Algebra Activities
For Eighth Grade Students
By: Jim Lubke
Bemidji Middle School
jlubke@bemidji.k12.mn.us
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Executive Summary:
Algebra has become law for all Eighth Grade math cirriculumn in Minnesota. Students who
achieve the objectives in Eighth grade algebra will have the best chance to advance through a
successful math career in high school.
The following lessons will provide meaningful material to help students learn algebra. The unit
activities will include work with linear functions, proportional relationships, non linear functions
arithmetic and geometric sequences.
The activities follow the design of the Minnesota Math Standards in 8.2 Algebra for 8 th grade.
Included are sample problems students will be able to accomplish upon completing the
activities.
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The following are Minnesota State Math Standards for 8th grade that will be covered with
the following lessons.
8.2.1 Understand the concept of function in real-world and mathematical
situations, and distinguish between linear and non-linear functions.
8.2.1.1 Understand that a function is a relationship between an independent
variable and a dependent variable in which the value of the independent variable
determines the value of the dependent variable. Use functional notation, such as
f(x), to represent such relationships.
8.2.1.2 Use linear functions to represent relationships in which changing the input
variable by some amount leads to a change in the output variable that is a
constant times that amount.
8.2.1.3 Understand that a function is linear if it can be expressed in the form
f(x)=mx+b or if its graph is a straight line.
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8.2.1.4 Understand that an arithmetic sequence is a linear function that can be
expressed in the form f(x)=mx+b, where x = 0, 1, 2, 3,....
8.2.1.5 Understand that a geometric sequence is a non-linear function that can be
expressed in the form f(x)=ab to the x power, where x = 0, 1, 2, 3,....
8.2.2 Recognize linear functions in real-world and mathematical situations;
represent linear functions and other functions with tables, verbal descriptions,
symbols and graphs; solve problems involving these functions and explain results
in the original context.
8.2.2.1 Represent linear functions with tables, verbal descriptions, symbols,
equations and graphs; translate from one representation to another.
8.2.2.2 Identify graphical properties of linear functions including slopes and
intercepts. Know that the slope equals the rate of change, and that the y-intercept
is zero when the function represents a proportional relationship.
8.2.2.3 Identify how coefficient changes in the equation f(x) = mx + b affect the
graphs of linear functions. Know how to use graphing technology to examine these
effects.
8.2.2.4 Represent arithmetic sequences using equations, tables, graphs and verbal
descriptions, and use them to solve problems.
8.2.2.5 Represent geometric sequences using equations, tables, graphs and verbal
descriptions, and use them to solve problems.
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Table of Contents
Lesson 1 Evaluate Linear Functions for Given Values………..page 7
Lesson 2 Linear Functions Completing Tables…………………….page 9
Lesson 3 Writing a Rule for a Linear Function…………………page 11
Lesson 4 Proportional Relationships…………………………………..page13
Lesson 5 Percentages and Proportions……………………………..page 15
Lesson 6 Linear Functions in F(x) = mx +b Form…………….page 17
Lesson 7 Arithmetic Sequences…………………………………………page 19
Lesson 8 Geometric Sequences………………………………………..page 21
Lesson 9 Arithmetic Sequences Two…………………………………page 23
Lesson 10 Geometric Sequence Two………………………………..page 25
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Lesson 1
Evaluate Linear Functions for given values
Hands on equations is a helpful tool for this lesson.
Objective: Students will replace a variable with a given value and evaluate
linear functions. Students will identify dependent and independent
variables.
Minnesota State Standard. 8.2.1.1 Understand that a function is a relationship
between an independent variable and a dependent variable in which the value of
the independent variable determines the value of the dependent variable. Use
functional notation, such as f(x), to represent such relationships.
Launch:
We are going to explore linear functions today. The first function we will
work with is F(x) = x + 1. In the linear function F(x) is considered to be the
dependent variable because it will depend on x to determine it’s value. The
x is the considered to be the independent variable. I want to know how
much F(x) is worth if x = 5.
