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Mathematical Architects · Geometry · Foundations
Get Ready: Module 1 — Reasoning with Shapes
Everyone starts somewhere. Before we reason with shapes, let's pour the floor the whole
module will stand on. This page walks through every skill you'll lean on in Module 1 —
slowly, plainly, one small step at a time. There's no rush and no quiz here. Go at your own pace,
reread anything twice, and come back as often as you need.
👋 A note before you begin. Feeling shaky on the basics doesn't mean you're
behind — it means you're being honest about where to start, which is exactly what good
architects do. Build the floor first, and the building goes up easily.
🧱
Skills to Build First
Six small foundations. Read the idea, follow the worked example, then try one yourself.
Skill 01
The Coordinate Plane & Ordered Pairs
What it is: a flat grid made of two number lines that cross at the
origin \((0,0)\). Across is the \(x\)-axis; up-and-down is the \(y\)-axis. Every
point gets an address called an ordered pair \((x,y)\): the first number is how
far right (or left, if negative), the second is how far up (or down).
Why you need it: all of Module 1 happens on this grid — you'll plot
triangles and slide, flip, and turn them, so you must read and place points with confidence.
Worked example — plot \((3,-2)\)
- Start at the origin \((0,0)\), the center where the axes cross.
- The first number is \(3\): move 3 to the right along the \(x\)-axis.
- The second number is \(-2\): from there move 2 down (negative means down).
- Mark the point. It sits in the lower-right region (Quadrant IV).
Try it
1. Where do you end up if you plot \((-4,\,1)\)? Describe the moves.
2. A point is 0 right/left and 5 up. Write it as an ordered pair.
Show answer
1. From the origin, move 4 left (because \(x=-4\)), then 1 up (because \(y=1\)). You land in the upper-left region, Quadrant II.
2. No left/right movement means \(x=0\); 5 up means \(y=5\). The point is \((0,\,5)\) — it sits right on the \(y\)-axis.
Skill 02
The Pythagorean Theorem
What it is: in a right triangle (one with a square \(90^\circ\) corner),
the two short sides \(a\) and \(b\) and the longest side \(c\) (the hypotenuse,
across from the right angle) always obey \(a^2+b^2=c^2\). Why you need it: the
distance between two points on the grid is really just the hypotenuse of a right triangle —
this theorem is the engine behind the distance formula you'll use all module long.
Worked example — legs 3 and 4, find the hypotenuse
- Write the rule: \(a^2+b^2=c^2\).
- Fill in the legs: \(3^2+4^2=c^2\).
- Square each: \(9+16=c^2\), so \(25=c^2\).
- Undo the square by taking the square root: \(c=\sqrt{25}=5\).
The hypotenuse is \(5\).
Try it
1. A right triangle has legs \(6\) and \(8\). Find the hypotenuse \(c\).
2. The legs are \(5\) and \(12\). Find \(c\).
Show answer
1. \(6^2+8^2=c^2 \Rightarrow 36+64=100 \Rightarrow c=\sqrt{100}=10\).
2. \(5^2+12^2=c^2 \Rightarrow 25+144=169 \Rightarrow c=\sqrt{169}=13\).
Skill 03
Slope
What it is: a number that says how steep a line is — "rise over run," or how
much the line goes up for every step it goes across. Between two points it is
\(m=\dfrac{y_2-y_1}{x_2-x_1}\). Why you need it: on the coordinate plane, slope
is how you tell whether sides are parallel (equal slopes) or
perpendicular (slopes that multiply to \(-1\)) — key for recognizing squares,
right angles, and parallelograms.
Worked example — slope through \((1,2)\) and \((4,8)\)
- Label the points: \((x_1,y_1)=(1,2)\) and \((x_2,y_2)=(4,8)\).
- Rise = change in \(y\): \(y_2-y_1 = 8-2 = 6\).
- Run = change in \(x\): \(x_2-x_1 = 4-1 = 3\).
- Divide: \(m=\dfrac{6}{3}=2\). The line rises 2 units for every 1 across.
