Notice Grade 12 Math & User's Guide Manuals 08th grade math
8 th Grade Blueprint
Assessed Quarter 1
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 1. Compare and order real
M08-S5C2-01. Analyze a problem
Students order real numbers in a variety of forms (fractions, decimals, simple
numbers including very large and
situation to determine the
radicals, etc.) on a number line. Students compare real numbers within and
small integers, and decimals and
question(s) to be answered.
among different subsets of the real number system.
fractions close to zero.
M08-S5C2-06. Communicate the
Connections
answer(s) to the question(s) in a
M08-S1C3-02
problem using appropriate
Estimate the location of rational and common
representations, including symbols irrational numbers on a number line. and informal and formal
mathematical language.
PO 2. Classify real numbers as
M08-S5C2-01. Analyze a problem
Students differentiate the definitions of rational and irrational numbers. They use
rational or irrational.
situation to determine the
the definitions to classify a list of real numbers.
question(s) to be answered.
8.NS.1. Know that numbers are not
Examples:
Strand 1: rational are called irrational. Concept 1: Convert 0.333… into a fraction. Understand informally that every
10n = 3.333…
Number number has a decimal expansion; for
n = 0.333…
Sense rational numbers show that the
9n = 3
decimal expansion repeats eventually,
and convert a decimal expansion,
n=
which repeats eventually into a
rational number.
Convert 0.1666… into a fraction.
100n = 16.666…
8.EE.2. Know that 2 is irrational.
10n = 1.666… 90n = 15
Connections
M08-S1C1-03
n=
Model the relationship between the subsets of
the real number system.
M08-S1C3-02 Estimate the location of rational and common irrational numbers on a number line.
8 th Grade Blueprint
Assessed Quarter 1
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 3. Model the relationship between
M08-S5C2-04. Represent a
Students can use graphic organizers to show the relationship between the
the subsets of the real number
problem situation using multiple
subsets of the real number system.
system.
representations, describe the process used to solve the problem,
Connections
and verify the reasonableness of
M08-S1C1-02
the solution.
Classify real numbers as rational or irrational.
Strand 1: Concept 1: Number
Sense
PO 4. Model and solve problems
M08-S5C2-04. Represent a
Students solve problems that include absolute values and graph their answers
involving absolute value.
problem situation using multiple
on a number line.
representations, describe the
Connections
process used to solve the problem,
M08-S1C2-05
and verify the reasonableness of
Simplify numerical expressions using the order
the solution.
of operations that include grouping symbols, square roots, cube roots, absolute values, and positive exponents.
8 th Grade Blueprint
Assessed Quarter 1
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 1. Solve problems with factors,
M08-S5C2-01. Analyze a problem
Examples:
multiples, divisibility or remainders,
situation to determine the
Use the rules of divisibility to classify numbers. Explain why some
prime numbers, and composite
question(s) to be answered.
numbers may be listed in more than one group.
numbers.
Compare the price of each of the jars of spaghetti sauce to determine
Connections
(none)
36 oz
24 oz
12 oz
the best deal.
You are planning a barbeque for 40 people. You will serve hot dogs.
Strand 1:
Concept 2:
Each of the packages of hot dogs contains 8 hot dogs and each of the
Numerical
packages of hot dog buns contains 6 buns. You want to buy the
Operations
minimum number of packages so that each hot dog is matched with a bun and there are no leftovers. How many packages of each must you
buy?
A florist has 56 roses, 42 carnations, and 21 daisies that she can use to create bouquets. What is the greatest number of bouquets she can make containing at least one of each flower, without having any flowers left over?
8 th Grade Blueprint
Assessed Quarter 1
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 2. Describe the effect of
M08-S5C2-06. Communicate the
Example:
multiplying and dividing a rational
answer(s) to the question(s) in a
Explain what happens to the number 2 when it is multiplied and divided
number by
problem using appropriate
by each of the real numbers listed below:
a number less than zero,
representations, including symbols
o -2
a number between zero and
and informal and formal
one,
mathematical language.
one, and
a number greater than one.
Numerical Operations
PO 3. Solve problems involving
M08-S5C2-01. Analyze a problem
Examples:
and simple interest rates. Gas prices are projected to increase 124% by April. A gallon of gas question(s) to be answered. costs $4.17. How much will a gallon of gas cost in April? A sweater is marked down 33%. Its original price was $37.50. What is
percent increase, percent decrease,
situation to determine the
Connections
M08-S5C2-08. Describe when to
the price of the sweater before sales tax?
M08-S1C3-01
use proportional reasoning to solve
Make estimates appropriate to a given
a problem.
situation.
M08-S1C2-05 Simplify numerical expressions using the order of operations that include grouping symbols, square roots, cube roots, absolute values, and positive exponents.
M08-S3C4-02 Solve problems involving simple rates.
