Grades 3–4 Sample Lesson Using Base Ten Blocks

Put Popular Manipulatives
to Work in Your Classroom!
The Super Source series of comprehensive teacher resources for
grades K–8 focuses on guided exploration with math manipulatives.
Through hands-on activities, students are able to construct
meaningful models that provide a link to mathematical
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The 21-volume Super Source series is
conveniently organized by manipulative
in grades K–2, 3–4, and 5–6:
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Inside this Super Source Sampler
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2
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3. Teacher Talk
• Where’s the Mathematics? offers thorough
explanations of concepts, skills, and
various strategies used in the activity
• Provides visual models of possible
solutions to activities
2. The Activity
• Introduces the manipulative and lesson concepts for a whole-group setting
• On Their Own contains engaging activities students can explore by themselves
• The Bigger Picture includes prompts for classroom discussion, writing ideas,
and extension activities
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5
Grades K–2 Sample Lesson Using Pattern Blocks
LOOK HOW
I’M GROWING
Getting Ready
What You’ll Need
Pattern Blocks, about 20-25 per pair
Crayons
Overhead Pattern Blocks (optional)
The Activity
6
PATTERNS/FUNCTIONS
• Using patterns
• Pattern recognition
• Counting
Overview
Children build the first two stages of a pattern using Pattern Blocks. Then
other children extend the pattern and describe how the shape is growing.
In this activity, children have the opportunity to:
◆
create and identify growing patterns
◆
use patterns to make predictions
◆
describe the patterns they see
Introducing
◆
Display this Pattern Block design. Explain that this
design is your beginning design and that you are
going to add blocks to make the design grow.
◆
Create the next stage of the design as
children watch.
◆
Call on volunteers to explain how your beginning
design and your growing design are alike and
different.
◆
Ask children to use their Pattern Blocks to create the next stage of
growth of the design. When they have finished, invite them to share
this third stage of the design, or “grown-up design.”
Grades K–2 Sample Lesson Using Pattern Blocks
On Their Own
Can you create a growing design and challenge your partner to make it
grow again?
• Work with a partner.
• One partner uses Pattern Blocks to create a design. Then that partner builds
the design again and adds 1 or more blocks to any part of it to make it grow.
For example:
Beginning
design
Growing
design
• The other partner tries to continue the same
growing pattern by adding 1 or more blocks to
the Growing design to make a Grown-Up design.
This partner should also describe how the pattern grows.
Grown-up
design
• Both partners trace the 3 designs and color the blocks used.
• Now switch roles and repeat the activity.
The Bigger Picture
Thinking and Sharing
Invite volunteers to display the three designs of one of their favorite growth patterns.
Children can use Pattern Blocks to predict what they think the fourth design (a more
grown-up design) would be. The partners who are displaying the original three designs can
then circulate and look for the fourth design they had in mind. Repeat this process with a
few other children’s three-stage designs. You may also wish to have children illustrate their
work with blocks on the overhead.
Use prompts like these to promote class discussion:
◆
What part of this design was growing? How was it growing?
◆
Look at the beginning design and the growing design of this pattern. Can anyone
think of a different grown-up design? What is it?
◆
What would the fourth and fifth stages of growth look like?
◆
Which was harder—making the first two designs or predicting the third design? Why?
◆
Did any of the designs have similar growth patterns? Which ones?
7
Grades K–2 Sample Lesson Using Pattern Blocks
Extending the Activity
1. Have children build patterns as they did in the lesson, but this time
predict the number of blocks used in each other’s next design.
Encourage children to record the number of blocks used in each of
the three stages and then extend that number pattern as far as
they can.
Teacher
Talk
Where’s the Mathematics?
Activities such as this help children to look for patterns and to learn to use
these patterns to make predictions. Some children can see patterns more
easily in the block designs than in the numbers generated by these designs.
For other children, the reverse may be true. The aim should be to have
children recognize patterns and use them to make predictions about what
comes next.
Many children will be able to build the next stage in the pattern but they
will not be able to verbalize why they are building it that way. Hearing
other children’s explanations of how patterns are growing will help these
children learn how to articulate what is going on in their heads. For
example, a child might look at this pattern and say that, “You keep
counting up…2,3,4.”
