Rebekah Lavigne
Topic: Finite Differences Equations
Grade Level: Algebra 2 (40 minute lesson)
Materials: Chalk/white board, ti-83/84 graphing calculator
Standards:
Algebra
Creating Equations
A-CED
Create equations that describe numbers or relationships.
1. Create equations and inequalities in one variable and use them to solve linear
and quadratic functions, and simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships
coordinate axes with labels and scales.
3. Represent constraints by equations or inequalities, and by systems solutions
as viable or nonviable options in a modeling context. For example, nutritional
and cost constraints on combinations of different foods.
4. Rearrange formulas to highlight a quantity of interest, using the same
example, rearrange Ohm’s law V = IR to highlight resistance R.
Linear, Quadratic, & Exponential Models
F-LE
Construct and compare linear, quadratic, and exponential models and solve
problems.
1. Distinguish between situations that can be modeled with linear functions and
with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals, and
that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit
interval relative to another.
Standards Continued: Standards for Mathematical Practice
1. Make sense of problems and preserve in solving them.


Select appropriate representations to solve problem situations
Interpret solutions within the given constraints of a problem
2. Reason abstractly and quantitatively.
 Understand how concepts, procedures, and mathematical results in one
area of mathematics can be used to solve problems in other areas of
mathematics
 Evaluate the relative efficiency of different representations and solution
methods of a problem
3. Construct viable arguments and critique the reasoning of others.
 Observe and explain patterns to formulate generalizations and conjectures
 Develop, verify, and explain an argument, using appropriate mathematical
ideas and language
 Devise ways to verify results, using counterexamples and informal indirect
proof
4. Model with mathematics.
 Understand how quantitative models connect to various physical models
and representations
 Model situations mathematically, using representations to draw conclusions
and formulate new situations
5. Use appropriate tools strategically
 Use multiple representations to represent and explain problem situations
(e.g., verbally, numerically, algebraically, graphically)
 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or
objects created using technology as representations of mathematical
concepts
6. Attend to precision.
 Understand and use appropriate language, representations, and terminology
when describing objects, relationships, mathematical solutions, and
rationale
 Communicate verbally and in writing a correct, complete, coherent, and
clear design (outline) and explanation for the steps used in solving a
problem
7. Look for and make use of structure.
 Communicate logical arguments clearly, showing why a result makes sense
and why the reasoning is valid

Understand and make connections among multiple representations of the
same mathematical idea
8. Look for and express regularity in repeated reasoning.
 Apply inductive reasoning in making and supporting mathematical
conjectures
 Understand the corresponding procedures for similar problems or
mathematical concepts
Anticipatory set:
Give three sets of points and ask the class if they can determine the equation of this
quadratic function that contains each of these points. When they realize that they
cannot do such a thing without any further information, I’ll give two points generated
from a linear graph and show them how to find “a” and “b” and determine the equation
from there. But first, I’ll create a table that includes these points in them and show that
since the differences in the Y values are the same on the first try, we see that it’s a linear
function (y=ax+b). Next I’ll give another three points from a separate linear function and
ask them to first determine if it’s a linear function and why and then find the equation
on their own, individually. Once this process is completed, we’ll move on to the lesson.
Learning Activity:
1. Given: (-2, 3), (1, 6) and (2, 15) as an introduction and ask to determine the equation
from them.
2. Given the table
x
-2
-1
0
1
2
y
-9
-2
5
12
19
What are the differences in the y values? We see that’s it +7 for each point in the table.
(-9+7=-2 where -2 is the next y value in the table)
Since the difference between all of the y values listed is the same on the 1st trial, this
determines that it’s a linear function. Now take two points from this table and figure
out the equation for this line. The equation form for a line is y=ax+b. So take two of
these points, substitute the y values and x values to generate two equations. We need
to find out what a and b are. Subtract one equation from the other, and find the values
(see attached sheet). The b’s will cancel each other and we will be able to solve for a.
Using the value of a, we can go back and solve for b. Then using both values, determine
the equation. I will be making sure that the students check to see that their solution is
correct.