8 th Grade Blueprint
Assessed Quarter 1
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 4. Convert standard notation to In addition to converting between standard notation and scientific scientific notation and vice versa
notation, also explore issues of magnitude, as it relates to real world (include positive and negative
8.EE.3. Use numbers expressed in the
Write the distance between the Earth and the Sun using scientific
form of a single digit times an integer
notation. The average distance between the Earth and the Sun is 150
power of 10 to estimate very large or
million kilometers.
very small quantities, and to express
What is the average size of a red blood cell in meters written in
how many times as much one is than
standard notation? The average size of a red blood cell is 7.0 x 10 -6
the other.
meters.
8 Estimate the population of the United States as 3 x 10 and the Strand 1: 9 Connections population of the world as 7 x 10 , and determine that the world
Concept 2:
(none)
population is more than 20 times larger.
Numerical
Operations
PO 5. Simplify numerical expressions
Students are expected to simplify expressions containing exponents, including
using the order of operations that
zero.
include grouping symbols, square roots, cube roots, absolute values,
Examples:
and positive exponents.
8.EE.2. Evaluate square roots of small
perfect squares and cube roots of
small perfect cubes.
Connections M08-S1C1-04 Model and solve problems involving absolute value.
Assessed Quarter 1
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 1. Make estimates appropriate to a
M08-S5C2-01. Analyze a problem
Students estimate using all four operations with whole numbers, fractions, and
given situation.
situation to determine the
decimals. Estimation skills include identifying when estimation is appropriate,
question(s) to be answered.
determining the level of accuracy needed, selecting the appropriate method of
8.NS.2. Estimate the value of
estimation, and verifying solutions or determining the reasonableness of
expressions (e.g. ).
situations using various estimation strategies.
Connections M08-S1C2-03
Estimation strategies for calculations with fractions and decimals extend from
Solve problems involving percent increase, percent decrease, and simple interest rates.
students’ work with whole number operations. Estimation strategies include, but are not limited to:
M08-S1C3-02 Estimate the location of rational and common
front-end estimation with adjusting (using the highest place value and
irrational numbers on a number line.
estimating from the front end making adjustments to the estimate by taking into account the remaining amounts),
M08-S2C1-02 Make inferences by comparing the same
clustering around an average (when the values are close together an
summary statistic for two or more data sets.
average value is selected and multiplied by the number of values to determine an estimate),
M08-S2C3-02 Solve counting problems and represent
rounding and adjusting (students round down or round up and then
counting principles algebraically including
adjust their estimate depending on how much the rounding affected the
Strand 1:
factorial notation. original values),
Concept 3:
M08-S3C3-02
using friendly or compatible numbers such as factors (students seek to
Estimation
Evaluate an expression containing one or two
fit numbers together - i.e., rounding to factors and grouping numbers
variables by substituting numbers for the
together that have round sums like 100 or 1000), and variables. using benchmark numbers that are easy to compute (students select
M08- S3C4-02
close whole numbers for fractions or decimals to determine an Solve problems involving simple rates. estimate).
M08-S4C1-02 Predict results of combining, subdividing, and changing shapes of plane figures and solids.
Specific strategies also exist for estimating measures. Students should develop fluency in estimating using standard referents (meters, yard, etc) or created
M08-S4C3-01
referents (the window would fit about 12 times across the wall).
Make and test a conjecture about how to find the midpoint between any two points in the
coordinate plane.
Connect to 1.2.5
Estimate 2 +1
M08-S4C4-01
Solve problems involving conversions within
the same measurement system.
M08-S5C1-01 Create an algorithm to solve problems involving indirect measurement, using proportional reasoning, dimensional analysis, and the concepts of density and rate.
8 th Grade Blueprint
Assessed Quarter 1
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 2. Estimate the location of rational and common irrational numbers on a
, 2 , and 3 are some examples of common irrational numbers that
number line.
students should be able to estimate.
8.NS.2. Use rational approximation of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram.
M08-S1C1-01 Compare and order real numbers including very large and small integers, and decimals and fractions close to zero.
M08-S1C1-02 Classify real numbers as rational or irrational.
M08-S1C3-01 Make estimates appropriate to a given situation.
Assessed Quarter 1
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 1. Solve problems by selecting,
M08-S5C2-07. Isolate and
Students calculate extreme values, mean, median, mode, range, quartiles, and
constructing, interpreting, and
organize mathematical information
interquartile ranges. They should approximate lines of best fit for scatter plots
calculating with displays of data,
taken from symbols, diagrams, and and analyze the correlation between the variables (positive, negative, and no
correlation). After creating the line of best fit, assess the model fit by Strand 2:
including box and whisker plots and
graphs to make inferences, draw
judging the closeness of the data points to the line. Concept 1:
scatter plots.
conclusions, and justify reasoning.
Data 8.SP.2. Know that straight lines are
M08-S5C2-10
Analysis widely used to model relationships
Solve logic problems involving
between two quantitative variables.
multiple variables, conditional
This concept
For scatter plots that suggest a linear
statements, conjectures, and
was introduced
association, informally fit a straight
negation using words, charts,
in Grade 7,
line, and informally assess the model
Quarter 4 and it
and pictures.
fit by judging the closeness of the data
will be
points to the line.
mastered in
Grade 8. Connections M08-S2C1-04 Determine whether information is represented
effectively and appropriately given a graph or a set of data by identifying sources of bias and compare and contrast the effectiveness of different representations of data. .
