Fractions with Pattern Blocks

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 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 = ? 1 red + 1 green=1/2 + 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