Mathematics Lesson Planning Guide IP3 | Sixth Grade
Instructional Period 3
Content
Strand 2: Data Analysis, Probability, and Discrete Math
Concept 1: Data Analysis
PO1. Solve problems by selecting, constructing, and interpreting displays of data, including
histograms and stem-and-leaf plots.
PO2. Formulate and answer questions by interpreting, analyzing, and drawing inferences from
displays of data, including histograms and stem-and-leaf plots.
PO3. Use extreme values, mean, median, mode, and range to analyze and describe the
distribution of a given data set.
PO4. Compare two or more sets of data by identifying trends.
Concept 2: Probability
PO1: Use data collected from multiple trials of a single event to form a conjecture about the
theoretical probability.
PO2. Use theoretical probability to:
 predict experimental outcomes,
 compare the outcome of the experiment to the prediction, and
 replicate the experiment and compare results.
PO3. Determine all possible outcomes (sample space) of a given situation using a systematic
approach.
Concept 3: Systematic Listing and Counting
PO1. Build and explore tree diagrams where items repeat.
PO2. Explore counting problems with Venn diagrams using three attributes.
Concept 4: Vertex-Edge Graphs
PO1. Investigate properties of vertex-edge graphs:
 Hamilton paths
 Hamilton circuits
 Shortest route
PO2. Solve problems related to Hamilton paths and circuits.
Process
Strand 5: Structure and Logic
Concept 2: Logic, Reasoning, Problem Solving, and Proof
PO1. Analyze a problem situation to determine the question(s) to be answered.
PO2. Identify relevant, missing, and extraneous information related to the solution to a problem.
PO3. Analyze and compare mathematical strategies for efficient problem solving; select and use
one or more strategies to solve a problem.
PO4. Apply a previously used problem-solving strategy in a new context.
PO5. Represent a problem situation using multiple representations, describe the process used to
solve the problem, and verify the reasonableness of the solution.
PO6. Communicate the answer(s) to the question(s) in a problem using appropriate
representations, including symbols and informal and formal mathematical language.
PO7. Isolate and organize mathematical information taken from symbols, diagrams, and graphs
to make inferences, draw conclusions, and justify reasoning.
PO8. Make and test conjectures based on information collected from explorations and
experiments.
Mathematical Practices
MP1. Make sense of problems and persevere in solving them.
MP2. Reason abstractly and quantitatively.
MP3. Construct viable arguments and critique the reasoning of others.
MP4. Model with mathematics.
MP5. Use appropriate tools strategically.
MP6. Attend to precision.
MP7. Look for and make use of structure.
MP8. Look for and express regularity in repeated reasoning.
Strand 3: Patterns, Algebra, and Functions
Concept 4: Analysis of Change
PO1. Determine a pattern to predict missing values on a line graph or scatter plot.
Strand 5: Structure and Logic
Concept 2: Logic, Reasoning, Problem Solving, and Proof
PO9. Solve simple logic problems, including conditional statements, and justify solution methods
and reasoning.
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Glendale Elementary School District | June 2011
Mathematics Lesson Planning Guide IP3 | Sixth Grade
Instructional Period 3
Topic: Data Analysis
Strand 2: Data Analysis, Probability, and Discrete Mathematics
Concept 1: Data Analysis (Statistics)
Understand and apply data collection, organization, and representation to analyze and sort data.
Concept 2: Probability
Understand and apply the basic concepts of probability.
Concept 3: Systematic Listing and Counting
Understand and demonstrate the systematic listing and counting of possible outcomes.
Concept 4: Vertex-Edge Graphs
Understand and apply vertex-edge graphs.
In Grade 6, students apply their understanding of fractions, decimals, and percents as they construct, analyze, and describe data. They are introduced to data displays and summary statistics to
analyze the distribution of data and compare two data sets.
Essential Questions: How can graphic displays of data help us answer questions? Which type of graph should be used to represent your data? Explain. Why, when collecting data does so
much depend on the questions you ask? How can central tendency impact the interpretation of data?
Big Ideas: Data is collected to investigate and find solutions to real-life situations. The problem determines the what, the how, the representation, and the interpretation of the data. A sample
is a ratio that reflects the population. The size and composition of the sample influences inferences being made from the data. Data can be discrete or continuous, and used to show trends.
Process Integration
Mathematical Practices
S5C2PO6. Communicate
the answer(s) to the
question(s) in a problem
using appropriate
representations, including
symbols and informal and
formal mathematical
Connections
Math: S2C1PO2, S2C1PO3, language.
S2C1PO4
A.V.
Science: S1C3PO1,
histogram
S1C3PO4, S1C4PO1,
stem and leaf plot
S1C4PO2
Social Studies: S1C1PO1,
S1C1PO2, S2C1PO1,
MP1. Make sense of
S2C1PO2, S4C1PO1,
problems and persevere
S4C1PO2
in solving them.
