Unit 1: Patterns
3 weeks
Unit Overview
Essential Questions:
What is a sequence?
How can patterns be represented?
What are the advantages and disadvantages of a recursive rule compared to an explicit rule?
Enduring Understandings:
Analyzing patterns and writing recursive and explicit algebraic rules provides a powerful way to extend patterns
and make predictions.
UNIT CONTENTS
Note: The bolded Investigations are model investigations for this unit.
Investigation 1:
Exploring Patterns with Hydrocarbons (2 days)
Investigation 2:
Modeling Algebraic Expressions (2 days)
Investigation 3:
Patterns with Fractals (2 days)
Mid-Unit Test (1day)
Investigation 4:
Arithmetic Sequences (2 days)
Investigation 5:
Building Bridges (1 day)
Investigation 6:
Geometric Sequences (2 days)
Performance Task: Honeycombs (2 days)
End-of-Unit Test (1 day)
Appendices:
Unit Performance Task – Sample Student Handout, Mid-Unit and End-of-Unit
Tests
Course Level Expectations
What students are expected to know and be able to do as a result of the unit
1.1.1 Express and analyze patterns and functions (including arithmetic and geometric sequences) drawn from
real-world contexts using tables, graphs, words and symbolic rules.
1.1.2 Determine the nth term of a sequence with and without the use of technology.
1.1.3 Translate a representation of a pattern or function into another representation.
1.1.4 Write both a recursive rule and an explicit rule for a sequence.
1.1.10 Make and justify predictions based on patterns.
2.1.1 Compare, locate, label and order integers, rational numbers and real numbers on number lines, scales
and graphs.
3.1.1 Use models to make, test and describe conjectures involving properties of two- and three-dimensional
figures.
3.3.1 Select and use appropriate units, scales, degree of precision to measure length, angle, area, and volume
of geometric models.
Vocabulary
arithmetic sequences
fossil fuels
mole
astrogeology
fractals
molecular structure
atom
geometric sequence
molecule
butane
hexagon
n-gon
combustion
honeycomb
nth term
degree of precision
hydrocarbons
pentagon
distributive property
interior angle
propcine (propanol, isopropanol)
energy
kilojoule
recursive rules
ethane (ethanol)
hendecagon
Rule of 72
explicit rules
methane (methonal)
symbolic algebraic expressions
truss style bridges
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Assessment Strategies
Performance Task(s)
Other Evidence
Authentic application in new context
Formative and Summative assessments
Honeycombs
 Warm-ups, class activities, exit slips, and
This performance task builds on the unit theme of
homework have been incorporated throughout the
patterns in nature. The task probes students’ skills in
investigations.
finding the nth term of an arithmetic sequence as well  Mid-Unit Test
as the recognition that not all patterns are linear.
 End-of-Unit Test
INVESTIGATION 1 – Exploring Patterns with Hydrocarbons (2 days)
In this investigation, students explore an application of patterns in the context of the molecular structure of
hydrocarbons. They have the opportunity to analyze and create different representations of patterns—from
tables to graphs to symbolic rules.
See Model Investigation 1
INVESTIGATION 2 - Modeling Algebraic Expressions (2 days)
If students had difficulty writing algebraic expressions to identify the explicit rules for the patterns in
Investigation 1, you may wish to have some, or all students complete this activity. Here students write
expressions in a variety of contexts related to health and fitness.
Suggested Activities
2.1 As a launch, you might bring in gram, ounce, pound and kilogram weights and/or household objects that
weigh about a kilogram and a pound. Have students compare the weights. Then pose this situation: “You
frequently go to the gym to work out lifting weights. You plan to gradually increase the size of the weights
over the next month. You always put two plates that appear to be the same weight on each side of the bar.