8 th Grade Blueprint
Assessed Quarter 1
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 2. Make inferences by comparing
M08-S5C2-07. Isolate and
Summary statistics include: extreme values, mean, median, mode, range,
the same summary statistic for two or
organize mathematical information
quartiles, and interquartile ranges. Students will include scatter plots, box and
more data sets.
taken from symbols, diagrams, and whisker plots, and all other applicable representations taught in previous grade graphs to make inferences, draw
levels. They will compare two different populations or two subsets of the same
Strand 2:
Connections
conclusions, and justify reasoning.
population.
Concept 1:
M08-S1C3-01
Data
Make estimates appropriate to a given
M08-S5C2-09. Make and test
Analysis
situation.
conjectures based on information
M08-S2C1-03
collected from explorations and
This concept was introduced
Describe how summary statistics relate to the
experiments.
in Grade 7,
shape of the distribution.
Quarter 4 and it
PO 3. Describe how summary
M08-S5C2-07. Isolate and
Summary statistics include: extreme values, mean, median, mode, range,
will be
statistics relate to the shape of the
organize mathematical information
quartiles, and interquartile ranges.
mastered in
distribution.
taken from symbols, diagrams, and Grade 8. graphs to make inferences, draw
Connections
conclusions, and justify reasoning.
M08-S2C1-02 Make inferences by comparing the same summary statistic for two or more data sets.
th
8 Grade Blueprint
Assessed Quarter 1
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 4. Determine whether information
M08-S5C2-06. Communicate the
Graphical displays include representations taught from kindergarten through
is represented effectively and
answer(s) to the question(s) in a
grade 8 (i.e., tally charts, pictographs, frequency tables, bar graphs (including
appropriately given a graph or a set of
multi bar graphs), line plots, circle graphs, line graph (including multi-line data by identifying sources of bias and representations, including symbols
problem using appropriate
graphs), histograms, stem and leaf plots, box and whisker plots, and
compare and contrast the
and informal and formal
scatterplots).
effectiveness of different
mathematical language.
representations of data.
Connections M08-S2C1-01
Solve problems by selecting, constructing, interpreting, and calculating with displays of data including box and whisker plots and scatterplots. SC08-S1C3-04
Strand 2:
Formulate a future investigation based on the
Concept 1:
data collected.
Data
SC08-S1C3-05
Analysis
Explain how evidence supports the validity and reliability of a conclusion.
This concept
was introduced SC08-S2C2-04 in Grade 7,
Explain why scientific claims may be Quarter 4 and it
questionable if based on very small samples of data, biased samples, or samples for which
will be there was no control. mastered in
Grade 8. SS08-S1C1-02 Interpret historical data displayed in graphs,
tables, and charts.
SS08-S1C1-06 Determine the credibility and bias of primary and secondary sources.
SS08-S2C1-02 Interpret historical data displayed in graphs, tables, and charts.
SS08-S2C1-06 Determine the credibility and bias of primary and secondary.
SS08-S4C1-03 Interpret maps, charts, and geographic databases using geographic information.
8 th Grade Blueprint
Assessed Quarter 1
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
Strand 2: PO 5. Evaluate the design of an
M08-S5C2-07. Isolate and
Students evaluate an experiment to determine if the design meets the intended
Concept 1: experiment.
organize mathematical information
purpose, is free of bias, and utilizes an appropriate sample.
Data
taken from symbols, diagrams, and
Analysis
Connections
graphs to make inferences, draw
Example:
SC08-S1C2-02
conclusions, and justify reasoning.
Students design an experiment to determine if there is a correlation
This concept
Design a controlled investigation to support or
between shoe size and height. All designs are evaluated to test for the
was introduced
reject a hypothesis. characteristics above (i.e., intended purpose, free of bias, and appropriate
in Grade 7,
sample size).
Quarter 4 and it will be
mastered in
Grade 8.
PO 1. Use directed graphs to solve
M08-S5C2-01. Analyze a problem
Example:
problems.
situation to determine the question(s) to be answered.
Four players (Dom, Nathan, Ryan, & Zachary) are playing in a round-
Connections
robin tennis tournament, where every player plays every other player.
(none)
M08-S5C2-04. Represent a
Strand 2:
problem situation using multiple
Dom beats Nathan and Ryan,
Concept 4:
representations, describe the
Nathan beats Zachary,
Vertex-Edge
process used to solve the problem,
Ryan beats Nathan and Zachary, and
Graphs
and verify the reasonableness of
Zachary beats Dom.
the solution.
This concept
o Represent this round-robin tournament using a directed graph.
was introduced
o How many matches are played in a round-robin tournament with
in Grade 7,
four players? Systematically list all the matches. Explain your
Quarter 4 and it
answer.
will be
o Find all Hamilton paths in this graph.
mastered in
Grade 8. o “A winner” can be defined as the first player in a Hamilton path. How many possible tournament “winners” are in this example?
What conclusions can you draw from this example?
