Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Patterns, Relationships, and Algebraic Thinking
Activity:
Fancy Footwork: Patterns and Relationships
TEKS:
(3.7) Patterns, relationships, and algebraic thinking. The student
uses lists, tables, and charts to express patterns and relationships.
The student is expected to:
(A) generate a table of paired numbers based on a real-life situation
such as insects and legs; and
(B) identify and describe patterns in a table of related number pairs
based on a meaningful problem and extend the table.
(3.6) Patterns, relationships, and algebraic thinking. The student
uses patterns to solve problems.
The student is expected to:
(A) identify and extend whole-number and geometric patterns to
make predictions and solve problems;
(C) identify patterns in related multiplication and division sentences
(fact families) such as 2 x 3 = 6, 3 x 2 = 6, 6 ÷ 2 = 3, 6 ÷ 3 = 2.
(3.4) Number, operation, and quantitative reasoning. The student
recognizes and solves problems in multiplication and division situations.
The student is expected to:
(B) solve and record multiplication problems (up to two digits times
one digit); and
(C) use models to solve division problems and use number
sentences to record the solutions.
(3.14) Underlying processes and mathematical tools. The student
applies Grade 3 mathematics to solve problems connected to everyday
experiences and activities in and outside of school.
The student is expected to:
(B) solve problems that incorporate understanding the problem,
making a plan, carrying out the plan, and evaluating the solution
for reasonableness;
(C) select or develop an appropriate problem-solving plan or
strategy, including drawing a picture, looking for a pattern,
systematic guessing and checking, acting it out, making a table,
working a simpler problem, or working backwards to solve a
problem; and
(D) use tools such as real objects, manipulatives, and technology to
solve problems.
(3.15) Underlying processes and mathematical tools. The student
communicates about Grade 3 mathematics using informal language.
The student is expected to:
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
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Mathematics TEKS Refinement 2006 – K-5
(A)
(B)
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explain and record observations using objects, words, pictures,
numbers, and technology; and
relate informal language to mathematical language and
symbols.
(3.16) Underlying processes and mathematical tools. The student
uses logical reasoning
The student is expected to:
(A) make generalizations from patterns or sets of examples and
nonexamples; and
(B) justify why an answer is reasonable and explain the solution
process.
Note: Portions of this lesson address TEKS at other grade levels as well;
however, the intent of the lesson fits most appropriately at the grade level
indicated.
Overview:
Using the book One is a Snail, Ten is a Crab as a springboard, students
will generate tables of paired numbers (based on the number of animal
feet/legs), identify and describe the patterns and relationships, and then
utilize the data in the tables to mathematically describe and solve
problems.
Materials:
One is a Snail, Ten is a Crab by April Pulley Sayre
Chart paper and markers
Fancy Footwork: Animal Trackers – Handouts/Transparencies 1a and 1b
Fancy Footwork: Patterns and Relationships – Handouts/Transparencies
2a, 2b, and 2c
Fancy Footwork: Multiple Feet – Handout/Transparency 3a
Fancy Footwork: On Equal Footing – Handouts/Transparencies 4a and
4b
Fancy Footwork: Watch Your Steps – Handouts/Transparencies 5a and
5c
Fancy Footwork: Creature Combinations – Handout/Transparency 6a
Fancy Footwork: Bulletin Boardwalk – Handout/Transparency 7a
Overhead hundreds chart and markers (optional)
Student hundreds charts and markers (optional)
For the teacher:
Fancy Footwork: Animal Trackers – Selected Sample Response is
provided on Sample Response 1c
Fancy Footwork: Patterns and Relationships – Teacher Key is provided
on Answer Keys 2d, 2e, and 2f
Fancy Footwork: Multiple Feet –Sample Responses are provided on
Sample Responses 3b
Fancy Footwork: On Equal Footing – Teacher Key is provided on Answer
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
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Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Keys 4c and 4d
Fancy Footwork: Watch Your Steps – Possible Responses is provided on
Sample Responses 5b
Fancy Footwork: Watch Your Steps – Teacher Key is provided on Answer
Key 5d
Fancy Footwork: Creature Combinations – Teacher Key is provided on
Answer Key 6b
Fancy Footwork: Bulletin Boardwalk – Teacher Key is provided on
Answer Key 7b
Grouping:
Whole group, pairs, or individual (teacher discretion)
Time:
Will vary according to student needs and teacher discretion.
Prerequisites:
Students should have been introduced to the conceptual models of
multiplication and division (2.4A,B), should be able to relate those
conceptual models to their symbolic representations (3.4A,C), and should
be familiar with the notion of fact families (3.6C).