Growth pattern
Another child might verbalize the pattern as “adding another row that has
one more in it.” Their explanations may be sketchy and you may want to ask
for clarification. Their ability to put a pattern into words will be a
valuable skill in mathematics, especially in algebra.
If any children are unable to see a pattern in other children’s designs, the
children involved might talk about the designs. In some cases, children may
come up with designs that don’t grow predictably, and you may have to
intervene to help children to see the problem in their design and get them
started on something that works.
Initially, some children may not understand that a growth pattern means that
the design is getting larger. Children may predict that the third stage will be
a repetition of the first stage because they confuse earlier work done with
“ababab” patterns with growth patterns. It may help to ask them to build the
second stage, or “Growing Design,” and then add on to it.
Growth pattern
8
Grades K–2 Sample Lesson Using Pattern Blocks
2. Show children the first two stages of a pattern that is “shrinking”
because blocks are removed in a predictable way. Ask children to
continue and describe the shrinking pattern.
Some children may make growth patterns that have more than one possible
answer for the third stage, or “Grown-up Design.” For example, in the
pattern below, the third stage could either have three squares in the bottom
step and follow the pattern 1, 2, 3, 4,... or it could have 4 squares in the
bottom layer and follow the pattern that each new step is a double of the
last step, or 1, 2, 4, 8, ...
Beginning
design
Growing
design
Grown-up design
possibilities
When asked if any of the designs have similar growth patterns, some
children may be so focused on the visual patterns that they will answer “no”
because none of the patterns looked identical. Other children may answer
“yes” because they have looked at the underlying counting patterns that are
the same. These children are demonstrating an ability to generalize when
they look at these two patterns. They will say that the patterns below are
similar because the first kept adding one more triangle and the second kept
adding one more square; or, even though the shapes—triangles and
squares—were different, the counting pattern was the same.
Beginning
design
Growing
design
Grown-up
design
Encourage this recognition of the generalization of patterning because good
problem solvers are those children who have learned to see what seemingly
unrelated problems can have in common. Many of the growth patterns that
children design will grow by using the counting numbers so there should be
many examples of common growth patterns for children to recognize.
9
Grades 3–4 Sample Lesson Using Base Ten Blocks
NIMBLE
NUMBERS
Getting Ready
What You’ll Need
Base Ten Blocks, 1 set of longs and
units per pair
Overhead Base Ten Blocks (optional)
The Activity
10
NUMBER • LOGIC
• Addition
• Counting
• Mental math
• Game strategies
Overview
In this game for two players, children take turns adding longs and units to a
pile in an effort to be the one who puts down the block that brings the value
of the pile to 100. In this activity, children have the opportunity to:
◆
count on by tens and ones
◆
use logical reasoning
◆
build mental math skills
◆
develop strategic thinking skills
Introducing
◆
Tell children that they are going to play a game called
NIMble Numbers.
◆
Go over the game rules given in On Their Own.
◆
Ask for a volunteer to demonstrate the game with you.
◆
Play a model game, working to reach a target number of 40.
◆
Tell children that they will now play this game using the target
number 100.
Grades 3–4 Sample Lesson Using Base Ten Blocks
On Their Own
Play NIMble Numbers!
Here are the rules.
1. This is a game for 2 players. The object is to put down the block that brings
the value of the pile of blocks to 100. Players decide who will go first.
2. The first player starts a pile by putting down:
1 or 2 units
or
1 or 2 longs
or
1 unit and 1 long.
3. The second player puts down 1 or 2 more blocks and then announces the total
value of the blocks in the pile.
4. Players take turns putting down any 1 or 2 blocks and saying each new total.
After every turn, the player checks the pile and makes trades from the set, if
necessary, to keep the number of blocks in the pile as small as possible.
5. Whoever reaches 100 exactly on a turn is the winner.
• Play the game several times. Take turns going first.
• Look for a winning strategy.
The Bigger Picture
Thinking and Sharing
Invite children to talk about their games and describe some of the thinking they did.
Use prompts like these to promote class discussion:
◆
Did winning have anything to do with going first? Explain.
◆
How did you decide which blocks and how many of each to play?
◆
Did you make any moves that you wanted to take back? Explain.
◆
Was there a turning point in the game? If so, what caused it?
◆
Did you find a winning strategy? Did it always work?
