Discrete Math in the Classroom
(Developmentally Appropriate Lessons for 3rd Grade Students)
So, Bobby Bear DOES have a large
combination of outfits he can wear!
Interesting...very interesting.
Amanda Anderson
Title I Teacher
Lincoln Elementary
[email protected]
1
Executive Summary
This nine day unit plan is constructed in a format that best fits my Title I teaching
schedule. I have, on average, a total of 14 classes a day. Each class fits in a 25 minute time
slot. Within these classes are students who range in age from kindergarten to fifth grade, and
developmentally from struggling to advanced. This unit plan teaches the concept of discrete
mathematics at a third grade level in 20 minute intervals. The Minnesota state standards that
this unit will address, for each grade level, can be found after the “Unit Plan Outline and
Overview” section located below. This unit plan starts with a pre-assessment at the beginning of
each week during this two week unit in order to find out the students' prior knowledge of the
following concepts: combinations, recursion, vertex, edges, and geoboards. Each concept will be
addressed over one or two days, and students will be tested through a post-assessment at the
end of each week that includes the concepts they learned. This assessment will be conducted
through activities, formal and informal assessments, and through the use of technology.
Standard Based Unit Plan Outline and Overview
Day 1
Pre-Assessment (First Part: Questions 1-5): Based off of similar MCA
questions.
Launch, Explore, Share, Summarize: Looking at Combinations (Bobby Bear
Outfit Options).
Day 2
Launch, Explore, Share, Summarize: Looking at Combinations (Bobby BearTree Diagram and Dice)
Day 3
Launch, Explore, Share, Summarize: Branching off of Number Patterns:
Recursion Formula.
Day 4
Launch Explore, Share, Summarize: Creating Recursion Formulas
Day 5
Learning Stations and Post-Assessment: Recursion Formulas and Combinations
Day 6
Pre-Assessment (Second Part: Questions 6-10): Based off of similar MCA
questions.
Launch, Explore, Share, and Summarize: Exploring Shapes through the
Geoboards (Looking at Vertices)
Day 7
Continued: Launch, Explore, Share, and Summarize: Exploring Shapes through
the Geoboards (Looking at Edges)
Day 8
Launch, Explore, Share, and Summarize: Attribute Blocks through a Venn
Diagram
Day 9
Learning Stations and Post Assessment: Vertices, Edges, and Formulas
2
Standard Addressed in this Unit
Third Grade:
Geometry and Measurement: Use geometric attributes to
describe and create shapes in various contexts.
3.3.1.2. Sketch polygons with a given number of sides or
vertices (corners), such as pentagons, hexagons, and
octagons.
Algebra: Use single-operation input-output rules to represent
patterns and relationships and to solve real-world and
mathematical problems.
3.3.1.2. Create, describe, and apply single-operation inputoutput rules involving addition, subtraction, and
multiplication to solve problems in various contexts.
3
Table of Contents
1. Third Grade Pre-Assessment.......................................
2. Day 1: Introduction.................................................
3. Day 2: Bobby Bear and Dice........................................
4. Day 3: Number Patterns Recursion Problem......................
5. Day 4: Continuing Recursion Formulas.............................
6. Day 5: Post-Assessment and Learning Stations..................
7. Day 6: Pre-Assessment and Intro (Geoboards).................
8. Day 7: Continuing with Geoboards.................................
9. Day 8: Working with Similar Attributes..........................
10. Day 9: Post-Assessment and Learning Stations..................
11.
Resource Citations................................................
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Page
5
9
12
15
21
26
30
33
36
39
43
Name_______________
Date:________
What Do You Know?
(Grade 3)
1. How many different outfit combinations could you make if
you have 5 shirts, 4 pants, and 3 types of shoes?
(You can use a calculator, but please tell me how you got that
answer.)
2. What is the pattern in this table? Please write this
pattern using words.
2
4
3
6
4
8
5
10
5
3. What is the pattern in this table? Please write this
pattern using words.
1
4
2
8
3
12
4
16
4. What is the pattern in this table? Please write this
pattern using words.
1
3
3
7
5
11
7
15
5. How many different ways could a six sided die, land four
times? (You can use a calculator.)
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6. What is the common attribute with these shapes?
7. What is a vertex?
8. What is an edge?
9. How many vertices and edges does this shape have?
7
10. Organize all of these shapes into the Venn Diagram
below.
You're Done!
Thank you for your hard work!
8
Day 1
Pre-Assessment and Introduction
Standard:
Algebra: Use single-operation input-output rules to represent
patterns and relationships and to solve real-world and mathematical problems.
3.3.1.2. Create, describe, and apply single-operation input-output rules involving
addition, subtraction, and multiplication to solve problems in various contexts.
Objectives:
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The students will complete a written pre-assessment which covers the concepts they will
be learning over a period of 5 days.
The students will represent, analyze, and solve a variety of counting and combination
problems by using arrays, and systematic lists.
The students will understand the relationships among arrays, and systematic lists.