Performance Objective
Big Ideas:
S2C1PO1. Solve problems by
selecting, constructing, and
interpreting displays of data,
including histograms and
stem-and-leaf plots.
MP6. Attend to
precision.
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Explanations and Examples
Resources
Students are expected to use appropriate labels, intervals, and title for an
appropriate visual representation of collected data. Students will use histograms and
stem-and-leaf plots in addition to all previously learned graphs. It is important that
students have opportunities to choose the appropriate display for the representation
of collected data.
MSM 1
13.5 Stem and Leaf Plots
DAU
Investigation 2
Using Graphs to Explore
Data
ITPS
pp. 135-141
CD: Student Activities, data
analysis and probability
ITRP
pp. 119-123
CD: Student Activities, data
analysis and probability
ITR
Pp 111-114
CD: Student Activities, data
analysis and probability
ITCM
pp., 137-142
MP1. In grade 6, students solve problems involving ratios and rates and discuss
how they solved them. Students solve real world problems through the application
of algebraic and geometric concepts. Students seek the meaning of a problem and
look for efficient ways to represent and solve it. They may check their thinking by
asking themselves, “What is the most efficient way to solve the problem?”, “Does
this make sense?”, and “Can I solve the problem in a different way?”.
MP6. In grade 6, students continue to refine their mathematical communication
skills by using clear and precise language in their discussions with others and in
their own reasoning. Students use appropriate terminology when referring to rates,
ratios, geometric figures, data displays, and components of expressions,
equations or inequalities.
Assessment
MSM 1
p. 665 #5-11
MSM 2
p. 133 #10-12
p. 119 #14-19
DAU
pp. 40-47 questions #1-15
Glendale Elementary School District | June 2011
Mathematics Lesson Planning Guide IP3 | Sixth Grade
S2C1PO2. Formulate and
answer questions by
interpreting, analyzing, and
drawing inferences from
displays of data, including
histograms and stem-and-leaf
plots.
S5C2PO1. Analyze a
problem situation to
determine the
question(s) to be
answered.
S5C2PO2. Identify
relevant, missing, and
Connections
extraneous information
Math: S2C1PO1, S2C1PO3, related to the solution to
S2C1PO4
a problem.
Science: S1C1PO2,
S5C2PO6.
S1C3PO4, S1C3PO6,
Communicate the
Social Studies: S1C1PO2,
answer(s) to the
S2C1PO2, S4C1PO2
question(s) in a problem
using appropriate
representations,
including symbols and
informal and formal
mathematical language.
Students are expected to make estimates and compute with a data set.
MSM 2
3.5 Histograms
3.6 Appropriate Data
Displays
3.3 Stem and Leaf Plots
Examples:
• The histogram below shows the number of DVDs students own:
o How many students own 20 or more DVDs?
o How many students own fewer than 30 DVDs?
o How many students own exactly 15 DVDs? (Students should notice that
histograms display intervals, not individual pieces of data.)
DAU
Investigation 1
Looking at Data
S5C2PO7. Isolate and
organize mathematical
information taken from • The line graph below shows the temperature of a can of juice over time, after
symbols, diagrams, and placing it in an ice and salt mixture. Describe any conclusions you can make about
the data. What are some possible questions you could ask using the data?
graphs to make
inferences, draw
conclusions, and justify
reasoning.
MP1. Make sense of
problems and
persevere in solving
them.
MP2. Reason
abstractly and
quantitatively.
MP6. Attend to
precision.
DAU
pp. 21-28 #1-47
MBL
Fantastic Book of 1,001
Lists
ITPS
pp. 135-141
CD: Student Activities,
data analysis and
probability
ITRP
pp. 119-123
CD: Student Activities,
data analysis and
probability
ITR
Pp 111-114
CD: Student Activities,
data analysis and
probability
ITCM
pp., 137-142
CD: Student Activities,
data analysis and
probability
MP1. In grade 6, students solve problems involving ratios and rates and
discuss how they solved them. Students solve real world problems through
the application of algebraic and geometric concepts. Students seek the
meaning of a problem and look for efficient ways to represent and solve it.
They may check their thinking by asking themselves, “What is the most
efficient way to solve the problem?”, “Does this make sense?”, and “Can I
solve the problem in a different way?”.
MP2. In grade 6, students represent a wide variety of real world contexts
3 of 15
Glendale Elementary School District | June 2011
Mathematics Lesson Planning Guide IP3 | Sixth Grade
through the use of real numbers and variables in mathematical expressions,
equations, and inequalities. Students contextualize to understand the meaning
of the number or variable as related to the problem and decontextualize to
manipulate symbolic representations by applying properties of operations.
