Download Activities for Algebra - Bemidji State University

Lynnea Salscheider
[email protected]
Jess Stuewe
[email protected]
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Executive Summary
This unit is designed to meet state and national standards concerning
algebra in the 8th grade. All of the lessons in this unit make use of three
main algebraic ideas: equality, symbols and variables, and patterns and
functions
Algebra topics include but are not limited to: concrete or pictorial
representations, verbal descriptions, tables, graphs, algebraic formulas,
linear and non-linear relationships, change, growth, and problem solving.
The Algebra Test that is included at the end of this unit is designed to be
used as both a pre and post test. The test will be administered at the
beginning of the unit, and then re-administered after the unit is taught. No
changes will be made to the test to allow for paired t-test scores for
comparison of student growth. Test answers will be reviewed after grading
for the unit is complete.
About the teachers:
Jessica Stuewe teaches seventh and eighth grade mathematics in Detroit
Lakes, Minnesota. Detroit Lakes Middle School schedules 43-minute class
periods. Jess teaches two sections of seventh grade mathematics, two
sections of eighth grade mathematics (which will be Eighth Grade Algebra
as of autumn, 2010) and one section of Algebra (which will be Eighth
Grade Advanced Algebra as of autumn, 2010).
Lynnea Salscheider teaches eighth grade mathematics in Piedmont, South
Carolina. She teaches on a 75-minute block schedule and has four sections
of eighth grade mathematics, one of which is a special education inclusion
class.
Both teachers wrote this unit while completing coursework at the Bemidji
State University in Bemidji, Minnesota during the summer institute of 2010
in pursuit of their master degrees in mathematics education.
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Table of Contents
Candy Boxes Problem--------------------------------------------------------------------------------pages 4 and 5
Building Houses on Islands-------------------------------------------------------------------------pages 6 and 7
Exploring Houses-------------------------------------------------------------------------------------pages 8 and 9
Building with Toothpicks------------------------------------------------------------------------pages 10 and 11
Letter Growth---------------------------------------------------------------------------------------pages 12 and 13
Reasoning about Growth------------------------------------------------------------------------------------page 14
Change the Pattern, Change the Growth?--------------------------------------------------------------page 15
Car Rental Problem-------------------------------------------------------------------------------------------page 16
Using Algebra Tiles--------------------------------------------------------------------------------pages 17 and 18
Going to the Movies and the Distributive Property---------------------------------------pages 19 and 20
Pre and Post Test---------------------------------------------------------------------------------------pages 21 - 24
Sample MCA & PASS Questions--------------------------------------------------------------------------page 25
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Candy Boxes Problem
Standards
8-1: The student will understand and utilize the mathematical processes of problem solving,
reasoning and proof, communication, connections, and representation. (SC)
8-3: The student will demonstrate through the mathematical processes an understanding of
equations, inequalities, and linear functions. (SC)
8-3.2 Generate equivalent numerical and algebraic expressions and use algebraic properties to
evaluate expressions. (MN)
Launch
Hold a box of candies in each hand. Tell students that the boxes have the exact same number of
pieces. One box has three pieces of candy on top. Have students guess how many pieces the
first box has in it (students may hold and shake the box if necessary).
Explore
Allow time for student or small group discussion. Next, have students show what they know by
writing it down on a piece of paper. Students may draw pictures, graphs, tables, etc.
Share
Have several volunteers share their ideas. Discuss key concepts with students. They should all
agree that one box has three more than the other box, and that there are a number of different
possibilities. On the smart board, record a prediction table with student names and guesses for
each box. Check to see that all guesses follow the rule that one box has 3 more than the other. If
a prediction does not follow the rule, have students help adjust the guess as necessary. Next,
discuss how a letter can stand in place of a value that is unknown. If we called the amount of
candy in box one N, what could we call the amount in box two that has three more pieces of
candy?
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Summarize/Extend
Today, we looked at a big mathematical idea called the variable.
After this activity, the students should have a concrete, real-world model of what N + 3 means.
