098i - Translations on the Cartesian Plane

Lesson 98 — Translations on the Cartesian Plane

Strand: Space | Descriptor: AC9M7SP03 | Duration: 45 minutes

Learning Intentions

• To describe and perform translations of points and shapes using coordinates.
• To express a translation as a rule acting on .

Success Criteria

I can:

1. Translate a point or shape by a given number of units right/left and up/down.
2. Write a translation as a coordinate rule, e.g. .
3. Find the translation that maps one shape onto another.
4. Explain what a translation preserves.

Warmup

(6 minutes — coordinate recall from Lesson 89, mini whiteboards)

On a grid from to :

1. Plot , , .
2. Start at and move right and down. Where do you land?
3. Start at and move left and up. Where do you land?
4. What happens to the -coordinate when you move right? Left? What about moving up and down?

Answers: 2. ; 3. ; 4. Right adds to , left subtracts; up adds to , down subtracts.

Q4 is the whole lesson’s grammar. Record it: right/up are positive; left/down are negative.

Activities

Activity 1 — Explicit Instruction: Translation as a Rule (14 min)

Definition. A translation slides every point of a shape the same distance in the same direction. Nothing turns, nothing flips, nothing changes size.

The rule notation:

where is the horizontal shift (right positive) and the vertical shift (up positive).

I do — translating a point. Translate by :

Notation to establish: the image is labelled with a dash — , read “P dash” or “P prime”. Original and image are distinguished in every answer.

I do — translating a triangle. , , under :

Point	Image

Plot both triangles. Ask what has changed and what has not.

What a translation preserves — record all four: side lengths, angles, area, and orientation (the shape does not turn or flip). Only position changes. The image is congruent to the original.

We do:

1. Translate by .
2. Translate by .
3. Describe in words: .
4. Write as a rule: ” left and up”.

(Answers: ; ; units up, no horizontal movement; .)

Activity 2 — Practice and Reversal (14 min)

Grid paper. Every task plotted as well as calculated.

Set A — apply the rule.

1. under
2. under
3. Square , , , under

Set B — find the rule. Each pair shows an original and its image:

7. Triangle

Set C — reverse the translation.

8. After , a point landed at . Where did it start?
9. Write the rule that undoes .

Socratic scaffolding for Set B:

Prompt	Purpose
Compare the two -coordinates. What was added?	Image minus original gives .
And the -coordinates?	Same subtraction gives .
Check on a second point of the shape.	One point suggests; all points must agree — that is what makes it a translation.
For Q7, do all three vertices give the same shift?	each time ✓ — confirms a genuine translation.

(Answers: 1. ; 2. ; 3. , , , ; 4. ; 5. ; 6. ; 7. ; 8. ; 9. .)

Activity 3 — Inquiry: Combining Translations (8 min)

Pairs.

1. Apply to the point , then apply to the result.
2. What single translation would have done both at once?
3. Does the order matter? Try the reverse.
4. What single translation returns a shape to where it started after ?

Socratic scaffolding:

Prompt	Purpose
Track the point through both steps.	.
Compare start and finish.	rose by ; fell by .
So the single rule is…?	— the shifts simply add.
Reverse the order and check.	— same destination.
Why does order not matter?	Adding or to gives the same total; translations commute.
Q4’s undo?	— negate both shifts.

Extension: a shape is translated by , then . Where does it end? (Exactly where it started — the two are inverses. This “identity” idea returns in Lesson 100 with rotations.)

Checks for Understanding

(5 minutes — exit ticket)

1. Translate by .
2. Write the rule for ” right and down”.
3. . Write the translation rule.
4. Name two things a translation does not change.
5. Reasoning. A student says the triangle maps to by a translation. Check the claim.

Answers: 1. ; 2. ; 3. ; 4. Any two of: side lengths, angles, area, orientation; 5. First two vertices shift by , but the third shifts by — not a translation, since the shifts disagree.

Common Misconceptions

Misconception	How to pre-empt it
Adding when moving left or down.	The warmup’s Q4 grammar, recorded and referred to all lesson.
Translating only one vertex and redrawing by eye.	Every vertex is computed and listed in a table.
Confusing the rule’s direction when finding it (original minus image).	Always image minus original; check on a second point.
Believing a translation can change size or orientation.	The four preserved properties, recorded explicitly.
Dropping the dash notation, so image and original are confused.	Marked in every answer.
Assuming any correspondence between two congruent shapes is a translation.	Exit Q5’s counterexample.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A point is translated by and lands at . Where did it start?

Answer

.

E2 (AMC Junior style). A shape is translated by , then by , then by . Find the single equivalent translation.

Answer

: ; : : .

E3 (Challenge). A square has vertices , , , . After a translation, one vertex is at . List all four possible translation rules and the resulting squares.

Answer

Any of the four original vertices could be the one that landed at : rules , , , . The problem is under-determined — a good reminder to check what is given.

E4 (Challenge). A translation maps to for the particular point . Write the rule, and explain why this rule does not swap coordinates in general.

Answer

needs . Applied to it gives , not — a translation adds fixed amounts; genuine coordinate-swapping is a reflection (Lesson 99), not a translation.

Homework

1. Translate each point by : (a) (b) (c) (d) .
2. Translate each by : (a) (b) (c) .
3. Write the translation rule for each: (a) (b) (c) (d) .
4. Triangle , , is translated by . (a) List the image vertices. (b) Plot both triangles. (c) State the side lengths of each — what do you notice?
5. A shape is translated by and then by . Find the single equivalent rule.
6. Write the rule that undoes .
7. After a translation, the point came from . Write the rule and apply it to .
8. Reasoning. Explain why a translation cannot change a shape’s area.
9. Reasoning. Two triangles are congruent but one is upside-down relative to the other. Can a translation map one to the other? Explain.
10. Challenge. A rectangle’s vertices are , , , . It is translated so that it sits entirely in the third quadrant (both coordinates negative for every vertex). Write one rule that achieves this, and explain the smallest shifts that would work.

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) (b) (c) . Q3 — (a) (b) (c) (d) . Q4 — (a) , , (c) identical side lengths — translations preserve them. Q5 — . Q6 — . Q7 — ; . Q8 — every point moves the same way, so all distances between points are unchanged; area depends only on those distances. Q9 — no: translations preserve orientation, so a flipped shape needs a reflection. Q10 — e.g. puts every vertex strictly negative; the smallest whole-number shifts are and , since must become negative in both coordinates.