Materials (located at the end of each continued lesson):
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Visualizer to Display Student Answers, Tables, Arrays, and Lists
Pre-Assessment: Questions 1-5 (pgs. 5 and 6)
“Bear Cutouts” Activity Sheet: Two per Student (For Source-See Reference List)
“Shoe Cutouts” Activity Sheet: One per Student (For Source-See Reference List)
Teacher's Bear Outfit Options for Visualizer (For Source-See Reference List)
Purple, yellow, and green crayons
Pencils
Scissors
Glue sticks
One large sheet of construction paper: 17 inches by 12 inches (One for each table of 3-4
students)
Online Bobby Bear Activity: http://illuminations.nctm.org/activitydetail.aspx?id=3
Laptop to Display Bobby Bear Activity through the Visualizer
Blank Paper to Record the Different Outfits
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Procedure:
Launch (3 min.):
The students will first come to class and complete a five question pre-assessment
(Questions 1-5). Once they finish their pre-assessment, they will turn their assessment in and
then see an online bear outfit activity displayed on the visualizer. The teacher will begin by
saying, “This is Bobby Bear and he is going on vacation, but he doesn't want to pack too much.
So, he is asking for our help. He wants to know how many different outfits or combinations he
can make with the different shirts and pants he has.” The teacher will then customize the
outfit list so that it starts off with Bobby bear having just one shirt and one pair of pants.
“How many outfits can he make here?” (Answer: One.) “Great, now let's see how many outfits he
can make with two shirts and one pair of pants. How many outfits are there?” (Answer: Two.)
“Fabulous.” The teacher will continue this activity and up the number of shirts and pants by one
each time until the students have collaborated the number of combinations for three shirts and
three pairs of pants. (Answer: Nine combination.) The teacher will then say, “Well, you have all
been a fabulous help for Bobby Bear. But, he has decided to go on a two week vacation. He is
only going to pack (As the teacher tells the students the clothes that Bobby Bear is going to
back, he/she will display the different clothing options on the board through the use of the
visualizer.) three different shirts, two different pairs of pants, and three different type of
shoes. But he wants to know how many different outfits or combinations he could wear using the
different clothing items. You will be given bear cutouts, outfit cutouts to color, and a large
piece of poster board to display the different outfits Bobby Bear can make.
Explore (10 min.):
You will be working together in your detective group to show the different outfits. There are
crayons in the same colors shown on the visualizer so that you can color your shoes, pants, and
shirts the same way. As you are making the different outfits, write down the outfit
combinations you find so that you know you didn't repeat any outfits. Remember, Bobby Bear
needs enough outfits to last at least two weeks, which is fourteen days. Have fun exploring!”
The teacher will then pass out the bear cutouts, outfit cutouts, and the rest of the materials
the students need in order to complete their boards.
Share (4 min.):
As the students finish they will display their findings on the bulletin board. At the end of class,
the students will present their poster to the class and explain how they came up with the
different combinations and how they organized their combinations in order to make sure they
did not have any repeating outfits.
Summarize (3 min.):
After the student have presented their findings and outfit combinations on their poster
boards, the teacher will highlight posters that organized Bobby Bear's outfit possibilities by
shirt, that organized his outfits by pants, and that organized his outfit by shows. This will help
10
stress the fact that there are a variety of ways to display and organize combinations and
information.
Additional Note: Extensions of this activity can be found in the book, Navigating through
Discrete Mathematics in Prekindergarten – Grade 5. (For Source-See Reference List)
11
Day 2
Bobby Bear and Dice
Standard:
Algebra: Use single-operation input-output rules to represent
patterns and relationships and to solve real-world and mathematical problems.
3.3.1.2. Create, describe, and apply single-operation input-output rules involving
addition, subtraction, and multiplication to solve problems in various contexts.
Objectives:
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The students will use their prior knowledge of combinations and lists, to learn the
concept of tree diagrams.
The students will create a tree diagram, based on the number of outfits Bobby Bear
could create to see if their tree diagrams match their visual poster diagrams.
The students will understand the relationships among arrays, and systematic lists.
Materials (located at the end of each continued lesson):
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Visualizer to Display Student Answers, Tables, Arrays, and Lists
Pencils
Calculators (One per Student)
Scissors
Glue sticks
Blank Paper to Record the Different Outfits
The Bobby Bear Posters the Students Created Yesterday
Procedure:
Launch (4 min.):
The students will first come to class and an online bear outfit activity will be displayed on
the visualizer and the class's posters from yesterday will be at their tables. Once the students
have found their spots and are seated the teacher will begin by saying, “Remember when we
created our Bobby Bear arrays yesterday? An array is a way of organizing information in order
to understand a group of possibilities. How do we know we have all of the possibilities?”
(Student responses: We counted, we worked really hard, we tried our best, etc.) “You're right,
you all worked very hard yesterday working on your posters. But let's say, Bobby Bear was
moving and he had 25 different shirts, 12 different pairs of pants, and 8 pairs of shoes. Would
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we want to sit and cut and paste all of the different possibilities or combinations? Our array
would be very, VERY large. But, for right now, let's look at the information we have. What's
another way we could organize the different outfits?” (Student responses may vary.) “Those
are all wonderful examples, let's try another way of showing our information through a graph
called a tree diagram. Grab a blank piece of paper and a pencil and let's create one together.”
