Topics Addressed • Fractional relationships • Measurement of area • Theoretical probability • Equivalent fractions • Addition and subtraction of fractions with unlike denominators • Multiplication and division of fractions • Lines of symmetry • Rotational symmetry • Connections among mathematical ideas
Pattern Block Pieces • Explore the relationships that exist among the shapes.
Sample Questions for Student Investigation • The red trapezoid is what fractional part of the yellow hexagon? • The blue rhombus is what fractional part of the yellow hexagon? • The green triangle is what fractional part of the yellow hexagon? the blue rhombus? the red trapezoid? • The hexagon is how many times bigger than the green triangle?
Pattern Block Relationships • is ½ of • is 1/3 of • is 1/6 of • is 2 times (twice)
More Pattern Block Relationships • is ½ of • is 3 times • is 3 times • is 1.5 times
Connections Among Mathematical Ideas • Suppose the hexagons on the right are used for dart practice. • If the red and white hexagon is the target, what is the probability that the dart will land on the trapezoid? Explain your reasoning. • If the green and white hexagon is the target, what is the probability that the dart will land on a green triangle? Why?
Sample Student Problems • Using only blue and green pattern blocks, completely cover the hexagon so that the probability of a dart landing on • blue will be 2/3. • green will be 2/3.
Equivalent Fractions I • Since one green triangle is 1/6 of the yellow hexagon, what fraction of the hexagon is covered by 2 green triangles? • Since 2 green triangles can be traded for 1 blue rhombus (1/3 of the yellow hexagon), then 2/6 = ? • Using the stacking model and trading the hexagon for 3 blue rhombi show 1 blue rhombus on top of 3 blue rhombi.
Equivalent Fractions II • If one whole is now 2 yellow hexagons, which shape covers ¼ of the total area? • Trade the trapezoid and the hexagons for green triangles. • The stacking model shows 3 green triangles over 12 green triangles or 3/12 = 1/4.
Equivalent Fractions III • If one whole is now 2 yellow hexagons, which shape covers 1/3 of the total area? • One approach is to cover 1/3 of each hexagon using 1 blue rhombus. • Trade the blue rhombi and hexagons for green triangles. • Then the stacking model shows 1/3 = 4/12.
Adding Fractions I • If the yellow hexagon is 1, the red trapezoid is ½, the blue rhombus is 1/3, and the green triangle is 1/6, then • 1 red + 1 blue is equivalent to ½ + 1/3 Placing the red and blue on top of the yellow covers 5/6 of the hexagon. This can be shown by exchanging (trading) the red and blue for green triangles.
Adding Fractions II • 1/3 + 1/6 = ? • Cover the yellow hexagon with 1 blue and 1 green. • ½ of the hexagon is covered. • Exchange the blue for greens to verify. • 1 red + 1 green=1/2 + 1/6=? • Cover the yellow hexagon with 1 red and 1 green. • Exchange the red for greens and determine what fractional part of the hexagon is covered by greens. • 4/6 of the hexagon is covered by green. • Exchange the greens for blues to find the simplest form of the fraction. • 2/3 of the hexagon is covered by blue.
Subtracting Fractions I • Use the Take-Away Model and pattern blocks to find 1/2 – 1/6. • Start with a red trapezoid (1/2). • Since you cannot take away a green triangle from it, exchange/trade the trapezoid for 3 green triangles. • Now you can take away 1 green triangle (1/6) from the 3 green triangles (1/2). • 2 green triangles or 2/6 remain. • Trade the 2 green triangles for 1 blue rhombus (1/3).
Subtracting Fractions II • Use the Comparison Model to find 1/2 - 1/3. • Start with a red trapezoid (1/2 of the hexagon). • Place a blue rhombus (1/3 of the hexagon) on top of the trapezoid. • What shape is not covered? • 1/2 - 1/3 = 1/6
Multiplying Fractions I • If the yellow hexagon is 1, then ½ of 1/3 can be modeled using the stacking model as ½ of a blue rhombus (a green triangle). Thus ½ * 1/3 = 1/6.
Multiplying Fractions II • If the yellow hexagon is 1, then 1/4 of 2/3 can be modeled as 1/4 of two blue rhombi. Thus 1/4 * 2/3 = 1/6 (a green triangle).
Multiplying Fractions III • If one whole is now 2 yellow hexagons, then 3/4 of 2/3 can be represented by first covering 2/3 of the hexagons with 4 blue rhombi and then covering ¾ of the blue rhombi with green triangles. • How many green triangles does it take? • The stacking model shows that ¾ * 2/3 = 6/12. • Trading green triangles for the fewest number of blocks in the stacking model would show 1 yellow hexagon on top of two yellow hexagons or 6/12 = ½.
Dividing Fractions 1 • How many 1/6’s (green triangles) does it take to cover 1/2 (a red trapezoid) of the yellow hexagon? 1/2 ÷ 1/6 = ?
Dividing Fractions 2 • How many 1/6’s (green triangles) does it take to cover 2/3 (two blue rhombi) of the yellow hexagon? 2/3 ÷ 1/6 = ?
Symmetry • A yellow hexagon has 6 lines of symmetry since it can be folded into identical halves along the 6 different colors shown below (left). • A green triangle has 3 lines of symmetry since it can be folded into identical halves along the 3 different colors shown above (right).
More Symmetry • How many lines of symmetry are in a blue rhombus? • Explain why a red trapezoid has only one line of symmetry.
Rotational Symmetry • A yellow hexagon has rotational symmetry since it can be reproduced exactly by rotating it about an axis through its center. • A hexagon has 60º, 120º, 180º, 240º, and 300º rotational symmetry.
Pattern Block Cake Student Activity • Caroline’s grandfather Gordy owns a bakery and has agreed to make a Pattern Block Cake to sell at her school’s Math Day Celebration. • This cake will consist of • chocolate cake cut into triangles, • yellow cake cut into rhombi, • strawberry cake cut into trapezoids, • and white cake cut into hexagons. • Like pattern blocks, the cake pieces are related to each other. • Adapted from NCTM Addenda Series/Grades 5-8 Understanding Rational Numbers and Proportions Activity 5
Pattern Block Cake Student Activity • If each triangular piece costs $1.00, how much will the other pieces cost? How much will the whole cake cost? • If each whole Pattern Block Cake costs $1.00, how much will each piece cost? Adapted from NCTM Addenda Series/Grades 5-8 Understanding Rational Numbers and Proportions Activity 5
Websites for Additional Exploration • National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/vlibrary.html • Online Pattern Blocks http://ejad.best.vwh.net/java/patterns/patterns_j.shtml