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Mathematical Architects · Geometry

Reasoning with Shapes — Module Hub & Visuals

TEKS G.3 · G.5 · G.6

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Lesson Slides & PDFs

Student-facing lesson slide PDFs are grouped by topic below for review, catch-up, and family support.

📖 Student Edition for this module (PDF) ✍ Skills Practice — Student Edition (PDF)

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Start Here — What Module 1 Is About

What this module is for, and what you will be able to do by the end of it.

Module 1 lives on the coordinate plane. Here you can move a triangle the four ways geometers do — translate, reflect, rotate, dilate — and watch the mapping rule, the side length, and the midpoint change in real time. This is the same idea behind every congruence and similarity argument you'll write all year.

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Key Vocabulary

The precise words for what you just investigated. The standards this module carries are in the next section.

Rigid motion (isometry)

A transformation that preserves distance and angle measure — translation, reflection, and rotation. The image is congruent to the pre‑image.

Translation

A slide by \((dx,dy)\): \((x,y)\to(x+dx,\,y+dy)\). Every point moves the same direction and distance.

Reflection

A flip across a line of reflection. Across the x‑axis: \((x,y)\to(x,-y)\); the y‑axis: \((x,y)\to(-x,y)\); the line \(y=x\): \((x,y)\to(y,x)\).

Rotation

A turn about a center (here, the origin). \(90^{\circ}\): \((x,y)\to(-y,x)\); \(180^{\circ}\): \((-x,-y)\); \(270^{\circ}\): \((y,-x)\).

Dilation

A resize by scale factor \(k\) about a center: \((x,y)\to(kx,ky)\). It preserves shape but not size — the image is similar, not congruent.

Distance & midpoint

For points \(P(x_1,y_1)\), \(Q(x_2,y_2)\): \(PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) and \(M=\left(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\right)\).

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Standards in This Module

The Texas Essential Knowledge and Skills this module is built to teach. Geometry is assessed locally — there is no state EOC for this course.

G.3A G.3B G.3C G.5A G.5B G.5C G.6A G.6C

G.3 — describe and perform transformations and their compositions on the coordinate plane. G.5 — use coordinate geometry (distance, midpoint, slope) to investigate figures. G.6 — prove congruence by mapping one figure onto another with a sequence of rigid motions.

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How to Read It

What's on the screen, and a few experiments worth running.

What you're seeing

The faded, dashed triangle is the pre‑image — the original figure, with vertices A, B, C. The solid triangle is the image — where the figure lands after the transformation, with vertices A′, B′, C′.

The Mapping rule row in the readout under the lab gives the transformation in coordinate notation, e.g. \((x,y)\to(x+3,\;y-2)\). On the plane itself, the thick highlighted segment is A′B′; its length and midpoint (the ringed dot at its centre) recompute every time you move a control. Notice that translate, reflect, and rotate keep the length the same — that's what makes them rigid motions — while dilate scales it.

Try this

  1. Set Translate to \(dx=3,\ dy=-2\). Watch A′B′ stay length 4 — the figure slid but did not change size.
  2. Switch to Reflect and choose y = x. The dashed guide line appears; the coordinates of every point swap.
  3. Use Rotate 90°, then 180°, then 270°. Track how \((x,y)\) becomes \((-y,x)\), then \((-x,-y)\), then \((y,-x)\).
  4. Open Dilate and slide \(k\) from \(1\) to \(2\). The midpoint and the length both scale — shape is kept, size is not. That's similarity, not congruence.
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Step 5 · 2D Precision Lab

The Transformation Studio

One live tool, hand-built right here in the studio — no Desmos or GeoGebra account needed.

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Want to See This in 3D?

Optional, and always second. The lab above is where you measure and calculate; this is the same figure as an everyday object you can turn over and inspect. Open it when the lab already makes sense — or skip it entirely.

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Keep Going

Back to the course, or into the planning documents.

Back to Geometry

Return to the full course page — all five modules, materials, and resources.

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Course Syllabus

Policies, the studio learning environment, grading, and the full itinerary.

Read the Syllabus

Pacing Guide

Every module mapped across the grading periods of the scholar calendar.

View the Pacing Guide

Module title, topic structure, and TEKS alignment adapted from TEA Bluebonnet Learning — Secondary Mathematics (Edition 1), itself adapted from Carnegie Learning, licensed CC BY-NC 4.0. Non-commercial classroom use.