Students should have grasped that no matter how many pieces of candy were in the first box,
there will have to be three more pieces in the second box. Ask students how the problem would
change if there were a different number of pieces on top of the box of candy. Students can make
up their own number of extra pieces, and make a table with ten guesses.
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Building Houses on
Islands
(Navigating through Algebra in Grades 3-5, NCTM, 2008)
Standards
8-3.2: Represent algebraic relationships with equations. (SC)
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. (MN)
Launch
Notify students that they will be part of a construction company building houses on three
different islands that are connected by bridges. On the site plan, the architect has written the
total number of houses between the two islands that the bridge connects. The student’s job is to
figure out how many houses need to be built on each of the three islands. Draw the illustration
from the top of page 52 on the board. Students may use beans to represent a house. Use the
guess and check method. Eventually, all should reach the conclusion that Sunny Isle has ten
houses, Pleasant Isle has five houses, and Tropical Isle has seven houses.
Explore
Print blackline master page 83 and distribute to students. Students may work in small groups
on the three new island problems. Students will keep track of all guesses. Students will also be
looking for a method for finding the number of houses on each island. Point out that a variable
can be used for each house!
Share
Students will share their findings with the class. There are many different methods for solving.
One way is to look at the differences between the bridges. Another way is to add up all the
bridges, then divide by two because of “double counting” the houses. A third approach is to use
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substitution to solve for each house. The substitution method is modeled nicely by the beans on
the islands.
Summarize/Extend
Observe student strategies for solving the island problems. Students should have noticed that
the relationship between the islands can be expressed by equations. As an extension and extra
challenge, four islands may be used.
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Exploring Houses
(Navigating through Algebra in Grades 6-8, NCTM, 2008)
Standards
8-3.1 Translate among verbal, graphic, tabular, and algebraic representations of linear functions.
(SC)
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. (MN)
Launch/Explore
Tell students they will be working with houses again today. A construction company needs
their help figuring out what the 5th house in the sequence would look like.
In order to teach this lesson, you will need a copy of page 74 for each student, and enough
pattern-block triangles and squares for each student to make the first four houses in the pattern.
Students should work individually on this activity. Students need to build the first four houses
from the sequence, and then complete the following tasks.
1. For each house, determine the total number of pieces needed. How many squares and
triangles are needed for a given house? Organize your information in some way.
2. Describe what house 5 would look like. Draw a sketch of this house.
3. Predict the total number of pieces you will need to build house 15. Explain your
reasoning.
4. Write a rule that gives the total number of pieces needed to build any house in this
sequence.
Share
Students should explain how they were able to find a pattern. Did they notice changes from one
pattern to the next? Did they organize information in a table? Did they add three each time to
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make a recursive formula? Do they talk about the number of blocks, or the number of squares
and triangles? Challenge students to describe the rule using words, then symbols. Can they
come up with an explicit formula?
Summarize/Extend
Remind students that constructing a table or chart is a great way to organize numerical
information from the patterns that were generated. Be sure that students relate the information
from the physical model to the table, and vice versa. Formulas can be recursive or explicit.
Challenge students to answer the same four questions above with the tiling patters for two
garden areas on page 10.
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Building with Toothpicks
(Navigating through Algebra in Grades
6-8, NCTM, 2008)
Standards
8-3 The student will demonstrate through the mathematical processes an understanding of
equations, inequalities, and linear functions. (SC)
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. (MN)
Launch
Tell students that their toothpick building skills will be challenged today! Students will be
constructing patterns with toothpicks and looking for relationships.
Pass out copies of black-line master page 75, and enough toothpicks for each student to be able
to construct the first four figures. Students will work in pairs. Draw the first four figures on
page 13 on the smart board. The students will need to build these patterns on their desks with
their toothpicks.
Explore
Students should look for patterns in the number of toothpicks used to make the perimeter of
each shape. The pairs should complete the following tasks:
1. Use a pattern from the sequence of shapes to determine the perimeter of the fifth shape
in the sequence. Show or explain how you arrived at your answer.
