

Summary: Students will learn to identify the relationship between patterns and functions.
Main Curriculum Tie: Mathematics Grade 6 Strand: EXPRESSIONS AND EQUATIONS (6.EE) Standard 6.EE.9 Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. Materials: Invitation to Learn
 Cookie ingredients
 Package of store cookies
Functional Machine
The Ins and Outs of Functions
Spaghetti Graphs
 Play Dough or other
clay
 Spaghetti
 Graph paper
 Spaghetti Graphs
equations
Additional Resources
Book
The Fly on the Ceiling, by Julie Glass; ISBN 0679886079
Attachments
Web Sites
 math.com
This free site has information and wellexplained tutorials on all math subjects
Background For Teachers:
There is a powerful pattern identified when any number is put
into an equation and consistently follows the rule. This is called a
function. When the rule is identified, each number does not have to be
solved, but one could simply skip to the input number desired, insert
it into the “rule” or equation, and the answer (output) will be given.
Functions are easily shown in tables, such as the example below:
Input
(X) 
Output
(Y) 
1 
3 
2 
4 
3 
5 
4 
6 
5 
7 
It is easy to see that the output increases by one each time. The
relationship between the input and output is the key, however. The
“rule” is to add 2. If 2 is added to the input number 1, the answer is 3.
Therefore, an equation can be formed. An equation is a mathematical
sentence that contains an equal sign. The equation for the above
example is x + 2 = y. Students should become proficient at spotting
the pattern, recognizing the rule, and creating an equation from that
rule. The rule or equation should be a one or twostep problem, or it
becomes really difficult to solve.
Another skill that students need to master is the ability to change
an equation back into a function table. If the equation is x + 2 =y, the
student can choose any number to represent the input (x). They will
then “plug in” that number to get the output (y). So, if 8 were chosen for the input, then 10 would be the output. Keep in mind that any
variable can and should be used, not just x and y each time.
Moving into the final skill, students need to first be able to graph
a function table. This is a simple plotting exercise (taught in 5th grade
and in 6th grade Standard III Objective 2). If x is 1 and y is 3, the
coordinates will be (1,3). At least 2 (preferably 3) coordinates must be
plotted, then connected to create a line (for these types of equations,
a straight line will be created). The goal is for students to be able to
graph an equation. In summary, here are the steps:
 Change the equation to a function table
 Graph the function table
 Connect the plots to form a line
Instructional Procedures: Invitation to Learn
Show students ingredients for cookies. Ask students if they
would like to eat each ingredient. They may want the sugar, but not
the salt, etc. Explain to students that these ingredients go through a
“magical” change from their separate ingredients until they are spit
out of a factory machine. The magic, of course, is the mixing of the
ingredients and chemical change when they are cooked together. Tell
students that today they will be putting numbers through a machine,
which will “magically” change the number. The magic, of course, is
the function rule. You may give the students a cookie, notifying them that this cookie may stimulate their brain and make them even better
mathematicians.
Instructional Procedures
Functional Machine
 Teach students the basics of functions: give examples of
function tables, discuss how to discover the rule, and how to
change that rule into an equation.
 It may take a few examples for students to catch on, but they
will begin to see this as a fun game. Allow students to come up
with their own functions with rules. Let a few of them try to
stump the class.
 Students will pair up. Student A will think of a function table,
rule, or equation and secretly write it down on a piece of
scratch paper. The rule or equation should be a one or two step
problem.
 Student B will write an input number (x) on the Functional
Machine worksheet and Student A will write the output (y).
The second student will then guess the function. If correct, they
switch. If incorrect, Student B guesses again.
 After 3 guesses, if Student B has not guessed correctly, Student
A will unveil their table, rule, or equation and explain it to
Student B.
 Students switch roles.
NOTE: You may want to allow students to choose to use a table,
rule, or equation, but eventually move students to using
equations.
The Ins and Outs of Functions
 Using the instructions in Background Information, ensure
that students understand the basics of functions.