MP6. In grade 6, students continue to refine their mathematical communication
skills by using clear and precise language in their discussions with others and
in their own reasoning. Students use appropriate terminology when referring
to rates, ratios, geometric figures, data displays, and components of
expressions, equations or inequalities.
S2C1PO3. Use extreme
values, mean, median, mode,
and range to analyze and
describe the distribution of a
given data set.
S5C2PO7. Isolate and
organize mathematical
information taken from
symbols, diagrams, and
graphs to make
inferences, draw
Connections
conclusions, and justify
Math: S1C3PO2, S2C1PO1, reasoning
S2C1PO2, S2C1PO4
.
Students use sets of data and graphical representations of data sets from real-world MSM 1
contexts.
2.8 Mean, Median &
Mode
Example:
• Use the stem and leaf plot below to determine the extreme values (maximum and MSM 2
minimum values represented), mean, median, mode and range. What do these
3.1 Mean, Median &
values show about the distribution of the data?
Mode
Key: 2 ç 3 = 23
A.V.
central tendency
extreme values
outlier
MSM 1
p. 96 #20-31
DAU
pp. 56-62 #1-23
DAU
Investigation 3
What Do We Mean By
Mean?
ITPS
CD: Student Activity;
Doing Homework
Peanut Ice Cream Bars
MP1. Make sense of
problems and
persevere in solving
them.
ITPR
CD: Student activity;
Don’t Change That Mean,
Median or Mode
MP2. Reason
abstractly and
quantitatively.
MP6. Attend to
precision.
MP1. In grade 6, students solve problems involving ratios and rates and
discuss how they solved them. Students solve real world problems through
the application of algebraic and geometric concepts. Students seek the
meaning of a problem and look for efficient ways to represent and solve it.
They may check their thinking by asking themselves, “What is the most
efficient way to solve the problem?”, “Does this make sense?”, and “Can I
solve the problem in a different way?”.
4 of 15
Glendale Elementary School District | June 2011
Mathematics Lesson Planning Guide IP3 | Sixth Grade
MP2. In grade 6, students represent a wide variety of real world contexts
through the use of real numbers and variables in mathematical expressions,
equations, and inequalities. Students contextualize to understand the meaning
of the number or variable as related to the problem and decontextualize to
manipulate symbolic representations by applying properties of operations.
MP6. In grade 6, students continue to refine their mathematical communication
skills by using clear and precise language in their discussions with others and
in their own reasoning. Students use appropriate terminology when referring
to rates, ratios, geometric figures, data displays, and components of
expressions, equations or inequalities.
S2C1PO4. Compare two or
more sets of data by
identifying trends.
S5C2PO7. Isolate and
organize mathematical
information taken from
symbols, diagrams, and
Connections
graphs to make
Math: S2C1PO1, S2C1PO2, inferences, draw
S2C1PO3
conclusions, and justify
Science: S1C3PO1
reasoning.
MP1. Make sense of
problems and
persevere in solving
them.
MP2. Reason
abstractly and
quantitatively.
MP6. Attend to
precision.
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Students analyze data to identify trends (increasing, decreasing, constant). Students SAP
also analyze two or more sets of data to determine how the trends in multiple sets of Investigation 1
data compare.
Comparing Data Sets
(excluding box plots)
MP1. In grade 6, students solve problems involving ratios and rates and discuss how
they solved them. Students solve real world problems through the application of
algebraic and geometric concepts. Students seek the meaning of a problem and look
for efficient ways to represent and solve it. They may check their thinking by asking
themselves, “What is the most efficient way to solve the problem?”, “Does this make
sense?”, and “Can I solve the problem in a different way?”.
SAP
pp. 17-24 #1-33
(excluding box plots)
TSCM
Chapter 11
Exploring Data Analysis
MP2. In grade 6, students represent a wide variety of real world contexts through the
use of real numbers and variables in mathematical expressions, equations, and
inequalities. Students contextualize to understand the meaning of the number or
variable as related to the problem and decontextualize to manipulate symbolic
representations by applying properties of operations.
MP6. In grade 6, students continue to refine their mathematical communication skills by
using clear and precise language in their discussions with others and in their own
reasoning. Students use appropriate terminology when referring to rates, ratios,
geometric figures, data displays, and components of expressions, equations or
inequalities.
Glendale Elementary School District | June 2011
Mathematics Lesson Planning Guide IP3 | Sixth Grade
Topic: Probability, Discrete Mathematics
Strand 2: Data Analysis, Probability, and Discrete Mathematics
Concept 2: Probability
Understand and apply the basic concepts of probability.