2. Write a formula that you could use to find the perimeter of any shape n. Explain how
you found your formula.
3. Organize information in a table and graph the ordered pairs.
Share
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After allowing enough time for groups to answer questions one and two from activity sheet,
have volunteers share their group’s findings. Students should share the different methods that
they used to arrive at their equation. Students should also discuss the recursive and explicit
formulas they found from the toothpick patterns.
Summarize
During this lesson, students were able to explore pattern development in a concrete way.
Students should begin to see the connection between models and formulas. Students should
have seen the usefulness of organizing information in a table, and using the ordered pairs to
make a graph representing the growth pattern.
Letter Growth
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Standards
8-3.5 Classify relationships between two variables in graphs, tables, and/or equations as either
linear or nonlinear. (SC)
8-2.1 Understand the concept of function in real-world and mathematical situations, and
distinguish between linear and nonlinear functions. (MN)
Launch
Students will be choosing a letter, and making it grow! Show example of an “L” on the smart
board as a demonstration. Students are to…
1. Choose a letter
L
Blocks
1
5
2
7
3
9
2. Make it grow in a pattern
3. Make a table of the stage and number
of boxes
4. Write a recursive and explicit formula
5. Graph the function.
Explore
Students will be creative and make up a pattern for their letter to grow. Allow time for students
to accomplish this task. When ready, each student will get to share his or her letter with the
class.
Share
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As each student shares his or her letter, the other students should copy down the table that
accompanies the letter. After the entire class has gotten to share, more time can be allowed to
write the equations for the growth patterns. Students may do this work in small groups.
Summarize
Students should notice the difference in graphs of the linear and non-linear equations. Remind
students that linear equations will form a straight line, and have a constant rate of change.
Students should also see the relationship of the growth from each letter, and how many lines
the letter has will be the same amount it grows by.
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Reasoning about Growth
(Navigating through Problem Solving and Reasoning in Grades 6-8, NCTM, 2008)
Standards
8-3.1 Translate among verbal, graphic, tabular, and algebraic representations of linear functions.
(SC)
8-2.2.1 Represent linear functions with tables, verbal descriptions, symbols, equations and
graphs; translate from one representation to another. (MN)
Launch
Today, we will continue to look at patterns and growth. You will have the opportunity to be
writers! There are several patterns that we will look at today, and what you see needs to be put
into words! I know you are up to the challenge. You will be working in small groups today.
Explore
Pass out blackline master pages 168-172. Small groups are to work together to complete the
activity sheets. The teacher will serve as a facilitator and questioner, and show support to
students as they reason mathematically to generalize about the patterns in the activity.
Share
Students will share what they have discovered about growing patterns. The students should
have moved through several stages of this activity. Students should share ideas of how they
proceeded through these stages. First, they drew the next growth pattern. Second, they move
from their drawing of stage 4 to mentally picture what stage 6 would look like. Students should
have used a combination of words and numbers to describe the number of tiles (black and
white) through the different stages. Look for recursive and explicit formulas for the number of
total blocks, and the number of black and white block at different stages. Listen to groups
explanations of how they answered questions 5 and 6. Clarify any confusion and address
student questions as necessary.
Summarize
Today, students gained important experiences with growing patterns, which can be very
effective in building students’ algebraic reasoning skills. Students should have noticed that their
patterns had a geometric context, and students can examine the patterns’ physical, visible
structures and organize numerical information about them in tables.
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Change the Pattern – Change
the Growth?
(Navigating through Problem Solving and Reasoning in Grades 6-8, NCTM, 2008)
Standards
8-3.1 Translate among verbal, graphic, tabular, and algebraic representations of linear functions.
(SC)
8-2.2.1 Represent linear functions with tables, verbal descriptions, symbols, equations and
graphs; translate from one representation to another. (MN)
Launch
Today, we will continue to look at patterns and growth. Our patterns have changed from
yesterday, but think back to the “Growing in Front of Your Eyes” to help you get started.