Lesson:
1.
2.
Procedures
Ask students: If 1 is a snail, and 2 is a
person, what must we be counting?
Have students brainstorm the number of
feet/legs for a variety of animals.
Notes
Answer: feet
This brainstorming session is
designed to get students engaged
in anticipation of the story.
Ask: What other animals have 2 feet? 4
feet? 6? 8? 10?
3.
Read the book One is a Snail, Ten is a Crab
by April Pulley Sayre.
You might ask students to predict
or fill in the blank before each of
the initial 10 pages until they begin
Ask students to listen for how many and
to see the numerical patterns and
what kind of feet relate to each number in the relationships being developed.
story.
For example, before reading page
3, ask students what might equal
3 feet?
4.
Elicit from the class the number of feet/legs
(1, 2, 4, 6, 8, 10) for each animal. Choose
one to focus their attention on, such as
“people feet.”
Tell students that although the
book uses the word “feet,” the
word “legs” is a more appropriate
term.
5.
Ask: If 1 person has 2 feet, how many feet
do 2 people have? 3 people? 4 people?
If you choose to focus on people,
you can use the problem solving
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
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Mathematics TEKS Refinement 2006 – K-5
Procedures
Chart students’ responses in a T-chart or
table:
people
feet
1
2
2
4
3
6
7.
Notes
strategy, “act it out,” to
demonstrate the initial pattern.
4
8
Ask: What patterns do you notice in the
chart?
6.
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Add additional steps (7, 24, 11,…) to the
chart as you guide the students through the
next few procedures.
Responses will vary.
people
feet
1 2 3 4 7
2 4 6 8
11
24
Ask: How many feet do 7 people have?
Answer: 14
How do you know?
At this point, abandon “act it out,”
so that students will utilize other
strategies. Have students
verbalize their strategies for the
class.
Possible responses might include
the following:
• skip count by 2, seven
times;
• add 2 seven times;
• the number of feet is 2
times the number of
people;
• add the number of feet for 3
and 4 people (decompose
7 into two parts, 3 and 4,
and add the chart results
for those parts); or
• utilize the distributive
property with other parts (2
and 5), etc.
8.
Ask: What number sentence(s) could we
use to find out how many feet 7 people
have?
Possible responses might include
the following:
• 2+2+2+2+2+2+2=14
• 7 x 2=14
• (3 x 2) + (4 x 2)=14, etc…
9.
Ask: If there are 24 human feet, how many
Answer: 12 people
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
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Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Procedures
people are there?
How do you know?
Notes
This “undoing” step connects the
inverse operation of division to
multiplication. Again, have
students verbalize their strategies
for the class.
Possible responses might include:
• use repeated subtraction or
measurement division;
• solve for the missing factor;
• use a basic division fact,
etc…
10.
11.
Ask: What number sentence(s) could we
use to find the number of people if we know
there are 24 human feet?
? x 2 = 24 or 24 ÷ 2 =12, etc….
Check for Understanding
Possible responses: (see
examples above and below)
If students have used triangular
flash cards, they can relate 24 to
the product (whole), 2 as one of
the factors (part), and 12 as the
missing factor (part) or quotient.
Ask: Based on the patterns in our chart and
the strategies just shared, if the number of
people was 11, how would you find the
number of feet?
11 x 2 = 22
If the number of feet is 16, how would you
determine the number of people?
16 ÷ 2 = 8
Based on the pattern, could there be a group
of people with 23 feet? Why?
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
No, based on the pattern there
cannot be a group of people with
23 feet.
Possible responses:
• If you divide 23 counters
into groups of 2, there is
one left over.
• When counting by 2s, you
get 22 or 24, not 23.
• 11 people would have 22
feet; 12 would have 24 feet.
• 23 is not a multiple of 2,
etc….
Grade 3
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Mathematics TEKS Refinement 2006 – K-5
12.
Procedures
(Optional Activity)
Show students how they can use a hundreds
chart and colored cubes or tiles to track the
number of feet for any number of people.
13.
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Notes
Mark each multiple of 2 on the
hundreds chart with an overhead
marker as you ask: How many
feet on 1 person? 2 people? 3
people? etc…
(Materials needed: hundreds chart
transparency and overhead color tiles,
cubes, or markers and/or student hundreds
charts and markers)
This whole group demonstration
may serve as segue into an
extension activity (Handouts/
Transparencies 1a and 1b –
Fancy Footwork: Animal Trackers)
for those students with special
needs who need further work or
assistance with skip counting and
multiples. A selected sample
response is provided on Sample
Response 1c.
