Grade 8 Math Curriculum Notebook

Assessment Tasks Used
Skill-based Task:
Identify and name sets of angles of parallel lines cut by a transversal and tell which are congruent
Problem Task:
The streets of 400 E and 900E run north and south. Euclid Drive cuts both streets at an angle from SE to NW. Pythagoras Way passes through all three streets SW to
NE. Are all possible triangles created by the intersection of the streets similar? Justify.
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Geometry 8G6
Core Content
Cluster Title: Understand and apply the Pythagorean Theorem
Standard 6. Explain a proof of the Pythagorean Theorem and its converse MASTERY Patterns of Reasoning:
Conceptual:
• Understand a proof of the Pythagorean Theorem
• Understand a proof of the converse of the Pythagorean Theorem
• Know that in a right triangle, a2 + b2 = c2 (the Pythagorean Theorem)
Procedural:
• Explain a proof of the Pythagorean Theorem
• Explain a proof of the converse of the Pythagorean Theorem Representational:
• Model a right triangle with a2 + b2 = c2 graphically Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand the relationship between a and a2, b and b2 and c and c2
• Understand the relationship between squares and square roots Procedural:
• Square a number Representational:
• Model the squaring of a number with manipulatives
Academic Vocabulary and Notation
Right triangle, leg, hypotenuse, square, Pythagorean Theorem, exponential notation
Instructional Strategies Used
• Consider introducing this with a application regarding distance
• Explore various proofs of the Pythagorean Theorem and discuss the logic behind each
Assessment Tasks Used
Skill-based Task:
Resources Used
Problem Task:
Explainthelogicalreasoningbehindaproofof Investigatethehistoricalcontextofoneofthe the Pythagorean Theorem proofs of the Pythagorean Theorem and present the proof in context to the class
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Geometry 8G7
Core Content
Cluster Title: Understand and apply the Pythagorean Theorem
Standard 7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions MASTERY Patterns of Reasoning:
Conceptual:
• Know the Pythagorean Theorem Procedural:
• Use the Pythagorean Theorem to solve for a missing side of a right triangle given the other two sides
• Use the Pythagorean Theorem to solve problems in real-world contexts, including three- dimensional contexts.
Representational:
• Use manipulatives to represent the Pythagorean Theorem to find missing sides of a right triangle
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know approximate values of irrational numbers Procedural:
• Solve an equation using squares and square roots
• Use rational approximations of irrational numbers to express answers Representational:
• Represent approximate values of irrational numbers on a number line
Academic Vocabulary and Notation
Right triangle, leg, hypotenuse, square, square root, Pythagorean Theorem, exponential notation, square root symbol
Instructional Strategies Used
• Find and solve right triangles in career situations such as construction
Assessment Tasks Used
Skill-based Task:
If the height of a cone is 10 meters and the radius is 6 meters, what is the slant height?
Problem Task:
TV’s are measured along their diagonal to find their dimension. How does a 52 inch HD (wide screen) TV compare to a traditional 52 in (full screen) TV?