In Grade 6, students begin to make and test conjectures about theoretical probability by predicting outcomes of experiments, performing experiments, comparing experimental outcomes to a
prediction, and replicating experiments for the comparison of results. They determine possible outcomes using a variety of systematic approaches.
Concept 3: Systematic Listing and Counting
Understand and demonstrate the systematic listing and counting of possible outcomes.
In Grade 6, students explore three attribute counting problems using Venn diagrams to build on prior learning about different counting problems. They learn to create and analyze tree diagrams
where data repeats and expand their prior learning of the multiplication principle of counting.
Essential Questions: How can we use experiments to formulate generalizations? How does understanding probability help you make decisions? What should predictions be based on? How can
systematic displays of data help us answer questions? What is the difference in the way you think about solving a problem using a Hamilton path vs a Hamilton circuit?
Big Ideas: Multiple experiments can be used to test theoretical probability. Combinations, arrangements, and shortest routes can be determined by listing possible outcomes in a systematic
way.
Process Integration
Mathematical Processes
S2C2PO1. Use data collected S5C2PO8. Make and test
from multiple trials of a single conjectures based on
event to form a conjecture
information collected from
about the theoretical
explorations and
probability.
experiments.
Performance Objective
Explanations and Examples
Resources
Example:
• Each group receives a bag that contains 4 green marbles, 6 red marbles, and 10
blue marbles. Each group performs 50 pulls, recording the color of marble drawn
and replacing the marble into the bag before the next draw. Students compile their
data as a group and then as a class. They summarize their data as experimental
probabilities and make conjectures about theoretical probabilities (How many
green draws would you expect if you were to conduct 1000 pulls? 10,000 pulls?).
Connections
MP1. Make sense of
Math: S1C1PO1, S2C2PO2, problems and persevere
S2C2PO3
in solving them.
Students create another scenario with a different ratio of marbles in the bag and
make a conjecture about the outcome of 50 marble pulls with replacement. (An
MP7. Look for and make example would be 3 green marbles, 6 blue marbles, and 3 blue marbles.)
use of structure.
Students try the experiment and compare their predictions to the experimental
outcomes to continue to explore and refine conjectures about theoretical
probability.
MSM 1
13.1 Conducting An
Experiment
13.1 Introduction to
Probability
Assessment
MSM 2
p. 633 #1-2
HLII
p. 13 #1-32
p. 28 #1-37
MSM 2
13.1 Investigating
Probability
13.1 Introduction to
Probability
HLII
Investigation 1
A First Look At Chance
Investigation 2
MP1. In grade 6, students solve problems involving ratios and rates and
Experimental and
discuss how they solved them. Students solve real world problems through Theoretical Probability
the application of algebraic and geometric concepts. Students seek the
meaning of a problem and look for efficient ways to represent and solve it. ATM
They may check their thinking by asking themselves, “What is the most
Probability and Statistics
efficient way to solve the problem?”, “Does this make sense?”, and “Can I pp. 85-106
solve the problem in a different way?”.
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Glendale Elementary School District | June 2011
Mathematics Lesson Planning Guide IP3 | Sixth Grade
MP7. Students routinely seek patterns or structures to model and solve
problems. For instance, students recognize patterns that exist in ratio tables
recognizing both the additive and multiplicative properties. Students apply
properties to generate equivalent expressions
(i.e. 6 + 2x = 2 (3 + x) by distributive property) and solve equations (i.e. 2c + 3
= 15, 2c = 12 by subtraction property of equality; c=6 by division property of
equality). Students compose and decompose two- and three-dimensional
figures to solve real world problems involving area and volume.
S2C2PO2. Use theoretical
S5C2PO7. Isolate and
Students need multiple opportunities to perform probability experiments and
probability to
organize mathematical
compare these results to theoretical probabilities. Critical components of the
• predict experimental
information taken from
experiment process are making predictions about the outcomes by applying the
outcomes,
symbols, diagrams, and
principles of theoretical probability, comparing the predictions to the outcomes of
• compare the outcome of the graphs to make inferences, the experiments, and replicating the experiment to compare results. Experiments
experiment to the prediction, draw conclusions, and
can be replicated by the same group or by compiling class data. Experiments can
and
justify reasoning.
be conducted using various random generation devices including, but not limited
• replicate the experiment and
to, bag pulls, spinners, number cubes, coin toss, and colored chips.
compare results.
MP1. Make sense of
problems and persevere MP1. In grade 6, students solve problems involving ratios and rates and
Connections
in solving them.
discuss how they solved them. Students solve real world problems through
Math: S1C1PO1, S1C3PO2,
the application of algebraic and geometric concepts. Students seek the
S2C2PO1, S2C2PO3
MP2. Reason abstractly meaning of a problem and look for efficient ways to represent and solve it.
and quantitatively.
