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    • ▾5th grade
      • ▸Decimal place value
        • •Decimal place value
        • •Hashi puzzles
        • •Decimals to thousandths
        • •Rounding decimals
      • ▸Multi-digit multiplication and division
        • •Multiplying whole numbers by 10, 100, and 1000
        • •Multiplying and dividing decimals by 10, 100, and 1000
        • •Convert fractions to decimals, 10, 100, or 1000 in the denominator
        • •Multiplication, partial products method
        • •Long division
      • ▸Decimal arithmetic
        • •Decimal arithmetic
        • •Dividing whole numbers to get a decimal
      • ▸Add and subtract fractions
        • •Add and subtract fractions, abstract
        • •Adding and subtracting mixed numbers with regrouping
        • •Add and subtract fractions, word problems
      • ▸Multiply fractions
        • •Multiplication as scaling
        • •Areas and perimeters, fractional side lengths
        • •Multiply fractions, abstract
        • •Multiply fractions, word problems
      • ▸Divide fractions
        • •Understanding fractions as division, word problems
        • •Dividing whole numbers by unit fractions
        • •Dividing unit fractions by whole numbers
        • •Dividing mixed numbers
      • ▸Volume
        • •Volume by counting unit cubes
        • •Volume of right rectangular prisms by multiplying
        • •Volume of composite right rectangular prisms
        • •Volume, word problems
      • ▸Coordinate plane
        • •Plot coordinates
        • •Plot coordinates from word problems
      • ▾Algebraic thinking
        • •Intro to parentheses
        • •Distributive property without variables
        • •Convert between words and arithmetic expressions
        • •Linear equations, tables
      • ▸Measurement and data
        • •Unit conversion
        • •Represent and reason with line plots
      • ▸Properties of shapes
        • •Geometric definitions
        • •Classifying shapes
        • •Classifying triangles
        • •Classifying polygons
        • •Symmetries of regular polygons
        • •Quantifiers
     › 5th grade › Algebraic thinking

    Distributive property without variables

    Students will learn how to use the distributive property to evaluate expressions. The distributive property works because multiplication is repeated addition. Consider that \(2 \cdot 3 + 2 \cdot 4\) is the sum of 3 2's and 4 2's, which is the same as 3 + 4 2's. Mathematically, \(2 \cdot 3 + 2 \cdot 4 = 2(3 + 4).\) Students will also use the distributive property to mentally evaluate certain expressions. For example, when looking at \(16 \cdot 43 + 16 \cdot 57,\) I don't want to figure out \(16 \cdot 43,\) nor \(16 \cdot 57.\) But I notice that \(43 + 57 = 100,\) so mentally I replace \(16 \cdot 43 + 16 \cdot 57\) with \(16 \cdot 100,\) which is obviously \(1{,}600.\) Thus, my original expression \(16 \cdot 43 + 16 \cdot 57\) must be equal to \(1{,}600.\) As another example, suppose I want to know \(9 \cdot 16.\) This isn't something I know off the top of my head, so I mentally replace this with \(9 \cdot 10 + 9 \cdot 6.\) That's \(90 + 54,\) I think to myself. And \(90 + 54 = 144.\) Thus, my original expression, \(9 \cdot 16,\) must be equal to \(144.\)

    The Common Core includes this as a grade 3 topic, but without understanding parentheses, the distributive property doesn't make sense. Parentheses are first mentioned in 5.OA.A.1, so I've placed this distributive property without variables in the 5th grade curriculum instead of the 3rd. Khan Academy includes this topic as part of their 6th grade curriculum, and mistakenly cites 6.EE.A.3, which is about the distributive property with variables.

    Conclude by leading this investigation:

    Full Integer Collatz
    by MathPickle

    Lessons and practice problems