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    • ▾6th grade
      • ▸Ratios, rates, and percentages
        • •Percentages
        • •Intro to ratios
        • •Unit rates
        • ▸Equivalent ratios
          • •Simplifying ratios
          • •Equivalent ratios (mathematical problems)
          • •Equivalent ratios (word problems)
          • •Find equivalent ratios using Cuisenaire rods
      • ▾Arithmetic operations
        • •Long division
        • •Decimal arithmetic
        • •Divide fractions
        • •Dividing whole numbers by fractions
        • •Raising numbers to whole number powers
        • •Mentally raising small numbers to small powers
        • •Powers of 10
        • •Multiplying and dividing by powers of 10
        • •Tower of Hanoi
        • •Counting in bases 5 and 2
        • •More on binary
        • •Remainders of large powers
      • ▸Negative numbers
        • •Intro to negative numbers
        • •Absolute value of rational numbers
        • •Plotting points in all four quadrants
      • ▸Factors and multiples
        • •Prime factorization
        • •Common factors
        • •GCD and LCM by making lists
        • •GCD and LCM by prime factorization
        • •GCD and LCM word problems
        • •Properties of the GCD and LCM
        • •Water pouring puzzles
        • •Factor with the distributive property (no variables)
        • •Relatively prime
      • ▸Variables and expressions
        • •Intro to variables
        • •Measurement with an unknown unit of length
        • •Convert phrases to algebraic expressions
        • •Identify parts of expressions
        • •Distributive property with variables
        • •Determining equivalence of algebraic expressions
      • ▸Equations and inequalities introduction
        • •Testing solutions to equations and inequalities
        • •Solving one-step equations, addition and subtraction
        • •Solving one-step equations, multiplication and division
        • •Model with one-step equations
        • •Modeling one-variable inequalities
        • •Dependent and independent variables
      • ▸Geometry
        • •Volume of right rectangular prisms with fractional lengths
        • •Area
        • •Number of diagonals in a convex polygon
        • •Counting vertices, edges, and faces
        • •Nets and surface area
        • •Surface area of rectangular prisms
        • •Coordinate plane
        • •Menseki Meiro puzzles
      • ▸Data and statistics
        • •Identifying statistical questions
        • •Plotting data
        • •Basic statistics
        • •Combining means
        • •Analyzing distributions
     › 6th grade › Arithmetic operations

    Raising numbers to whole number powers

    Students will understand exponentiation as repeated multiplication. Then they'll use this understanding to evaluate whole numbers, decimals, or fractions, raised to whole number exponents. Ask your students to evaluate some expressions like \(6^2\) or \(3^3.\) Also ask students to evaluate expressions like \(50^2,\) \(200^3,\) etc. That is, some number for which it's easy to determine its square, plus some zeros tacked on at the end. If it's easy to determine \(2^3,\) then it's pretty easy to determine \(200^3\) as well. At some point in the lesson, show students the wheat and chessboard problem, or a variation of it. You can alter the problem as tripling, quadrupling, etc. each day, if you like. Allow students to use a calculator, as the numbers will quickly become too large for the student to multiply with. Next, give your students this challenge.

    I decided to put the wheat and chessboard problem here because that's where Core Curriculum places it.

    After your students are comfortable with squaring and cubing whole numbers, lead this investigation. To learn how to lead this activity, it's necessary to also watch this and this.

    Watch these Khan Academy videos:

    • Intro to exponents
    • Intro to order of operations
    • Order of operations example
    • Worked example: Order of operations (PEMDAS)
    • Order of operations example
    • The zeroth power

    Do these Khan Academy exercises:

    • Powers of fractions
    • Exponents
    • Exponents (basic)
    • Order of operations challenge
    • Variable expressions with exponents

    Next, give your students this challenge:

    Make the equation below true by replacing each letter with a unique digit (0-9).

    $$BA = A \times A \times A$$

    Here's the solution:

    \(A \times A \times A = A^3.\) Thus, \(BA = A^3.\) Hence, \(BA\) is a two-digit cubed number, ending in \(A.\) By listing the first few cubes, we find \(3^3 = 27\) and \(4^3 = 64\) are the only two-digit cubed numbers, and only \(64 = 4^3\) fits \(BA = A^3.\) In conclusion, our equation is \(64 = 4 \times 4 \times 4.\)

    Conclude by leading this investigation:

    Square Sardine Packing (percentages, algorithm)
    by MathPickle

    6.EE.A.1: Write and evaluate numerical expressions involving whole-number exponents.

    Lessons and practice problems