They may check their thinking by asking themselves, “What is the most
efficient way to solve the problem?”, “Does this make sense?”, and “Can I
MP6. Attend to precision. solve the problem in a different way?”.
TSCM
Chapter 12 Exploring
Concepts of Probability
ITPS
pp. 135-141
CD: Student Activities;
Weather Watcher
MSM 1
13.1 Conducting An
Experiment
13.1 Introduction to
Probability
MSM 2
13.1 Investigating
Probability
13.1 Introduction to
Probability
HLII
Investigation 1
A First Look At Chance
Investigation 2
Experimental and
Theoretical Probability
ATM
MP2. In grade 6, students represent a wide variety of real world contexts
Probability and Statistics
through the use of real numbers and variables in mathematical expressions, pp 85-106
equations, and inequalities. Students contextualize to understand the
TSCM
meaning of the number or variable as related to the problem and
Chapter 12 Exploring
decontextualize to manipulate symbolic representations by applying
Concepts of Probability
properties of operations.
ITRP
MP6. In grade 6, students continue to refine their mathematical
CD: Student Activities;
communication skills by using clear and precise language in their
Fair or Not Fair
discussions with others and in their own reasoning. Students use
Predict That Spin
appropriate terminology when referring to rates, ratios, geometric figures,
Dartboard Probabilities
data displays, and components of expressions, equations or inequalities.
ITR
CD: Student Activities;
Take a Chance
7 of 15
MSM 1
p. 629 #1-2
HLII
p. 13 #1-32
p. 28 #1-37
Glendale Elementary School District | June 2011
Mathematics Lesson Planning Guide IP3 | Sixth Grade
S2C2PO3. Determine all
possible outcomes (sample
space) of a given situation
using a systematic approach.
S5C2PO5. Represent a
problem situation using
multiple representations,
describe the process used
to solve the problem, and
Connections
verify the reasonableness
Math: S2C2PO1, S2C2PO2, of the solution.
S2C3PO1
MP8. Look for and
express regularity in
repeated reasoning.
Systematic approaches may include, but are not limited to, frequency tables, tree
diagrams, charts/tables, ordered pairs, and matrices.
Example:
• What are all of the outcomes of flipping a coin three times?
TTT
HHT
TTH
CP:PS 6
pp. T-27, T-37
pp. P-6, P-8
THH
THT
MP8. In grade 6, students use repeated reasoning to understand algorithms
and make generalizations about patterns. During multiple opportunities to
solve and model problems, they may notice that a/b ÷ c/d = ad/bc and
construct other examples and models that confirm their generalization.
Students connect place value and their prior work with operations to
understand algorithms to fluently divide multi-digit numbers and perform all
operations with multi-digit decimals. Students informally begin to make
connections between covariance, rates, and representations showing the
relationships between quantities.
8 of 15
HLII
p. 13 #1-32
p. 28 #1-37
TSCM
Chapter 12 Exploring
Concepts of Probability
Systematic List
HHH
HTH
HTT
ATM
Probability and Statistics
pp. 85-106
MBL
Anno’s Hat Trick
ITR
CD: Student Activities;
Train Cars
Ice Cream Combinations
Map It Out
Glendale Elementary School District | June 2011
Mathematics Lesson Planning Guide IP3 | Sixth Grade
S2C3PO1. Build and explore S5C2PO5. Represent a
tree diagrams where items
problem situation using
repeat.
multiple representations,
describe the process used
Connections
to solve the problem, and
Math: S2C2PO3
verify the reasonableness
of the solution.
A.V.
arrangements
systematic
approach
MP8. Look for and
express regularity in
repeated reasoning.
Students have had opportunities to build tree diagrams in balanced situations, that
is, when a consistent outcome happens at every step. They will be challenged by
counting problems where an item is repeated. This seemingly little twist in the
problem requires students to count the outcomes differently and makes the
problem harder to solve. For example, how many ways can you arrange the letters
in the word “FREE.” Although you have a total of four letters in the word, there are
only three possible choices for the first letter (F, R, or E); the repeated letter E
throws a different twist into the construction of the tree diagram, namely it makes it
“unbalanced.” Look at the tree diagram below. Can you find where a different
number of options are possible?
NDM 6-12
Chapter 1 Systematic
Listing and Counting
NDM 6-12
pp 17-25
Flag Trademarks
ITPS
CD: Student Activities;
Order of the Evening
ITR
CD: Student Activities;
Train Cars
Students should notice that after the first choice of a letter “F,” there will only be
two possible letters that could come next – namely, either R or E. But if their
choice for a first letter was “E,” they would have three possible letters for their
second choice, namely, F, R, or E. When students look at the three subgroups in
this tree, they will notice that the structure of the “E” subgroup is different from the
structure of “R” subgroup, and from the structure of the “F” subgroup. The tree is
not balanced.
