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 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 thehexagons for green triangles.

• The stacking model shows 3green triangles over 12 greentriangles 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 hexagonsfor 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 berepresented 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