Resources Used
53


Geometry 8G8
Core Content
Cluster Title: Understand and apply the Pythagorean Theorem
Standard 8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system
MASTERY Patterns of Reasoning:
Conceptual:
• Recognize the right triangle associated with finding the distance between two points on a coordinate system
Procedural:
• Calculate the distance between two points on a coordinate system using the Pythagorean Theorem
Representational:
• Graphically represent the right triangle associated with the distance between two points on graph paper
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know the Pythagorean Theorem Procedural:
• Use the Pythagorean Theorem to find the hypotenuse of a right triangle Representational:
• Represent the location of two points on graph paper
Academic Vocabulary and Notation
Right triangle, leg, hypotenuse, square, square root, Pythagorean Theorem, exponential notation, square root symbol
Instructional Strategies Used
• Overlap a map with a coordinate grid and use the Pythagorean Theorem to find the distance between two locations
• Investigate the relationship between the Pythagorean Theorem and the distance formula
• Use the Pythagorean Theorem to explore and categorize triangles and quadrilaterals on a coordinate system
Resources Used
54


Assessment Tasks Used
Skill-based Task:
Using the Pythagorean Theorem, find the distance between (4, 2) and (7, 10)
Problem Task:
List 3 coordinate pairs that are 5 units away from the origin in the first quadrant. Describe how to find the points and justify your reasoning. (Note: points on the axes are not in the quadrant)
55


Geometry 8G9
Core Content
e re Standard 9. Know the formulas for the volumes of cones, cylinders, and spheres and use them
to solve real-world and mathematical problems
MASTERY Patterns of Reasoning:
Conceptual:
• Know the formulas for the volumes of cones, cylinders, and spheres Procedural:
• Use the formulas for volume to find the volumes of cones, cylinders, and spheres Representational:
• Use manipulatives to represent the volumes of cones and cylinders Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know what pi is and how to derive it
• Understand that volume is measured in cubic units
• Understand exponential notation for squares and cubes
Procedural:
• Solve equations involving square roots and cube roots Representational:
• Represent rational approximations of irrational numbers such as pi Academic Vocabulary and Notation
Pi, the symbol for pi, radius, slant height, volume, hemisphere, diameter
Cluster Title: Solve real-world and mathematical problems involving volume of cylinders, con
s, and sphe
Instructional Strategies Used
• Use physical models to compare volumes of cones and cylinders using water or rice
• Derive the formula for cylinders using physical models
• Explore questions such as “why is the volume of a cone 1/3 the volume of a cylinder of the same base?”
• Compare and contrast the formulas for cones, cylinders, and spheres
Resources Used
56
s


Statistics and Probability 8SP1
Core Content
Cluster Title: Investigate patterns of association in bivariate data
Standard 1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association
MASTERY Patterns of Reasoning:
Conceptual:
• Understand clustering patterns of positive or negative association, linear association, and nonlinear association
• Know what outliers are Procedural:
• Collect, record, and construct a set of bivariate data using a scatter plot
• Interpret patterns on a scatter plot such as clustering, outliers, and positive, negative or
not association
Representational:
• Graphically represent the values of a bivariate data set with a scatter plot Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand graphing of linear values and points
• Understand the meaning of linear and nonlinear Procedural:
• Graph points on a coordinate system Representational:
• Represent linear relationships graphically
Academic Vocabulary and Notation
Bivariate data, scatter plot, outlier, clustering, positive association, negative association, linear, nonlinear
Instructional Strategies Used
• Use data from multiple sources to construct a scatter plot
• Compare and contrast scatter plots with various degrees of association
Resources Used
http://www.learnzillion.com/lessons/363-describe- patterns-in-scatterplots
http://www.learnzillion.com/lessons/364-fit-a-linear- function-to-a-scatterplot
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Assessment Tasks Used
Skill-based Task:
Construct a scatter plot and describe any association you observe
Problem Task:
Compare class test scores to hours of television watched
• Predict whether there is a positive, negative or no association
• Collect data and make a scatter plot
• Compare your prediction to the scatter plot
result
• Describe any association you observe
• Interpret your findings and explain your
reasoning
Height
Hand span
70 in 72 in 61 in 62 in 68 in
10 in 9.5 in 8in 9.5 in 9in
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Statistics and Probability 8SP2
Core Content
Cluster Title: Investigate patterns of association in bivariate data
Standard 2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line
MASTERY Patterns of Reasoning:
Conceptual:
• Know that straight lines are widely used to model relationships between two quantitative variables
Procedural:
• Judge how well an informal trend line fits the data by looking at the closeness of the data points
Representational:
• Informally fit a straight line to represent the trend of data points Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand graphing of linear values and points
• Understand the meaning of linear and nonlinear Procedural:
• Plot linear points Representational:
• Represent linear relationships graphically Academic Vocabulary and Notation
Linear association, scatter plot, trend line
Instructional Strategies Used
• Given a scatter plot and an uncooked spaghetti noodle, have students place the noodle on the scatter plot to create a trend line