Explore
Students will work in small group on activity pages 173 – 177. The teacher will serve as a
facilitator and questioner, and show support to students as they reason mathematically to
generalize about the patterns in the activity.
Share
Students should share some of the following ideas. This pattern removes two white corner tiles
from opposite corners at each stage, along with the two white tiles that neighbor each of these
corner tiles. Each stage 6 has few white tiles than the pattern in the original sequence. The
formula for white tiles in the previous growth pattern was 4n + 4 and students should have
discovered that the new formula is 4n – 2. Students should also find that the number of black
tiles remain the same as before, as always being a perfect square.
Summarize
Today, students gained important experiences with growing patterns, which can be very
effective in building students’ algebraic reasoning skills. Students should have noticed that their
patterns had a geometric context, and students can examine the patterns’ physical, visible
structures and organize numerical information about them in tables.
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Car Rental Problem
Standards
8-3.5 Classify relationships between two variables in graphs, tables, and/or equations as either
linear or nonlinear. (SC)
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. (MN)
Launch
Your teacher is going on a road trip, needs to rent a car, and can’t decide which company to
choose from. Option A has a cost per day of $25 plus $0.45 per mile. Option B has a cost per day
of $40 plus $0.25 per mile. Help me decide which option to go with!
Explore
Allow time for students to explore the problem. Encourage students to make tables for the two
plans with several ordered pairs. Students should graph their findings on a sheet of graph
paper. Students should write a recursive and explicit formula for both options. When students
have had ample time to work, more questions can be posed. For one day, how many miles must
be driven for the cost to be the same?
Share
Volunteers will share their findings on renting a car. Students should have discovered the
number of miles traveled will have an influence on which rental company to rent from. Discuss
the constant rate of change from the formulas and the graph.
Summarize
After this activity, students should recognize the linear equation in this real-world situation of
renting a car. Students should see the connection between tables and graphs, and be able to
move comfortably between the two. Graphing calculators can be used as an extension to this
activity.
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Using Algebra Tiles
(Adapted from Pre-Algebra, Glencoe, 2009)
Standards
8-3.3 Use commutative, associative, and distributive properties to examine the equivalence of a
variety of algebraic expressions. (SC)
8-2.3.2 Justify steps in generating equivalent expressions by identifying the properties used,
including the properties of algebra. Properties include the associative, commutative and
distributive laws, and the order of operations, including grouping symbols. (MN)
Launch
My little brother, Nathan, scored x points in his first basketball game. He scored three times as
many points in his second game. In his third game, he scored 6 more points in the second game.
I want to figure out how many points my little brother scored! Write an expression in simplest
form that represents the number of points Nathan scored. If x = 10, how many points did he
score in his first three games?
Today, students will be exploring how to simplify expressions using algebra tiles. Explain the
meaning of each tile. The small white and red squares represent a positive and negative unit,
and have the dimensions of one by one. The rectangles represent positive and negative
variables, such as x, and have dimensions of one by x units. Students should each have a set of
algebra tiles. Before students get into groups, go over a few examples as a whole group on the
smart board. For example, model with the algebra tiles 2x + 4 + 4x + 1. Use the commutative
property to regroup the like terms, and then simplify. Also, model with the algebra tiles x + 6 +
3x - 3. Use the commutative property to regroup using like terms. Make zero pairs with the
units, and simplify.
Explore
Have the following expressions on the board. Students should model, draw, and simplify each
expression with their partner.
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1. 3x + 4 + x + 3
2. 2x + 3 + 2x + 3
3. x + 7 + 5x
4. 4x – 1 + 2x + 5
5. 3x + 2x – 4
6. 2x + 2 + 2x – 2
7. Which mathematical property allows you to sort the algebra tile by their shapes?
8. Which mathematical property allows you to remove zero pairs?
Students can use the skills they have learned simplifying expressions to simplify real-world
expressions. Here are a few challenge problems.
9. Marena is using a certain number of blue beads in a bracelet design. She will use 7 more
red beads than blue beads. Write an expression in simplest form that represents the total
number of beads in her bracelet design.