The plates are not labeled but you do know the bar weighs 20 kg. How can we express the total weight you
lifted on any day?” This problem requires students to identify that the weight of the bar is a constant, but the
weight of the added plates changes from day to day. Ask students to try to represent this situation with a
drawing. What should we label the bar? What should we label the plates? Have students compare their
labeled drawings. What expressions might we write that model the drawing? Students might suggest w + w
+ 20 + w + w, which clearly corresponds to the picture. Other students might write 2w + 20 + 2w, or 2w +
2w + 20, or perhaps 4w + 20. Discuss the role of the coefficient, and the commutative and associative
properties when combining like terms. Which expression is the most succinct? Use substitution to explore
why all these expressions are equivalent. As an alternative, if we construct a table, beginning with a weight
of 1 kg per plate, and continue to select plates that increase by a kg, the situation would look like this:
Weight of one plate (kg)
1
2
3
w
Weight of Plates & Bar
4(1) + 20
4(2) + 20
4(3) + 20
4(w) + 20
Total Weight Lifted (kg)
24
28
32
4w + 20
Suppose each weight was 5 kg. What would be the value of each expression? Next, explore the value of the
expression when plates of various weights are substituted. The variable does not need to be a whole number.
For instance, the plates could each weigh 8.2 kg or they could weigh 2 ½ kg.
As an extension, you may use the weights problem to explore substituting values for a variable and
recursive patterns with the graphing calculator. Remind students that they were working with the expression
4w + 20, where w represented the weight, in kg, of each of the four weights. Start with students in the home
screen. Enter the fixed weight of the bar, 20 kg. If each plate weighs one kg, then adding four plates would
result in the addition of 4 kg. Simulate this on the home screen of the graphing calculator using the method
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of recursive addition shown below:
20 + 4 ENTER, then 2nd ANS + 4 ENTER, then recursively touch the ENTER key.
What happens each time you touch the Enter key? (Another four kg is added.) Suppose
each plate weighs 3.5 kg. How would you set this up on the calculator? Students may
explore the Y-editor menu and table setup features.
2.2 Continue to explore writing expressions using variables whose values may be represented by real world
data. To stay with the fitness and health theme, you may offer students the following factoid: Someone
weighing 130 pounds burns about 6.6 calories per minute while dancing (found at
http://www.coolnurse.com/calories.htm). Students may write expressions using time (in minutes) as the
variable. This site provides information on a variety of activities other than dancing.
Another context involving health and fitness is the Nintendo Wii TM. When it first came out, proponents
believed that it was the answer to childhood obesity problems. Studies have shown, however, that children
burn only an extra 60 calories per hour versus stationary games. You can use this information to develop
algebraic expressions and substitute for the variables. http://bmimedical.blogspot.com/2008/03/doesplaying-nintendo-wii-really-burn.html
Another interesting extension involves pedometers. For an explanation of how pedometers work, go to
http://www.ideafit.com/fitness-library/pedometer-power. Explore the variable steps per mile. Ask students
to think of other situations in which variables such as time, distance, weight, cost, or other rates or measures
are involved. Write expressions and substitute realistic values.
Assessment
By the end of this investigation students should be able to:
 define a variable; and
 evaluate algebraic expressions.
INVESTIGATION 3 – Patterns with Fractals (2 days)
Students will be able to recognize, describe, and extend sequences and create an algebraic expression to model
the rule.
See Model Investigation 3.
Mid-Unit Test (1 day)
INVESTIGATION 4 – Arithmetic Sequences (2 days)
Students will write both the recursive rule and the explicit rule for the nth term of an arithmetic sequence.
Suggested Activities
4.1 You may choose to begin with a visual, geometric problem such as, “The
following pentagon patterns are made up of toothpicks. One pentagon takes 5 toothpicks to create, two take
9, etc.” Have students construct a table for the data. How many toothpicks will it take to build 8 pentagons?
For differentiation purposes, students may draw, or use toothpicks to create the pentagons and write the
values in a table to find the answer. Look at the data in the table. “How would you use the data to predict the
number of toothpicks needed to build 10 pentagons? 12 pentagons?”. This question can be answered by
looking for a pattern and stating a recursive rule. Students should see that once the first pentagon is built,
the others are created by adding four more toothpicks each time. Ask, “Would you like to use this rule to
find the number of toothpicks for 1000 pentagons?”
It would be helpful to find a general rule that predicts the number of toothpicks needed to construct any
number of pentagons. Let students know that this general rule is called an explicit rule. For this example, we
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need a formula that will get us from the number of pentagons to the number of toothpicks.