What I would like you to is replace F(x) with an empty box. The box will be
like a birthday gift and you are trying to figure out what will be in the box.
Since I have chosen x to be = to 5. We will replace x with an open ( ) and
place the 5 inside.
So we have BOX = (5) + 1. What would happen if we replaced x with the
number 6? How much would we have in the box? What would happen if we
changed x to 2x, or 3x? Change the linear equation form to BOX = 2( 5) +
1. Now we have to multiply the 5 by 2 then add the one. What does our
box equal now?
Explore:
You are going to work alone and come up with your own linear function. I
would like you to start by writing F(x) = x + some number of your choice.
Next I would like you to replace F(x) with an empty box and x with open ( )
and write BOX = (a number of your choice) + 1. Then, figure out how much
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you would have in the box. What would happen if we changed x to 2x, 3x?
Lets write two more linear functions using 2x and 3x. When changing the x
to 2x we need to multiply the replaced value by 2, then add the 1. Use the
form BOX = 2( ) + 1. Identify the dependent and independent variables.
Share:
When most of the students are finished with the explore assignment ask for
their examples. Write all examples on the board. Can anyone come up with
an idea that this linear function can be used for? What are we working with
today? Why did we replace the F(x) with an empty box? What did we
replace the variable x with? How do we evaluate linear functions? What
happened when we changed x to 2x, 3x?
Summarize:
We are working with linear functions today. Linear functions can be
evaluated for different values for the variable. We replaced F(x) with an
empty box so we could figure out what number would go in the box. We
replace x with ( ) and place our chosen number inside. Next we evaluated
the linear function. We changed the coefficient of x to 2 then 3 and
evaluated the function. The main purpose of the lesson for today was to
introduce linear functions and evaluate them for a given and chosen variable
value and identify the dependent and independent variable.
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Lesson 2
Linear Functions Completing Tables
Hands on equations is a helpful tool for this lesson
Objective: Students will be able to identify the dependent and
independent variables and make tables of values for linear functions.
Minnesota State Standard. 8.2.1.1 Understand that a function is a relationship
between an independent variable and a dependent variable in which the value of
the independent variable determines the value of the dependent variable. Use
functional notation, such as f(x), to represent such relationships.
Launch:
In lesson 2 students will identify dependent and independent variables and
find values for linear functions by completing tables with four given values.
Today we are going to be working with tables, used, when working with
linear functions. Tables are used to keep track of a linear function’s value
when several different numbers are being used to replace x.
We need to start by choosing a linear function. Lets choose the same
function we started with yesterday. F(x) = x + 1. Today we are going to
replace the x variable with several different numbers. We are going to use a
table to keep track of our work. The table of values will have two columns.
The first column will be for our (x) values. The second column will be for our
F(x) or BOX values. The values we are going to choose to represent x are 0,
1,2, and 3. We are going to take turns putting these different numbers in
for x and finding how much we have in the box.
First write F(x) = x + 1. Next, we will replace the F(x) with a BOX, and the
x with ( ). So we have BOX = ( ) + 1. Now we will take turns replacing x
with our chosen values 0, 1, 2, and 3. How much is in the box if I replace x
with 0? Keep track of your work in the table. First put 0 in the x column
then since BOX = 1 we will put a 1 in the F(x) column. Now it is 1’s turn to
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go in for x. How much is in the BOX now. Keep track in your table.
Complete the steps with 2 then 3.
Explore:
Now it is time for you to choose your own linear function. For today we will
keep the x values in our tables 0,1,2,and 3. You may work with a partner
and create your tables as you go. What would happen if we changed x to 2x,
3x? Lets make two more tables using 2x and 3x.
Share:
Ask students to share their linear functions and tables with the class. Write
all the linear functions and tables on the board. Ask questions about the
tables. Can you tell what numbers would come next in the tables? Why do
you think we are finding different values for our function? What happened
when we changed x to 2x, 3x?
Summarize:
Today we worked with linear functions. We chose several values to replace
our variable x with. We made a table of values to keep track of our work.
We changed the coefficient for x to 2x then 3x and made tables of values.
