Coordinate Choreography
Students choreograph a dance for a figure on the coordinate plane, writing the coordinate rule for every move and challenging classmates to reproduce it.
Common Core standards covered
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Teacher-facing overview
Big idea: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Student target: “I can write and apply coordinate rules for translations, reflections, rotations and dilations, and say which of them change a figure's size.”
Success criteria: I can write the coordinate rule for a translation, reflection, rotation and dilation in the form (x, y) → ( , ). · I can apply a coordinate rule to every vertex of a figure and list the image coordinates. · I can sort the six moves into rigid motions and dilations, and say what happens to side lengths in each. · I can write a sequence of moves precisely enough for a partner to reproduce it exactly.
Essential question
How can you predict exactly where a figure will land without drawing it first?
Learning target: I can write and apply coordinate rules for translations, reflections, rotations and dilations, and say which of them change a figure's size.
Success criteria
- I can write the coordinate rule for a translation, reflection, rotation and dilation in the form (x, y) → ( , ).
- I can apply a coordinate rule to every vertex of a figure and list the image coordinates.
- I can sort the six moves into rigid motions and dilations, and say what happens to side lengths in each.
- I can write a sequence of moves precisely enough for a partner to reproduce it exactly.
Why this matters
Coordinate rules are what turn a transformation from something you do with tracing paper into something you can compute, predict and program. Every time a phone rotates a photo, a game moves a sprite, or a CAD program mirrors a part, it is applying these rules to coordinates. A student who can only transform by hand is limited to the size of the paper; a student who can write the rule can transform anything, including figures far too large to draw.
Students should already be able to
- Plot and read ordered pairs in all four quadrants (6.NS.C.8).
- Recognize translations, reflections and rotations informally — Unit 1 of this sequence.
- Multiply integers and negate a number.
Vocabulary to define explicitly
- Preimage
The original figure, before any move is applied.
Example: Rectangle KLMN is the preimage in this lesson.
Not an example: K′L′M′N′ is not the preimage — it is what a move produced.
🔒 5 more terms with examples and non-examples — Image, Prime notation, Center of dilation, Scale factor, Rigid motion — are defined in the full lesson. See plans →
Where this shows up outside the classroom
Assessment plan
Formative — for feedback during the lesson
- The rules chart, checked during the mini-lesson: is each rule written in general (x, y) form rather than for one point only?
- The partner choreography swap: a mismatch pinpoints exactly which rule is not yet secure.
- Circulating during the dilation task: watch for students who double only one coordinate.
Summative — the graded evidence: The exit ticket, collected as the graded evidence for 8.G.A.3.
The task students work on
Rectangle KLMN has vertices K(2, 1), L(6, 1), M(6, 3) and N(2, 3). You will build a rules chart that predicts where any point lands, then choreograph a routine for a partner to reproduce from the rules alone.
Complete the rule for each move, then apply it to the point K(2, 1).
| Transformation | Rule for (x, y) | Image of K(2, 1) |
|---|---|---|
| Translate 3 left, 2 up | ||
| Reflect across the x-axis | ||
| Reflect across the y-axis | ||
| Rotate 180° about the origin | ||
| Rotate 90° counterclockwise about the origin | ||
| Dilate by a factor of 2 about the origin |
Anticipated student strategies
Share concrete before abstract: the ray drawing shows what dilation means, the coordinate rule generalizes it, and the structure shortcut rewards students who see the shape rather than four separate points. If no student produces the first strategy, demonstrate it yourself before discussing the rule.
- Graphical: stretch along the ray from the origin. The student draws a ray from the origin through K and marks K′ twice as far out, then repeats for L, M and N. The image lands at K′(4, 2), L′(12, 2), M′(12, 6), N′(4, 6) without any arithmetic on paper.
Share this first — it grounds what a dilation physically does before any rule appears.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
Students answer (3, −4). Draw out that x was untouched. Record the pattern in words before any symbols appear, and introduce the words preimage and image for the point before and after.
Listen for: The y-coordinate changed sign; the x-coordinate stayed exactly the same. The point is the same distance from the x-axis, on the other side.
Work through the five rigid motions as a class, testing each on one point before generalizing to (x, y). Each rule must earn its place in the chart by being checked on the grid. Post the vocabulary cards where every student can see them.
- Establish counterclockwise as the positive direction of rotation and hold to it all year.
- Do the 180° rotation before the 90°, since negating both coordinates is easier to see.
- Name prime notation explicitly: K becomes K′, and after a second move K″.
Listen for: One point could fit several different rules. A second point — especially one in another quadrant — rules the wrong ones out.
Listen for: The point ends up on the opposite side of the origin in both directions — left-right and up-down — so x and y both flip.
🔒 The remaining 5 steps — fully scripted with teacher-says lines, discourse questions, look-fors, and interventions — are in the full lesson. See plans →
Misconceptions and how to respond
- “Reflecting across the x-axis changes the x-coordinate.” Test a point that is not on an axis, such as (3, 4) → (3, −4). Name the axis you reflect across as the one that stays fixed.
- “A dilation is just another rigid motion.” — the teacher response is in the full lesson.
- “Rotating 90° clockwise and counterclockwise give the same result.” — the teacher response is in the full lesson.
- “Doubling the side lengths doubles the area.” — the teacher response is in the full lesson.
🔒 Every misconception ships with a tested classroom response. See plans →
Scaffolded practice
The workbook follows gradual release rather than a page of blank prompts: 1 fully worked, 2 partially completed, then 4 independent.
- I doReflect K(2, 1) across the x-axis.
🔒 6 more prompts across all three release tiers are in the full lesson. See plans →
Worked solutions and scaffolds for every prompt are in the downloadable kit, included with membership. See plans and pricing.
Exit ticket
- Point A(-3, 5) is reflected across the y-axis. Write the coordinates of A′.
🔒 2 more exit-ticket questions are in the full lesson. See plans →
Worked answers, the completed answer tables, and the scoring rubric are in the downloadable kit, included with membership. See plans and pricing.
Printable handouts in the kit
All handouts are in Printable_Handouts.pdf inside the kit download.
Materials and classroom resources
Student experience
Students enter through a low-floor prompt, build or sketch a model, test a strategy, revise their thinking, collaborate, and present a final solution. The task remains accessible while allowing for sophisticated reasoning.
Detailed facilitation notes
- Step 1: Build a class chart of coordinate rules for common reflections, rotations, and translations. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Students choreograph a four-move routine for a starting figure and record the rules. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Partners perform each other's routine using only the written rules. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Compare final positions and debug any mismatched steps. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Keep the dilation in the lesson for every student. Scaffold with a partially completed rules chart, the vocabulary reference cards on each table, color-coding each move on the grid, and a two-move routine instead of three. Extend by composing a dilation with a rotation and asking whether changing the order changes where the figure lands.
Language support: “I started by…”, “My model shows…”, and “I know my answer is reasonable because…”.
Extension: Require a second solution, alter a condition, or have students write a challenge for another group.
Accessibility: Manipulatives, enlarged visuals, oral response choices, partner reading, and reduced-copy student pages.
Assessment plan
Formative checks: Strategy choice, accuracy, representation, collaboration, and explanation.
End evidence: Students apply a given coordinate rule to a figure and state the transformation it describes.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.