Do students notice that the first value in the table is one more toothpick than the difference between each of
the other values, where the number increases by 4 each time? How can this be written as an expression?
Test out any expressions that students write. Does the expression support the values in the table? Does it
predict the next few values? Students should see this pattern as an example of an arithmetic sequence with
each term increasing by a fixed amount—in this case by 4.
You also may choose to develop, with the students, the expression a + (n-1)d to find the nth term of the
arithmetic sequence. Here a is the starting amount and d is the difference between each value of the
variable. You may need to review the distributive property here.
You may also explore a decreasing sequence. That is, the d, or fixed amount, will be negative. Here is an
example. “You are president of your school’s chapter of Students for Open Space. It is your job to advocate
for more open space in your town. That is, you want more parks and open areas. Your club did some
research and found that the town started with 520 acres of open space but has been losing an average of 15
acres per year. How much open space will be left in 20 years?” You might pair students and have them start
with a table, find the recursive rule, and then try to write the explicit rule.
4.2 Another possible activity involves measuring rope or thick string. Cut several pieces of rope all to the same
length. Students may work in cooperative groups. Each group should get a piece of rope which they will
measure in centimeters using a tape measure. They should then tie one knot in the rope and measure its
length again. Ask them what happened to the length of the rope. Continue two more times to tie another
knot in the rope and measure the new length each time. The data they collect will approximate an arithmetic
sequence. Groups of students might compare their data to account for differences in the accuracy of the
measurements. Students should then write the recursive rule and the explicit rule.
The next activity can begin in class and be completed for homework. It is a geometry extension that
explores the sum of the interior angles of a polygon. Depending on the level of students, this can either be
done as a teacher-led exploration or totally student-driven. Have students use a straight edge to draw some
triangles, quadrilaterals, pentagons and hexagons (not necessarily regular polygons), or give them shapes
such as pattern blocks, or pre-made cardboard polygons. Have students use protractors to measure the
interior angles and write the sum of the interior angles of each shape. Students should notice that the sums
are the same for a particular n-gon. Use the pattern in a data table to write the recursive rule and then find
the sum of the interior angles in a hendecagon (11-sided figure). Have students suggest what the explicit
formula might be for the interior angle sum of any n-gon.
Assessment
By the end of this investigation students should be able to:
 write the recursive rule and the explicit rule for an arithmetic sequence; and
 use the rules to extend patterns and find the specific element in an arithmetic sequence.
INVESTIGATION 5 – Building Bridges (1 day)
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Students will further examine arithmetic sequences in the context of bridge design. This activity will provide
more opportunities to recognize and write the explicit rule for the sequence.
Suggested Activities
5.1 You may want to begin class with a video, such as this one, which shows a class building popsicle stick
bridges: http://www.youtube.com/watch?v=T2ZyqNODJpU.
Truss bridges are built using a pattern of steel beams. This pattern has great strength due to weight
distribution in the beams. This activity explores the pattern of steel beams in a truss bridge design.
Three beams are joined together in the triangular shape shown below.
One Section
Two Sections
Three Sections
The bridge is made wider by adding more sections as shown above.
Total number Total number
bridge sections
of beams
1
3
2
3
Ask students to use the information in the table to predict how many
beams a bridge with 4 sections would have. Most students should be able to determine this result by looking
at the arithmetic sequence formed. Some students might need to draw the fourth bridge to be sure of their
answer. Students may find it helpful to extend the table, including drawing more sections, in order to have
enough data to describe the recursive and explicit rules.
There is a pattern in this bridge design. The total number of beams is
related to the number of bridge sections. Have the students complete
the first three rows in the table to explore the pattern.
Next ask students to determine the total number of beams of the nth number of bridge sections. Check that
students understand what the nth term in this case represents. The expression, formula or rule that they get
will give them the total number of beams needed for any given number of bridge sections. You might have
students work in pairs to develop the explicit rule and describe their thinking. They may notice that the total
number of beams for a given number of sections is one less than four times the number of sections, or 4  1.