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Lesson 3
Writing a rule for a linear function
Hands on equations is a helpful tool for this lesson
Objective: Students will be able to write a rule for linear functions using
table of values.
Minnesota State Standard. 8.2.1.1 Understand that a function is a relationship
between an independent variable and a dependent variable in which the value of
the independent variable determines the value of the dependent variable. Use
functional notation, such as f(x), to represent such relationships.
Launch:
Students will be using tables of values to write rules for linear functions. In
lesson 3 we will begin by looking at a few tables of values similar to the
table we created yesterday. Today the tables will start off by having the
form F(x) = ax + b where a, and x, could equal, 0, 1, 2, 3, 4, or 5 and b =
0. We will start by looking at a table of values together. In the table the x,
independent variable, values will be 0, 1, 2, and 3. The F(x), dependent
variable will have values of 0, 2, 4, and 6. We will start by looking at how
the variables are changing. How much is x changing. Is x going up or
down? How much is F(x) changing? Is F(x) going up or down? What is the
relationship between the dependent variable and the independent variable?
If the change in F(x) is 2 and the change in x is 1 what is the F(x) when x =
4, 5? If we know how a linear function is changing we can write a rule for
the function. The rule for this function is F(x) = 2x. I will provide the
students with 4 tables of values with the established criteria. Students can
work in pairs to find the rules of the linear functions. Next students can
challenge each other by making tables of values following from the form F(x)
= ax + b. Now a, and b, can be any integer value between -5 and 5 and x
will go from 0-5.
Explore:
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I will provide the students with 4 tables of values with the established
criteria. Students can work in pairs to find the rules of the linear functions.
Next students can challenge each other by making tables of values following
from the form F(x) = ax + b. Now a, and b, can be any integer value
between -5 and 5 and x will go from 0-5.
Share:
Give students time to work on each problems and after most have
completed a few rules ask for examples to be put on the board.
What happens in the table when the rules change? Does anyone have a
table they can’t find a rule for? What is the dependent variable?
Summarize:
Today we wrote rules for linear functions given tables of values. Students
will be able to look at tables of values and find the changes of the dependent
and independent variables and write a rule for the linear functions.
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Lesson 4
Proportional Relationships
Students will be able to solve proportional relationships and
learn graphs of linear proportions pass through the origin.
Objective:
Minnesota State Standard 8.2.1.2 Use linear functions to represent relationships in
which changing the input variable by some amount to a change in the output
variable that is a constant times that amount.
Launch:
The eighth graders at Bemidji Middle School are selling calendars to raise
money for a class trip. The amount of money they raise is proportional to
the number of calendars they sell. As with any proportional relationship, the
graph of the relationship between the calendars sold and the dollars raised is
a line that passes through the origin. If the graph of the relationship
contains the points (1,6),(3, 18),(5,30) can we find the relationship
between the points? Today we need to introduce a few new terms. The
dependent variable is also called the output or the range. The dependent
variable is always in the y position of an ordered pair, (x,y). If you
remember RYVO you will know the Range are the Y Values which are the
Output values of a linear function. The Independent variable is also called
the input or the domain. The independent variable always in the x position
of an ordered pair, (x,y). If you remember DXVI you will know the Domain
are the X Values which are the Input values of a linear function.
The input values of our three points are 1, 3, and 5. The output values of
our three points are 6, 18 and 30. The inputs are the number of calendars
we sold and the outputs are the money we raised. How did the money
change? How did the number of calendars, sold change? What is the rule
for the linear equation? Make a table of values for the inputs and outputs.
Can you find a rule for the proportion? The rule is called the constant of
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proportionality. Make a graph of the linear function. Does the graph contain
the point (0,0), the origin? Is the graph a proportion?
Explore:
The price of potatoes is 25cents a pound. Find three different amounts of
potatoes you could buy and the total cost of each amount. Identify the input
and the output values. Students will make a table of values. Students will
write a rule for the linear function from the table and make a graph.