Share (3 min.):
The teacher will then say, “How many different shirts did Bobby Bear have? (Three.)
What were they? (A green shirt, a yellow shirt, and a purple shirt.) How many different pairs of
pants did he have? (Two: dotted pants and striped pants.) How many pairs of shoes? (Three:
boots, sneakers, and sandals). As the students are sharing their answers, the teacher is drawing
a tree diagram on the visualizer that is projected onto the board. The students are writing as
the teacher is writing. The tree diagram should look like the following:
(The teacher should write, the three
types of shirts, two types
of pants and three
types of shoes on
the different branches.)
“So, how many outfits is Bobby Bear able to make?” (Answer: 18)
Explore (7 min.):
“Very good! Now, let's look at the number patterns we've seen and written about the
number of clothing items and the number of outfits that have occurred each time. Talk with
your neighbor to see if there are any patterns.” The teacher will be there for guidance. The
point of this activity is for the students to eventually see that you can multiply the number of
clothing items together to get the total number of outfits possible. If the students do come up
with this for an answer the teacher will continue by saying, “Another way to think about this is
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multiplying the three shirts, two pants, and 3 pairs of shoes together or 3 times 2 times 3 and
you get the number 18. Now, Let's say that Bobby Bear has 4 pairs of shirts, 3 pairs of pants,
and 4 types of shoes. What would the tree diagram look like? Try and draw it on the blank
piece of paper in front of you. You can work with the other detectives at your table.”
Share (4 min.):
As the students finish the teacher will select a few volunteers to show their tree
diagrams on the board through the use of a visualizer. “So, how many outfits are there this
time?” (Answer: 36) “Very good! Another way to say this is 4 pairs of shirts, times 3 pairs of
pants, times 4 types of shoes equals 36. This is the same as saying 4 times 3 times 4 equals 36.
Now what if I had a different problem entirely? What if I had a six sided die, and I wanted to
roll it 4 times. How could I use my calculator to find the number of combinations I could get?”
(Student responses may vary and guidance may need to occur.) “How many numbers are on a die?
(1, 2, 3, 4, 5, 6) So there are six different numbers. How many times am I rolling it? (Four)
Exactly! So what is six times six, times six, times six? Let's find out!” The teacher will then
have the students guide him/her on how to use the calculator. The students should try using the
calculator too.
Summarize (2 min.):
Towards the end of class, the teacher will highlight different tree diagrams that the
students have presented to show how they organized their information. The teacher will also
highlight how students were able to correctly multiply the different combination formulas into
their calculators in order to show that the tree diagrams and the multiplication and problems
relate to each other.
Additional Note: Extensions of this activity can be found in the book, Navigating through
Discrete Mathematics in Prekindergarten – Grade 5. (For Source-See Reference List)
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Day 3
Branching Off of Number Patterns: Recursion Formula
Standard:
Algebra: Use single-operation input-output rules to represent
patterns and relationships and to solve real-world and mathematical problems.
3.3.1.2. Create, describe, and apply single-operation input-output rules involving
addition, subtraction, and multiplication to solve problems in various contexts.
Objectives:
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The students will participate in looking for numeric patterns in a grid and triangle chart.
The students, as a class, will develop an understanding of number patterns through the
understanding of recursion formulas using the concepts of Next and Now.
The students will use their prior knowledge of number patterns to create recursion
formulas with their classmates through the guidance of the teacher.
Materials:
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Question of the Day to Display on the Visualizer (page 19)
Visualizer to Discuss Pattern Creations
Blank Grid Activity Sheet: One per Student (page 20)
Triangle Activity Sheet: One per Person (page 18)
Blank Paper
Crayons or Markers
Pencils
Procedure:
Launch (5 min.):
The students will first come to class and and a question and directions will be displayed
on the board through the use of a visualizer. The question and directions are, “Remember when
we learned about number patterns? Try to find out what the number pattern for the outlined
parts of this grid. Talk to your neighbor to see if they have the same pattern. If it is
different, talk with your neighbor about how you came up with that number pattern. Don't
forget the equals sign in your math problem. Have fun!” There will be blank paper, pencils, and
crayons available at a separate spot in the classroom for the students to use to create their
math pattern ideas.
15
Once their math pattern theories have been created and shared, the teacher will get the
class's attention to continue today's lesson. “So, who would like to share the ideas they created
in order to solve this math problem?” (Volunteers will be selected to bring their patterns up to
the visualizer and share what they learned with the class. The teacher will ask the students how
they came up with their pattern, or how they solved the different number patterns. “So, what is
a pattern again?” (Possible student responses: something that repeats over and over, something
that starts again and again, etc.) “Very good, detectives! Where might we see number patterns
in real life?” (Possible student responses: When we're counting, on rulers, etc.) “Excellent!