5.2 Once students are comfortable with the two-dimensional representations, you may wish to challenge the
class to examine drawings of a three-dimensional bridge structure. The illustration below shows one bridge
section and then what it looks like when we start adding in more bridge sections. Students may use
toothpicks and mini-marshmallows or gumdrops to build the models.
One section
Two sections
Three sections
Again there will be a mathematical relationship between the number of bridge sections and the number of
beams. Some students who are less proficient with spatial visualization might have difficulty counting the
number of beams. Reinforce the idea that it is important to take their time when counting. For these
students, marking the model as they count will be helpful.
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Have students complete a table for the three-dimensional models. They should be able to describe the
recursive and explicit rules.
For some students it might be helpful to go through the
exercise of predicting what the bridge with four sections
will look like and then have them check the answer.
Number of
bridge sections
1
2
3
Total number
of beams
9
Note: Four sections have 39 beams, and the explicit rule is10n-1.
5.3 You may wish to offer different extension and enrichment opportunities. Students may work in small groups
at different centers. Students may choose to investigate three-dimensional bridges with two lanes. These are
fascinating because they still follow the linear patterns of the arithmetic sequences. It gets difficult, though,
to keep track of the numbers and to ensure proper counts. Some students, especially tactile learners or
students interested in engineering, etc. might really get into this problem. Below is an example of what the
first couple of bridges would look like.
As another means of differentiation, students may elect to investigate how much mass a bridge can hold.
There are several variations of this classic “experiment”. Below are some examples:
http://www.education-world.com/a_lesson/worksheets/TCM/pdfs/010413is.pdf (bridges and pennies)
http://coeweb.eb.uah.edu/cee/exp6.html (bridges and paperclips)
http://www.go-explore-trans.org/2007/nov-dec/bridge_loading.cfm (both pennies and paperclips)
Here is a website showing a class project involving building bridges solely out of pennies:
http://www.fincher.org/Misc/Pennies/index.shtml
Students may go to the following websites to design and test their own virtual bridges:
www.jhu.edu/virtlab/bridge/truss.htm
www.eecs.usma.edu/bridgecontest/pdfs/appenda.pdf
Students may view some notable bridge collapses, and discover design flaws and physical and
mathematical explanations, at:
http://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge_(1940) or
www.ketchum.org/bridgecollapse.html.
Assessment
By the end of this investigation students should be able to:
 Further determine formulas to find nth terms of arithmetic sequences.
INVESTIGATION 6 – Geometric Sequences (2 Days)
Students will be examining geometric sequences through real world applications.
Suggested Activities
6.1 There are many contexts in which to introduce geometric sequences. One context is that of money
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and finances. Whatever the context you choose, it is important to stress that geometric sequences
are similar to arithmetic ones in that you are doing the same thing to each term. By comparing the
two types of sequences, students should discover that there is a difference in that, instead of adding
by a fixed amount for an arithmetic sequence, you are multiplying by the same factor in a
geometric sequence. In keeping with a finance context, you may want to launch the investigation
by offering students the following situation:
“You just won first prize in a poetry writing contest. If you take the $500 you won and
invest it in a mutual fund earning 8% interest per year, about how long will it take for
your money to double? You might suggest that students construct a table that shows the
growth from year to year.”
There will most likely be multiple ways that students chose to solve the problem. You may have students
work in pairs and then share their various answers and strategies. If no student (or group of students)
chooses to use the graphing calculator, you may want to review how to use the recursive function of the
home screen to keep multiplying by 1.08.
A nice extension to this problem is to discuss the “Rule of 72”. To find the amount of time it takes for your
money to double, you divide 72 by your interest rate. So, in this case, 72 ÷ 8 = 9 years. The rule of 72 is just
an estimate. Here is an online calculator that compares the estimate with the “real” answer:
http://www.moneychimp.com/features/rule72.htm.
As an extension, students may explore ways to calculate the rules for tripling (114) and quadrupling (144)
their money. Information can be found here: http://allfinancialmatters.com/2007/05/14/the-rule-of-72-114and-144/.