Student may work with a partner? Create a proportion relationship and do
another problem. Repeat all previous steps. Identify the constant of
proportionality. Find the cost of 50 pounds of potatoes.
Share:
After 15-20 minutes ask students for their graphs and tables. Can you make
a table that is not proportional? How would the graph look?
Summarize:
Students learned about RYVO and DXVI. They made tables and graph linear
proportions. They learned that linear proportions have to pass through the
origin. Student also learned about the constant of proportionality.
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Lesson 5
Percentages and Proportions
Objective: Students will understand percents
Minnesota State Standard 8.2.1.2 Use linear functions to represent relationships in
which changing the input variable by some amount leads to a change in the output
variable that is a constant times that amount.
Launch:
Franklin Middle School is putting on a play. The eighth grade had 300
tickets to sell and the 7th grade had 250 tickets to sell. One hour before the
play the eighth grade had sold 225 tickets and the 7th grade had sold 200
tickets. Which grade was closer to the goal of selling all its tickets? Explain
your answer. To compare ratios change them into percents. The ratios are
225:300 and 200:250. Percent means out of 100. To change ratios into
percents we can set up proportions. 225/300 = X/100 and 200/250 = x/100.
Put fractions in lowest terms then cross multiply. What is 225/300 in lowest
terms? How many quarters is 225, 300? 9 and 12. Is 9/12 in lowest
terms? What is a common factor of 9 and 12? Is ¾ in lowest terms? ¾ =
X/100. Solve using cross products. 3 x 100 = 4 x X. Now divide by 4. X =
75%. Next we will do the same to 200/250. What number is a common
factor? Lowest terms 4/5 = X/100. 400 = 5X. Divide by 5. X is 80%.
Who sold a higher percent of their tickets?
Explore:
Johnson Middle School has 600 students. Madison Middle School has 450
students. For each of the problems below solve for the percents.
1. A survey found 300 Johnson students watch more than 1
hour of tv every night, while 270 Madison students watch
more than 1 hour.
2. What percents of each school watch more than 1 hour?
3. Which school has a higher percent of students watching tv?
4. Johnson has 275 girls and Madison has 250 girls.
5. Which school has a higher percent of girls?
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Share:
Give students several minutes to answer questions. Ask students
to share their answers. How could you figure what percent your
grade is if I give you your points and the total points?
Summarize:
Students learned how to figure percents by using a ratio and
cross products.
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Lesson 6
Linear functions in F(x) = mx + b Form
Objective: Students will be able to recognize a Linear function in
F(x) = mx + b form.
Minnesota Math Standards 8.2.1.3 Understand that a function is linear if it can be
expressed in the form f(x)=mx+b or if its graph is a straight line.
8.2.2.3 Identify how coefficient changes in the equation f(x) = mx + b affect the
graphs of linear functions. Know how to use graphing technology to examine these
effects.
Launch:
In this lesson students will be working with linear functions in the
F(x) = mx + b, slope intercept form. In previous lessons we
have found rules for linear functions when given tables of values.
In this lesson we are going to learn when we found our rule in
previous lessons it is the same as the slope intercept form of a
function. The change in the dependent value compared to the
change in the independent value is known as the slope, or the
rate of change. The b value tells where the line crosses the y
axis. We will begin by exploring several examples together.
Then you will partner up, with graph paper and draw several lines
with your partner. First you will graph without a calculator. Then
use the TI-73 to experiment with different graphs.
The first linear function we will look at is F(x) = 1x+ 1. We will
make a table of values to keep track of our work as we input the
values 0, 1, 2 and 3. We will replace the F(x) with a BOX and the
x with ( ) and write BOX = 1( ) + 1. Next we will substitute in
our inputs for x one at a time keeping track in our table of values.
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What are the output values? What is the rate of change? Graph
the line. Where does it cross the y axis?
Explore:
Now it is your turn. The linear functions you and your partner will
make a table of and graph are:
1.
2.
3.
4.
F(x)
F(x)
F(x)
F(x)
=
=
=
=
2x
2x
3x
3x
+1
+2
–1
-2
Remember to use 0, 1, 2, and 3 for your input values. When you
are finished making tables and graphing these functions you may
make one for your partner to do.