Let's look at this grid again and write a table showing how the outlined parts change. Why don't
you write it along with me so that you have the same clues I have.” The table should look like
the following:
The Row Number
Number of Blocks
1
1
2
3
3
5
4
7
5
9
“As you can see I made sure to write an equals sign between the two columns to help me
remember that the type of square is on the left, and the number of blocks each square has is on
the right. Look at the column on the right that shows the number of blocks each shape has.
What is happening with the numbers in this column?” (Possible student responses: It gets bigger
by two each time, it gets bigger, etc.) “Right! If we were to find the difference between each
of the numbers, we would see that it gets bigger by 2 each time. So, for our next shape, could
we say the word Next equals Now?” (Yes.) “Right! What is happening right now? How many
more blocks are added each time?” (Two.) “Excellent. So, another way I could right this is the
next shape with have the blocks we have now, plus two more blocks. Or, we could say it as Next
equals Now + 2.” The teacher would then write the equation up on the board as the following:
Next = Now + 2.
Explore (5 min.):
“I will now give you another activity sheet and I would like you to try and find out the
pattern, and write the pattern using words just like we did together here using the words Next
and Now. Try making a table so that the pattern becomes even clearer to you. You may work
with a partner. Have fun exploring, detectives!”
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The students will then take time to explore the triangle pattern chart for different
patterns.
Share (10 min.):
“Once you find at word pattern, write this pattern up on the board for us to explore as a
class.” Once everyone has written down their patterns, the teacher will then go through the
different patterns found and ask the students how they came up with their different number
patterns. The students will be expected to show their reasoning and help any students who do
not understand how a particular pattern was made. “As we can see there are a few of us who
had the same pattern, others looked at the pattern in a different way, but we all came up with
great math formulas using the words Next and Now.”
Summarize (2 min.):
Towards the end of class, the teacher will highlight different Next and Now formulas
that the students have created and show that it is possible to write a variety of math equations
using the concepts of Next and Now. The teacher will also check to see if basic addition and
subtraction formulas were solved correctly.
Additional Note: There is an extended activity regarding the triangle and square patterns that
can be found in the book: Navigating through Discrete Mathematics in Grades Pre-Kindergarten
-5. (For Source-See Reference List)
1. Triangle Activity Sheet (page 18)
2. Question of the Day Grid Activity Sheet(page 19)
3. Blank Square Activity Sheet (page 20)
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Triangle Activity
18
Eight By Eight Grid
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Eight By Eight Grid
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Day 4
Continuing with the Recursion Formula
Standard:
Algebra: Use single-operation input-output rules to represent
patterns and relationships and to solve real-world and mathematical problems.
3.3.1.2. Create, describe, and apply single-operation input-output rules involving
addition, subtraction, and multiplication to solve problems in various contexts.
Objectives:
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The students will participate in looking for numeric patterns in a shape, and grid letter
chart.
The students, as a class, will use their prior knowledge to develop a deeper understanding
of number patterns through the different recursion formulas using the concepts of Next
and Now.
The students will use their prior knowledge of number patterns to create recursion
formulas with their classmates through the guidance of the teacher.
Materials:
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Question of the Day to Display on the Visualizer (page 25)
Visualizer to Discuss Pattern Creations
Letter Grid Activity Sheet: One per Student (page 24)
Blank Paper
Crayons or Markers
Pencils
Procedure:
Launch (5 min.):
The students will first come to class and and a question and directions will be displayed
on the board through the use of a visualizer. The question and directions are, “Remember when
we learned about number patterns? Try to find out what the number pattern is for this shape
that keeps getting bigger. Talk to your neighbor to see if they have the same number pattern.
If it is different, talk with your neighbor about how you came up with that number pattern.
Don't forget the equals sign in your math problem. Have fun!” There will be blank paper, pencils,
and crayons available at a separate spot in the classroom for the students to use to create their
math pattern ideas.
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Once their math pattern theories have been created and shared with each other, the
teacher will get the class's attention to continue today's lesson. “So, who would like to share
the ideas they created today to solve this math problem?” (Volunteers will be selected to bring
their pattern up to the visualizer and share what they learned with the class. The teacher will
ask the students how they came up with their pattern, or how they solved the different number
patterns. “So, what is a pattern again?” (Possible student responses: Something that repeats
over and over, something that starts again and again, etc.) “Very good, detectives! Where might
we see number patterns in real life?” (Possible student responses: When we're counting, on
rulers, etc.) “Excellent! Let's look at this grid again and write a table showing how the shape
changes. Why don't you write it along with me so that you have the same clues I have. The
table should look like the following:
The Type of Shape
The Number of Blocks
1
1
2
4
3
7
4
10
5
?
“As you can see I made sure to write an equals sign between the two columns to help me
remember that the type of square is on the left, and the number of blocks for each shape is on
the right. Another way to remind myself is I need to think about is the number of blocks my
next shape will have, depends on the number of blocks a certain shape has now. So, when writing
a math equation or problem using the words next and now, what pattern is happening with the
numbers in 'Number of Blocks' column?” (Possible student responses: It gets bigger by 3 each
time, it gets bigger, etc.) “Right! If we were to find the difference between each of the
numbers in the 'Number of Blocks' column, we would see that it gets bigger by 3 each time. So,
if we are looking for the next shape, and we see that now each shape has three blocks added
each time, then another way I could say it, is Next equals Now + 3.” The teacher would then
write the equation up on the board as the following: Next = Now + 3. “What would the answer
be then for the next letter T? How many blocks would there be?” (Three.)