Solution: (Continued on next page)
8 th Grade Blueprint
Assessed Quarter 1
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
(PO 1 continued)
Dom
Nathan
Strand 2: Concept 4: Vertex-Edge Graphs
This concept was introduced in Grade 7,
Zachary
Ryan
Quarter 4 and it will be
o There are six matches played in a round-robin tournament with four
mastered in
players. These “matches” are represented by each edge in the graph Grade 8. above. One possible systematic list is below:
MATCH #1 – Dom plays Nathan MATCH #2 – Dom plays Ryan MATCH #3 – Dom plays Zachary MATCH #4 – Nathan plays Ryan MATCH #5 – Nathan plays Zachary MATCH #6 – Ryan plays Zachary
8 th Grade Blueprint
Assessed Quarter 1
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
(PO 1 continued) o Following the edges in the direction of the arrows, one can find a Hamilton path that starts with Nathan to Zachary to Dom to Ryan. Thus we can say that “Nathan” is a winner!
Another Hamilton path can start with Ryan to Nathan to Zachary to Dom (or Ryan to Zachary to Dom to Nathan). In both such cases, we can call “Ryan” a winner!
Strand 2:
A third type of Hamilton path can start with Dom to Ryan to Nathan to
Concept 4:
Zachary, so we can call “Dom” a winner!
Vertex-Edge Graphs
And finally, the last type of Hamilton path can start with Zachary to Dom to Ryan to Nathan; we can call “Zachary” a winner! Therefore, in this
This concept
tournament, we can have four different tournament winners!
was introduced in Grade 7,
o How do we decide who is the tournament winner? On the basis of the
Quarter 4 and it
Hamilton paths, there is no clear winner in this tournament. In one
will be mastered in
Hamilton path, Nathan wins, in another Hamilton path, Ryan wins; Grade 8. another Dom wins and yet another Zachary wins! Who is the overall
winner? Unfortunately, there is no clear winner -- the ranking of these players is ambiguous. Students should enjoy deciding who should be ranked first and why that player should be ranked first! For tournament situations that can be modeled where one Hamilton path exists in the graph, the ranking is unambiguous.
th
8 Grade Blueprint Assessed Quarter 2
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 1. Recognize, describe, create, and
M08-S5C2-07. Isolate and
Given an equation, students should create a table, graph the points on a
analyze numerical and geometric
organize mathematical information
coordinate grid, and describe the sequence.
sequences using tables, graphs,
taken from symbols, diagrams, and
Example:
words, or symbols; make conjectures
graphs to make inferences, draw
Given a sequence such as 1, 4, 9, 16 … students need to create a table, graph the points on a coordinate grid, and describe algebraically the rule.
about these sequences.
conclusions, and justify reasoning.
Connections Note the different representations of a sequence of blocks below: M08-S3C2-02
o Graphical:
Determine if a relationship represented by a graph or table is a function.
M08-S3C2-03 Write the rule for a simple function using algebraic notation.
M08-S3C2-05 Demonstrate that proportional relationships are
Strand 3:
linear using equations, graphs, or tables.
Concept 1:
o Table:
Patterns
Step
Number Blocks
2n-1
Written description: Begin with a square, add 2 squares on each step. o Physical Models:
Equation: y = 2n – 1
8 th Grade Blueprint Assessed Quarter 2
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 1. Sketch and interpret a graph that M08-S5C2-07. Isolate and Use graphs of experiences common to students. Students are expected to both
models a given context; describe a
sketch and interpret graphs. They are also expected to explain the context that is modeled by a given
organize mathematical information
taken from symbols, diagrams, and relationship between two quantities and their impact on each other. graph.
graphs to make inferences, draw conclusions, and justify reasoning.
Key Vocabulary:
8.F.5. Describe qualitatively the
Increasing relationship
functional relationship between two
M08-S5C2-05. Apply a previously
Decreasing relationship
quantities by analyzing a graph (e.g.
used problem-solving strategy in a
Linear
where the function is increasing or
new context.
Nonlinear
decreasing, linear or nonlinear).
Sketch a graph that exhibits the
Example:
qualitative features of a function that Strand 3:
Sketch a graph of someone riding a bike to school that starts at home,
Concept 2: has been described verbally.
travels two blocks at a constant speed, travels one block up a hill at a
Functions
decreasing speed, then travels one block at a constant speed to reach
and Connections
school.
M08-S3C2-04
Relation-
Identify functions as linear or nonlinear and
ships
Students describe the relationship between two quantities:
contrast distinguishing properties of functions
using equations, graphs, or tables.
“As ________ increases, ___________ decreases.” “As ________ increases, ___________ increases.”
M08-S3C2-05
“As ________ decreases, __________ decreases.”
Demonstrate that proportional relationships are linear using equations, graphs, or tables.
M08-S3C3-01
Write or identify algebraic expressions, or inequalities that represent a situation.
M08-S3C3-04 Translate between different representations of linear equations using symbols, graphs, tables, or written descriptions.