Example:
• All
possible arrangements of
the letters in the word FREE.
MP8. In grade 6, students use repeated
reasoning to understand algorithms and make generalizations about
patterns. During multiple opportunities to solve and model problems, they
may notice that a/b ÷ c/d = ad/bc and construct other examples and models
that confirm their generalization. Students connect place value and their
prior work with operations to understand algorithms to fluently divide multidigit numbers and perform all operations with multi-digit decimals. Students
informally begin to make connections between covariance, rates, and
representations showing the relationships between quantities.
9 of 15
Glendale Elementary School District | June 2011
Mathematics Lesson Planning Guide IP3 | Sixth Grade
S2C3PO2. Explore counting S5C2PO5. Represent a
problems with Venn diagrams problem situation using
using three attributes.
multiple representations,
describe the process used
Connections
to solve the problem, and
Math: S5C2PO7
verify the reasonableness
of the solution.
Example:
• Ms. Taft’s class has 35 students. Ms. Taft surveyed her students to find out the
games they like to play in class.
o said they liked to play only dodge ball.
o said they like to play only basketball.
o said they like to play only soccer.
o said they liked to play dodge ball, basketball and soccer.
MP2. Reason abstractly
and quantitatively.
Record the results in a Venn diagram that shows the fraction of students and
MP6. Attend to precision. number of students in each group. What is the total number of students who said
they enjoy each sport?
Ride
2s
3
4
2
6
1
Conce
4
rt
Game
s 1
1
9
NDM 6-12
Chapter 1 Systematic
Listing and Counting
NDM 6-12
pp. 26-33
Counting the Kids
CP:PS 6
pp. T-77, T-79
CP: PS6
P17 problem 81
P24 problem 96
P26 problem 99
P30 problem 108
P33 problem 114
CP:PS 8
pp. P-4, P-10. P-24
ITRP
CD: Student Activities;
Chicken, Beef, or Fish
ITR
CD: Student Activities;
Sandwich Survey
5
MP2. In grade 6, students represent a wide variety of real world contexts
through the use of real numbers and variables in mathematical expressions,
equations, and inequalities. Students contextualize to understand the
meaning of the number or variable as related to the problem and
decontextualize to manipulate symbolic representations by applying
properties of operations.
MP6. In grade 6, students continue to refine their mathematical
communication skills by using clear and precise language in their
discussions with others and in their own reasoning. Students use
appropriate terminology when referring to rates, ratios, geometric figures,
data displays, and components of expressions, equations or inequalities.
MP8. In grade 6, students use repeated reasoning to understand algorithms
and make generalizations about patterns. During multiple opportunities to
solve and model problems, they may notice that a/b ÷ c/d = ad/bc and
construct other examples and models that confirm their generalization.
Students connect place value and their prior work with operations to
understand algorithms to fluently divide multi-digit numbers and perform all
operations with multi-digit decimals. Students informally begin to make
connections between covariance, rates, and representations showing the
relationships between quantities.
10 of 15
Glendale Elementary School District | June 2011
Mathematics Lesson Planning Guide IP3 | Sixth Grade
S2C4PO1. Investigate
properties of vertex-edge
graphs
• Hamilton paths,
• Hamilton circuits, and
• shortest route.
Connections
Math: S2C4PO2
S5C2PO5. Represent a
problem situation using
multiple representations,
describe the process used
to solve the problem, and
verify the reasonableness
of the solution.
A.V.
Hamilton path
Hamilton circuit
shortest route
A Hamilton path in a vertex-edge graph is a path that starts at some vertex in the
graph and visits every other vertex of the graph exactly once. Edges along this
path may be repeated. A Hamilton circuit is a Hamilton path that ends at the
starting vertex. The shortest route may or may not be a Hamilton path. Depending
upon the constraints of a problem, each vertex may not need to be visited.
GESD Portals
See resource section for
website.
GESD Portals
See resource section for
website.
NDM K – 5
NDM K – 5
Example
• If the park ranger is required to visit every location on the vertex-edge graph
below, what route should he take? Where should he begin and end his trip?
NDM 6 – 12
NDM 6 – 12
CP:PS 6
p. T-25
p. P-9
MP8. Look for and
express regularity in
repeated reasoning.
• One possible Hamilton path is: Prospector-Tent-Coyotes-Snakes-JavelinasWatering Hole-Cacti-Cave Creek Canyon. Can you find other Hamilton paths?
• Is it possible to start at one vertex (site) on the vertex-edge graph and visit every
other vertex just once and return to the starting vertex? If it is possible, name that
circuit.
• What is the shortest route between Cave Creek Canyon and the Tent?
