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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

    Remainders of large powers

    Students will learn how to find the last digit or last two digits of a whole number raised to a very large whole number power. Start by demonstrating that remainders repeat. This happens because the last digits of a product are only dependant on the last digits of the multiplicands. Here's an example demonstration. Once your students are comfortable with this idea, give them a couple practice problems. Include at least one problem asking for the last digit and at least one problem asking for the last two digits. Asking for more digits would be tedious, so I advise against it. Next, give your students this challenge. Finally, give them this one.

    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

    Conclude by leading this investigation:

    12 Days (addition, logic, factoring)
    by MathPickle

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

    Lessons and practice problems