8 th Grade Blueprint Assessed Quarter 2
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 2. Determine if a relationship
M08-S5C2-02. Analyze and
Students justify their reasoning about why a graph or table is a function, or why a
represented by a graph or table is a
compare mathematical strategies
graph or table is not a function. Students use strategies such as graphing the
function.
for efficient problem solving; select
ordered pairs from a table, applying the vertical line test, or analyzing the
and use one or more strategies to
patterns in a table to determine if each value of the independent variable has a
8.F.1. Understand that a function is a
solve a problem.
unique value for the dependent variable.
rule that assigns to each input exactly Strand 3:
one output. The graph of a function is
M08-S5C2-07. Isolate and
Concept 2: the set of ordered pairs consisting of
organize mathematical information
Functions
an input and the corresponding output. taken from symbols, diagrams, and
and
graphs to make inferences, draw
Relation-
Connections
conclusions, and justify reasoning.
ships
M08-S3C1-01 Recognize, describe, create, and analyze
numerical and geometric sequences using tables, graphs, words, or symbols; make conjectures about these sequences.
M08-S3C2 –05 Demonstrate that proportional relationships are linear using equations, graphs, or tables.
8 th Grade Blueprint Assessed Quarter 2
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 3. Write the rule for a simple Students are expected to write the rule for a simple function from both a function using algebraic notation.
table and from a graph. All representations should be tied to context, and students should be able to explain the rate of change and the initial value
8.F.4. Construct a function to model a
of the function in terms of the contextual situation.
linear relationship between two quantities. Determine the rate of
Key Vocabulary:
change and initial value of the function
rate of change
from a description of a relationship or
initial value
from two (x, y) values, including reading these from a table or from a
Example:
graph. Interpret the rate of change and
Write a rule for the function illustrated by the table of values below.
initial value of a linear function in
terms of the situation it models, and in
X 2 3 5 8 12
terms of its graph or a table of values.
M08-S3C1-01
Concept 2:
Recognize, describe, create, and analyze
Functions
numerical and geometric sequences using
and
tables, graphs, words, or symbols; make
Relation-
conjectures about these sequences.
ships
M08-S3C2 –05 Demonstrate that proportional relationships are linear using equations, graphs, or tables.
PO 4. Identify functions as linear or
M08-S5C2-03. Identify relevant,
Properties of functions include increasing, decreasing, and constant growth and
nonlinear and contrast distinguishing
missing, and extraneous
minimum and maximum values.
properties of functions using
information related to the solution
equations, graphs, or tables.
to a problem.
Students use strategies to determine linearity such as creating a table and graph from an equation or looking for patterns in equations and tables.
8.F.3. Interpret the equation y = mx + b
M08-S5C2-12. Make, validate, and
as defining a linear function, whose
Students must give examples of functions that are not linear. For instance, graph is a straight line; give examples
justify conclusions and
the function A = s 2 giving the area of a square as a function of its side of functions that are not linear.
generalizations about linear
relationships.
length is not linear because its graph contains the points (1, 1), (2, 4), and (3, 9), which are not on a straight line.
Connections M08-S3C2-01
Sketch and interpret a graph that models a given context; describe a context that is
modeled by a given graph.
8 th Grade Blueprint Assessed Quarter 2
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 5. Demonstrate that proportional
M08-S5C2-08. Describe when to
Students model direct and indirect variation.
relationships are linear using
use proportional reasoning to solve
equations, graphs, or tables. a problem.
Example:
Graph and/or make a table of these equations:
Connections
M08-S1C2-03
M08-S5C2-12. Make, validate, and
o y = 2x
Solve problems involving percent increase, percent
justify conclusions and decrease, and simple interest rates. generalizations about linear
M08-S3C1-01
relationships.
o y=
Recognize, describe, create, and analyze numerical
and geometric sequences using tables, graphs,
words, or symbols; make conjectures about these sequences.
y=
M08-S3C2-01
Sketch and interpret a graph that models a given context; describe a context that is modeled by a given graph.
M08-S3C2-02
Strand 3:
Determine if a relationship represented by a graph or
Concept 2:
table is a function.
Functions M08-S3C2-03
and
Write the rule for a simple function using algebraic
M08-S3C3-03 Analyze situations, simplify, and solve problems involving linear equations and inequalities using the properties of the real number system.
M08-S3C3-04 Translate between different representations of linear equations using symbols, graphs, tables or written descriptions.
M08-S3C4-01 Interpret the relationship between a linear equation and its graph, identifying and computing slope and intercepts. M08-S3C4-02
Solve problems involving simple rates.
M08-S5C1-01 Create an algorithm to solve problems involving indirect measurement, using proportional reasoning, dimensional analysis, and the concepts of density and rate.
8 th Grade Blueprint Assessed Quarter 2
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 1. Write or identify algebraic
M08-S5C2-04. Represent a
Example:
expressions, equations, or inequalities
problem situation using multiple
Florencia has at most $60 to spend on clothes. She wants to buy a pair
that represent a situation.
representations, describe the
of jeans for $22 dollars and spend the rest on t-shirts. Each t-shirt costs
process used to solve the problem,
$8. Write an inequality for the number of t-shirts she can purchase.
Connections
and verify the reasonableness of
M08-S3C2-01
the solution.
Sketch and interpret a graph that models a
Strand 3:
given context; describe a context that is
Concept 3:
modeled by a given graph.
