Coordinates and translation
🎯 In this topic you will
- Use coordinates to compare positions on a grid
- Use coordinates to plot points that form 2D shapes
- Translate 2D shapes accurately on a coordinate grid
🧠 Key Words
- coordinates
- translation
Show Definitions
- coordinates: A pair of numbers that describe the exact position of a point on a grid, usually written as $(x, y)$.
- translation: A movement of a shape to a new position on a grid without rotating or flipping it; every point moves the same distance and direction.
Understanding Translation
Translation makes it possible to describe the direction something moves in and how far it moves. In this section you will learn how to describe how a shape moves on a grid.
Looking at the Grid
Look at this grid.
Describing Movement
Can you describe how each rabbit can move through the grid to get to the carrot?
❓ EXERCISES
1. Draw a coordinate grid on squared paper with the $x$-axis and the $y$-axis labelled from $0$ to $5$.
For each pair of coordinates, write the coordinates of the point that is closer to $(0,0)$.
a. $(3,0)$ or $(5,0)$
b. $(0,2)$ or $(0,4)$
c. $(0,2)$ or $(3,0)$
d. $(4,1)$ or $(3,5)$
e. $(2,3)$ or $(3,1)$
👀 Show answer
The point closer to $(0,0)$ has the smaller value of $x^2 + y^2$.
a. $(3,0)$ is closer than $(5,0)$.
b. $(0,2)$ is closer than $(0,4)$.
c. $(0,2)$ is closer than $(3,0)$.
d. $(4,1)$ is closer than $(3,5)$.
e. $(3,1)$ is closer than $(2,3)$.
2. On the coordinate grid, there is a tree at the position with coordinates $(2,4)$. Sarah is standing at the position $(1,3)$. Pasha is at the position $(3,1)$. Who is closer to the tree?
👀 Show answer
Distance squared from the tree to Sarah: $(2-1)^2 + (4-3)^2 = 1 + 1 = 2$.
Distance squared from the tree to Pasha: $(2-3)^2 + (4-1)^2 = 1 + 9 = 10$.
Since $2 < 10$, Sarah is closer to the tree.
3. The numbers are not labelled on this coordinate grid. The cross marks the position $(3,5)$. Which letter is closest to the position $(4,2)$?
👀 Show answer
By reading from the grid, the letter whose position is nearest to $(4,2)$ is C.
4.
a. Estimate the coordinates for the position of letter $Z$.
b. Write down what you would say to convince your partner that your estimate is correct.
c. Discuss your answer with your partner. How could you or your partner improve your estimates? Alternatively, how could you make your argument more convincing?
👀 Show answer
a. From the diagram, a reasonable estimate is that $Z$ is at about $(6,6)$ (any close estimate with good reasoning is acceptable).
b. You might say: “The cross is at $(10,10)$, and $Z$ is a little over halfway from $(0,0)$ to that cross along both axes. Half of $10$ is $5$, so a little more than $5$ on each axis gives about $(6,6)$.”
c. Estimates could be improved by marking a clear scale on each axis, counting equal steps carefully, or using a ruler to compare distances. Explaining how the position of $Z$ relates to $(0,0)$ and $(10,10)$ makes the argument more convincing.
5. $(1,3)$, $(1,2)$ and $(4,2)$ are the coordinates of three vertices of a rectangle. Plot these on a grid. What are the coordinates of the other vertex?
👀 Show answer
Points $(1,3)$ and $(1,2)$ lie on a vertical side, and $(1,2)$ and $(4,2)$ lie on a horizontal side. The missing vertex must be directly opposite $(1,3)$ across the rectangle, so it has $x$-coordinate $4$ and $y$-coordinate $3$.
The other vertex is at $(4,3)$.
6. $(2,4)$ and $(4,2)$ are the coordinates of two vertices of a square. Plot these on a grid. What could the other vertices of the square be? Find all the possible solutions.
👀 Show answer
The distance between $(2,4)$ and $(4,2)$ is $2\sqrt{2}$, so they can be either adjacent vertices or opposite vertices of a square.
Square 1 (adjacent vertices, leaning up-right):
Other vertices: $(6,4)$ and $(4,6)$.
Square 2 (adjacent vertices, leaning down-left):
Other vertices: $(2,0)$ and $(0,2)$.
Square 3 (given points are opposite vertices):
Other vertices: $(4,4)$ and $(2,2)$.
So there are three possible squares, with the pairs of remaining vertices:
$(6,4)$ and $(4,6)$; $(2,0)$ and $(0,2)$; $(4,4)$ and $(2,2)$.
🧠 Reasoning Tip
There are three possible solutions to question $6$. Have you found them all? Think about which solutions you found quickly and which took more time. How can you remember to look for different solutions in the future?
7. Look at the shapes on this grid.
Name the shape that has been translated and how its colour has changed. For example, you could write ‘Orange square to red square’ or ‘square B to square A’.
a. $4$ squares to the right
b. $3$ squares down
c. $1$ square left and $2$ squares up
d. $1$ square right and $1$ square up
👀 Show answer
a. Purple triangle $A$ translated $4$ squares to the right becomes the red triangle $B$ (colour changes from purple to red).
b. Red rectangle $A$ translated $3$ squares down becomes the orange rectangle $B$ (colour changes from red to orange).
c. Yellow arrow-shaped quadrilateral $B$ translated $1$ square left and $2$ squares up becomes the orange arrow-shaped quadrilateral $A$ (colour changes from yellow to orange).
d. Dark blue square block $A$ translated $1$ square right and $1$ square up becomes the turquoise square block $B$ (colour changes from dark blue to turquoise).
8. Copy this shape onto squared paper. Translate the shape $3$ squares to the right and $2$ squares down.
👀 Show answer
Every vertex of the shape should be moved $3$ squares to the right (in the positive $x$-direction) and $2$ squares down (in the negative $y$-direction). The translated shape is congruent to the original and lies exactly $3$ squares right and $2$ squares below it on the grid.
🧠 Think like a Mathematician
Task: Make a repeating pattern using translations of a 2D shape.
Method:
- Choose a 2D shape and draw it on squared paper. Label this shape “1”.
- Choose a number of squares to translate your shape horizontally and a number of squares to translate your shape vertically.
- Translate the shape using these values.
- Translate the new shape again by the same number of squares horizontally and vertically.
- Continue translating each new shape to make a repeating pattern.
Challenge: After creating your pattern, identify which translation you used to generate it. Then study your pattern closely and describe the translations that appear. You are characterising when you identify and describe the translations.
Follow-up Questions:
Show Answers
- 1: My translation used h squares horizontally and v squares vertically (for example, $3$ squares right and $2$ squares up). Each copy was moved by exactly the same amounts.
- 2: You can check that the translation is consistent by measuring the horizontal and vertical distances between corresponding vertices of consecutive shapes—these distances must be identical each time.
- 3: Increasing the translation values spaces the pattern out more, creating a larger gap between each repeated shape. Decreasing the values makes the pattern denser and the shapes closer together.