Tell students they will be working in pairs to
generate or complete the rest of the animal
legs T-charts, tables, and discussion
questions. (Use Handouts/Transparencies
2a, 2b, and 2c – Fancy Footwork: Patterns
and Relationships)
Based on your students’ needs,
you might want to:
• work on each chart
together as a warm-up
patterning activity over
several days;
• assign pairs of students
different tables to complete;
• ask students to complete
the entire handout; or
• have students generate
their own charts for each
animal before completing
the handout tables and
questions.
IMPORTANT: Be sure to point out
that students must rely on the
relationships between the paired
sets rather than the vertical or
horizontal patterns, because the
steps (inputs) are not sequential.
14.
In accordance with your method of
assignment (see procedure 13 above), ask
students to share strategies for completing
the tables. Use Transparencies 2a, 2b, and
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Adding a process column to each
table allows you to record the
students’ various strategies or
processes as symbolic
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Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Procedures
2c - Fancy Footwork: Patterns and
Relationships to record students’ solution
strategies.
Notes
representations. Sharing and
recording these symbolic methods
of solution allows students to see
multiple (and sometimes more
sophisticated and efficient) ways
of arriving at the same answer.
(See example on Answer Key 2d Fancy Footwork: Patterns and
Relationships – Teacher Key.)
15.
Discuss student responses to questions for
each table on Handouts/Transparencies 2a,
2b, and 2c.
(See possible responses and
teacher notes on Answer Keys 2d,
2e, and 2f – Fancy Footwork:
Patterns and Relationships –
Teacher Key.)
16.
Handout/Transparency 3a - Fancy Footwork:
Multiple Feet allows students additional
opportunity to work with multiples.
Use the handout to debrief this
activity with the class. (Sample
responses are included on 3b.)
Items 4 and 5 are TAKS formatted
questions. To avoid errors,
students may need to generate
the entire set of multiples for the
target number (i.e., feet/legs) first,
and then select the appropriate
subset or answer choice.
(Special-needs students can
utilize the Animal Tracker
handout, 100s chart, and cubes or
tiles.)
17.
Draw a simple pan balance on the board.
Review with students how a balance
functions.
The pan balance helps to develop
the meaning of equivalence (=) as
well as inequalities (< and >).
Share with students the pages for the
number forty (40) in the book.
Ask: What number sentence can we write to
represent the legs on 4 crabs?
Possible responses:
10+10+10+10; 4 x 10; etc…
Write one of those number sentences on the
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
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Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Procedures
left side of the balance.
Then ask: What number sentence can we
write to represent the legs on 10 dogs?
Notes
Possible responses:
4+4+4+4+4+4+4+4+4+4; 10 x 4,
etc…
Choose one to write on the right side of the
balance.
Ask: Do we have the same number of legs
on each side of the balance?
Yes, 40.
Then, we can say that 4 sets of crab legs is
equal to 10 sets of dog legs, or
4 x 10 = 10 x 4.
18.
Continue with more examples. Include some
sets of animal legs that balance and some
that do not.
For example:
5 sets of insect legs > 3 sets of
spider legs
5x6>3x8
30 > 24
19.
Give students Handouts/Transparencies 4a
and 4b - Fancy Footwork: On Equal Footing.
Possible strategies might include
the following:
• drawing a picture;
• using objects;
• making lists/tables;
• using inverse relationships;
• etc….
Read the introduction and example together.
Tell students that their task is to balance the
equations. They can use any problem
solving strategy they choose. Students
should be prepared to share their work.
Items 5 and 6 require computation
beyond basic facts. (Once again,
special-needs students can utilize
the Animal Tracker handout, 100s
chart, and cubes or tiles.)
A teacher key is provided on 4c
and 4d.
20.
Have students share their strategies with the
entire class.
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Students may use concrete,
pictorial, and/or symbolic methods
to solve each equation. You can
use the various strategies to
provide scaffolding to the
symbolic, as well as the algebraic
solution.
Grade 3
Page 8
Mathematics TEKS Refinement 2006 – K-5
Procedures
Tarleton State University
Notes
Example (See problem 1)
• Elicit how to find the
number of legs on 12
people and the legs on any
number of spiders
• Record 12 x 2 = ? x 8
• Rewrite 24 = ? x 8
• Find the missing factor by
using the inverse operation
of division, 24 ÷ 8 = ?
Once again, you can use
triangular flash cards to remind
students of this inverse operation
while reinforcing fact families.