Explore (10 min.):
“Great! Now I will give you a grid sheet and I would like you to make, by coloring in the
squares, any letter in the alphabet. Try to make the letter bigger and bigger each time by
adding the same number of squares. Use a table like the one on the visualizer to write down your
numbers if it will help you. Since the letter T has already been used, try to think of a different
letter and then write your math problem using the words Next and Now. Have fun exploring,
detectives!”
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The students will then take time to explore their alphabet letters for different patterns.
Share (5 min.):
“Once you find at number pattern, share your pattern with a partner and see if they can
guess what your Next and Now math problem is. See if you can guess theirs as well. Then write
one of the two patterns up on the board for us to explore as a class.” Once everyone has
written down their patterns, the teacher will then go through the different patterns found and
ask the students how they came up with their different number patterns. The students will be
expected to show their reasoning and help any students who do not understand how a particular
pattern was made. “As we can see there are a few of us who had the same pattern, others who
looked at the pattern in a different way, but we all came up with great math formulas using the
words Next and Now.”
Summarize (2 min.):
Towards the end of class, the teacher will go through the students' presentations and
highlight groups who have created unique shapes or alphabet equations that fit the concept of
using Next and Now. The teacher will also be checking for correct computations and showing
that there can be a variety of Next and Now equations for some of the letters displayed.
Additional Note: There is an extended activity regarding the alphabet letter patterns that can
be found in the book: Navigating through Discrete Mathematics in Grades Pre-Kindergarten -5.
(For Source-See Reference List)
1. Blank Grid Paper (page 24)
2. Question of the Day for Visualizer (page 25)
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Graph Paper
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Question of the Day
Remember when we learned about number patterns?
Try to find out what the number pattern is for this shape that
keeps getting bigger. Talk to your neighbor to see if they have
the same number pattern. If it is different, talk with your
neighbor about how you came up with that number pattern.
Don't forget the equals sign in your math problem.
Have fun!
1.
2.
3.
4.
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Day 5
Post-Assessment and Learning Stations with Discrete Math
Standard:
Algebra: Use single-operation input-output rules to represent
patterns and relationships and to solve real-world and mathematical problems.
3.3.1.2. Create, describe, and apply single-operation input-output rules involving
addition, subtraction, and multiplication to solve problems in various contexts.
Objectives:
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The students will be using their prior knowledge of number patterns, Next vs. Now, and
combinations to re-explore the games and activities learned this week through the use of
learning stations and manipulatives.
The students will be using their prior knowledge of number patterns, Next vs. Now, and
combinations to complete a post-assessment that focuses on the mathematical concepts
learned this week.
Materials:
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Post-Assessment: Same as pre-assessment found on page 5 and 6 (Questions 1-5).
Post-Assessment Chart (page 29)
Triangle Activity Sheet (page 18)
Bobby Bear Computer Online Activity: http://illuminations.nctm.org/activitydetail.aspx?
id=3
Directions for Today's Learning Stations (page 28)
Markers or Crayons
Computers (4-6)
Alphabet Grid Sheet Activity (page 24)
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Procedure:
1. The students will be working at four learning stations. (One station will be focusing on
the students creating a letter or symbol on a grid sheet and then adding a certain number
of squares to that shape each time in order to find out the formula of the shape using
the words Next and Now, a second station will allow the students to play on the computer
in order to find the different combinations of outfits Bobby Bear could have, a third
station allows the students to find more number patterns using the words Next and Now
through the Triangle Activity sheet they played with earlier this week, and the forth
station will be designated for the students to complete their five question postassessment. The students will be rotating among these two stations approximately every
5 minutes. This way every group has a chance to explore every station. With there being
16-18 students at a time, these four projects will be spread out over four tables so that
approximately 4-5 students can sit at each table.
2. While the students are working at the different learning stations, the teacher will be
walking around with a checklist to informally and formally assess the students on their
comprehension of combinations, Next vs. Now, and number patterns This form of
assessment takes about 5 minutes to complete for each student.
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Directions For Today's Learning Stations
Station 1:
1. You will be playing with different outfit combinations at the
computer station.
2. First, try to make a guess on the number of combinations there
are for the different types of outfits.
3. Then, move around the shirts and trousers to actually make the
different outfits to see how close your guess was to the actual
answer.
Station 2:
1. Create different number patterns by making a letter or symbol of
your choice.
2. Make it bigger and bigger by adding the same number of blocks
each time.
3. Write down your Next and Now patterns as you find them.
Station 3:
1. Create different number patterns by circling different triangle
groups on the triangle activity sheet.
2. See how many Next and Now patterns you can make using the
different triangles.