6.2 Another finance investigation is the penny problem. Students may have seen this problem before, but you
can put a twist on it by using the graphing calculator to find the answer or even better, by using Microsoft
Excel. “You have your choice between getting a million dollars or a penny on the first day, two pennies on
the second day, four on the third, etc. for 30 days.” Here is a site showing what collections of pennies look
like: http://www.kokogiak.com/megapenny/default.asp.
A natural extension, especially if using Excel, is the King’s Chessboard problem in which a grain of rice or
wheat is placed on the first square of a chessboard, two on the second, etc. for all 64 squares. After
examining this problem, students can visit www.freerice.com to answer questions and get free rice donated
to those in need around the world.
Another possibility is the story of the Devil and Daniel Webster. The problem comes from NCTM’s
Illuminations collection of lessons and can be found at:
http://illuminations.nctm.org/LessonDetail.aspx?id=L288.
All of these patterns represent geometric sequences and can be modeled very effectively on either a
graphing calculator or via a software program. Additionally, with Excel, you can have students create scatter
plots showing that, unlike arithmetic sequences, the graph will be non-linear.
Assessment
By the end of this investigation students should be able to:
 Find a recursive rule to calculate terms of geometric sequences. Note: students are not necessarily expected
to find the explicit rule for geometric sequences; and
 Explain the difference between an arithmetic and a geometric sequence.
Unit 1 PERFORMANCE TASK – Honeycombs (2 days)
Honeycombs are a network of hexagons that bees create for their hives. They are a wonderful natural shape
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because they efficiently use material for the amount of area they create and they are strong. The patterns formed
tie in nicely with what students have been studying in this introductory unit. Students may work in pairs or
small groups during this performance task. See the Sample Student Handout – Honeycombs.
Suggested Activities
To introduce this task, you might show a video about bees and honeycombs. Here is a link that describes the
honeycomb: http://www.pbs.org/wgbh/nova/bees/hive.html. Here is more information on the reason for the
hexagon shape: http://www.microscopy-uk.org.uk/mag/artsep98/bees.html. If time permits, have students share
answers to the questions about the recursive and explicit rules they found.
End-of-Unit Test (1 day)
Technology/Materials/Resources/Bibliography
Technology:
 Classroom set of graphing calculators
 Microsoft Excel software
 Whole-class display for the graphing calculator
 Computer and/or classroom set of computers for students to use spreadsheet software
 Overhead projector with view screen or computer emulator software that can be projected to whole class
On-line Resources:
 Geology of our solar system: http://astrogeology.usgs.gov/
 ChemPad Software (free download): http://graphics.cs.brown.edu/research/chempad/home.html
 Calorie calculator: http://www.coolnurse.com/calories.htm
 Calories and video games: http://bmimedical.blogspot.com/2008/03/does-playing-nintendo-wii-reallyburn.html
 How pedometers work: http://www.ideafit.com/fitness-library/pedometer-power
 Popsicle bridge video: http://www.youtube.com/watch?v=T2ZyqNODJpU
 Building bridges with pennies: http://www.fincher.org/Misc/Pennies/index.shtml
 Another penny and bridge activity: http://www.educationworld.com/a_lesson/worksheets/TCM/pdfs/010413is.pdf
 Bridges and paperclips: http://coeweb.eb.uah.edu/cee/exp6.html
 More bridges: http://www.go-explore-trans.org/2007/nov-dec/bridge_loading.cfm
 Investment calculator: http://www.moneychimp.com/features/rule72.htm
 Rules for doubling, tripling, quadrupling money: http://allfinancialmatters.com/2007/05/14/the-rule-of-72114-and-144/
 Freerice.com: www.freerice.com
 MegaPenny Project: http://www.kokogiak.com/megapenny/default.asp
 NCTM Devil and Daniel Webster activity: http://illuminations.nctm.org/LessonDetail.aspx?id=L288
 virtual bridges: www.jhu.edu/virtlab/bridge/truss.htm and
www.eecs.usma.edu/bridgecontest/pdfs/appenda.pdf
 notable bridge collapses, design flaws and physical and mathematical explanations, at:
 http://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge_(1940)
 Honeycombs: http://www.pbs.org/wgbh/nova/bees/hive.html
 Reason for honeycomb shape: http://www.microscopy-uk.org.uk/mag/artsep98/bees.html.