Algebraic PO 2. Evaluate an expression
Any rational number (whole numbers, integers, fractions, and decimals) can be
Represen- containing variables by substituting
used as the value for a variable.
tations rational numbers for the variables.
Example:
Connections M08-S1C3-01
b – 4ac , where b=2 , a=
and c = –4
Make estimates appropriate to a given
situation. 2
8 th Grade Blueprint
Assessed Quarter 2
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 3. Analyze situations, simplify, and
M08-S5C2-02. Analyze and
The properties of real numbers and properties of equality include but are not
solve problems involving linear
compare mathematical strategies
limited to the following: associative, commutative, distributive, identity, zero,
equations and inequalities using the
for efficient problem solving; select
reflexive, and transitive. The property of closure is not expected at this grade
properties of the real number system.
and use one or more strategies to
level.
solve a problem.
8.EE.7.b. Solve linear equations with
Example:
Strand 3: rational number coefficients, including
Steven saved $25 dollars. He spent $10.81, including tax, to buy a new
Concept 3:
equations whose solutions require
DVD. He needs to set aside $10.00 to pay for his lunch next week. If
Algebraic expanding expressions using the
peanuts cost $0.38 per package including tax, what is the maximum
Represen- distributive property and collecting like
number of packages that Steven can buy?
tations terms.
Write an equation or inequality to model the situation. Explain how you
Connections
determined whether to write an equation or inequality and the properties of the
M08-S3C2 –05
real number system that you use to find a solution.
Demonstrate that proportional relationships are
linear using equations, graphs, or tables.
8 th Grade Blueprint
Assessed Quarter 2
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 4. Translate between different
M08-S5C2-04. Represent a
Example:
representations of linear equations
problem situation using multiple
Given one representation, students create any of the other
using symbols, graphs, tables, or
representations, describe the
representations that show the same relationship. Representations of
written descriptions.
process used to solve the problem,
linear equations include tables, graphs, equations, or written
and verify the reasonableness of
descriptions.
Connections
the solution.
M08-S3C2-01
o Equation: y = 4x + 1
Sketch and interpret a graph that models a given context; describe a context that is
modeled by a given graph.
o Written description: Susan started with $1 in her savings. She plans to add $4 per week to her savings.
M08-S3C2 –05 Demonstrate that proportional relationships are
o Table:
linear using equations, graphs, or tables. x
Strand 3: Concept 3:
Algebraic
Represen- tations
o Graph
8 th Grade Blueprint
Assessed Quarter 2
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 5. Graph an inequality on a number
Example:
line.
Graph x ≤ 4.
PO 1. Interpret the relationship
Students use their understanding from 3.2.2, 3.2.5, 3.3.1 and 3.3.4, to derive between a linear equation and its
M08-S5C2-07. Isolate and
organize mathematical information
the formula for a linear equation. Emphasis should be placed on solving
graph, identifying and computing slope taken from symbols, diagrams, and problems in context, such as scatter plots of bivariate data. and intercepts.
graphs to make inferences, draw conclusions, and justify reasoning.
Students determine the slope, x- and y- intercepts given an equation in slope intercept form. Students graph an equation given in slope intercept form.
8.EE.5. Graph proportional
M08-S5C2-12. Make, validate, and
relationships, interpreting the unit rate
Students should understand the slope in terms of unit rate and its as the slope of the graph.
justify conclusions and
generalizations about linear
connections to triangle similarity.
relationships.
8.EE.6. Uses similar triangles to
Key Vocabulary:
explain why the slope m is the same
Unit rate
Strand 3:
between any two distinct points on a
Rate of change
Concept 4:
non-vertical line in the coordinate Analysis of Initial value plane; derive the equation y = mx from
Bivariate data
Change
a line through the original and the equation y = mx + b for a line
Examples:
intercepting the vertical axis at b. In a linear model for a biology experiment, interpret a slope of 15 cm/her as meaning that an additional hour of sunlight each day is
8.SP.3. Use the equation of a linear associated with an additional 15 cm in mature plant height. model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
Connections M08-S3C2 –05
Demonstrate that proportional relationships are linear using equations, graphs, or tables.
8 th Grade Blueprint
Assessed Quarter 2
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 2. Solve problems involving simple M08-S5C2-08. Describe when to Simple rates include interest, distance, and percent change.
rates. use proportional reasoning to solve
a problem. Emphasize how the unit rate can help to make sense of problems. 8.EE.5. Graph proportional relationships, interpreting the unit rate
Key Vocabulary:
as the slope of the graph.
Rate of change Initial value
M08-S1C2-03
Examples:
Analysis of Solve problems involving percent increase,
Mark deposits $120 into a savings account that earns 4% interest
percent decrease, and simple interest rates.
Change
annually. The interest does not compound. How much interest will Mark M08-S1C3-01
earn after 2 years?
Make estimates appropriate to a given
Linda traveled 110 miles in 2 hours. If her speed remains constant, how
situation.
many miles can she expect to travel in 4.5 hours?
At the end of the first quarter, Robin’s overall grade percentage was 74%.