3. Write down your Next and Now patterns as you find them.
Station 4:
1. Try your best to answer these five questions.
2. Hand your paper in when you are done, and you can visit, or
revisit, any other station you would like. Enjoy Exploring!
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Name of Student
Date:
Classroom Teacher
Grade Level: 3
Discrete Math
Mastery Level
Yes, fully
mastered!
Understands the Needs Additional
majority of this Help and Guidance
concept.
Provide Example:
1. Is the student able to use the
clues in today's activities, to
write their own number pattern?
2. Is the student able to create a
number pattern using the words
Next and Now?
3. Is the student able to help
others figure out the day's
activities?
4. Is the student able to create
different number patterns from
just looking at pictures?
5. Are the number patterns
clear? Do they make sense?
6. Are the student's addition,
subtraction, and multiplication
examples correct?
7. Is the student able to come up
with ideas to answer the
combination problems in the preassessment?
8. Does the child use different
strategies to write down different
combination possibilities?
9. Would the student be able to
explain the concepts of
combinations, number patterns,
and Next. vs. Now equations?
29
Additional
Comments
Day 6
Pre-Assessment and Introduction
Standard:
Geometry and Measurement: Use geometric attributes to describe and create shapes in
various contexts.
3.3.1.2. Sketch polygons with a given number of sides or vertices (corners), such as
pentagons, hexagons, and octagons.
Objectives:



The students will complete a written pre-assessment which covers the concepts they will
be learning over a period of 4 days.
The students will represent, analyze, and solve a variety of vertex, edge, and counting
problems by using pictures and geoboards.
The students will understand the relationships among shapes, their edges, and their
vertices.
Materials (located at the end of each continued lesson):







Visualizer to Display Student Answers, Shapes, Patterns
Pre-Assessment: Questions 6-10 (pgs. 7 and 8)
Geoboards (One per Student)
Vertices and Edge Table to Display on Visualizer (page, 32)
Geoboard Rubber Bands (One Pack for Each Student)
Blank Paper
Pencils
Procedure:
Launch (4 min.):
The students will first come to class and complete a five question pre-assessment
(Questions 1-5). Once they finish their pre-assessment, they will turn their assessment in and
see directions displayed on the board through the use of the visualizer. The directions are as
followed, “Use the geoboard at your spot and rubber bands to make as many different shapes as
you can. Then, I want you to pick two of the shapes you made that you liked the best and make
those again to share with the other detectives at your table. After the students have had
enough time to make several different shapes and share them, the teacher will continue with
today's activities.
The teacher will then say, “Well, you have all had to time to create different shapes and
I see wonderful examples at your spots. I made a shape too, my shape just happens to be a star.
Looking at my shape I have edges, which are the lines that make up the outside of my shape, and
30
I have vertices, which are where any two lines meet to form a corner or point. If I have only
one point or corner, then I just have one vertex, but I don't have a shape do I? Looking at my
star, how many edges does it have?” (Ten.) “Very good! How many vertices, or points where the
lines meet are there?” (Ten.)
Explore (8 min.):
“Excellent. Now I want you to look at the shapes you've made and see how many lines and
vertices you have. Then, I would like you to try making shapes that have the following (table
listed below).” The students will then practice making different shapes with a certain number of
edges and vertices using their geoboards. “As you are making the different shapes, try using
the words edges and vertices.”
Number of Edges
Number of Vertices
2
1
7
7
5
5
3
3
6
6
4
4
9
9
8
8
10
10
Share (6 min.):
As the students are working, volunteers will be called to show the shapes they made that
fit a certain number of edges or vertex category. “Did you notice that the shapes you made had
the same number of sides and they did vertices? Why do you think that is?” (Possible student
responses: You can't have two sides and three vertices, etc.) “Right. A vertex needs two lines
to connect. So, no matter how many lines I use after two lines, eventually, those lines have to
connect to form that last vertex. So, the number of vertices equals, or is the same as, the
number of edges of a shape.”
Summarize (2 min.):
After the students have displayed their different shapes, the teacher will highlight
shapes that look different but still have the same number of sides and vertices as other shapes
in the same category. The teacher will looking at the different shapes to stress the point that
although shapes my look different, they still have some of the same numbers of vertices and
edges.
1. Vertices and Edge Table to Display on Visualizer (page, 32)
31
Number of
Edges
Number of
Vertices
2
1
7
7
5
5
3
3
6
6
4
4
9
9
8
8
10
10
32
Day 7
Continuing with Geoboards
Standard:
Geometry and Measurement: Use geometric attributes to describe and create shapes in
various contexts.
3.3.1.2. Sketch polygons with a given number of sides or vertices (corners), such as
pentagons, hexagons, and octagons.
Objectives:



The students will represent, analyze, and solve a variety of vertex, edge, and counting
problems by using pictures and geoboards.
The students will understand the relationships among shapes, their edges, and their
vertices.
The students will use their prior knowledge to identify the edges and vertices of the
shapes they make and create.