• Use technology to create scatter plots with a trend line, and then observe changes to the trend line as data points are deleted or added
Resources Used
www.gapminder.org
gapminder foundation
www.nlvm.usu.edu
Utah State University National Library of Virtual Manipulatives
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Assessment Tasks Used
Skill-based Task:
Draw a trend line and describe the closeness of the fit for the following scatter plot
Problem Task:
Which line is the best fit for the data? Justify your answer
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Statistics and Probability 8SP3
Core Content
Cluster Title: Investigate patterns of association in bivariate data
Standard 3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height
MASTERY Patterns of Reasoning:
Conceptual:
• Know that a linear model can be used to explain bivariate relationships Procedural:
• Use the equation of a linear model to solve problems in the context of bivariate measurement data
• Interpret the meaning of the slope as a rate of change and the meaning of the y- intercept in context given bivariate data
Representational:
• Model problems of bivariate data with graphs to explain the context Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand graphing of linear values and points
• Understand the meaning of linear and nonlinear
Procedural:
• Plot linear points Representational:
• Represent linear relationships by graphing the y-intercept and using the slope
Academic Vocabulary and Notation
Rate of change, slope, intercept
Instructional Strategies Used
• Find linear models of approximately linear data in newspapers, magazines, or on the internet and discuss the meaning of the slope and intercepts
• Create linear models using data from other disciplines such as science, social studies, or careers and describe the meaning of their slopes and intercepts
Resources Used
http://www.learnzillion.com/lessons/237- compare-linear-and-nonlinear-functions
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Assessment Tasks Used
Skill-based Task: Problem Task:
Find and interpret the slope and y-intercept of the Create a story problem that uses a line with trend line. Create an equation and use it to predict a slope of 2/5 and y-intercept of
how much a tree will grow in three years 3. Describe the meaning of the slope and
y-intercept in the context of the problem.
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Assessment Tasks Used
Skill-based Task: Problem Task:
A silo has 1500 ft3 of grain. The grain fills the What does the height of the cone need to be so silo to 20 fit in height. What is the radius of thethat one spherical scoop of ice cream with the
silo?
What is the relationship between the volume of a cylinder and a cone with the same radius and height?
same radius as the cone won’t overflow if it melts?
A Christmas tree is 7 feet tall with a 5 foot diameter at the base, with one foot between the floor and the lowest branch. How far up the tree should your first two strings of lights end so that you will have enough lights to evenly fill the Christmas tree?
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Statistics and Probability 8SP4
Core Content
Cluster Title: Investigate patterns of association in bivariate data
Standard 4. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? MASTERY Patterns of Reasoning:
Conceptual:
• Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table
Procedural:
• Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects
• Interpret and describe relative frequencies for possible associations from a two-way table
Representational:
• Model the relationships of bivariate categorical data with graphs and tables Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand data recorded in tables with columns and rows Procedural:
• Read and interpret data recorded in rows and columns Representational:
Academic Vocabulary and Notation
Relative frequency, categorical data, frequency, two-way table, associations
Instructional Strategies Used
• Explore questions such as:
• o Are honor students more likely to
wear athletic shoes?
• o Is gender related to video console
ownership?
• o Are eighth graders more or less likely
to have a cell phone based on birth order (youngest, middle, oldest)?
Resources Used
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Assessment Tasks Used
Skill-based Task:
Are boys or girls more likely to be in band?
Girls
Boys Total
Problem Task:
Construct a two-way table to display data from two or more categories. Explain why you believe there is or is not an association between the two variables.
Gender
Band
No Band
Total
10
7
17
9
2
11
19
9
28
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Pacing Guide
(Note, Bold text indicates an “essential” standard)
Term 1
Equations and inequalities
Describe and analyze scatterplots
Informally fit a trendline
Using a trendline to predict and interpret data
Identify linear/nonlinear
Find slope/graph lines
Write y=mx+b equations from table/graph; interpret slope Similar triangles (slope)/derive y=mx+b and y=msxx
8EE7 8SP1 8SP2 8SP3 8F3 8EE5 8F4 8EE6
Term 2
Definition of function
Switch between representations of functions
Story of function (increasing/decreasing)
Convert rationals/know irrational
Use square roots
Exponent rules
Compare and order numbers/place on number line Solve x2 = 9/ know root 2 is irrational
8F1
8F2
8F5
8NS1 8NS3 8EE1 8NS2 8EE2
Term 3
Solve systems graphically including no solution and infinite many solutions Interior/exterior angles, parallel lines/transversals, AA similarity
Apply Pythagorean Theorem
Proofs of Pythagorean
Distance Formula
Volumes of cylinders, cones spheres
8EE8 8G5 8G7 8G6 8G8 8G9
Term 4
Write scientific notation
Operations with scientific notation
Verify transformations preserve angles and lengths Congruence with transformations
Perform transformations
Similarity with transformations
Two-way tables
8EE3 8EE4 8G1 8G2 8G3 8G4 SP4
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Detailed Pacing Guide
Note: Bold text indicates an “essential” standard. Italic text indicates a “good to know” standard.