Next you and your partner are to solve and graph this word
problem.
Mario has $3.00 in his piggy bank. Mario wants to save his
money for an Ipod. Mario gets $12.00 a week for allowance.
Write a linear function in the F(x) = mx + b, Slope Intercept
Form. Make a table of values using 0, 1, 2, and 3 for the number
of weeks and make a graph to monitor Mario’s savings.
Share:
Ask students to share their work with the class. Can anyone tell
why we call F(x) = mx + b Slope Intercept form of a line? Where
can you find the slope? Where can you find the y intercept? How
much does an IPOD cost? How long will Mario have to save?
Summarize:
Students made tables and graphed linear functions in Slope
Intercept form. Students solved a word problem. Made a table
and graphed the function.
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Lesson 7
Arithmetic Sequences
Ojective: Students will recognize and write arithmetic
sequences as linear functions in Slope Intercept form.
Minnesota State Standard 8.2.1.4 Understand that an arithmetic sequence is a
linear function that can be expressed in the form f(x)=mx+b, where x = 0, 1, 2,
3,....
Launch:
Sequences of numbers that follow a pattern of adding a fixed number from one term to
the next are called arithmetic sequences. The following sequences are arithmetic
sequences:
Sequence A: 5 , 8 , 11 , 14 , 17 , ...
Sequence B: 26 , 31 , 36 , 41 , 46 , ...
Sequence C: 20 , 18 , 16 , 14 , 12 , ...
For sequence A, if we add 3 to the first number we will get the second number. This
works for any pair of consecutive numbers. The second number plus 3 is the third
number: 8 + 3 = 11, and so on.
For sequence B, if we add 5 to the first number we will get the second number. This also
works for any pair of consecutive numbers. The third number plus 5 is the fourth
number: 36 + 5 = 41, which will work throughout the entire sequence.
Sequence C is a little different because we need to add -2 to the first number to get the
second number. This too works for any pair of consecutive numbers. The fourth number
plus -2 is the fifth number: 14 + (-2) = 12.
We are going to make a table of values for sequence a. For this set of problems we are
going to start with 1 for our first input. When we put in 1 our output is 5. What is the
second term of the sequence? We need to keep track on a table of values. Our table of
values will have inputs 1, 2, 3, 4, and 5 and outputs that match our sequence 5, 8, 11, 14,
and 17. With sequences our slope is called a common difference. What is the common
difference between the numbers? We need to find out what our sequence value would
be if we put 0 in. What would our output be. Arithmetic Sequences are linear functions.
The Slope is 3 and the intercept is 2.
Explore:
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Students may work in partners. Make a table of values for sequence b and
c. Find the rate of change between the sequence numbers. This will be
your Slope. Next, figure out what the sequence value if our input is 0.
Write the sequences in Slope intercept form.
Next create a sequence for you partner to solve. Repeat previous steps.
Share:
Ask students to share their answers with the class. Ask questions of the
students on how they found the intercept. What is the common difference?
Summarize:
We used arithmetic sequences to make tables and write linear functions in
Slope Intercept form. We found out the common difference is the same as
the slope
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Lesson 8
Geometric Sequences
Objective: Students will express geometric sequences in F(x) =
ab^x. Students will learn that a common difference is the 2nd
term divided by the 1st term.
Minnesota State Standard 8.2.1.5 Understand that a geometric sequence is a nonlinear function that can be expressed in the form f(x)=ab to the x power, where x
= 0, 1, 2, 3,....
Sequences of numbers that follow a pattern of multiplying a fixed number from one term
to the next are called geometric sequences. The following sequences are geometric
sequences:
Sequence A: 1 , 2 , 4 , 8 , 16 , ...
Sequence B: 0.01 , 0.06 , 0.36 , 2.16 , 12.96 , ...
Sequence C: 16 , -8 , 4 , -2 , 1 , ...
For sequence A, if we multiply by 2 to the first number we will get the second number.
