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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 › Geometry

    Volume

    Students will learn how to find the volume of cylinders, spheres, and cones. They will solve word problems and mathematical ones. Khan Academy covers this topic completely. Next, give your students the following two challenges:

    The setup to the problem is here. Call the width \(w\) and the height \(h.\) Ask students to express \(w \mathop{?} h\) as a formula. After a bit of play, students should see the dots exist on two grids, one shifted. The more numerous set of dots is \(wh,\) while the less numerous is \((w - 1)(h - 1).\) Thus, the answer is

    $$w \mathop{?} h = wh + (w - 1)(h - 1)$$

    Then use the setup found here, calling depth \(d.\) Ask for the formula \(w \mathop{@} h \mathop{@} d.\) The answer here is similar, it's

    $$w \mathop{@} h \mathop{@} d \ = whd + (w - 1)(h - 1)(d - 1)$$

    Finally, give your students this Pythagorean theorem puzzle.

    Conclude by leading this investigation:

    Treefolk Tribes (symmetry, sorting, sequence)
    by MathPickle

    Lessons and practice problems