Term 1
Equations and inequalities: Solve linear equations and inequalities in one variable.
A. Giveexamplesoflinearequationsinonevariablewithone solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
B. Solvesingle-variablelinearequationsandinequalitieswith rational number coefficients, including equations and inequalities whose solutions require expanding expressions using the distributive property and collecting like terms.
C. Solvesingle-variableabsolutevalueequations.
Describe and analyze scatterplots:
Informally fit a trendline:
Using a trendline to predict and interpret data:
Identify linear/nonlinear:
association between two quantities. Describe patterns such as
clustering, outliers, positive or negative association, linear
association, and nonlinear association.
Know that straight lines are widely used to model
relationships between two quantitative variables. For scatter plots that
suggest a linear association, informally fit a straight line, and informally
assess the model fit by judging the closeness of the data points to the
line.
linear model to solve problems in the context of bivariate
measurement data, interpreting the slope and intercept. For
Use the equation of a
example, in a linear model for a biology experiment, interpret a slope
of 1.5 cm/hr as meaning that an additional hour of sunlight each day
is associated with an additional 1.5 cm in mature plant height.
(Calculating equations for a linear model is not expected in grade 8.)
Interpret the equation y = mx + b as
defining a linear function, whose graph is a straight line; give
examples of functions that are not linear. For example, the function
A = s2 giving the area of a square as a function of its side length is
not linear because its graph contains the points (1,1), (2,4) and (3,9),
which are not on a straight line.
Find slope/graph lines: Graph proportional relationships,
Construct and interpret scatter
plots for bivariate measurement data to investigate patterns of
8.EE.7
8.SP .1
8.SP .2
8.SP .3
8.F .3
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interpreting the unit rate as the slope of the graph. Compare two
different proportional relationships represented in different ways.
For example, compare a distance-time graph to a distance-time
equation to determine which of two moving objects has greater
speed.
Write y=mx+b equations from table/graph; interpret slope: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Similar triangles (slope)/derive y=mx+b and y=mx: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
8.EE.5
8.F .4
8.EE.6
Term 2
Definition of function: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in grade 8.)
Switch between representations of functions: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
Story of function (increasing/decreasing):
Convert rationals/know irrational:
functional relationship between two quantities by analyzing a graph (e.g.,
Sketch a graph that exhibits the qualitative features of a function that has
Describe qualitatively the
where the function is increasing or decreasing, linear or nonlinear).
been described verbally.
rational are called irrational. Understand informally that every
Know that numbers that are not
number has a decimal expansion; for rational numbers show that the
decimal expansion repeats eventually, and convert a decimal
expansion which repeats eventually into a rational number
8.F .1
8.F .2
8.F .5
8.NS.1
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Use square roots: Understand how to perform operations and simplify radicals with emphasis on square roots.
Exponent rules:
Compare and order numbers/place on number line:
Solve x^2=9/know root 2 is irrational: Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
= 3–3 = 1/33 = 1/27.
Know and apply the properties of integer exponents
to generate equivalent numerical expressions. For example, 32 × 3–5
estimate the value of expressions (e.g., π2). For example, by truncating
Use rational
approximations of irrational numbers to compare the size of irrational
numbers, locate them approximately on a number line diagram, and
the decimal expansion of √2, show that √2 is between 1 and 2, then
between 1.4 and 1.5, and explain how to continue on to get better
approximations.
8.NS.3 8.EE.1
8.NS.2
8.EE.2
Term 3
Solve systems graphically including no solution and infinitely many solutions: Analyze and solve pairs of simultaneous linear equations.