MP8. In grade 6, students use repeated reasoning to understand algorithms
and make generalizations about patterns. During multiple opportunities to
solve and model problems, they may notice that a/b ÷ c/d = ad/bc and
construct other examples and models that confirm their generalization.
Students connect place value and their prior work with operations to
understand algorithms to fluently divide multi-digit numbers and perform all
operations with multi-digit decimals. Students informally begin to make
connections between covariance, rates, and representations showing the
relationships between quantities.
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Glendale Elementary School District | June 2011
Mathematics Lesson Planning Guide IP3 | Sixth Grade
S2C4PO2. Solve problems S5C2PO7. Isolate and
related to Hamilton paths and organize mathematical
circuits.
information taken from
symbols, diagrams, and
Connections
graphs to make inferences,
Math: S2C4PO1
draw conclusions, and
justify reasoning.
Example:
• The Clark family is vacationing in the southwestern part of the United States.
They are going to visit every location on the graph below. What is the shortest
route they can take? Where should the first vacation stop be for the Clark family?
The last stop?
GESD Portals
See resource section for
website.
GESD Portals
See resource section for
website.
NDM K – 5
NDM K – 5
NDM 6 – 12
NDM 6 – 12
MP1. Make sense of
problems and persevere
in solving them.
MP2. Reason abstractly
and quantitatively.
MP6. Attend to precision.
MP1. In grade 6, students solve problems involving ratios and rates and
discuss how they solved them. Students solve real world problems through
the application of algebraic and geometric concepts. Students seek the
meaning of a problem and look for efficient ways to represent and solve it.
They may check their thinking by asking themselves, “What is the most
efficient way to solve the problem?”, “Does this make sense?”, and “Can I
solve the problem in a different way?”.
MP2. In grade 6, students represent a wide variety of real world contexts
through the use of real numbers and variables in mathematical expressions,
equations, and inequalities. Students contextualize to understand the
meaning of the number or variable as related to the problem and
decontextualize to manipulate symbolic representations by applying
properties of operations.
MP6. In grade 6, students continue to refine their mathematical
communication skills by using clear and precise language in their
discussions with others and in their own reasoning. Students use
appropriate terminology when referring to rates, ratios, geometric figures,
data displays, and components of expressions, equations or inequalities.
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Glendale Elementary School District | June 2011
Mathematics Lesson Planning Guide IP3 | Sixth Grade
Topic: Reasoning Through Patterns, Algebra, Functions
Strand 3: Patterns, Algebra, and Functions
Concept 4: Analysis of Change
In Grade 6, students extend prior learning about patterns of change to predict missing values on line graphs or scatterplots.
Essential Questions: How can you determine the value of a variable? How can you identify patterns in systems? How might you describe a rule used in a function?
Big Ideas: Algebra is determining the unknown.
Patterns are all around us and help us understand our world.
Relationships between numbers can be expressed as patterns and mathematical equations.
S3C4PO1. Determine a
S5C2PO7. Isolate and Example:
pattern
to predict
missing
organize
mathematical
A pattern
of change
can be described
using
a function. • Use the graph below to determine how much money a person makes after working
values on a line graph or
information taken from exactly 9 hours.
scatterplot.
symbols, diagrams, and
graphs to make
Connections
inferences, draw
Math: S1C3PO2, S3C2PO1, conclusions, and justify
Science: S1C3PO1
reasoning.
MSM 2
MSM 2
6.8 The Coordinate Plane p. 296, #29-30
MSM 3
11.2 Scatter Plots
MSM 3
p. 547, #8-9
MP1. Make sense of
problems and
persevere in solving
them.
MP2. Reason
abstractly and
quantitatively.
MP6. Attend to
precision.
MP1. In grade 6, students solve problems involving ratios and rates and discuss how
they solved them. Students solve real world problems through the application of
algebraic and geometric concepts. Students seek the meaning of a problem and look for
efficient ways to represent and solve it. They may check their thinking by asking
themselves, “What is the most efficient way to solve the problem?”, “Does this make
sense?”, and “Can I solve the problem in a different way?”.
MP2. In grade 6, students represent a wide variety of real world contexts through the
use of real numbers and variables in mathematical expressions, equations, and
inequalities. Students contextualize to understand the meaning of the number or
variable as related to the problem and decontextualize to manipulate symbolic
representations by applying properties of operations.
MP6. In grade 6, students continue to refine their mathematical communication skills by
using clear and precise language in their discussions with others and in their own
reasoning. Students use appropriate terminology when referring to rates, ratios,
geometric figures, data displays, and components of expressions, equations or
inequalities.