21.
Introduce Transparency 5a - Fancy
Footwork: Watch Your Steps.
See possible responses on Sample
Responses 5b - Fancy Footwork: Watch
Your Steps.
Assign the problems on Handout 5c - Fancy
Footwork: Watch Your Steps.
22.
Have students share their solution strategies
with the entire class.
23.
Handout/Transparency 6a – Fancy
Footwork: Creature Combinations provides
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Note: You may want to skip the
introductory transparency and let
students use their own choice of
problem solving strategies to work
the problems on the handout.
Be sure that students are familiar
with the 4-step problem solving
model and can apply it to one-step
problems before proceeding to
multi-step problems. Multi-step
problems can be taught by
introducing a one-step problem,
asking students to create a
second problem using the answer
to the first problem, etc... (see
Huinker, 1994, cited in Van de
Walle). The problems on this
handout include some understood
information (number of legs on
specified animals) and hidden
questions.
These problems allow students to
choose which animal combination
Grade 3
Page 9
Mathematics TEKS Refinement 2006 – K-5
Procedures
an alternate or additional set of problems
which are TAKS formatted.
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Notes
matches the specified number of
legs. However, students must still
carry out multiple steps and follow
the order of operations.
Homework:
Students can write and solve their own animal leg problems at home.
Then, they can share their problems with parents or classmates.
Assessment:
During class discussions and student sharing, you can assess
students’ levels of development by observing behaviors and
responses. For example:
1. Do students skip count, add, or multiply when calculating
multiples?
2. Are students able to extend patterns when the input is not
sequential?
3. Are students able to verbalize the patterns and relationships
between ordered pairs?
4. Are students able to represent multiplication concretely,
pictorially, and/or symbolically?
5. Do students relate division to multiplication?
6. Do students understand the equal sign as an indicator of
equivalence?
Extensions: 1. Use Transparency 7a – Fancy Footwork: Bulletin Boardwalk as an
extension to this lesson. Decorate a bulletin board with
characters from the book One is a Snail, Ten is a Crab by April
Pulley Sayre. Explain the riddle to children and share one solution
with them in picture and symbolic form. Ask students if they think
there are other solutions to the riddle. Provide the appropriate
materials (perhaps in a center) so students can generate solutions
and post them on the bulletin board. (A teacher key is provided on
7b.)
2. The author, April Pulley Sayre, has developed worksheets to
accompany her book. You can find and download these
worksheets at the following web site:
http://www.aprilsayre.com/pages%20books/onesnailpage.htm
3. In 3rd grade science, students learn that species have different
adaptations that help them survive and reproduce in their
environment. Have students research how various animals use
their legs (or other body parts) to survive in their environment. The
following web sites may provide some ideas related to this
investigation.
http://www.nsta.org/main/news/stories/science_and_children.php?
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
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Mathematics TEKS Refinement 2006 – K-5
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news_story_ID=51821
http://www.suzy.co.nz/suzysworld/Factpage.asp?FactSheet=251
http://www.k12.de.us/warner/feet.html
http://www.smithsonianeducation.org/educators/lesson_plans/
lewis_clark/lesson1_a.html
http://www.arthurdorros.com/escape/activities-animaltracks.html
4. Many non-fiction books support the learning expectation outlined in
Extension 3. The book Claws, Coats and Camouflage: The Ways
Animals Fit into Their World by Susan E. Goodman is one such book.
Resources:
Art used with permission from www.aprilsayre.com.
Sayre, A. (2003). One is a snail, Ten is a crab: A counting by feet
book. Cambridge, MA: Candlewick Press.
Van de Walle, J. A. (2004). Elementary and middle school
mathematics: Teaching developmentally. Boston: Pearson
Education, Inc.
Modifications: As noted in procedures 12, 16, and 19, Handouts/Transparencies 1a
and 1b – Fancy Footwork: Animal Trackers may provide support for
special-needs students.
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 11
Mathematics TEKS Refinement 2006 – K-5
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Fancy Footwork: Animal Trackers
people
spiders
dogs
crabs
insects
1. Assign a different color of linking cube, color tile, etc. to each animal listed above.
Then, use that color to track the number of legs for various groups/sets (1, 2, 3,
and so on) of that animal on the 100s chart.
2. Based on the pattern for people, could a group of people have 23 feet? How do
you know?