10. Kevin bought 3 CD’s that cost x dollars, 2 DVD”s that cost $10 dollars each, and a book
that costs $15. Write an expression in simplest form that represents the total amount that
Kevin spent.
Share
Students will share the different ways they approached simplifying the expressions. Check to be
sure that students have simplified each expression. Clarify any questions that students may
have.
Summarize
Today, we learned how to simplify algebraic expressions. An easy way to do this is to use a
model. Remind students that they can group the x-terms together using the commutative
property, and remove zero pairs to simplify.
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Going to the Movies and the
Distributive Property
(Adapted from Pre-Algebra, Glencoe, 2009)
Standards
8-3.3 Use commutative, associative, and distributive properties to examine the equivalence of a
variety of algebraic expressions. (SC)
8-2.3.2 Justify steps in generating equivalent expressions by identifying the properties used,
including the properties of algebra. Properties include the associative, commutative and
distributive laws, and the order of operations, including grouping symbols. (MN)
Launch
I am going to see a movie this weekend, and need to figure out how much I will spend. I am
paying for myself and two friends to have a snack, drink, and movie ticket. If the tickets are
$6.50, snacks are $3.00, and drinks are $3.00, how much will I spend?
Explore
Allow time for students in small groups to figure out how much it would cost for 3 people to
see a movie and get a snack and drink.
Share
Have groups share how they arrived at their answer of $37.50. If none of the groups used the
distributive property, take the time to introduce it now. Be sure to emphasize the equivalence of
$37.50 = 3($6.50 + $3 + $3) = 3($6.50) + 3($3) + 3($3). Show the students that adding the cost of a
movie, drink, and snack and then multiplying by three is the same as adding up the cost of
three tickets, three drinks, and three snacks. Algebra tiles may also be used to show this
equivalent relationship.
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Summarize /Extend
The main idea from today is to show students equivalence. In their small groups, students need
to solve problems 19, 20, 40, and 41 on page 174 of textbook.
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Pre and Post Test
Name __________________________
1. Give a real-world example of what the algebraic expression N + 3 means. Make a table
with five ordered pairs.
2. Using variables and showing your work, find how many houses are on each island.
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3. Based on the pattern below, draw the 5th house in the sequence. Make a table with five
ordered pairs, and construct a graph.
4. Choose a letter, make it grow, and draw the first few stages. Construct a table of ordered
pairs, then write a recursive or explicit formula, and identify which kind of formula you
chose.
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5. Using the pattern below, draw the 4th stage of growth. Write a recursive and explicit
formula, and explain how you got them.
6. Admission to the state fair is $8 for adults and $7 for students. Write two equivalent
expressions if two adults and two students go to the fair. Then find the total admission
cost.
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7. You used P minutes on month on your cell phone. The nets month you used 75 fewer
minutes. Write an expression in simplest form.
8. A community center offers a canoeing day trip. The canoeing fee is $80 per person. The
cost of food is an additional $39 per person. Find the total cost for a family of four.
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Sample MCA & PASS Test Questions
1. A caterer charges a $50 fee plus an additional $6.25 per person. She
determines the total cost, C, for x people by using the equation shown
above. How much will a party cost for 75 people?
A. $125.00
B. $469.25
C. $518.75
D. $718.25
2. Use the equation below to answer question 23.
F = 9/5 C + 32
What is 180° Celsius in degrees Fahrenheit?
A. 82°F
B. 212°F
C. 324°F
D. 356°F
3. The Eaton family rented a car for a weekend trip. The rate to rent the car
was $30 per day plus $0.25 for each mile driven.
Part A What is the cost of renting a car for 4 days and driving a total of 200
miles? Show or explain how you got your answer.
Part B Write an expression to show the cost of renting a car for d days and
driving m miles.
Part C Show 2 different solutions that allow the Eatons to rent the car and
not spend more than $150. For both solutions, explain the number ofdays
they rent the car and the number of miles they drive.
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