A. Understandthatsolutionstoasystemoftwolinearequations
in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
B. Solvesystemsoftwolinearequationsintwovariables graphically, approximating when solutions are not integers and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
C. Solvereal-worldandmathematicalproblemsleadingtotwo linear equations in two variables graphically. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
Interior/exterior angles, parallel lines/transversals, AA similarity :Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for
8.EE.8
8.G.5
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similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
Apply Pythagorean Theorem: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Proofs of Pythagorean:
Distance Formula: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Volumes of cylinders, cones, spheres:
Explore and explain a proof of the Pythagorean
Theorem and its converse.
Know the formulas for the
volumes of cones, cylinders, and spheres and use them to solve
real-world and mathematical problems.
8.G.7
8.G.6
8.G.8 8.G.9
Term 4
Write scientific notation:
Operations with scientific notation:
Verify transformations preserve angles and lengths:
Congruence with transformations:
Use numbers expressed in the form of a single
digit times a whole-number power of 10 to estimate very large or very
small quantities, and to express how many times as much one is than the
other. For example, estimate the population of the United States as 3
times 108 and the population of the world as 7 times 109, and determine
that the world population is more than 20 times larger.
Perform operations with numbers
expressed in scientific notation, including problems where both decimal
and scientific notation are used. Use scientific notation and choose units
of appropriate size for measurements of very large or very small
quantities (e.g., use millimeters per year for seafloor spreading). Interpret
scientific notation that has been generated by technology.
Verify experimentally
the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line
segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
Understand that a two-dimensional
figure is congruent to another if the second can be obtained from the first
by a sequence of rotations, reflections, and translations; given two
congruent figures, describe a sequence that exhibits the congruence
between them.
8.EE.3
8.EE.4
8.G.1
8.G.2
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Perform transformations:
Similarity with transformations:
Two-way tables: Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
Observe that orientation of the plane is
preserved in rotations and translations, but not with reflections. Describe
the effect of dilations, translations, rotations, and reflections on two-
dimensional figures using coordinates.
Understand that a two-dimensional figure is
similar to another if the second can be obtained from the first by a
sequence of rotations, reflections, translations, and dilations; given two
similar two-dimensional figures, describe a sequence that exhibits the
similarity between them.
8.G.3
8.G.4
8.SP .4
71


Units
Planning Guide: Jay McTighe, an expert in unit planning and author of Understanding by Design, has written four point to consider when planning units. They are presented below.
UbD Design Standards Stage 1 – To what extent does the design:
1. focus on the “Big ideas” of targeted content? Consider: are . . .
– the targeted understandings enduring, based on transferable, big ideas at the heart of the
discipline and in need of “uncoverage”?
– the targeted understandings framed as specific generalizations?
– the “big ideas” framed by questions that spark meaningful connections, provoke genuine
inquiry and deep thought, and encourage transfer?
– appropriate goals (e.g., content standards, benchmarks, curriculum objectives) identified? – valid and unit-relevant knowledge and skills identified?
Stage 2 – To what extent do the assessments provide:
2. fair, valid, reliable and sufficient measures of the desired results? Consider: are . . .
– students asked to exhibit their understanding through “authentic” performance tasks? – appropriate criterion-based scoring tools used to evaluate student products and
performances?
– a variety of appropriate assessment formats provide additional evidence of learning? Stage 3 – To what extent is the learning plan:
3. effective and engaging? Consider: will students . . .
– know where they’re going (the learning goals), why (reason for learning the content), and
what is required of them (performance requirements and evaluative criteria)?
– be hooked – engaged in digging into the big ideas (e.g., through inquiry, research, problem- solving, experimentation)?
– have adequate opportunities to explore/experience big ideas and receive instruction to equip them for the required performance(s)?
– have sufficient opportunities to rethink, rehearse, revise, and/or refine their work based upon timely feedback?
– have an opportunity to self-evaluate their work, reflect on their learning and set future goals? Consider: the extent to which the learning plan is:
– tailored and flexible to address the interests and learning styles of all students?
– organized and sequenced to maximize engagement and effectiveness?
