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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

    Mentally raising small numbers to small powers

    First, students will mentally evaluate small whole numbers to whole number powers \(\le 3.\) For example, evaluate \(4^3.\) If students have memorized \(4^2,\) then \(4^3\) is \(4^2 \cdot 4 = 16 \cdot 4.\) Because I don't know \(16 \cdot 4\) off the top of my head, I break it down into \((10 + 6) \cdot 4,\) which I mentally distribute to obtain \(40 + 24,\) which is easy to add mentally to get \(64.\) So \(4^3 = 64.\) Next, challenge your students to mentally evaluate \(0^4, 1^4, 2^4, 3^4, 4^4, 5^4,\) one at a time, in that order. To find \(5^4\) I think \(25 \cdot 25.\) Since I can't do \(25 \cdot 25\) in my head, I break it up into \(25(20 + 5).\) Since I can't easily do \(25 \cdot 20,\) I think \(25 \cdot 10 \cdot 2,\) so doubling \(250\) gives me \(500.\) I remember I must still find \(25 \cdot 5.\) If you think about quarters, five quarters is \($1.25,\) so \(25 \cdot 5\) must be \(125.\) Adding this to \(500\) gives me \(625.\) So \(5^4 = 625.\) There's lots of different ways you could find \(5^4.\) Let students figure out their own strategies. Then have students explain their strategies orally. They probably won't recall exactly what they did, so you'll have to ask probing questions as they attempt to communicate their strategy.

    Next, give your students these challenges:

    • Intersecting Squares by NRICH
    • 2017 Math Kangaroo Levels 1-2 Problem #14 by STEM4all
    • 2020 Math Kangaroo Levels 5-6 Problem #25 by STEM4all

    Conclude by leading this investigation:

    Using Least 1s by MathPickle