2. 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
4. 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?
6. More Pattern Block Relationships
• is ½ of
• is 3 times
• is 3 times
• is 1.5 times
7. 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?
8. 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.
9. 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.
10. 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.
11. 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.
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.
13. 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.
14. 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).
15. 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
16. 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.
17. 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).
18. 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.
19. 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 = ?
20. 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 = ?
21. 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).
22. More Symmetry
• How many lines of symmetry are in a
blue rhombus?
• Explain why a red trapezoid has only one
line of symmetry.
23. 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.
24. 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
25. 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
26. 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/patte
rns_j.shtml