This works for any pair of consecutive numbers. The second number times 2 is the third
number: 2 × 2 = 4, and so on.
For sequence B, if we multiply by 6 to the first number we will get the second number.
This also works for any pair of consecutive numbers. The third number times 6 is the
fourth number: 0.36 × 6 = 2.16, which will work throughout the entire sequence.
Sequence C is a little different because it seems that we are dividing; yet to stay
consistent with the theme of geometric sequences, we must think in terms of
multiplication. We need to multiply by -1/2 to the first number to get the second number.
This too works for any pair of consecutive numbers. The fourth number times -1/2 is the
fifth number: -2 × -1/2 = 1.
Because these sequences behave according to this simple rule of multiplying a constant
number to one term to get to another, they are called geometric sequences. So that we
can examine these sequences to greater depth, we must know that the fixed numbers
that bind each sequence together are called the common ratios. Mathematicians use the
letter r when referring to these types of sequences.
We are going to look at sequence A. First we will make a table we are going
to start with 0,and use 1, 2, 3, and 4 for the rest of our inputs. The outputs
are just the sequence values. Next, we will find the common ratio. To find
the common ratio take the second term of the sequence and divide it by the
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first term. 2 / 1 is 2. Our common ratio is 2. Our common ratio is the b in
our function F(x) = ab^x. Now we can write F(x) = a(2) raised to the x.
next, I input 0 for x. 2 raised to the zeo is one so that means a is one. We
will write F(x) = 2^x
Explore:
Students may work with a partner. You need to find the function
that makes sequences B and C. First make a table. Remember
to start with 0 for you first input. Next find you common ratio.
Next write a geometric sequence for your partner to solve.
Share:
Ask students for their functions for sequences B and C. What are
the common ratios? What would the 5th term be? Ask students
for their partner solve functions.
Summarize:
We wrote geometric sequences in F(x) = ab^x form. We made
tables and found a common ratio.
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Lesson 9
Arithmetic Sequences Two
Objective: Students will represent arithmetic sequences using
equations, tables, graphs, and verbal descriptions and use them
to solve problems
Minnesota State Standards 8.2.2.4 Represent arithmetic sequences using
equations, tables, graphs and verbal descriptions, and use them to solve problems.
8.2.2.1 Represent linear functions with tables, verbal descriptions, symbols,
equations and graphs; translate from one representation to another.
8.2.2.2 Identify graphical properties of linear functions including slopes and
intercepts. Know that the slope equals the rate of change, and that the y-intercept
is zero when the function represents a proportional relationship.
Launch:
In this lesson we will continue our work with arithmetic
sequences. Recall arithmetic sequences can be written in
F(x)=mx + b, Slope Intercept Form. The Slope, m, is the
constant change between the terms. We will work with several
sequences, making tables, graphs, writing equations, verbal
descriptions and solve problems.
Sequence A -2, 3, 8, 13, 18……..
First we will make a table. Leave a space at the top of the inputs
for 0. Sequences start with term 1 so 1 is our first input followed
by 2, 3, 4, and 5. The outputs are the terms of the sequence, -2,
3, 8, 13, and 18. The change in sequence terms can be found by
subtracting any two consecutive numbers. I would suggest t3 –
t2. We find the common difference is 5. We know the Slope of
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the line is 5. Now we need to work back to 0. If we take -2 and
subtract 5 we will end up at -7. Now we can write our Slope
intercept form of the sequence. F(x) = 5x -7. Now we can make
a graph of our sequence. We know we are going to cross the y
axis at -7. Next, we will go up 5 and over to the right 1. We can
verbalize the situation by saying sequence A crosses the F(x) axis
at -7 and has a positive Slope of 5.
Explore:
Here are two sequences for you to work with a partner on.
Sequence B 3, 7, 11, 15, 19
Sequence C 5, -2, -9, -16, -21
Make tables, Graph, Find the common difference, write a linear
function to represent the sequences. Remember to leave a space
for 0 in the input column of your table of values. Write a
sentence or two describing the sequence. Find the 15th term of
the sequence.