Overall Design – to what extent is the entire unit:
4. coherent, with the elements of all 3 stages aligned?
Grant Wiggins and Jay McTighe 2005
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Assessment Standards
Utah SAGE Secondary Blueprints
Math 8
45 Operational Items
Domain
Min
Max
Functions
20%
24%
Expressions and Equations
20%
24%
The Number System
With Geometry
Geometry and The Number System
34%
40%
Statistics and Probability
16%
20%
DOK1
20%
30%
DOK2
40%
50%
DOK3
20%
26%
Disclosure: Depth of Knowledge (DOK) and Elements of Rigor are essential components of the Utah Mathematics Core Standards. As such, DOK and Elements of Rigor are integrated into the Student Assessment of Growth and Excellence (SAGE) assessment items. All students will see a variety of DOK and Elements of Rigor on the SAGE summative assessment. For more information about DOK and Elements of Rigor please see: http//www.schools.utah.gov/assessment/Criterion-Referenced-Tests/Math.aspx
Or http://static.pdesas.org/content/documents/M1-Slide_22_DOK_Hess_Cognitive_Rigor.pdf
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Purpose of Testing (from USBE testing ethics training)
The purpose of statewide assessment is for accountability.
When administered properly, standardized assessments allow students to demonstrate their abilities, knowledge, aptitude, or skills (see R277 – 404). Valid and reliable results from uniform assessments provide information used by:
Students, to determine how well they have learned the skills and curriculum they are expected to know;
Parents, to know whether their student is gaining the skills and competencies needed to be competitive and successful;
Teachers, to gauge their students’ understanding and identify potential areas of improvement in their teaching;
LEAs (districts or charter schools), to evaluate programs and provide additional support;
State, for school accountability; and
Public, to evaluate schools and districts.
As educators, we are obligated to provide students with an opportunity to demonstrate their knowledge and skills fairly and accurately.
Educators involved with the state – wide assessment of students must conduct testing in a fair and ethical manner (see Utah Code 53A-1-608; R277-404).
The best test preparation a teacher can provide is good instruction throughout the year that covers the breadth and depth of the standards for a course, using varied instructional and assessment activities tailored to individual students.
74


Ethical Assessment Practices (USBE ethics training) Licensed Utah Educators should:
• Ensure students are enrolled in appropriate courses and receive appropriate instruction
• Provide instruction to the intended depth and breadth of the course curriculum
• Provide accommodations throughout instruction to eligible students as identified by an
ELL, IEP, or 504 team.
• Use a variety of assessments methods to inform instructional practices
• Introduce students to various test-taking strategies throughout the year
• Provide students with opportunities to engage with available training test to ensure that
they can successfully navigate online testing systems, and to ensure that local
technology configurations can successfully support testing.
• Use formative assessments throughout the year using high-quality, non-secure test
questions aligned to Utah Standards.
Licensed Utah Educators shall ensure that:
• An appropriate environment reflective of an instructional setting is set for testing to limit distractions from surroundings or unnecessary personnel.
• All students who are eligible for testing are tested.
• A student is not discouraged from participating in state assessments, but upon a
parent’s opt-out request (follow LEA procedures), the student is provided with a
meaningful educational activity.
• Tests are administered in-person and testing procedures meet all test administration
requirements.
• Active test proctoring occurs: walking around the room to make sure that each
student has or is logged into the correct test; has appropriate testing materials
available to them; and are progressing at an appropriate pace.
• No person is left alone in a test setting with student tests left on screen or open.
• The importance of the test, test participation, and the good faith efforts of all
students are not undermined.
• All information in the Test Administration Manual (TAM) for each test administered
is reviewed and strictly followed (see 53A-1-608; R277-404).
• Accommodations are provided for eligible students, as identified by an ELL, IEP, or
504 team. These accommodations should be consistent with accommodations
provided during instruction throughout the instructional year.
• Any electronic devices that can be used to access non-test content or to
record/distribute test content or materials shall be inaccessible by students (e.g., cell phones, recording devices, inter-capable devices). Electronic security of tests and student information must not be compromised.
• Test materials are secure before, during and after testing. When not in use, all materials shall be protected, where students, parents cannot gain access.