3. Could a pack of dogs have 48 feet? How do you know?
4. Could a group of crabs have 96 feet? How do you know?
5. Stacy raises ants in her ant farm. Which list shows the possible number of
feet/legs she might see?
a) 6, 12, 18, 26, 30
b) 12, 18, 22, 30, 42
c) 18, 24, 30, 36, 42
d) 6, 12, 18, 24, 32
Handout/Transparency 1a
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 12
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Fancy Footwork: Animal Trackers
(continued)
Use the Hundreds Chart below to track the number of legs
for various groups of animals.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
Handout/Transparency 1b
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 13
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Fancy Footwork: Animal Trackers
Selected Sample Response
people
spiders
dogs
crabs
insects
1. Assign a different color of unifix cube, color tile, etc. to each animal listed
above. Then, use that color to track the number of legs for various
groups/sets (1, 2, 3, ...) of that animal on the 100s chart.
Example:
Red
(R)
people
1
2
3
4
5
6
7
8
9 10
(R)
(R)
(R)
(R)
(R)
11 12 13 14 15 16 17 18 19 20
(R)
(R)
(R)
(R)
(R)
21 22 23 24 25 26 27 28 29 30
(R)
(R)
(R)
(R)
(R)
2.
Based on the pattern, could a group of people have 23 feet? No.
How do you know? When counting by 2’s (people feet), you cannot get 23
feet. Twenty-three (23) is not a multiple of 2. You cannot multiply 2 by any
whole number and get 23.
3. Could a pack of dogs have 48 feet? How do you know?
4. Could a group of crabs have 96 feet? How do you know?
5. Stacy raises ants in her ant farm. Which list shows the possible number of
feet/legs she might see?
a) 6, 12, 18, 26, 30
b) 12, 18, 22, 30, 42
c) 18, 24, 30, 36, 42
d) 6, 12, 18, 24, 32
Sample Response 1c
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 14
Mathematics TEKS Refinement 2006 – K-5
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Fancy Footwork: Patterns and Relationships
1. Complete the table.
Snails Feet
1
1
2
3
4
5
96
What do you notice about the relationship between the number of snails and the
number of feet?
________________________________________________________________
Why do you think this happens?_______________________________________
________________________________________________________________
2. Complete the table.
People Feet
1
2
2
4
8
12
10
How would you find the number of feet for any number of people?
_________________________________________________________________________________________________
Handout/Transparency 2a
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 15
Mathematics TEKS Refinement 2006 – K-5
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3. Complete the table.
Dogs
Feet
1
4
2
8
3
5
10
28
Based on the pattern, what could you do to find the number of feet on 12 dogs?
________________________________________________________________
4. Complete the table.
Insects Feet
1
6
2
3
30
6
10
If the number of insect feet/legs is 18, how could you find the number of insects?
________________________________________________________________
Handout/Transparency 2b
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 16
Mathematics TEKS Refinement 2006 – K-5
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5. Complete the table.
Spiders Feet
1
8
2
4
40
6
64
If the number of spider feet/legs is 88, how could you find the number of spiders?
________________________________________________________________
6. Complete the table.
Crabs
Feet
1
10
2
4
6
30
10
80
How can you use the data in the table to find the number of feet/legs on 16 crabs?
________________________________________________________________
Handout/Transparency 2c
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 17
Mathematics TEKS Refinement 2006 – K-5
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Fancy Footwork: Patterns and Relationships
(Teacher Key)
1. Complete the table.
Snails Process (possible representations) Feet
1
1 or (1 x 1)
1
2
1+1 or (2 x 1)
2
3
1+1+1 or (3 x 1)
3
4
? x 1 or 4÷1
4
5
? x 1 or 5÷1
5
96
96 x 1
96
What do you notice about the relationship between the number of snails and the
number of feet?
The number of feet is the same or equal to the number of snails.
Why do you think this happens?
Each snail only has one foot. To find the number of feet, you can count by 1’s,
repeatedly add 1, or multiply the number of snails by 1. For every snail there is exactly
one foot.
NOTE: This table and pattern provides an opportunity to review or clarify the identity
property of multiplication and/or division.
2. Complete the table.
People Process (possible representations) Feet
1
2 or (1 x 2)
2
2
2+2 or (2 x 2)
4
4
2+2+2+2 or (4 x 2)
8
6
? x 2 or 12÷2
12
10
10 x 2
20
How would you find the number of feet for any number of people?
Multiply the number of people by two (2). Number of feet = Number of people times 2.
f=px2
NOTE: Students at this grade are not expected to use variables.
Answer Key 2d
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
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Page 18
Mathematics TEKS Refinement 2006 – K-5
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3. Complete the table.