 Students will be in heterogeneous groups of 3 or 4. Each
group will be given 20 Ins and Outs of Functions cards.
The first card is the simplest, with each card becoming more
difficult. They will work as a group to answer each question on
the card:
 What’s the rule?
 What are three more examples?
 What’s the equation?
 When they have finished a card, they may check with you to see
if their answers are correct. If they are incorrect, they need to
go back to their group to understand where they went wrong. If
a group is consistently getting answers wrong, determine what
they are missing and reteach the group.
 When they are working, walk around and make sure that
everyone is participating. They may split up the cards, but they
should also help each other.
 The first group to finish all 20 cards is declared the winner and
will become “experts” that will go around to help the other
groups (not give answers, but assist). You could even give them
stickers to put on to show they are the experts.
Spaghetti Graphs
 FIRST THINGS FIRST: The prerequisite to this is that students
need to have the ability to change an equation to a function
table, like this:
2x + 3 = y
If x = 1, then y = 5
If x = 2, then y = 7
If x = 3, then y = 9
Then, they need to plot a function table.
Step 1: Students will put an equation into a function table (at least
3 sets to plot).
7 + x = y
Step 2: Students will graph the above table using play dough “dots.”
Step 3: They will put the dried spaghetti in all 3 dots. This will
ensure that the points are straight.
Step 4: They will repeat the process with the rest of the equations
in the set.
Step 5: If done correctly, the 3 lines will intersect with at least one
other line on the same graph.
Equation sets to use:
SET A:
X + 1=Y
X – 2=Y
2X + 3=Y
SET B:
X + 3=Y
3X – 2=Y
X ÷ 2=Y
SET C:
X  1=Y
X ÷ 3 + 1=Y
5X  3=Y
SET D:
X•X=Y
X=Y
3X ÷ 2=Y
SET E:
3X + 7=Y
X – 7=Y
6X ÷ 3=Y
Strategies For Diverse Learners: The equations used for the spaghetti graphs were positive slopes
(lines that go from right to left). Your advanced learners can be
exposed to negative slopes, which are lines that go from left to
right. If the equation has both a negative number before X and
the second number, it will be a negative slope. For example, 5X
– 3=Y is a negative slope.
Extensions:
 For an introduction to coordinate grids, a great picture book
is The Fly on the Ceiling about Rene` Descartes creating the
Cartesian coordinate system.
 Include ideas for integration for other curricular areas (use
appropriate subject area headings).
Family Connections
 Using spaghetti, clay, and graph paper, students will show a
parent or older sibling how to graph equations.
 Students will take home the Ins and Outs of Functions cards,
or create their own function tables. They will show family
members how to figure out the rule and create an equation.
Assessment Plan:
 Give students 6 equations. If they can accurately plot the
equations on a graph, they are proficient. This may also be done
with function tables and many other math concepts.
02 correct: InterventionThese students need direct reteaching
instruction
34 correct: PracticeThese students need extra practice
56 correct: ProficientThese students have mastered the
content. Give them an enrichment/extension activity to do
 In a gym, have students create a coordinate grid, using masking
tape as the x and y axes. Students will be the points, and they
may use a broomstick, etc. to create a line by connecting the
“points.”
Bibliography:
Cwikla, J. (2004). Less experienced mathematics teachers report what is wrong with their
professional support system. Teachers & Teaching, 10(2), 181197.
When lessexperienced mathematics teachers interviewed, they
expressed disappointment that many of their more experienced
colleagues lacked content knowledge. Overall, they were not satisfied
with the mentoring or collaboration offered by fellow teachers because
they often knew more content than their more experienced peers.
Holly, K. R. (1997). Patterns and functions. Teaching Children Mathematics, 3, 312313.
This article gives many ideas and activities for teaching patterns and
functions in elementary grades K6. Venn diagrams, function machines,
and building cubes are some ideas presented.
Author: Utah LessonPlans
Created Date : Jul 06 2007 16:28 PM