Materials (located at the end of each continued lesson):







Visualizer to Display Student Answers, Shapes, Patterns
Geoboards (One per Student)
Vertices, Edge, and Shape Table to Display on the Visualizer and for the Students to Use
(page, 35)
Geoboard Rubber bands (One Pack for Each Student)
Blank Paper
Pencils
Crayons or Markers
Procedure:
Launch (2 min.):
The students will first come to class and see directions displayed on the board through
the use of the visualizer. The directions are as followed, “Use the geoboard at your spot and
rubber bands to make as many different shapes that you want. Then, I want you to work with
the other detectives and record the edges and vertices for each shape that you create at your
table. Make sure to draw the shape as well. This can all be done on the activity sheet found at
your spot.”
Explore: (8 min.):
The students will then explore making different shapes using the geoboards and rubber bands.
After the students have had enough time to make and record several different shapes, the
33
teacher will then continue with today's activities.
Share (8 min.):
The teacher will then say, “Well, you have all had to time to create different shapes and
I see wonderful examples at your spots. Let's look at the shapes made at each table and have
them share the shapes they made.” The teacher will then call on the different tables and have
the different small groups show the shapes they made. The teacher will guide the discussion
and ask each group how many edges and vertices the particular shapes had. While the students
are presenting their shapes, the rest of the class will be expected to try the different shapes
as well to see the same number of edges and vertices that the different groups are presenting.
This will take the majority of the time.
Summarize (2 min.):
Towards the end of class, the teacher will highlight several of the shapes that show the
same number of sides and vertices. The teacher will stress how in order to have a complete
shape, the number of sides must equal the number of vertices. The students will be able to
spend the rest of the time in class freely playing with the geoboards to make different designs
and shapes.
1. Vertices, Edge, and Shape Activity Sheet (page, 35)
34
Shape
Edges
Shape:
1
Shape:
2
Shape:
3
Shape:
4
Shape:
5
35
Vertices
Day 8
Working with Similar Attributes
Standard:
Geometry and Measurement: Use geometric attributes to describe and create shapes in
various contexts.
3.3.1.2. Sketch polygons with a given number of sides or vertices (corners), such as
pentagons, hexagons, and octagons.
Objectives:


The students will represent, analyze, and solve a variety of vertex, edge, and counting
problems by using pattern blocks and shapes.
The students will use their prior knowledge of shapes, their edges, and their vertices to
compare and contrast different shapes through the use of a Venn Diagram.
Materials (located at the end of each continued lesson):







Visualizer to Display Student Answers, Shapes, Patterns
Venn Diagram Activity Sheet: One per Student (page 38)
Colored Shape Blocks
Blank Paper
Pencils
Crayons or Markers
Venn Diagram Problem During the Lesson's Summary (page 8)
Procedure:
Launch (4 min.):
The students will first come to class and see pattern blocks and shapes displayed on the
board through the use of the visualizer. The teacher will then ask the students, “What do these
shapes have in common? What do they all have that's the same?” (The shapes are a square, a
rhombus, and trapezoid and a parallelogram) The teacher will wait for possible student
responses. (Possible student responses: They all have four sides, they all have four vertices,
etc.) “Wonderful job! Now, what about these shapes?” (The shapes are a blue circle, a blue
triangle, a blue square, and a blue rhombus.) (Possible student responses: They are all blue.)
“Excellent! What these different shapes have in common is called an attribute. So, the
attribute for these shapes are that they are all blue. Before, the attribute was they all had for
sides, or they all had four vertices.”
“Let's look at a Venn Diagram. Where the circles overlap are where the shapes have the
same attribute, where the circles do not overlap is where the shapes are not the same or have
different attributes.” The teacher will then put blue shapes in one Venn Diagram circle, Orange
36
Squares in the other Venn Diagram circle, and blue squares in the middle section of the diagram.
“What is the attribute I am looking for here? What is the same and different?” (Possible
student responses: One circle has orange squares, the other part has blue shapes, and the part
in the middle shows a square that is a blue shape.) “That's exactly right!”
Explore:
“Now I am going to give you time to play with the different pattern blocks and compare
and contrast what is the same and different about the different blocks. Try to come up with
three different ways that some of the shapes have a similar attribute. Keep one of them in
mind because you are going to share with the other detectives in class the attribute you found.
Have fun exploring!” The teacher will walk around to provide guidance as needed.
Share:
After everyone has had time to explore the different blocks, the teacher will have the
students come to the visualizer to show their Venn Diagram and how they organized their
shapes. While the students are presenting their shapes, the rest of the class will try to
predict which attribute the students decided to try. This will take the majority of the time.
“So, how could I organize these shapes on the Venn Diagram so that none of the shapes are left
out?” (triangle, square, trapezoid, circle, and rectangle) (Possible student responses: Do the
shapes with four sides or vertices in one, and the rest of the shapes in the other, etc.) “Very
good! So, all the shapes with four sides or vertices go in one circle, and the shapes with less
than four sides or vertices go in the other. In this case do I have any shapes that share an
attribute in the middle?” (No.) “That's right!”