Find the missing term of the sequence.
Sequence D .5, -2.5, __, -5.5, -8.5…………..
Make 3 sequences for your partner to solve.
Share:
Collect answers from all students. Ask about common differences
and intercepts. What shape do arithmetic sequence graph to be?
Summarize:
In this lesson we did some work with arithmetic sequences. We
discover arithmetic sequences were disguised linear functions.
We needed to solve for our 0 input to find the F(x) intercept and
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the common difference to find the slope. We made tables,
graphs, wrote linear functions, described the sequences and
solved problems.
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Lesson 10
Geometric Sequences Two
Objective: Students will represent geometric sequence using
equations, tables, graphs, and verbal descriptions, and solve
problems.
Minnesota State Standard8.2.2.5 Represent geometric sequences using equations,
tables, graphs and verbal descriptions, and use them to solve problems.
Launch:
In this lesson we are continuing our work with geometric
sequences. We know we can represent geometric sequence in
the F(x) = ab^x Form. We learned b is the common ratio
between the numbers of the sequence. We find the common
ratio by dividing consecutive terms. When we work with
geometric sequences the 1st term has an input of 0.
Sequence A 3, 6, 12, 24, 48……………
First find the common ratio. Divide t2 by t1. 6/3 = 2. We now
know b = 2. Next we will solve for a. F(x)= a(2)^x. We are
going to replace the x with our first input, 0. The first term of the
sequence is 3 so we will replace the F(x) with a 3. 3 = a(2)^0
2^0 is 1. We have 3= a(1). Now we know a = 3. We will write
as F(x) = 3(2)^x. This is our function equation. Now we will
make a table. Our inputs are 0, 1, 2, 3, and 4. The outputs are
the terms of the sequence, 3, 6, 12, 24, and 48. Now we will
make our graph. Is our graph linear? Lets talk about geometric
sequences. Explain about the 1st and 2nd differences from the
table. What if I wanted to know the 10th term of the sequence?
Remember the first term had an input of 0. F(9) = 3(2)^9 so the
functions value is 1536.
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Explore:
Here are two geometric sequences for you to solve.
Sequence B 2,4, 8, 16, 32………
Sequence C 5, 15, 45, 135, 405……..
Work with a partner. Make a table, find the common ratio. Solve
for a. Write a function. Make a graph. Describe the sequence.
Solve for the missing term
Sequence D -3, 6, __, 24, -48
Write a Geometric sequence for your partner to solve.
Share:
Collected information from students. What is the common ratio?
How did you solve for a? What is the first input of a geometric
sequence? Talk about the graphs.
Summarize:
In this lesson we solve for geometric sequences. We made
tables, graphs, we wrote function to represent our sequences.
We found common ratios, and solved for our constant, a. We
discovered Geometric sequences are not linear.
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Lesson 11:
Linear Equations
Objective:
Minnesota Standard 8.2.21 and 8.2.2.3 Find the slope of a line from an equation
and graph a line from an equation.
Launch:
In this lesson we are going to be working with two forms of linear equations.
The first form is called standard form. It follows the formula Ax + By = C.
The second form is called slope intercept form. It follows the formula y = Mx +b.
We are going to change standard form into slope intercept form. Solving for y we
get y = -(A/B)x + (C/B). This is slope intercept form.
Next, we will plot two points and find the equation of the line in both forms. We
will make a table of values and a picture using the table we made.
Example graph ( 0, 3) (4, 15). Find the slope. Write standard form. Convert to
slope intercept form. Make a Table. Make a picture.
Explore:
1.
2.
3.
4.
5.
6.
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Graph the points (0, 4) and (3, 10).
Find the slope between the points by making a triangle.
Write the standard form of the equation.
Change the equation to slope intercept form.
Make a table of values.
Make a picture from the table.
Share:
Student will share their pictures with the class. Are all the pictures the
same? Do the pictures represent the same table?
Summarize:
Students found the slope of a line from an equation. Students also graphed
a line from an equation. We learned how to convert standard form of lines
to slope intercept form.
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