Dogs 1 2 3 5 7 10
Feet 4 8 12 20 28 40
Based on the pattern, what could you do to find the number of feet on 12 dogs?
Skip count by 4 twelve times, add 4 twelve times, multiply 12 times 4, etc.
NOTE: Students need to construct and work with horizontal tables. However, to
facilitate a process column, record the table vertically.
4. Complete the table.
Insects Feet
1
6
2
12
3
18
5
30
6
36
10
60
If the number of insect feet/legs is 18, how could you find the number of insects?
You can divide 18 by 6, or ask yourself, “What can I multiply by 6 to get 18?”
Answer Key 2e
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
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Mathematics TEKS Refinement 2006 – K-5
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5. Complete the table.
Spiders Feet
1
8
2
16
4
32
5
40
6
48
8
64
If the number of spider feet/legs is 88, how could you find the number of spiders?
You can divide 88 by 8, or ask yourself, “What can I multiply by 8 to get 88?”
6. Complete the table.
Crabs 1
2 3 4 6 8 10
Feet
10 20 30 40 60 80 100
How can you use the data in the table to find the number of feet/legs on 16 crabs?
You could add the number of feet/legs on 10 crabs to the number of feet/legs on 6
crabs. Or, you could add the number of legs on 2 crabs, the number of legs on 4 crabs,
and the number of legs on 10 crabs, etc.
Answer Key 2f
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 20
Mathematics TEKS Refinement 2006 – K-5
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Fancy Footwork: Multiple Feet
Animals often live or gather in groups. Can you determine the number of legs for each
of the animal groups below? Use any problem solving strategy you choose to answer
each question. (Make a table, draw a picture, use manipulatives, etc.)
1. Could a pack of dogs have 48 feet/legs? Explain your answer.
2. Could a bed of crabs have 96 feet/legs? Explain your answer.
3. Could a swarm of bees have 88 feet/legs? Explain your answer.
4. Stacy raises ants in her ant farm. Which list shows the possible number of
feet/legs she might see?
a.
b.
c.
d.
6, 12, 18, 26, 30
12, 18, 22, 30, 42
18, 24, 30, 36, 42
6, 12, 18, 24, 32
5. Mrs. Washington is decorating a Charlotte’s Web bulletin board with baby
spiders. Which list below shows the possible number of feet/legs that might be
on the web?
a.
b.
c.
d.
8, 16, 24, 30, 48
8, 16, 24, 36, 42
16, 24, 32, 42, 48
16, 24, 32, 40, 48
Handout/Transparency 3a
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 21
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Fancy Footwork: Multiple Feet
Sample Responses
Animals often live or gather in groups. Can you determine the number of legs for each
of the animal groups below? Use any problem solving strategy you choose to answer
each question. (Make a table, draw a picture, use manipulatives, etc.)
1.
Could a pack of dogs have 48 feet/legs? Explain your answer.
Yes, a pack of dogs could have 48 feet/legs. Each dog has 4 legs, and 48 is a
multiple of 4.
2.
Could a bed of crabs have 96 feet/legs? Explain your answer.
No, a bed of crabs could not have 96 feet/legs. Each crab has 10 legs, and 96 is
not a multiple of 10.
3.
Could a swarm of bees have 88 feet/legs? Explain your answer.
No, a swarm of bees could not have 88 feet/legs. A bee is an insect and has 6
legs. 88 is not a multiple of 6.
4.
Stacy raises ants in her ant farm. Which list shows the possible number of
feet/legs she might see?
a.
b.
c.
d.
6, 12, 18, 26, 30
12, 18, 22, 30, 42
18, 24, 30, 36, 42
6, 12, 18, 24, 32
List “c” shows the possible number of feet/legs she might see.
5.
Mrs. Washington is decorating a Charlotte’s Web bulletin board with baby spiders.
Which list below shows the possible number of feet/legs that might be on the web?
a.
b.
c.
d.
8, 16, 24, 30, 48
8, 16, 24, 36, 42
16, 24, 32, 42, 48
16, 24, 32, 40, 48
List “d ” shows the possible number of feet/legs that might be on the web.
Sample Responses 3b
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 22
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Fancy Footwork: On Equal Footing
Four
have the same number of feet/legs as five
We can say that the number of feet/legs on 4
.
equals the number of
. Look at the equations below.
feet/legs on 5
4
=5
4 groups of ten legs = 5 groups of 8 legs
4 x 10 = 5 x 8
40 = 40
Use any problem solving strategy you choose to solve each of the following
problems. Show your work.