No one may enter a student’s computer-based test to examine content or alter a student’s response in any way either on the computer or a paper answer document for any reason.
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Unethical Assessment Practices (USBE ethics training)
It is unethical for educators to jeopardize the integrity of an assessment or the validity of student responses.
Unethical practices include:
• Providing students with questions from the test to review before taking the test.
• Changing instruction or reviewing specific concepts because those concepts appear on
the test.
• Rewording or clarifying questions, or using inflection or gestures to help students
answer.
• Allowing students to use unauthorized resources to find answers, including dictionaries,
thesauruses, mathematics tables, online references, etc.
• Displaying materials on walls or other high visibility surfaces that provide answer to
specific test items (e.g., posters, word walls, formula charts, etc.).
• Reclassifying students to alter subgroup reports.
• Allowing parent volunteers to assist with the proctoring of a test their child is taking or
using students to supervise other students taking a test.
• Allowing the public to view secure items or observe testing sessions.
• Reviewing a student’s response and instructing the student to, or suggesting that the
student should, rethink his/her answers.
• Reproducing, or distributing, in whole or in part, secure test content (e.g., taking
pictures, copying, writing, posting in a classroom, posting publically, emailing).
• Explicitly or implicitly encouraging students to not answer questions, or to engage in
dishonest testing behavior.
• Administering tests outside of the prescribed testing window for each assessment.
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Intervention Standards
PCSD MTSS/RTI Model
Provo City School District's Academic MTSS (Multi-Tiered Systems of Support) details the system for providing Tier 1, 2, and 3 instruction; interventions; and assessment to help each student receive appropriate support. It is detailed below.
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Unpacking the Complexity of MTSS Decision Making
Successful MTSS implementation is a highly complex process that involves the following tasks:
• Gathering accurate and reliable data
• Correctly interpreting and validating data
• Using data to make meaningful instructional changes for students
• Establishing and managing increasingly intensive tiers of support
• Evaluating the process at all tiers to ensure the system is working
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Utah’s Multi-Tiered System of Supports USBE website:
http://www.schools.utah.gov/umtss/UMTSS-Model.aspx
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Supplemental Resources
Provo City School District’s Instructional Model
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• Student focus
• Educator credibility
• Meeting norms
• Professional Learning Communities (PLC)/Collaboration
• Civility policy
• Appearance and interactions
• Continual Leaning
• Testing ethics
• Research orientation
• Policy adherence
• Culture
• Safety–emotional and physical
• Physical classroom space
• Relationships
• Family connections
• Procedures
• Classroom management
• Student artifacts
• Student focus
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• Formative evaluation
• Summative evaluation
• Feedback:
• Performance of understanding
• Self-reported grades
• Student self-evaluation
• Testing ethics
• Differentiation
• Data analysis
• Response to interventions (RTI)/Multi-tiered system of success (MTSS)
• Lesson design
• Teacher clarity: share LT, share SC, share PoU
• Evidence-based instructional strategies
• Based on data
• Student engagement
• DOK – Depth of Knowledge
• Differentiation
• Student ownership of learning
• Curriculum notebook
• RTI/MTSS
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• State standards
• Curriculum map/pacing guide
• Units
• Objectives
• Curriculum Notebooks
• Course essentials
• Current
• Planning
Professional Association
The National Council of Teachers of Mathematics NCTM is the largest professional association for mathematics teachers.
Their website is at: https:// http://www.nctm.org/
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Evidence-based Pedagogical Practices
Hattie's Visible Learning
John Hattie, creator of Visible Learning, is a leading education researcher who has analyzed meta analyses in order to rank education practices (and factors) from most effective to least effective.
Hattie's list of highest ranking factors can be found at: https://visible-learning.org/hattie-ranking-influences-effect-sizes-learning-achievement/
or
https://visible-learning.org/nvd3/visualize/hattie-ranking-interactive-2009-2011-2015.html
Hattie's original book on the topic can be found at:
https://www.amazon.com/Visible-Learning-Synthesis-Meta-Analyses- Achievement/dp/0415476186
Definitions of Hattie's factors can be found at:
https://www.amazon.com/Visible-Learning-Synthesis-Meta-Analyses- Achievement/dp/0415476186
National Reading Panel Research
The federal government commissioned a National Reading Panel to review and compile the best evidence of effective practices for reading instruction.