M08-S3C2 –05
Demonstrate that proportional relationships are
At the end of the second quarter her grade percentage was 88%. linear using equations, graphs, or tables. Calculate the percent change in her grade from first and second quarter.
8 th Grade Blueprint
Assessed Quarter 3
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 1. Determine theoretical and
Conditional probability is limited to situations with and without replacement.
experimental conditional probabilities
in compound probability experiments.
Connections (none)
PO 2. Interpret probabilities within a
M08-S5C2-07. Isolate and
Students predict the outcomes of an experiment with and without replacement by
given context and compare the
organize mathematical information
calculating the theoretical probability. They compare the results of the
outcome of an experiment to
taken from symbols, diagrams, and experiment to their predictions.
predictions made prior to performing
graphs to make inferences, draw
the experiment.
conclusions, and justify reasoning.
Example:
Tyrone takes two coins at random from his pocket, choosing one and
Connections
setting it aside before choosing the other. Tyrone has 2 quarters, 6 (none) dimes, and 3 nickels in his pocket. Make a prediction based upon the
Strand 2:
theoretical probability that he chooses a quarter followed by a dime. Try
Concept 2:
Tyrone’s experiment by performing 50 trials. What is the experimental
Probability
probability of drawing a quarter followed by a dime? How does the experimental probability compare to your prediction (theoretical probability)?
PO 3. Use all possible outcomes
Independent events are two events in which the outcome of the second event is
(sample space) to determine the
not affected by the outcome of the first event (e.g., rolling two number cubes,
probability of dependent and
tossing two coins, rolling a number cube and spinning a spinner). Dependent
independent events.
events are two events such that the likelihood of the outcome of the second event is affected by the outcome of the first event (e.g., bag pull without
Connections
replacement, drawing a card from a stack without replacement, two cars parking
M08-S2C3-01
in a parking lot).
Represent, analyze, and solve counting
problems with or without ordering and
repetitions.
* Strand 5, Concept 1 (Algorithms and Algorithmic Thinking) and Strand 5, Concept 2 (Logic, Reasoning, Problem
Solving, and Proof) will be assessed Third Quarter.
See the Process Integration column in Quarters 1-3 for these Strand 5 concepts.
8 th Grade Blueprint
Assessed Quarter 3
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 1. Represent, analyze, and solve
M08-S5C2-04. Represent a
By the end of Grade 8, students are able to solve a variety of counting problems
counting problems with or without
problem situation using multiple
using both visual and numerical representations. They should have had varied
ordering and repetitions.
representations, describe the
counting experiences that, over time, have helped to build these understandings.
process used to solve the problem,
Initially, they begin by randomly generating all possibilities and then they begin to
Connections
and verify the reasonableness of
organize their thinking through visual representations such as charts, systematic
M08-S2C2-03
the solution.
listing, and tree diagrams. Finally, they are able to make connections from these
Use all possible outcomes (sample space) to
visual representations to build numeric solutions.
determine the probability of dependent and independent events.
Through this process of connecting numeric representations with visual representations, even if they cannot be completely drawn but rather are mentally
Strand 2:
visualized, students are now able to solve a variety of counting problems
Passwords are often a sequence of letters and numbers. A 6-character password is composed of 4 digits and 2 letters.
o If no repetition of letters is allowed, how many passwords are there? o If no repeating letters or digits are allowed, how many passwords are there? o If repeating both letters and digits are allowed, how many passwords are allowed?
8 th Grade Blueprint
Assessed Quarter 3
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
(PO 1 continued)
Solution:
Students should be able to represent the general counting problem as:
---- ---- ---- ---- ---- ---- digit digit digit digit letter letter
and mentally visualize a tree diagram which, from some starting vertex, that spans either ten edges (if the initial position is a digit) or twenty-six edges (if the initial position contains a letter) and where each branch of the tree diagram has six levels that represent the next possible options for that position. Their visualization of this problem should convince students that the solution will involve many possibilities, that actually drawing the tree diagram will be hard work, and thus motivate them to find a numerical way to count all possibilities.
Strand 2:
Concept 3:
o If no repetition of letters is allowed, students should count the number
Systematic
of possible passwords as 10 x 10 x 10 x 10 x 26 x 25 (or some
Listing
equivalent arrangement of this multiplication problem, for example, 26 x
and
25 x 10 x 10 x 10 x 10).
Counting
o If no repeating letters or digits are allowed, students should count the number of possible passwords as 10 x 9 x 8 x 7 x 26 x 25. o If repeating letters and digits are allowed, students should count the number of possible passwords as
10 x 10 x 10 x 10 x 26 x 26.
8 th Grade Blueprint
Assessed Quarter 3
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 2. Solve counting problems and
M08-S5C2-04. Represent a
Example:
Strand 2: represent counting principles
problem situation using multiple
Concept 3: algebraically including factorial
representations, describe the
Five athletes are entered in a race, and five places are awarded
Systematic notation.
process used to solve the problem,
ribbons. In how many different possible ways might they finish?
Listing
and verify the reasonableness of
and
Connections
the solution
Solution: W= 5! Or 5 x 4 x 3 x 2 x 1
Counting
M08-S1C3-01 Make estimates appropriate to a given
situation.