1) 12
=
?
2) 3
=
?
Handout/Transparency 4a
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Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 23
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Fancy Footwork: On Equal Footing (cont.)
?
3) 12
=
4) 9
=
?
Challenge:
5) 7
=
?
6) 12
=
?
Handout/Transparency 4b
Art used with permission from www.aprilsayre.com
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 24
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Fancy Footwork: On Equal Footing
(Teacher Key)
Four
have the same number of feet/legs as five
We can say that the number of feet/legs on 4
.
equals the number of
. Look at the equations below.
feet/legs on 5
4
=5
4 groups of ten legs = 5 groups of 8 legs
4 x 10 = 5 x 8
40 = 40
Use any problem solving strategy you choose to solve each of the following
problems. Show your work.
1) 12
=
?
2) 3
12 groups of 2 legs = 3 groups of 8 legs
12 × 2 = 3 × 8
24 = 24
=
?
3 groups of 8 legs = 6 groups of 4 legs
3×8 = 6×4
24 = 24
Answer Key 4c
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Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 25
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Fancy Footwork: On Equal Footing (cont.)
(Teacher Key)
?
3) 12
=
4) 9
12 groups of 4 legs = 8 groups of 6 legs
12 × 4 = 8 × 6
48 = 48
=
?
9 groups of 8 legs = 12 groups o 6 legs
9 × 8 = 12 × 6
72 = 72
Challenge:
5) 7
=
?
6) 12
7 groups of 8 legs = 14 groups of 4 legs
7 × 8 = 14 × 4
56 = 56
=
?
12 groups of 10 legs = 20 groups of 6 legs
12 × 10 = 20 × 6
120 = 120
Answer Key 4d
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Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 26
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Fancy Footwork: Watch Your Steps
How can we solve this Fancy Footwork Problem using multiple steps?
Ben and two of his friends take their dogs to the park to play Frisbee.
There are exactly 26 legs among them. How many dogs did they take to
the park?
What do we know?
What do we need to find out?
If we solve this problem one step at a time, it’s easy!
Try this.
• How many people are there?
• If there are __ people, how many of the legs are “people legs?”
• If __ of the 26 legs are “people legs,” how many legs belong to dogs?
• If there are __ dog legs, how many dogs did they take to the park?
Handout/Transparency 5a
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 27
Mathematics TEKS Refinement 2006 – K-5
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Fancy Footwork: Watch Your Steps
Possible Responses
How can we solve this Fancy Footwork Problem using multiple steps?
Ben and two of his friends take their dogs to the park to play Frisbee. There
are exactly 26 legs among them. How many dogs did they take to the
park?
What do we know?
(3 people took their dogs to the park, there are 26 legs in all)
What do we need to find out?
(how many legs belong to people, how many legs belong to dogs, how
many dogs they took to the park)
If we solve this one step at a time, it’s easy!
Try this:
• How many people are there?
(3, Ben and 2 friends)
• If there are 3 people, how many of the legs are “people legs?”
(6 of the legs are “people legs,” 3 x 2 = 6)
• If 6 of the 26 legs are “people legs,” how many legs belong to dogs?
(20 legs belong to dogs, 26 – 6 = 20)
• If there are 20 “dog legs,” how many dogs did they take to the park?
(5 dogs, 20 ÷ 4 = 5, or 5 x 4 = 20)
Sample Responses 5b
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 28
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Fancy Footwork: Watch Your Steps
Solve these multi-step problems. Be ready to share your strategy.
1. Shelly and her two friends are on the bed playing with their hermit
crabs. There are 36 legs on the bed. How many hermit crabs are on
the bed?
2. Two kinds of animals have 72 feet/legs. Six of the animals are
insects. The rest are dogs. How many dogs are there?
3. Two kinds of animals have 102 feet/legs. Six of the animals are
crabs. The rest are insects. How many insects are there?
4. Two kinds of animals have 96 feet/legs. Ten of the animals are dogs.
The rest are spiders. How many spiders are there?
5. Todd has a collection of insects and spiders that he made for science
class. There are 116 legs/feet in his collection. He has six insects.
How many spiders does he have?
Handout/Transparency 5c
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 29
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Tarleton State University
Fancy Footwork: Watch Your Steps
(Teacher Key)
Solve these multi-step problems. Be ready to share your strategy.
1. Shelly and her two friends are on the bed playing with their hermit
crabs. There are 36 legs on the bed. How many hermit crabs are on
the bed?
There are 3 hermit crabs on the bed.
