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    • ▾8th grade
      • ▸Numbers and operations
        • •Irrational numbers
        • •Evaluating square roots of small perfect squares
        • •Approximating square roots
        • •Evaluating cube roots of small perfect cubes
        • •Negative bases
        • •Negative exponents
        • •Approximating with powers of 10
        • •Proving properties of even, odd, and square numbers
        • •Divisibility with sums and differences
        • •Proving the divisibility rules for 7 and 13
        • •1089 trick
      • ▸Properties of exponents
        • •Power of a power rule for exponents
        • •Power of a product rule for exponents
        • •Power of a quotient rule for exponents
        • •Product of powers rule for exponents
        • •Quotient of powers rule for exponents
        • •Multi-step problems, properties of exponents
        • •Simplifying square roots of decimals and fractions
      • ▸Scientific notation
        • •Intro to scientific notation
        • •Converting between scientific notation and standard form
        • •Comparing numbers in scientific notation
        • •Adding and subtracting in scientific notation
        • •Multiplying and dividing in scientific notation
        • •Scientific notation, multi-step word problems
      • ▸Linear functions and equations
        • •Mentally solving certain multi-step equations
        • •Number of solutions to linear equations
        • •Finding the equation of a line
        • •Graphing proportional relationships
        • •Analyzing graphs of step functions
        • •Intro to functions
        • •Slope-intercept form
        • •Slope-intercept form from word problems
        • •Understanding slope with similar triangles
        • •Solving multi-step equations
      • ▸Linear systems of equations in two variables
        • •Testing solutions to linear systems of equations in two variables
        • •Solving linear systems of equations in two variables by graphing
        • •Solving linear systems of equations in two variables by substitution
        • •Solving linear systems of equations in two variables by elimination
        • •Number of solutions to a system of equations algebraically
        • •Solving linear systems of equations in two variables by any method
        • •Age problems
      • ▸Geometry
        • •Deriving the surface area and volume formulas for spheres
        • •Changing linear dimensions
        • •Congruent figures
        • •Corresponding angle theorems
        • •Converse of the corresponding angle theorems
        • •Proving lines are parallel
        • •Pythagorean theorem
        • •Missing square puzzle
        • •Finding the diagonal length of a rectangle
        • •Pythagorean triples
        • •Proving the inverse Pythagorean theorem
        • •Pythagorean inequality theorem
        • •Distance formula
        • •Volume
        • •Midpoint formula
        • •Symmetry
      • ▾Geometric transformations
        • •Dilating lines
        • •Dilating polygons
        • •Scaling along an axis
        • •Reflecting across axes
        • •Rigid transformations
      • ▸Data and modeling
        • •Estimating the line of best fit
        • •Making and describing scatter plots
     › 8th grade › Geometric transformations

    Rigid transformations

    Students will learn how to draw points, line segments, and polygons rotated about the origin. Specifically, clockwise and counterclockwise rotations which are multiples of \(90^\circ.\) Here's an example. Students will also learn the relation between clockwise and counterclockwise rotations. Students should be shown that any combination of rigid transformations can be done solely by reflection, as shown in this video. As an interesting aside, demonstrate how an affine transformation can be used to divide an oval pizza equally amongst any number of people, as seen here.

    Next, give your students these challenges:

    • Fitted by NRICH
    • A Patchwork Piece by NRICH

    2017 AMC 10B Problems/Problem 18 by AoPS: My solution is based upon Solution 1. Denote corner as C and side as S. I started by thinking about all the ways I could place the 3 blues. There is 1 way for CCC, 2 for CCS, 2 for CSS, and 1 for SSS. Here's how I drew the 1 CCC and 2 CCS:

    If you're very careful, you can also solve this problem by placing the 1 green first, as seen here. My way is a bit nicer, because after placing the blues you can figure out all possible positions for the green in your head. That is, you will end up drawing the 6 arrangements for the blues instead of all 12 arrangements for the blues and greens.

    Conclude by leading this investigation:

    Round Tower – (drill and kill multiplication)
    by MathPickle

    Lessons and practice problems