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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

    Proving properties of even, odd, and square numbers

    Students will learn how certain types of numbers can be represented algebraically. First, students will learn that a number is even if and only if it can be represented as \(2k,\) where \(k\) is some integer. Likewise, a number is odd, if and only if it can be represented as \(2k + 1,\) where again, \(k\) is some integer. After that, students will learn how to prove properties involving even and odd numbers, together with addition or multiplication. See below for a list of such properties. Have students prove them until they get the gist.

    • even + even = even
    • even + odd = odd
    • odd + odd = even
    • even * even = even
    • even * odd = even
    • odd * odd = odd
    • even^n = even, for \(n \ge 1\)
    • odd^n = odd

    Notice the last two properties can be proven by using induction informally. We can prove even^n = even, for \(n \ge 1,\) by using even * even = even, \(n - 1\) times. Likewise, we can prove odd^n = odd, by using odd * odd = odd, \(n - 1\) times.

    We can also ask if statements involving even and odd numbers are always true, sometimes true, or never true, as seen here.

    Next, students will generalize this idea of representing multiples and non-multiples, to numbers which may or may not be \(2.\) That is, even numbers are exactly the multiples of \(2,\) and odd numbers are exactly the non-multiples of \(2.\) As an example of generalizing this concept, if a whole number is one more than a multiple of \(4,\) then it can be represented as \(4n + 1,\) where \(n\) is some whole number.

    After that, give your students this challenge:

    What Numbers Can We Make? by NRICH

    After that, students will learn that a number is square, if and only if it can be represented as \(k^2,\) for some integer \(k.\) Then students will learn how to prove some basic theorems using such representations. Theorems should include those involving addition, multiplication, and squaring. For example, prove the sum of two even numbers is even. As another example, is an odd number raised to a whole number power even or odd? Make a conjecture, then prove it. These proofs require nothing more than distributing, combining like terms, and applying properties of exponents. Conclude by giving your students this challenge.

    Next, give your students this challenge:

    Board Block Challenge by NRICH

    Conclude by leading this investigation:

    Destroying Democracy! (counting, fractions, shapes)
    by MathPickle

    Lessons and practice problems