11
Grades 3–4 Sample Lesson Using Base Ten Blocks
Teacher
Talk
Writing
1. Have children play the game again, recording the blocks they put down
on each turn. When the game is over, have children look back at the
blocks they played and tell what they might do differently if they could
replay a turn.
2. Have children describe a strategy for winning at NIMble Numbers. Their
strategy may be one that they developed for themselves or one that they
learned during class discussion.
Where’s the Mathematics?
The name of this activity, NIMble Numbers, reflects its connection to the
ancient Chinese game NIM, for which there is a strategy for winning. Most
children will begin to play this game without a strategy in mind, randomly
choosing combinations of blocks to contribute to the pile. After playing the
game several times, however, children may come to realize that there are
important numbers to reach along the way. Once having reached one of
those numbers, a thoughtful player should be able to win the game.
Children will notice that there are five possible block values that they can
contribute to the pile on a turn:
Number of Blocks
1
1
2
2
2
Blocks Used
1 unit
1 long
1 unit and 1 long
2 units
2 longs
Block Values
1
10
11
2
20
As they discuss and analyze the game as it progresses, children communicate
their mathematical thinking and solidify their reasoning. They may think that
they have identified a good strategy for winning because it seemed to have
worked once. Point out that is necessary to play the game several times to see
whether or not a strategy works every time.
12
Grades 3–4 Sample Lesson Using Base Ten Blocks
Extending the Activity
Ask children to suggest some ideas for making up a new strategy game by
changing a few of the rules for NIMble Numbers.
Some children may realize that one way of gaining control of the game is by
making sure that strategic values are reached along the way. They may note
that there is a winning follow-up play for each strategic value. (And, they
must try to keep their opponents from being able to make these plays.) A
few children will understand why these values are strategic. Others may
have a notion of why but be unable to explain it.
A player will be able to win on one play if it is his or her turn to put down
blocks when the pile has one of the following strategic values. The strategic
values given are listed working backward from the target number, 100:
Strategic Values
99
98
90
89
80
Winning Plays
1 unit
2 units
1 long
1 long and 1 unit
2 longs
This Works Because:
99 + 1 = 100
98 + 2 = 100
90 + 10 = 100
89 + 11 = 100
80 + 20 = 100
Children may extend this list, finding other strategic values by continuing
backward from 80.
13
Grades 5–6 Sample Lesson Using Tangrams
IT’S WHAT’S INSIDE
THAT COUNTS!
Getting Ready
What You’ll Need
Tangrams, 1 set per pair
Protractor, 1 per pair
Calculator, 1 per pair (optional)
Overhead Tangram pieces (optional)
The Activity
If children have not had experiences
finding the degree measure of angles,
you may want to have children do the
lesson called “What’s Your Angle?”
(page 86) before this one.
14
GEOMETRY • MEASUREMENT
• Properties of geometric figures
• Angles
Overview
Children make a variety of Tangram polygons, measure their interior
angles, and find the sum of their angles. In this activity, children have the
opportunity to:
◆
create and compare polygons
◆
measure angles in degrees
◆
identify interior angles
◆
discover the relationships between number of sides in a polygon and
the sum of its interior angles
Introducing
◆
Make a shape with the two small Tangram triangles and
the parallelogram, like the one shown.
◆
Have the children count the angles of the shape with you.
Point out that it has six interior angles. Explain that any angle inside
a shape is called an interior angle.
◆
Point out several angles which are not interior
angles, such as the ones indicated with an
arrow.
◆
Have children make several more shapes and
count the interior angles.
◆
Finally, ask children how they would find the number of degrees in
each of the interior angles.
Grades 5–6 Sample Lesson Using Tangrams
On Their Own
How can you find the sum of the interior angles of any polygon without
measuring each angle?
• Work with a partner. Each of you pick any 2 or more Tangram pieces and make a
shape that has the same number of sides but looks different. Record the outline
of your shape.
• Measure the interior angles of your shape. Find their sum. Record it.
• Make two more sets of shapes. Each time, make sure the shapes have a different
number of sides than the set of shapes you just made.
• Study your shapes and the data. Be ready to talk about what you notice.
The Bigger Picture
Thinking and Sharing
Have children post their shapes, grouping them according to their number of sides. Give
children time to remove any duplicates. Label each grouping with the appropriate number
of sides and the corresponding interior-angle sum.