The full report and executive summary can be accessed at:
https://lincs.ed.gov/communications/NRP
Learning Targets
Provo City School District employs the use of learning targets, success criteria, formative assessment, and feedback. A basis of study on these topics is the book, Learning Targets, by Connie Moss and Susan Brookhart, can be found at: https://www.amazon.com/Learning-Targets-Helping-Students-Understanding- ebook/dp/B008FOKP5S.
The district has produced four videos that demonstrate elements of learning target instruction and can be found at:
http://provo.edu/teachingandlearning/learning-targets-videos/
Teacher Resource Guide
Provo City School District's Teacher Resource Guide helps teachers meet the Utah Effective Teaching Standards and includes effective teaching practices. It can be found at: http://provo.edu/teachingandlearning/wp-content/uploads/sites/4/2016/01/11182016-TRG- fixed.pdf
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Glossary
Assessment Standards
College and Career Readiness
Curriculum Resources
ELA
Essential Learning Standards
Evidence-based Pedagogical Practices
Intervention Standards
Language Standards Math Content
A set of criteria to guide the assessment of student learning in a course that is based on Standards/Essentials of the course; this might include formative assessment practices, summative assessments/practices, common assessment plans, feedback practices, and a schedule for testing.
The College and Career Readiness (CCR) anchor standards
and grade-specific standards are necessary complements—the former providing broad standards, the latter providing additional specificity—that together define the skills and understandings that all students must demonstrate.
The materials teachers use to plan, prepare, and deliver instruction, including materials students use to learn about the subject. Such materials include texts, textbooks, tasks, tools, and media. Sometimes organized into a comprehensive program format, they often provide the standards, units, pacing guides, assessments, supplemental resources, interventions, and student materials for a course.
English Language Arts, includes components of Reading, Writing, Speaking and Listening, and Language.
These are also known as power standards. They are particular standards/objectives/indicators that a school/district defines as critical for student learning. In fact, they are so critical that students will receive intervention if they are not learned. Essentials are chosen because they: 1. have endurance, 2. have leverage, and 3. are important for future learning.
A list of teaching strategies that are supported by adequate, empirical research as being highly effective.
A set of criteria to guide teachers to provide additional instruction to students who did not master the content in Tier 1 instruction. This might include: commercial intervention programs, teacher- developed intervention materials, diagnostic testing, RTI/MTSS processes, and a list of essential knowledge/skills that will prompt intervention if the student does not demonstrate mastery.
(L) A component of ELA Standards that focus on conventions of standard English grammar, usage, and mechanics as well as learning other ways to use language to convey meaning effectively.
(MC) Math Content Standards identify the knowledge of concepts
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Standards
Mathematical Practice Standards
MTSS
Pacing Guide
Pathways of Progress
Performance of Understanding.
Provo Way Instructional Model
Reading Standards: Foundational Skills
SAGE
and the skills students need for college and career readiness.
The 8 Mathematical Practice Standards describe varieties of expertise that mathematics educators at all levels seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. All 8 mathematical practice standards are essential standards.
Multi-Tiered Systems of Support is an approach to academic and behavioral intervention. It is part of the intervention standards.
The order and timeline of the instruction of standards, objectives, indicators, and Essentials over the span of a course (semester or year).
(POP) An evaluation of individual student growth or improvement over time compared to other students with the same level of initial skills. It empowers educators to set goals that are meaningful, ambitious, and attainable.
(PoU). Student results that provide compelling evidence that the student has acquired the learning target. (Brookhart, 2012).
The five areas of expectations for successful instruction identified by Provo City School District.
(RF) A component of ELA Standards that focus on helping students gain a foundation where curriculum is intentionally and coherently structured to develop rich content knowledge within and across grades as well as acquiring the habits of reading independently and closely, which are essential to their future success.
Student Assessment of Growth and Excellence. This is the state end of level test for ELA and Math grades 3 – 8, and Science grades 4 – 8.
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