Download Recipes for Functions – Guess My Rule!

Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Basic Understandings
Activity:
Recipes for Functions – Guess My Rule!
TEKS:
(a) Basic Understandings.
(3) Function concepts. A function is a fundamental mathematical
concept; it expresses a special kind of relationship between two
quantities. Students use functions to determine one quantity from
another, to represent and model problem situations, and to
analyze and interpret relationships.
(5) Tools for algebraic thinking. Techniques for working with
functions and equations are essential in understanding underlying
relationships. Students use a variety of representations
(concrete, pictorial, numeric, symbolic, graphic, and verbal), tools,
and technology (including, but not limited to, calculators with
graphing capabilities, data collection devices, and computers) to
model mathematical situations to solve meaningful problems.
(A.2) Foundations for functions. The student uses the properties and
attributes of functions.
The student is expected to:
(B) identify mathematical domains and ranges and determine
reasonable domain and range values for given situations, both
continuous and discrete;
(A.4) Foundations for functions. The student understands the
importance of the skills required to manipulate symbols in order to solve
problems and uses the necessary algebraic skills required to simplify
algebraic expressions and solve equations and inequalities in problem
situations.
The student is expected to:
(C) connect equation notation with function notation, such as y = x +
1 and f(x) = x + 1.
(A.5) Linear functions. The student understands that linear functions
can be represented in different ways and translates among their various
representations.
The student is expected to:
(C) use, translate, and make connections among algebraic, tabular,
graphic, or verbal descriptions of linear functions.
Overview:
The purpose of this activity is for students to work on algebraic
representations of functions and connections to function tables. Students
will create their own linear functions and challenge others to discover the
underlying rule.
Basic Understandings
Recipes for Functions – Guess My Rule
Algebra I
Page 1
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Materials:
Transparency 1/Handout 1 Guess My Rule! Input/Output Table
Transparency 2/Handout 2 Guess My Rule! Find the Rule
Grouping:
Part 1: no groups, teacher works with entire class
Parts 2-4: 3 or 4 students
Time:
1 to 2 class periods
Lesson:
Procedures
Part 1: Mapping Notation
Show the rule f : N → N × 4 + 1 . Describe
that this means the rule f says to take the
input value, multiply it by four, add one, and
that will be the output value. Ask students
for more input values and calculate the
output values, recording them in a table.
Then give some output values and ask
students to figure out what the input values
must have been.
Notes
Input is another name for domain.
Output is another name for range.
It is important that students get
comfortable with the ideas of input
and output using “friendly” terms,
and then move to the algebraic
terminology of domain and range.
Beginning with the terms input and
output helps students understand
the basic concepts without
intimidating them.
Show the rule f : a → a × 4 + 1 ; ask students
how this is different from the rule above.
Show several more examples, such as
f : z → z × 4 + 1 , f : G → G × 4 + 1 , and
f : t → t × 4 + 1 , to help students see that the
variable name is arbitrary.
It is important that students
understand the name of the
variable is just that – a name.
There is nothing significant in the
use of any particular letter.
Teachers might even use
symbols, like smiley faces or
stars, to help illustrate that this is
just a “place holder” to direct
where to put the input value.
3.
Ask students to plug in input values for
f : N → N × 2 − 3 . Then ask for output
values, and find the corresponding input
values. Record all pairs in a table.
For input values include positive,
negative, large, and small values,
fractions and decimals, and
variables and variable
expressions.
4.
Do not give students the rule, but provide
output values for any input values they give.
If they choose, allow them to give you output
values, and give the corresponding input
values. Write the pairs in a table. Then ask
students to guess your rule.
Use several different rules that
involve fractions or decimals, not
just integers, for the slope and
intercept of the linear function.
Help students come up with some
techniques to find the rule (such
1.
2.
Basic Understandings
Recipes for Functions – Guess My Rule
Algebra I
Page 2
Mathematics TEKS Refinement 2006 – K-5
Procedures
Tarleton State University
Notes
as finding the difference between
the outputs of successive
integers).
5.
Part 2: Creating a Rule
Divide students into small groups (three or
four). Have each group create a rule that
they do not share with other groups (be sure
that it is linear at this point).
6.