After this is completed, the students will be asked to find an equation on their own
individually with two given point from another example. I will create a table of values
for X and Y where the students do not know the equation they came from. First I’ll set
the table up and see when the differences are equal. We’ll see that they’re not equal on
the first try, but the second try. This shows that it’s a quadratic function rather than a
linear (y=ax2+bx+c). With the points given at the beginning of class, we can now
substitute them into the quadratic form and solve for a, b and this time c as well
(check!). Once we find an equation for these points, I’ll ask them to set up the table of
values and see what they notice about the differences in the y values. Once the first
example is completed I will provide another example and expect the students to be able
to work individually to generate an equation by finding the values of a, b and c.
A discussion will follow about the meaning of the differences in the y values.
The lesson will contain visual learning activities (writing on the board and working out
the problems on their own) and auditory learning (lecture as well as discussion).
The lesson will be taught by limited investigation (students finding an equation for given
points, differences in y values) instruction (the quadratic and linear equation form) and
exploration (The relationship between the y values).
My role will be to guide them through the process while letting them figure it out for
their own.
Throughout the lesson, the NCTM Standards for Teaching Mathematics will be
paramount, i.e.,
The students will be engaging in worthwhile mathematical tasks both in class
work and assigned work. All assigned work is relevant to the NYS standards and
will involve realistic applications.
The teacher’s role in discourse will be positive, engaging, challenging, and
inspiring.
The students’ role in discourse will be encouraged, important, and mathematical
and will fully satisfy the communication process strand.
Multiple tools and methods will be used for enhancing discourse including
manipulatives, technology and hands-on activities.
The learning environment will constantly be one that will foster each student’s
mathematical power. Students will be encouraged and assisted in active problem
solving, making connections, and in understanding and creating representations
while employing strong reasoning and proof skills.
The teacher will engage in a constant analysis of teaching and learning pre- and
post-lessons to ensure that all objectives are met. Strengths of the lesson will be
identified as well as areas which may need adjustment.
Technology Integration:
At the end of the examples, I would like the students to get their TI-83/84 calculators
out and check that the equation they came up with was correct.
Provision for Diversity:
Gearing Down: This lesson is a stepping stone for following lessons. I would say that
there is little need to gear down. There shouldn’t be many questions. For the students
who have trouble and need extra help, a copy of the lessons notes will be available if
needed. A page explaining the equation forms and y differences will be available as
well.
Gearing Up: Students who understand this lesson better than others will be expected
to help those students who need it, and will be encouraged to assist in the lesson. They
will also be encouraged to assist in the explanation.
Questions for Understanding:
Knowledge: What is the difference between a linear function and a quadratic function?
Comprehension: Give an example of a linear function and a quadratic and explain why
they are linear or quadratic.
Application: Given a table of values for a function, find the differences in the y values;
are the differences the same the first try? Second?
Analysis: Is there significance in the differences? Why? How many times would you have
to determine differences to reach a degree of 5? How many points would you need to
determine the equation of the function?
Synthesis: Create a proof of why we can determine that the 3rd row of differences in
values is an equation is a function of degree 3.
Evaluation: What is the advantage to knowing that finite differences equations
determine the degree of a polynomial? Do you think this is an effective way of
determining an equation with given points?
Practice:
Guided: Students will be working on problems given in class.
Independent: students will be asked to complete a worksheet of given problems on
their own.
Closure:
To summarize, finite differences equations can be used to determine the equation of a
function when given only a select number of points. From there we can complete the
graph, and analyze. This activity is done by solving two-variable and three-variable
system of equations. They will be able to understand patterns and relationships in the
values.
Assessment:
Immediate: Students will be expected to take notes. Each student will be observed
when working on in class problems. Each student will be given the opportunity to
determine relationships and patterns as well as stating the next step that should be
taken.
Long Range: There will be assignments to help the students practice and strengthen
their understanding of the material. Assigned homework will be collected and graded.
At the end of the section, students will be given a test.