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