2. Two kinds of animals have 72 feet/legs. Six of the animals are
insects. The rest are dogs. How many dogs are there?
There are 9 dogs.
3. Two kinds of animals have 102 feet/legs. Six of the animals are
crabs. The rest are insects. How many insects are there?
There are 7 insects.
4. Two kinds of animals have 96 feet/legs. Ten of the animals are dogs.
The rest are spiders. How many spiders are there?
There are 7 spiders.
5. Todd has a collection of insects and spiders that he made for science
class. There are 116 legs/feet in his collection. He has six insects.
How many spiders does he have?
Todd has 10 spiders.
Answer Key 5d
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 30
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Fancy Footwork: Creature Combinations
1. There are exactly 26 feet/legs. Which combination of creatures totals
26 feet/legs?
a)
b)
c)
d)
6 people and 5 dogs
2 people and 5 dogs
5 people and 4 dogs
10 people and 4 dogs
2. There are exactly 72 feet/legs. Which combination of creatures totals
72 feet/legs?
a)
b)
c)
d)
5 insects and 7 spiders
4 insects and 6 spiders
3 insects and 6 spiders
6 insects and 4 spiders
Challenge:
Anna counts 70 feet/legs in the aquarium tank. There are crabs and spiders
in the tank. How many crabs and spiders could be in the tank?
Handout/Transparency 6a
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Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 31
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Fancy Footwork: Creature Combinations
(Teacher Key)
1. There are exactly 26 feet/legs. Which combination of creatures totals
26 feet/legs?
a)
b)
c)
d)
6 people and 5 dogs
2 people and 5 dogs
5 people and 4 dogs
10 people and 4 dogs
c) 5 people and 4 dogs totals 26 feet/legs.
2. There are exactly 72 feet/legs. Which combination of creatures totals
72 feet/legs?
a)
b)
c)
d)
5 insects and 7 spiders
4 insects and 6 spiders
3 insects and 6 spiders
6 insects and 4 spiders
b) 4 insects and 6 spiders totals 72 feet/legs.
Challenge:
Anna counts 70 feet/legs in the aquarium tank. There are crabs and spiders
in the tank. How many crabs and spiders could be in the tank?
There could be 3 crabs (30 feet/legs) and 5 spiders (40 feet/legs) in the tank for a total
of 70 feet/legs.
Answer Key 6b
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Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
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Tarleton State University
Fancy Footwork: Bulletin Boardwalk
Help our class build a bulletin board based on the book One is a Snail, Ten
is a Crab by April Pulley Sayre. Solve the riddle below and record your
solution in picture and number sentence form. Post your solution on the
bulletin board. Look at the posted solutions so that you don’t duplicate one
that has already been posted.
Riddle:
There are 12 creature feet/legs in the backyard. What animals are in
the backyard? (List all the combinations you can find.)
Your teacher, ___________________, has done this one for you.
3 sets of people legs and 1 set of insect legs equal 12 legs
(3 x 2) + (1 x 6) = 12 legs
6 + 6 = 12 legs
Reminder
3 sets of people legs + 1 set of insect legs
is the same as
1 set of insect legs + 3 sets of people legs
(3 x 2) + (1 x 6) = (1 x 6) + (3 x 2)
Handout/Transparency 7a
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Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
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Fancy Footwork: Bulletin Boardwalk
(Teacher Key)
Riddle:
There are 12 creature feet/legs in the backyard. What animals are in the
backyard? (List all the combinations you can find.)
Possible Combinations: (For teacher use and information only. These are not in bulletin
board format)
12 snails (sn)
6 people (p)
3 dogs (d)
2 insects (i)
Students may represent the following combinations in a different order.
10 sn + 1 p (this is the same as 1 p + 10 sn)
8 sn + 2 p
8 sn + 1 d
6 sn + 3 p
6 sn + 1 i
6 sn + 1 p + 1 d
4 sn + 4 p
4 sn + 2 d
4 sn + 1 sp (spider)
4 sn + 1 p + 1 i
4 sn + 2 p + 1 d
2 sn + 5 p
2 sn + 1 c (crab)
2 sn + 1 p + 2 d
2 sn + 1 p + 1 sp
2 sn + 2 p + 1 i
2 sn + 3 p + 1 d
2 sn + 1 d +1 i
4p+1d
3p+1i
2p+2d
2 p + 1 sp
1p+1c
1p+1d+1i
1 d + 1 sp
Answer Key 7b
Patterns, Relationships, and Algebraic Thinking
Fancy Footwork: Patterns and Relationships
Grade 3
Page 34