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Glendale Elementary School District | June 2011
Mathematics Lesson Planning Guide IP3 | Sixth Grade
Topic: Structure and Logic
Strand 5: Structure and Logic
Concept 2: Logic, Reasoning, Problem Solving, and Proof
In Grade 6, students continue to use a variety of problem-solving strategies, and analyze them for efficiency and appropriateness for contextual situations. They communicate their thinking using
multiple representations, synthesize and organize information from multiple sources to make inferences, draw conclusions, and justify their reasoning. Students begin to solve logic problems using
conditional statements.
Essential Questions: How do I know my solution is accurate? How can I defend my answer? What is the purpose of approximation?
Big Ideas: Use evidence and proper terminology to verify solutions to problems. Problems have multiple representations. Estimation can be used to predict or verify the reasonableness of a
solution.
Performance Objective
S5C2PO9. Solve simple
logic problems, including
conditional statements,
and justify solution
methods and reasoning.
Process Integration
S5C2PO3. Analyze and
compare mathematical
strategies for efficient
problem solving; select and
use one or more strategies
to solve a problem.
Explanations and Examples
Resources
Example:
• In a magic square below, if the sum of every row and column is the same, then
what values can be placed in the empty boxes? Explain how you know your answer
is correct.
6
7
S5C2PO7. Isolate and
organize mathematical
information taken from
8
3
4
symbols, diagrams, and
graphs to make inferences,
draw conclusions, and justify
reasoning.
MP1. In grade 6, students solve problems involving ratios and rates and discuss how
ATM
p. 130-143
Assessment
ATM
p. 130-143
CP:PS 6
pp. T-31, T-41, T-43,
T-49, T-51
pp. P-2, P-8
they solved them. Students solve real world problems through the application of
algebraic and geometric concepts. Students seek the meaning of a problem and look for
MP1. Make sense of
problems and persevere in efficient ways to represent and solve it. They may check their thinking by asking
themselves, “What is the most efficient way to solve the problem?”, “Does this make
solving them.
sense?”, and “Can I solve the problem in a different way?”.
MP2. Reason abstractly
and quantitatively.
MP6. Attend to precision.
MP 2. In grade 6, students represent a wide variety of real world contexts through the
use of real numbers and variables in mathematical expressions, equations, and
inequalities. Students contextualize to understand the meaning of the number or
variable as related to the problem and decontextualize to manipulate symbolic
representations by applying properties of operations.
MP6. In grade 6, students continue to refine their mathematical communication skills by
using clear and precise language in their discussions with others and in their own
reasoning. Students use appropriate terminology when referring to rates, ratios,
geometric figures, data displays, and components of expressions, equations or
inequalities.
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Glendale Elementary School District | June 2011
Mathematics Lesson Planning Guide IP3 | Sixth Grade
Key for Resources
Adopted Text
Code
BAP 1 (Grade 6)
BAP 2 (Grade 6)
CAS (Grade 7)
MSM 1
MSM 2
ML Pre-Algebra
DAU (Grade 6)
MSA (Grade 7)
HLII (Grade 6)
PT (Grade 6)
SIWS (Grade 8)
TWMM (Grade 8)
SAP (Grade 8)
BAP 1 (Grade 6)
BAP 2 (Grade 6)
CAS (Grade 7)
ML Pre-Algebra
MSA (Grade 7)
Resource
Connected Math Bits and Pieces 1
Connected Math Bits and Pieces 2
Connected Math Comparing and Scaling
Middle School Math Course 1
Middle School Math Course 2
McDougal Little Pre-Algebra
Connected Math Data About Us
Connected Math Moving Straight Ahead
Connected Math How Likely Is It?
Connected Math Prime Time
Connected Math Say It With Symbols
Connected Math Thinking With Mathematical Models
Connected Math Samples and Populations
Connected Math Bits and Pieces 1
Connected Math Bits and Pieces 2
Connected Math Comparing and Scaling
McDougal Little Pre-Algebra
Connected Math Moving Straight Ahead
Additional Resources (ask Achievement Advisor)
Code
Resource
TSCM
Teaching Student-Centered Mathematics
MMK-6
Math Matters K-6
NDM 6-12
Navigating Through Discrete Mathematics Grades 6-12
NDM K-5
Navigating Through Discrete Mathematics Grades K-5
ATM
About Teaching Mathematics
CP:PS 6
Creative Publications: The Problem Solver 6
CP:PS 8
Creative Publications: The Problem Solver 8
MLB
Marilyn Burns Classroom Math Library
ITPS
The Math Process Standards Series: Introduction to Problem Solving
ITR
The Math Process Standards Series: Introduction to Representation
ITRP
The Math Process Standards Series: Introduction to Reasoning and Proof
ITCM
The Math Process Standards Series: Introduction to Communication
ITCN
The Math Process Standards Series: Introduction to Connections
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Glendale Elementary School District | June 2011