Summarize:
Towards the end of class, the teacher will look at the different Venn Diagrams that the
students displayed, and highlight certain comparisons that show higher level thinking when
comparing and contrasting the different shape blocks. The teacher will be looking for a variety
of strategies the students use in order to challenge students even further when looking at
common attributes among different shapes.
1. Venn Diagram Activity Sheet: One per Student (page 38)
2. Venn Diagram Problem During the Lesson's Summary (page 8)
37
38
Venn Diagram
Day 9
Post-Assessment and Learning Stations with Discrete Math
Standard:
Geometry and Measurement: Use geometric attributes to describe and create shapes in
various contexts.
3.3.1.2. Sketch polygons with a given number of sides or vertices (corners), such as
pentagons, hexagons, and octagons.
Objectives:


The students will be using their prior knowledge of geoboards, vertices, edges, and
number patterns to re-explore the games and activities learned this week through the
use of learning stations and manipulatives.
The students will be using their prior knowledge of geoboards, vertices, edges, and
number patterns to complete a post-assessment focused on the mathematical concepts
learned this week.
Materials:










Post-Assessment: Same as pre-assessment found on page 7 and 8 (Questions 6-10).
Post-Assessment Chart (page 42)
Directions for Today's Learning Stations (page 41)
Markers or Crayons
Computers (4-6)
Geoboards
Pattern Blocks
Venn Diagram Activity Sheet(page 38)
Computers (4-6)
Online Geoboard Activity: http://nlvm.usu.edu/en/nav/frames_asid_282_g_3_t_3.html?
open=activities
39
Procedure:
1. The students will be working at four learning stations. (One station will be focusing on the
students creating different patterns and comparing the shapes of different pattern blocks.
They will be looking at similar attributes among the shapes and using a Venn Diagram to organize
their shapes however they would like. A second station will be spent having the students create
different shapes and patterns on the geoboards. A third station will have a geoboard activity on
the computer that the students can explore. The fourth station will be designated for the
students to complete their five question post-assessment. The students will be rotating among
these two stations approximately every 5 minutes. This way every group has a chance to explore
every station. With there being 16-18 students at a time, these two projects will be spread out
over four tables so that approximately 4-5 students can sit at each table.
2. While the students are working at the different learning stations, the teacher will be walking
around with a checklist to informally and formally assess the students on their comprehension of
vertices, edges, geoboards, and number patterns. This form of assessment takes about 5
minutes to complete for each student.
40
Directions For Today's Learning Stations
Station 1:
1. You will be creating different patterns and comparing the shapes
of different pattern blocks.
2. Try to find attributes that are the same with the different
shapes.
3. Use the Venn Diagram to organize your shapes however you would
like.
Station 2:
1. You will create different shapes and patterns on the geoboards.
2. Try creating shapes that have different edges and vertices.
3. See if you can make a pattern with those shapes.
Station 3:
1. You will create different shapes and patterns on the geoboards on
the computer.
2. Try creating shapes that have different edges and vertices.
3. See if you can make a pattern with those shapes.
Station 4:
1. Try your best to answer these five questions.
2. Hand your paper in when you are done, and you can visit, or
revisit, any other station you would like.
3. Enjoy exploring!
41
Name of Student
Date:
Classroom Teacher
Grade Level: 3
Discrete Math
Mastery Level
Yes, fully
mastered!
Understands the Needs Additional
majority of this Help and Guidance
concept.
Provide Example:
1. Is the student able to use the
clues in today's activities, to
create and find the vertices of a
shape?
2. Is the student able to use the
clues in today's activities, to
create and find the edges of a
shape?
3. Is the student able to help
others figure out the day's
activities?
4. Is the student able to create
different shapes with different
edges and vertices?
5. Is the students able to find
common attributes among
different shapes?
6. Is the student able to draw
different shapes?
7. Would the student be able to
explain the concept of vertices,
edges, geoboards, and patterns to
someone else?
42
Additional
Comments
Source Citations:
Day 1 and 2 of Unit Plan:
DeBellis, Valerie A., Eric W. Hart, Margaret J. Kenney, and Joseph G. Rosenstein. Navigating through
Discrete Mathematics in Prekindergarten-Grade 5. The National Council of Teachers of Mathematics
Inc., 2009, pp. 35-45, 170-173.
Day 3 and 4 of Unit Plan:
DeBellis, Valerie A., Eric W. Hart, Margaret J. Kenney, and Joseph G. Rosenstein. Navigating through
Discrete Mathematics in Prekindergarten-Grade 5. The National Council of Teachers of Mathematics
Inc., 2009, pp. 129-138.
Illuminations: National Council of Teacher of Mathematics. “Bobby Bear.” [Online] Available
http://illuminations.nctm.org/ActivityDetail.aspx?ID=3, July 10, 2010.
National Library of Virtual Manipulatives. “Geoboard.” [Online] Available
http://nlvm.usu.edu/en/nav/frames_asid_172_g_2_t_3.htmlopen=activities&from=category_g_2_t_3.htm
l , July 10, 2010.
43