Try it
1. Find the slope through \((2,1)\) and \((6,9)\).
2. Find the slope through \((0,5)\) and \((4,5)\). What does the answer tell you about the line?
Show answer
1. \(m=\dfrac{9-1}{6-2}=\dfrac{8}{4}=2\).
2. \(m=\dfrac{5-5}{4-0}=\dfrac{0}{4}=0\). A slope of \(0\) means the line is perfectly horizontal (flat) — it never rises.
Skill 04
Solving Simple Equations
What it is: finding the value of an unknown (like \(x\)) by keeping the equation
balanced — whatever you do to one side, you do to the other, until \(x\) is alone.
Why you need it: geometry constantly hands you a setup like "these two angles are
equal" or "the perimeter is 40" and asks you to solve for a missing measure. Clean equation-solving
is the tool that finishes the job.
Worked example — solve \(2x+5=13\)
- Goal: get \(x\) by itself. First peel off the \(+5\).
- Subtract \(5\) from both sides: \(2x+5-5 = 13-5\), giving \(2x=8\).
- Now undo the "times 2" by dividing both sides by \(2\): \(\dfrac{2x}{2}=\dfrac{8}{2}\).
- Result: \(x=4\). (Check: \(2(4)+5=13\). \(\checkmark\))
Try it
1. Solve \(3x-7=11\).
2. Two angles are equal: \(x+10\) and \(2x-30\). Solve \(x+10=2x-30\).
Show answer
1. Add \(7\) to both sides: \(3x=18\). Divide by \(3\): \(x=6\).
2. Subtract \(x\) from both sides: \(10=x-30\). Add \(30\): \(x=40\).
Skill 05
Basic Shape & Angle Vocabulary
What it is: the everyday names geometers use — point (a
location), line / segment (straight path), vertex
(a corner), and angle sizes: an acute angle is less than \(90^\circ\), a
right angle is exactly \(90^\circ\) (the square corner), and an
obtuse angle is between \(90^\circ\) and \(180^\circ\). A
triangle has 3 sides; a quadrilateral has 4.
Why you need it: Module 1 describes shapes precisely, so knowing the words
lets you follow — and write — the reasoning.
Worked example — name the angle
- An angle measures \(120^\circ\). Compare it to the landmarks.
- Is it less than \(90^\circ\)? No. Exactly \(90^\circ\)? No.
- Is it between \(90^\circ\) and \(180^\circ\)? Yes — \(90 < 120 < 180\).
- So it is an obtuse angle.
Try it
1. What kind of angle is \(45^\circ\)?
2. A shape has exactly 4 straight sides and 4 corners. What is the general name for it?
Show answer
1. Since \(45^\circ < 90^\circ\), it is an acute angle.
2. A four-sided figure is a quadrilateral (squares, rectangles, and parallelograms are all special quadrilaterals).
Skill 06
Integer Operations (Positives & Negatives)
What it is: adding, subtracting, and multiplying with negative numbers.
Two quick rules: subtracting is the same as adding the opposite
(\(a-b = a+(-b)\)), and a negative times a negative gives a positive.
Why you need it: coordinates, slope, and distance are full of subtractions like
\(y_2-y_1\) where the numbers are often negative. One sign slip changes the whole answer, so
getting comfortable here pays off everywhere in the module.
Worked example — compute \(-3 - (-8)\)
- Subtracting a negative is adding its opposite: \(-3-(-8) = -3 + 8\).
- Now you're adding numbers with different signs — subtract sizes: \(8-3 = 5\).
- Keep the sign of the bigger size (the \(+8\)): the answer is positive.
- So \(-3 - (-8) = 5\).
Try it
1. Compute \(4 - 9\).
2. Compute \((-2)\times(-6)\).
Show answer
1. \(4-9 = 4+(-9)\). Different signs, so subtract sizes \(9-4=5\) and keep the negative: \(-5\).
2. Negative times negative is positive: \((-2)\times(-6)=12\).