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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

    Multiplying and dividing by powers of 10

    Students will learn how to multiply whole numbers, and decimal numbers, by powers of 10. They'll also learn how to describe how the decimal point moves. For example, in \(2.8 \cdot 10^{-3},\) the decimal moves \(3\) places to the left.

    Watch these Khan Academy videos:

    • Multiplying and dividing by powers of 10
    • Exponents and powers of 10 patterns
    • Exponents and powers of 10 patterns by Khan Academy

    Multiply and divide by powers of 10

    Next, give your students these challenges:

    • Red Express Train by NRICH

    5 on the Clock by NRICH: Note the solution provided by NRICH is wrong. Here's the solution: The hour will read 5 for 60 mins. Likewise, the hour will read 15 for 60 mins. We've got 22 hours left to account for. There will be a 5 in the 10 minute column for 10 minutes per hour. Over the 22 hours we've left to account for, that's 10 * 22 = 220 times. Now we've got 22 hours left to account for, and 50 of those minutes to account for per hour. A 5 occurs in the 1 minute column once per 10 minutes. Thus, over 50 minutes, a 5 will occur in the 1 minute column 5 times. Over 22 hours, that's 5 * 22 = 110 times. We've considered when a 5 appears in the 1 hour column, the 10 minute column, and the 1 minute column, so we must be done. Adding everything up, we have 60 + 60 + 220 + 110 = 450 times. So over 24 hours, at least one 5 will be present 450 times. The answer is the same for a 12 hour clock. On a 12 hour clock you will have 5 AM and 5 PM instead of 5 and 15, which doesn't affect our count. Here's a small Python program you can use to verify my answer is correct:

    h, m = 0, 0
    n = 0
    while h < 24:
    	if '5' in str(h) + ':' + str(m):
    		n += 1
    	if m < 59:
    		m += 1
    	else:
    		m = 0
    		h += 1
    print(n)

    Conclude by leading this investigation:

    Integral Centrifuge II (multiplication of negatives)
    by MathPickle