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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 › Linear functions and equations

    Mentally solving certain multi-step equations

    Certain multi-step equations, where \(x\) appears exactly once, as the innermost and leftmost operand, can be solved mentally. These are equations like

    $$\begin{align} & [(x \cdot a) + b] / c = d \\ & [(x - a) / b] \cdot c = d \\ & \ldots \end{align}$$

    Notice that to solve such equations, we start with \(d,\) the output, then repeatedly whittle down until only \(x\) remains. So to solve the equation

    $$[(x - a) \cdot b] + c = d$$

    we'd first subtract \(c\)

    $$(x - a) \cdot b = d - c$$

    Then we'd divide by \(b\)

    $$x - a = (d - c) / b$$

    Then we'd add \(a\)

    $$x = (d - c) / b + a$$

    Thus, equations of the form

    $$[(x - a) \cdot b] + c = d$$

    can be solved mentally exactly when \((d - c) / b + a\) can be evaluated mentally. Now you try. Derive a similar formula for equations of the form \((x \cdot a + b) / c = d.\)

    Here's the solution:

    To solve the equation

    $$(x \cdot a + b) / c = d$$

    we'd first multiply by \(c\)

    $$x \cdot a + b = d \cdot c$$

    Then we'd subtract \(b\)

    $$x \cdot a = d \cdot c - b$$

    Then we'd divide by \(a\)

    $$x = (d \cdot c - b) / a$$

    This strategy of repeatedly doing the inverse can be used to mentally solve both word problems and mathematical ones. Here's a word problem for you to try:

    If I multiply my number by 3, then subtract 5, then multiply that by 4, I get 6 less than 202. What's my number?

    Answer: [(202 - 6) / 4 + 5] / 3 = 18

    For more problems like this, go here. Refresh that page for even more.

    Next, give your students these challenges:

    • Area and Perimeter by NRICH
    • 2018 Math Kangaroo Levels 3-4 Problem #24 by STEM4all

    Find the perimeter:

    Here's the solution: Refer to the two figures below.

    Notice that in the left figure, the lengths of the red segments sum to the length of the blue segment. Likewise, in the right figure, the lengths of the green segments sum to the length of the magenta segment. After making these two observations, finding the length of each segment is easy. Adding them all up gives us the perimeter, \(24.\)

    Conclude by leading this investigation:

    Square Dance (patterns)
    by MathPickle