Use prompts such as these to promote class discussion:
◆
How did you measure the angles?
◆
Were some angles easier to measure than others? Why?
◆
What do you notice about the posted shapes?
◆
What patterns do you see?
Writing and Drawing
Ask children to make three different convex polygons. Have them trace, measure, and
record the interior angles and the sum of the angles. Then have them explain whether they
think it is possible to predict the sum of the interior angles in a convex polygon. Have them
refer to their tracings when explaining their answer.
Extending the Activity
Have children work with a partner to make a shape from Tangram pieces whose interior
angles have the greatest sum possible. Then have then trace, measure, and record the
interior angle measures and their sum.
15
Grades 5–6 Sample Lesson Using Tangrams
Teacher
Talk
Some children may have
trouble deciding which
angle to measure in a
concave polygon. To help
them, you may suggest
that children use crayons
or markers to color in the
interior of the region after
it is traced, then measure
only the angles that are
colored in.
Where’s the Mathematics?
It is possible to use a protractor to find the number of degrees in each angle,
but it is easier to use the Tangram pieces themselves. Children can use the
90°-angle of the square as a benchmark. The acute angles of the Tangram
triangles are each 45°, which can be verified by putting two Tangram
triangles together on top of the square.
Putting the 90° angle and the 45°- angle together gives the larger angle
measurement of the parallelogram—135°.
Children may notice that the degree measures of all their polygons end in
either 0 or 5. They may also notice that their shapes have only six different
size angles, the smallest being 45° and the largest being 315°. This makes
sense once children realize that the Tangram pieces only have the following
angle sizes: 45°, 90°, or 135°. The following shapes are representative of
what children might post.
16
Grades 5–6 Sample Lesson Using Tangrams
Many children are surprised that the number of sides, not the shape of the
polygon, determines the interior angle sum. All the polygons with the same
number of sides have the same interior angle sum. Triangles have the
smallest sum, 180°. Quadrilaterals, or four-sided figures, have sums of 360°.
As the shapes get bigger, the number of degrees increases. In fact, every
time a side is added, the sum increases by 180°.
This 180°-increment can be explained by partitioning a convex polygon
into triangles. Pick a vertex. From that vertex draw a diagonal to every other
vertex. The number of triangles formed is two less than the number of sides
in the polygon. Since the sum of the interior angles of a triangle is 180°,
multiplying the number of triangles by 180 yields the interior angle sum. For
example, the hexagon, no matter what its shape, has an interior angle sum
of 720° because it can be partitioned into 4 triangles, each having a sum of
180°.
A 12-sided shape (dodecagon) can be partitioned into 10 triangles; thus, the
sum of its interior angles, is 1800°. An 18-sided shape has an interior angle
sum of 2880°.
17
Grades 7–8 Sample Lesson Using Geoboards
GLASS
TRIANGLES
• Area
• Congruence
• Spatial visualization
Getting Ready
What You’ll Need
Overview
Geoboards, 1 per student
Students search to find all the possible areas of triangles that can be
made on a Geoboard. They then investigate combinations of different
triangles that can be used to completely cover the Geoboard. In this
activity, students have the opportunity to:
Rubber bands
Scissors
Geodot paper, pages 119 and 123
• develop strategies for finding the areas of triangles
Activity Master, page 109
• learn that triangles with the same area are not necessarily
congruent
• strengthen spatial reasoning skills
Other Super Source activities that explore these and related concepts are:
The Squarea Challenge, page 63
Colorful Kites, page 72
The Activity
On Their Own (Part 1)
Ernie designs stained glass. He wants to create a stained-glass window
using only triangular pieces. If he uses a Geoboard as a template, how
many different triangles can he make?
• Work with a group. Each of you should make a Geoboard triangle that has a different
area. Use only one rubber band to make each triangle. Record your triangles on
geodot paper and label the areas.
• Continue to make and record triangles until you have at least one for each of the
possible areas a Geoboard triangle can have.
• Cut apart and organize your recordings.
• Be ready to explain how you conducted your search and organized your work.
18
Grades 7–8 Sample Lesson Using Geoboards
Thinking and Sharing
Invite students to share their triangles and post recordings until there are several examples posted
for each possible area. Discuss the methods students used to find all the possible areas and the
different triangles.