Materials:
 Toothpicks
 Mini-marshmallows or gumdrops
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Unit Plan 1, 8 13 09
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Unit 1 Performance Task
Sample Student Handout, p. 1 of 4
Honeycombs
Name:__________________________________________Date:__________________________
Honeycombs are a network of hexagons that bees create for their hives. They are a wonderful natural shape
because they efficiently use material for the amount of area they create and they are strong. Let’s build rows of
honeycomb shapes starting with a single hexagon.
1. The diagrams below show the first two stages of a honeycomb row. At each stage another hexagon is added
to the sequence. Work with your group to draw the next four stages in one row of honeycombs.
#1
#2
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Unit 1 Performance Task
Sample Student Handout, p. 2 of 4
2. Notice that the first hexagon has six edges, so it took 6 segments to build it. Complete the table to show the
number of segments in the next five stages of a honeycomb row.
Stage
1
2
3
4
5
6
Honeycomb Edges
Total # of Segments
6
3. By now, we have talked a lot about the difference between explicit rules and recursive rules. Explain briefly
the difference between the two types of rules. Be sure to give examples of when one might be more useful
than the other.
4. What is the recursive rule for finding the number of segments in the honeycomb row?_________________
Explain how your group arrived at your answer. Also, show all the mathematics you used.
5. What is the explicit rule for finding the number of segments in the honeycomb row? __________________
Again, explain how you arrived at your answer. Also, show all the mathematics you used.
6. How many segments will be in the 10th stage of the honeycomb row? _________The50thstage?__________
Briefly explain why.
7. You were not asked to graph this, but if you were, what type of graph would you expect? ____________
Why?
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Unit 1 Performance Task
Sample Student Handout, p. 3 of 4
8. Now, let’s see how a honeycomb shape can be built starting with a single hexagon and building out:
Now we add new hexagons all around the outside as we start to build our hive:
How many new hexagons did we add? ___________
How many do we have all together? _____________
9. How many honeycombs do you predict will be in the next formation? _______________
Explain how you arrived at your answer.
10. Your group should now either model (using either pattern blocks or hexagon cut-outs) or draw the next
three honeycomb formations. Was your prediction in Question 8 right or wrong? _______________
Why?
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Unit 1 Performance Task
Sample Student Handout, p. 4 of 4
11. Now, complete the table below to show the total number of hexagons that are added for each stage and the
total number of hexagons in each stage. Note that we are calling the first, single hexagon Stage 0.
Second Honeycomb Exploration
Honeycomb Stage
Number of hexagons
Total number of
added
hexagons
0
1
1
1
6
7
2
3
4
5
12. What is the recursive rule to find the total number of hexagons in each successive stage? _____________
Explain how you arrived at your answer. Use the rule to predict the number in the fifth stage.
13. Graph the relationship between the honeycomb stage and the total number of hexagons. Would your graph
be similar to the graph you were asked about in question 7? _______________
If not, what would you expect this graph to look like? _____________________
Why?
14. What is the explicit rule to find the total number of hexagons in any stage? __________________________
This may be difficult to determine, so make sure your group has a detailed discussion. What ideas did you
try?
15. Using your rule, what would be the number of hexagons in stage 10?
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UNIT 1 MID-UNIT TEST
PATTERNS
(After Investigation 3)
NAME ______________________________ DATE __________
1.
Phase 1
Phase 2
Phase 3
a. Extend the pattern and sketch Phase 3 in the space provided above
a. Complete the following table.
Phase
Number of Squares
c. Write the recursive rule for the pattern.___________________________________
2. Physical activities burn different numbers of calories. Mannie walked to a youth club to dance to his
favorite rock band. The walk burned 145 calories. He burns 6 calories per minute dancing. The total calories
he burns for the two activities can be found by the equation: C = 6 M + 145.
Taquisha rides her bike to the park to play soccer with her friends. The bike ride burns 326 calories. She
burns 8 calories per minute playing soccer. The total calories she burns for those two activities can be found
by the equation: C = 8 M + 326.
a. What are the variables in this problem? ________________________________________
b. Use the equations to complete the tables below for total calories burned.
