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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 › Numbers and operations

    Evaluating square roots of small perfect squares

    First, students will learn how to evaluate square roots of small perfect squares by drawing dots, as seen here. In my opinion, this method becomes tiresome for the square root of any number larger than 25. After that, students will learn the abstract method. That is, to find the square root by thinking in terms of square numbers. Students can practice the abstract method here. Students will also learn how to evaluate nested square roots. For example

    $$\sqrt{\sqrt{16}} = \sqrt{4} = 2$$

    As an interesting aside, show your students how to find the square roots of larger numbers using dots, as shown here. Before attempting this section, you should be excellent at multiplying whole numbers. Next, give your students this challenge. Tell students they're only allowed to use +, -, *, /, and sqrt. This greatly reduces the difficulty of the puzzle. Here's my solution:

    $$\begin{align} & 6 = 2 + 2 + 2 \\ & 6 = 3 * 3 - 3 \\ & 6 = \sqrt{4} + \sqrt{4} + \sqrt{4} \\ & 6 = (5 / 5) + 5 \\ & 6 = 6 + 6 - 6 \\ & 6 = 7 - (7 / 7) \\ & 6 = 8 - \sqrt{\sqrt{8 + 8}} \\ & 6 = \sqrt{9} * \sqrt{9} - \sqrt{9} \end{align}$$

    Keep in mind, there are likely other solutions, some of which may be better than my own. Conclude by giving your students this challenge. Let your students know they can use a calculator to help them solve this challenge.

    Next, give your students these challenges:

    • 2018 Math Kangaroo Levels 5-6 Problem #17 by STEM4all

    2021 Math Kangaroo Levels 1-2 Problem #23: Julia only eats fruits. She eats apples on Mondays, Wednesdays, and Fridays; she eats mangoes on Tuesdays and Thursdays. She eats either 2 apples or 3 mangoes a day. On Saturdays and Sundays she eats nothing. How many pieces of fruit does Julia eat in two weeks? Here's the solution.

    Conclude by leading this investigation:

    Tetracide (percentages, algorithm)
    by MathPickle

    Lessons and practice problems