Each group should fill in a table with input
and output values.
Have students include at least 6
input values. They should include
positive, negative, and rational
input/output values in their tables.
7.
.Each group should also plot their
input/output pairs on a rectangular
coordinate system.
This is a good time to hold a
discussion about discrete vs.
continuous data.
Ask, “Should we connect these
ordered pairs? Why or why not?”
Discussion should identify
situations in which it is appropriate
to connect and situations in which
it would not be appropriate.
8.
9.
10.
Part 3: Figuring Out the Rule
This is played like 20 questions. In turn,
each group will come to the front of the
classroom to present their input and output,
and the rest of the class tries to guess their
rule.
The class should have a limited
number of guesses before
additional information is given.
If the class cannot successfully guess the
group’s rule, one member of the group will
select a member of the class. The class
member identifies whether he/she is giving
input or output, and gives a value. The
group then gives the corresponding output or
input value. This should continue until the
class guesses the rule or gives up.
The teacher should carefully
monitor that each group gives
correct input/output values for
those the class gives them.
Points are awarded as follows:
The group gets one point for every input or
Be careful not to let guessing get
out of hand. To prevent shouting
Basic Understandings
Recipes for Functions – Guess My Rule
Algebra I
Page 3
Mathematics TEKS Refinement 2006 – K-5
Procedures
output value they are given.
The group gets one point for every incorrect
guess given by a member of the class.
The group loses a point if they give an
erroneous input or output value. (This should
be monitored by the teacher.)
Tarleton State University
Notes
out randomly, guessers should be
selected by the group presenting
their rule by raising their hands.
(Groups who guess out of turn can
have points deducted if it
becomes problematic.) Also
watch that the students selecting
the guessers are not selecting
only their friends.
The group that correctly guesses the rule
receives one point.
When the rule is guessed correctly, scoring
ends, and that group returns to their seats
and another group comes to the front.
The group with the most points after all
groups have presented their rules is declared
the winner.
11.
12.
13.
Part 4: Making Connections
After all rules have been guessed, have each
group put the rule and the graph on the
board.
Show the relationships of each graph to the
In the discussion, include the
rule that generated it.
ideas of slope and y-intercept,
positive and negative slope,
parallel lines (rules that have the
same slope but different
intercepts), etc.
Plot the input and output from an unknown
rule on the board. Have students guess the
rule without seeing the input/output table.
Show how the slope and y-intercept relate to
the rule of a linear function.
Students should be able to identify
the slope and y-intercept from a
graph and connect the graph to
the table and to the rule.
Homework:
Assign appropriate homework from the text to graph linear functions
from their rules and find the rules from the graphs. Homework should
also include problems that ask students to generate tables.
Extensions:
Activity can also be used for any other family of functions to teach
general characteristics of that family: i.e., quadratic functions,
exponential and logarithmic functions, square root functions, etc.
Basic Understandings
Recipes for Functions – Guess My Rule
Algebra I
Page 4
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Modifications: Alternate scoring method: Awarding one point to the student who
correctly guesses the rule makes the competition more individual.
Each student earns one point for each point the group earns, but also
earns points for correctly guessing the rules of other groups. This
method of scoring gives an individual winner, rather than a group
winner.
Basic Understandings
Recipes for Functions – Guess My Rule
Algebra I
Page 5
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Name ______________________________________
Guess My Rule!
1.
In your group, choose a rule and write it down on the back of this worksheet.
Do not let other students outside your group see your rule. It should be in the
form f(N)=N x 4 + 1.
2.
Fill in the table below, using your secret rule.
Input
3.
Output
Plot your table values on a rectangular coordinate system using the xcoordinate for the input value and the y-coordinate for the output value.
Connect the points with a line.
Basic Understandings
Recipes for Functions – Guess My Rule
Algebra I
Page 6
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Name _________________________________
Guess My Rule!
Use the space below to figure out the rule for each group. Then write the rule on the
blank provided.
Rule: ___________________________ Rule: _____________________________
Rule: ___________________________ Rule: _____________________________
Rule: ___________________________ Rule: _____________________________
Rule: ___________________________ Rule: _____________________________
Basic Understandings
Recipes for Functions – Guess My Rule
Algebra I
Page 7