Mannie:
M
10
20
30
40
50
60
C
Taquisha:
M
10
20
30
40
50
60
C
c. Mannie danced for two hours. How many total calories did he burn that evening?
____________________
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d. Taquisha plays soccer for one hour and fifteen minutes. How many total calories did she burn?
__________
3. Stages 0 to 2 of a fractal design are completed below.
Phase 0
Phase 1
Phase 2
Phase 3 (Incomplete)
a. Complete Phase 3 of the fractal.
b. Complete the following table:
# of Triangles
Total # of
Total # of
Total # of squares and
Phase
Added
Triangles
Squares
Triangles
0
0
1
0
1
1
2
3
1
4
2
3
4
c. Write the recursive rule for the pattern from the column titled # of Triangles Added.
____________________________________________________________________
d. Use the grid on the next page to graph any one of the column’s patterns. Label your graph with a title,
then label the axes and choose a scale for each axis. Note: Let phase # be the label of the x variable.
e. Why did you choose the column you did? Explain.
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UNIT 1 TEST
PATTERNS
NAME ________________________________ DATE ________
1. Tomeka placed decorative boxes in a window display in the following pattern:
a. Extend the pattern by sketching the next two displays in the space above.
b. Complete the table below:
Display
1
2
3
4
5
Number of boxes
1
3
c. Write the recursive rule for the pattern.
d. How many boxes would be needed to make a display that has 10 boxes in the bottom row?
____________________ Explain and show your work.
2. Brett and Yolanda each have a favorite video game.
a. Bert receives points according to the pattern below. Extend the pattern of points.
5, 15, 45, ______, ______, ______
b. Yolanda gets points on a similar game according to the following pattern.
5, 25, 65, 125, _____ , ______
c. Which scoring pattern would you choose, a or b? ____________
Explain why.
CT Algebra One for All
Unit Plan 1, 8 13 09
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3. Marcus hires a taxi to take him to the airport. The fare is $4.50 plus $0.75 for each mile.
a. Complete the table for 5- mile intervals.
# of
Miles
5
10
15
20
Cost of Trip
b. Is this pattern an arithmetic or geometric sequence? ____________ Explain your answer.
c. Write the explicit rule for this pattern.
d. If the distance to the airport is 44 miles, how much will Marcus pay for the taxi? __________________
Show and label your work to explain how you found the answer.
4. The Sundae Zone organizes birthday parties. They charge $3.00 for each sundae, plus a $25.00 fee to cover
the cost of the balloons, paper goods and clean-up. You plan to invite 10 of your friends to your birthday
party.
a. Write an explicit rule.
b. How much will it cost for a party for you and your 10 friends? _________________
Show your work to explain how you found the answer.
5. The diagonal of a geometric polygon is a line segment that connects two non-adjacent vertices. Bill and
Hector need to find a rule for the number of diagonals in any polygon when given the number of sides. They
drew several polygons to see how many diagonals each one had. Then they looked for a pattern. Using the
polygons below, draw the diagonals and complete the table.
Diagonals of Polygons
# of Sides
# of Diagonals
CT Algebra One for All
Unit Plan 1, 8 13 09
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a. Is the number of diagonals an arithmetic or geometric sequence? ________________
Explain your answer.
b. Write a recursive rule.
5. At the beginning of the baseball season, the manager of the Bridgeport Bluefish decided to reward two of
his potential home run hitters with incentives to hit more home runs. The previous season each player had
hit 25 home runs. They will receive the reward in addition to their regular salary.
The manager gave them two choices.
For Plan A the player would receive $1000 for each home run he hits.
For Plan B the player would receive $1 for the first home run, then twice as much money as he had
received for each successive home run he hits.
a. Organize the information in each plan in the tables below to show how much money a player would
receive for the first six home runs hit.
Plan A
Plan B
a. Write the recursive rule for each plan.
Plan A:
Plan B:
c. Write the explicit rule for each plan.
Plan A:
Plan B:
c. Which plan you would choose? ____________
Why?
CT Algebra One for All
Unit Plan 1, 8 13 09
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