or position.
- translate: To move a shape on a grid without rotating or resizing it, keeping the shape exactly the same.
🎮 Games and Coordinates
There are many people, both young and old, who enjoy playing computer games. Some games are played on a coordinate grid, where players must hit exact points or move shapes to different positions.
❓ EXERCISES
1. Match each point on the grid to its correct coordinates. One is done for you: A and ii.
i. $(-3, 2)$
ii. $(4, -2)$
iii. $(-2, -3)$
iv. $(2, 1)$
👀 Show answer
B → iv $(2, 1)$
C → i $(-3, 2)$
D → iii $(-2, -3)$
2. Here is a treasure map. Write down the coordinates of:
a. the volcano
b. the treasure chest
c. the shark
d. the pirate ship
👀 Show answer
b. Treasure chest → $(-3, 2)$
c. Shark → $(-3, -1)$
d. Pirate ship → $(4, -3)$
3. Draw axes from $-5$ to $+5$ on squared paper. Draw a trapezium with vertices at $P(-2, 1)$, $Q(-1, 3)$, $R(2, 3)$ and $S(3, 1)$.
a. Translate trapezium $PQRS$ $2$ squares right and $1$ square up. Label the trapezium $P'Q'R'S'$ and write down the coordinates of its vertices.
b. Translate trapezium $PQRS$ $2$ squares left and $5$ squares down. Label the trapezium $P''Q''R''S''$ and write down the coordinates of its vertices.
👀 Show answer
b. $P''(-4, -4)$, $Q''(-3, -2)$, $R''(0, -2)$, $S''(1, -4)$
🧠 Think like a Mathematician
The diagram shows points A, B and C on a coordinate grid.
Read what Zara says.
Zara says that point A is $(-2, \frac{5}{2})$.
a. Is Zara correct? Explain your answer. Think about different ways that you can write the coordinates of the point A.
b. Write down the coordinates of point B and C in as many different ways as you can.
Show Answers
- a: Yes, Zara is correct. The coordinate $(-2,\frac{5}{2})$ means the point is at $x=-2$ and $y=2.5$. This can also be written as $(-2,2.5)$.
- b:
B can be written as $(1,-2)$ or $(1,-\frac{4}{2})$.
C can be written as $(3,\frac{3}{2})$ or $(3,1.5)$.
❓ EXERCISES
4. Draw axes from $-6$ to $+6$ on squared paper. Plot the points $J(-2, 2)$, $K(-2, -1)$ and $L(1, -1)$.
a. Write down the coordinates of $M$ so that $J$, $K$, $L$ and $M$ are the vertices of a square.
b. Write down two possible coordinates of $M$ so that $M$ is a point on the line segment $JL$.
c. Write down two possible coordinates of $M$ so that $J$, $K$, $L$ and $M$ are the vertices of a parallelogram.
d. Write down two possible coordinates of $M$ so that $J$, $K$, $L$ and $M$ are the vertices of a kite.
e. In which parts $\mathrm{a}$ to $\mathrm{e}$ are there more than two answers for the coordinates of $M$? Explain why.
👀 Show answer
b. Two possible coordinates are, for example, $(-1, 1)$ and $(0, 0)$.
c. Two possible coordinates are, for example, $(1, 2)$ and $(-5, 2)$.
d. Two possible coordinates are, for example, $(1, 2)$ and $(-5, 2)$.
e. There are more than two answers in $\mathrm{b}$, $\mathrm{c}$ and $\mathrm{d}$. In part $\mathrm{b}$, any point on the line segment $JL$ is possible. In parts $\mathrm{c}$ and $\mathrm{d}$, different valid positions of $M$ can make a parallelogram or a kite.
🧠 Think like a Mathematician
The diagram shows triangles A to I placed on a coordinate grid.
Here are nine translation cards:
- A → C
- E → G
- C → B
- C → E
- I → H
- E → H
- B → H
- C → D
- F → G
a. Sort the cards into groups of equivalent translations. Describe the translation for each group.
b. Write two more translation cards using four of the triangles A to I that would form a different group. Describe the translation for this new group.
Show Answers
- a: Cards that move shapes the same distance and direction form equivalent groups. For example, cards such as A → C and E → G represent the same translation (moving left by the same number of squares). Another group may include C → B and B → H if they translate the triangle in the same direction and distance.
- b: Example of another group: A → B and C → H. These might represent a translation moving several squares to the right and up. The exact translation can be described by counting the horizontal and vertical movement on the grid.
🧠 Think like a Mathematician
Lukman is making a pattern by translating a kite. He uses the same translation every time. The diagram shows the 1st and 2nd kites.
a. What translation does he use?
b. Copy the diagram and draw the 3rd and 4th kites in the pattern.
c. Lukman marks a cross on the same vertex of the kite every time he translates the kite. Copy and complete this table showing the coordinates of this vertex.
| Kite | 1st | 2nd | 3rd | 4th |
|---|---|---|---|---|
| Coordinates | $(1,7)$ | $(3,6)$ | ? | ? |
d. What do you notice about the $x$-coordinate as the pattern continues?
e. What do you notice about the $y$-coordinate as the pattern continues?
f. Without drawing the 5th and 6th kites in the pattern, write down the coordinates of the vertex marked with a cross for each kite. Show how you worked out your answer.
g. Draw your own pattern on a coordinate grid using your own shape and translation. Investigate what happens to the coordinates of one vertex of your shape as the pattern continues.
h. Think about the patterns you found and explain what happens to the coordinates as the shape keeps translating.
Show Answers
- a: The translation moves the kite $2$ units to the right and $1$ unit down each time.
- b: The 3rd and 4th kites are created by repeating the same translation: each kite moves $2$ squares right and $1$ square down from the previous one.
- c: 3rd kite: $(5,5)$, 4th kite: $(7,4)$.
- d: The $x$-coordinate increases by $2$ each step.
- e: The $y$-coordinate decreases by $1$ each step.
- f: 5th kite: $(9,3)$, 6th kite: $(11,2)$.
- g: Different shapes will follow a similar coordinate pattern depending on the translation vector used.
- h: Each translation adds a constant amount to the $x$-coordinate and subtracts a constant amount from the $y$-coordinate.