8 th Grade Blueprint
Assessed Quarter 3
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 1. Identify the attributes of circles:
M08-S5C2-11. Identify simple valid Example:
radius, diameter, chords, tangents,
arguments using if… then
Students will draw a circle and identify and label attributes or identify
secants, inscribed angles, central
statements.
attributes from a diagram.
angles, intercepted arcs, circumference, and area.
Connections (none)
Strand 4: Concept 1: Geometric Properties
PO 2. Predict results of combining,
M08-S5C2-09. Make and test
Students need multiple opportunities to engage in activities such as paper
subdividing, and changing shapes of
conjectures based on information
folding, tiling, rearranging cut up pieces, modeling cross sections of solids, and
plane figures and solids.
collected from explorations and
constructing Frieze patterns and tessellations to accurately predict and describe
experiments.
the results of combining and subdividing two- and three-dimensional figures.
Connections M08-S1C3-01 Make estimates appropriate to a given situation.
M08-S4C2-02 Describe the transformations that create a given tessellation.
th
8 Grade Blueprint
Assessed Quarter 3
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 3. Use proportional reasoning to
Proportional reasoning includes consideration of conservation of angle and determine congruence and similarity of use proportional reasoning to solve proportionality of side length.
M08-S5C2-08. Describe when to
triangles.
a problem.
Example:
The triangles shown in the figure are similar. Find the length of the sides
Connections
M08-S5C2-13. Verify the
M08-S4C4-02
Pythagorean Theorem using a valid
labeled x and y.
Solve geometric problems using ratios and
3 y 8 3 x 10
2 1 y 8 2 x 10 3
Slope triangles are an application of this concept.
8 th Grade Blueprint
Assessed Quarter 3
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 4. Use the Pythagorean Theorem to M08-S5C2-02. Analyze and Students should be familiar with the common Pythagorean triples.
solve problems.
compare mathematical strategies for efficient problem solving; select
Students should be able to explain a proof of the Pythagorean Theorem
and its converse. Converse: If the sum of the square of two sides is equal Pythagorean Theorem and its
8.G.6. Explain a proof of the
and use one or more strategies to
to the square of the third side of a triangle, then the triangle is a right converse.
solve a problem.
triangle.
M08-S5C2-06. Communicate the
8.G.7. Apply the Pythagorean Theorem answer(s) to the question(s) in a
Examples:
Strand 4: to determine unknown side lengths in
problem using appropriate
Is a triangle with side lengths 5 cm, 12 cm, and 13 cm a right triangle?
Concept 1: right triangles in real-world and
representations, including symbols
Why or why not?
Geometric mathematical problems in two-
and informal and formal
Determine the length of the diagonal of a rectangle that is 7 ft by 10 ft.
Properties dimensions.
mathematical language.
Connections M08-S4C3-02 Use the Pythagorean Theorem to find the distance between two points in the coordinate plane.
M08-S5C2-13 Verify the Pythagorean Theorem using a valid argument.
8 th Grade Blueprint
Assessed Quarter 3
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 1. Model the result of rotations in
M08-S5C2-02. Analyze and
Figures may be rotated with the origin at the center or another point on the figure
multiples of 45 degrees of a 2-
compare mathematical strategies
or using the origin as the point of rotation where the figure does not contain the
dimensional figure about the origin.
for efficient problem solving; select
origin.
and use one or more strategies to
Connections
solve a problem.
(none) M08-S5C2-05. Apply a previously used problem-solving strategy in a
new context.
Strand 4: Concept 2: Transforma tion of
Shapes
8 th Grade Blueprint
Assessed Quarter 3
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 2. Describe the transformations
Students will look at a tessellation or Frieze pattern. They will identify the original
that create a given tessellation.
figure and the transformation(s) used to create the tessellation or Frieze pattern.
Connections
Example:
M08-S4C1-P02 Predict results of combining, subdividing, and Look at the pattern below. What figure was used to create the pattern? What transformation(s) did the figure undergo?
changing shapes of plane figures and solids.
Strand 4: Concept 2:
Transforma
tion of
Shapes
PO 3. Identify lines of symmetry in
Students are expected to classify figures by symmetry including rotational
plane figures or classify types of
symmetry and reflection symmetry and differentiate between them.
symmetries of 2-dimensional figures.
Connections (none)
8 th Grade Blueprint
Assessed Quarter 3
Explanation, Examples, and Resources Concept
Strand Performance Objectives
Process Integration
Strand 5
PO 1. Make and test a conjecture
M08-S5C2-09. Make and test
Students are expected to find the midpoint between any two points including
about how to find the midpoint
conjectures based on information
points that are not horizontal or vertical from each other as shown in the model
between any two points in the
collected from explorations and
below. Students should not be given the formula, but rather create a formula or
coordinate plane.
experiments.
process with which to find the midpoint. Students test their conjecture and the conjecture of others to determine their validity. Students can then compare their
Connections
conjectures to the formula or to the graphical algorithm for finding the midpoint of
M08-S1C3-01
a line segment.
Make estimates appropriate to a given
situation.
Strand 4: Concept 3:
Coordinate Geometry