Use prompts like these to promote class discussion:
• How did you go about searching for triangles with different areas?
• How did you find the areas of your triangles? Did you use more than one method?
If so, describe the methods you used.
• How did you organize your work?
• How did you know you found every possible area?
• Did you see any patterns in the data you collected?
• What discoveries did you make about Geoboard triangles?
What if... Ernie wants to create a square window containing at least five
glass triangles, each having a different area? If he uses a Geoboard to
model the window, what designs can he make?
• Using your Geoboard as a frame and your triangles from Part 1, create a
stained-glass design that completely fills the frame and uses at least five
triangles, each with a different area.
• Be sure that there are no “holes” in your designs. The design must contain only
triangles, attached side to side.
• If you find a triangle that was not on your list from Part 1, you may add it to the
list and use it in your design. Record your design on geodot paper.
• Investigate other possible square window designs that could be made with your
triangles. Record them on geodot paper. Be ready to discuss your work.
Thinking and Sharing
Invite students to share their designs. You may want to allow them to cut out the triangles in their
designs from colored transparencies and display them using an overhead projector.
Use prompts like these to promote class discussion:
• How did you go about choosing your triangles? creating your designs?
• What was difficult about the activity? What was easy?
• How are your designs alike? How are they different?
• Did you find any triangles that were not on your list from Part 1? If so, why do you think
you missed them in Part 1?
• What generalizations can you make about the possible square windows that Ernie could
make with his stained-glass triangles?
19
Grades 7–8 Sample Lesson Using Geoboards
Suppose Ernie’s design must include a triangle that covers an area of 8 square
units. How many possible designs can be created if the other 4 triangles must have
different areas? Write a letter to Ernie detailing his design choices and explaining
how you found them.
Teacher Talk
Where’s the Mathematics?
In Part 1, students may be surprised to discover that it is possible to make Geoboard triangles with
16 different areas. One triangle for each area is shown below.
square unit
1 square unit
1 square units
2 square units
2 square units
3 square units
3 square units
4 square units
4 square units
5 square units
5 square units
6 square units
6 square units
7 square units
7 square units
8 square units
Note: Some students may point out that it is possible to make triangles
with other areas by crossing the rubber band over itself, as in the
example shown here.
20
2 triangles, each
with an area of
square units
Grades 7–8 Sample Lesson Using Geoboards
As students collect and organize their triangles, they may notice that the area of the smallest
possible triangle (1 x 1) is
square unit and the area of the largest (4 x 4) is 8 square units.
They may hypothesize that they should be able to make triangles with areas that range from
square unit to 8 square units in increments of
square units. This may prompt them to search
for triangles having particular areas that they may have missed, and help them to determine or
verify the areas of some of the triangles they were unsure about.
There are several methods students may use to find the area of their triangles. Some students may
try to count the number of unit squares contained in the interior of their triangles, piecing together
the partial squares. This may become difficult to do for many Geoboard triangles.
Some students may apply the formula for area of a triangle (Area =
base x height). This method
works well for triangles that have one or two sides parallel to the sides of the Geoboard.
x 1 x 3
x 3 x 3
x 2 x 1
Area = 1 square units Area = 4 square units
x 2 x 3
Area = 1 square unit
Area = 3 square units
For other triangles, it may be easier to enclose the triangle in a rectangle whose sides are parallel
to the sides of the Geoboard. Students can then find the area of their triangle by first finding
the area of the rectangle, and then subtracting the areas of the right triangles surrounding the
original triangle.
A
B
Area of rectangle = 3 x 3 = 9 square units
Area of A =
Area of B =
C
Area of C =
x1x3=1
square units
x 2 x 2 = 2 square units
x3x1=1
square units
Area of original triangle = 9 – (1
+2+1
) = 4 square units
Pick’s theorem is another method that can be used to find the area
of Geoboard shapes. The formula states that Area =
B
+ I – 1, where
B represents the number of pegs on the boundary of the shape, and I
represents the number of pegs in the interior.
For example, the triangle shown here has 8 boundary pegs and
3 interior pegs. Applying Pick’s theorem, the area of this triangle
is
+ 3 – 1, or 6, square units.
21
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