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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 › Properties of shapes

    Classifying polygons

    First, student will learn how to identify triangles, quadrilaterals, pentagons, hexagons, etc. Here's practice. After that, students will learn what it means for a polygon to be equilateral or equiangular.

    Next, students will see a proof that equilateral doesn't imply equiangular, and vice-versa. This is easily proven by counterexample. In the figure below, the pentagon on the left is equilateral but not equiangular, and the rectangle on the right is equiangular but not equilateral. Thus, equilateral doesn't imply equiangular, and vice-versa. There are infinitely many counterexamples, so you don't necessarily need to use the two I've provided.

    After that, students will learn that polygons which are equilateral and equiangular are called regular. Any polygon which is not regular can be called irregular. What's commonly referred to as an equilateral triangle could also be characterized as a regular triangle, or the triangle which is both equilateral and equiangular. All of these phrases would refer to the same shape.

    After that, students should learn the definition of concave and convex. Demonstrate that concave polygons can be equilateral but not equiangular. They cannot be equiangular because all concave polygons have at least one angle greater than \(180^\circ\) and at least one angle less than \(180^\circ.\) Students can practice classifying polygons here.

    Pictures of equiangular and non-equiangular, regular and irregular, etc.

    Next, give your students this challenge:

    Hexagon Transformations by NRICH

    Conclude by leading this investigation:

    Three Color Equilateral Triangle by MathPickle

    Lessons and practice problems