ps37r.altervista.orgps37r.altervista.org/nysaacc/Math M5 8.docx · Web viewD 75 Alternate Assessment Curriculum Grade 8 Math Module 5: Mathematical Practices. Page

Alternate Assessment Curriculum FrameworkIntroduction

The D75 Alternate Assessment Curriculum Framework was developed in response to schools’

requests for instructional expectations connected to the Common Core Learning Standards

(CCLS) for students in Alternate Assessment classes. Groups of teachers, administrators, and

district content area coaches gathered for four weeks during the summer of 2013, and

participated in a collaborative process to create an Alternate Assessment Curriculum

Framework. The process included a workshop at the beginning of each week to train the group

in the leveled learner concept (Levels B, C, and D), resources available (developmental math

skills progressions, Webb’s Depth of Knowledge, Common Core Essential Elements and

Alternate Achievement Descriptors for Mathematics from the State Members of the Dynamic

Learning Maps Alternate Assessment Consortium and Edvantia, Inc.), and final product

expectations. Subsequently, small groups collaborated to develop the leveled learning plans

and activities, culminating performance tasks, and the introductory contexts for the different

modules.

The structure of the framework provides four modules in ELA, Math, Science, and Social

Studies created in grade bands (K-2, 3-5, 6-8, and High School). Four math modules have been

developed as grade specific modules for K-8, while High School modules reflect specific

conceptual categories.

Each module consists of:

a context overview

culminating performance tasks for each level

Common Core Learning Standards connections

Career Development and Occupational Studies (CDOS) standards connections

Content standards connections

essential questions

key vocabulary

lesson strands with leveled learning plans and activities for each

Resources list

D 75 Alternate Assessment Curriculum Grade 8 Math Module 5: Mathematical Practices

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materials lists

Underlying the development of the activities included in this document is the profound

belief that students with significant intellectual disabilities need high standards that are

reasonable and achievable given sufficient and appropriate opportunities to learn. All students

who participate in Alternate Assessment classes are expected to be provided with access and

exposure to the content learning expectations of their general education peers at a reduced

depth, breath and complexity. The presented tasks, while not reflecting the degree of higher

order skills and comprehensiveness of expectations established for students participating in the

general assessment system, do reflect reasonable and achievable expectations for students

with significant intellectual disabilities. In addition, they maintain a necessarily broad

connection with the Common Core Standards through a concentrated focus on salient features

of specific Standards. These content area sample learning plans and activities are designed not

only to elicit performances of content area thinking skills/behaviors but also to provide

opportunities for students to engage with, read and/or use content understandings that are

imbedded within the tasks.

The sample learning plans and activities for each strand have been divided into three distinct

levels of student expectations based on cognitive abilities: Level D, Level C, and Level B.

Level D learning plans and activities are reflective of students who experience the most

significant cognitive disabilities within our district. These students are typically working at the

engagement level. Instruction is typically focused on developing the accessing skills that a

student needs to possess. It is understood that for additional information processing to take

place, engagement is a necessary first step. (Please refer to the Essential Thinking Skills and

Behaviors Explanatory Notes document for further information regarding the concept of

Engagement).

Level C learning plans and activities are reflective of students who demonstrate the

essential thinking skill of conceptualization. These students can form mental representations

of a concept and apply this knowledge. They exhibit intentional behavior in response to

situations. They rely heavily on objects, picture cues, a print rich environment, and an exposure

to content in multiple and modified formats to facilitate learning. These students typically work


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within Level one and two in Webb’s Depth of Knowledge. (Please refer to the Essential

Thinking Skills and Behaviors Explanatory Notes document for further information regarding

the concept of conceptualization, and Webb’s Depth of Knowledge).

Level B learning plans and activities are reflective of students who demonstrate skill abilities

closest to meeting the CCLS and content standards expectations as they are written. These are

typically students who may participate in inclusion settings and students who may return to

community based instruction programs. These students would be expected to work in all levels

of Webb’s Depth of Knowledge.

The Revision of Modules

The Alternate Assessment Curriculum Framework was developed to serve as a guide for

schools. It is expected to be modified and adjusted in order to meet school-specific instructional

goals and objectives.

To assist schools with understanding what the revision process entails, the district gathered

a small group of teachers and administrators during the summer of 2014 to revise Math module

2 for third grade, sixth grade, and High School. These modules serve as guiding examples for

schools to refer to as they consider revisions to the additional modules in all content areas.

Along with these examples, a general revision protocol and a sample reflections document

from the summer revision group regarding the revision process can be found at the end of this

introduction.

Each revised Math module 2 (grades 3, 6, and HS) now consists of:

a context overview

culminating performance tasks for each level

sample rubric designs for the performance task at the varied levels

An IEP goal tracking rubric format

Common Core Learning Standards connections

Career Development and Occupational Studies (CDOS) standards connections

Content standards connections

essential questions

key vocabulary


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Sequenced lesson strands with leveled learning plans and sequenced activities

Resources list

materials lists

A sample lesson written related to one activity in one strand

It is hoped that the D75 Alternate Assessment Curriculum Framework provides teachers and

schools with a resource to better understand how students can be provided with opportunities

to develop targeted skills through content-based instructional experiences that are also applied

in the context of functional activity experiences.


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Revision Protocol

The following is a step-by-step process that schools can reference when they

begin the process of revising a module for their own use. These are generic

expectations in the order they should occur to ensure an efficient and effective

revision of a module. This is by no means the only way in which a module can be

revised, but is intended to provide the essence of what the revision process

should include and be focused around.

1. Understand the standards for the learners in your class/school.

2. Ensure the connection between the standards, the learning strands and the

performance task.

3. Ensure that the learning strands and activities within the activities are

sequenced correctly for your students.

4. Ensure that the learning activities are appropriate for each level (B, C, and D).

5. Determine and agree upon the specific considerations that must be

accounted for when creating a rubric against the performance task for Level B,

C, and D.


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A reflection Sample on “How to” Revise an Alternate

Assessment Curricular Framework Module of Study (AACF) based

on the guiding protocol. 1. How do you ‘unpack’ or understand the standards for the learners in your class?Read the standards listed in the module and isolated the key nouns and verbs. Determined what the standard asking the students to know and do. Came to consensus regarding what the performance of these standards would look like for the students in alternate classes. Finally, the group translated the standard into actionable skills for the learners.2. How do you ensure connection between the standards, the learning strands and the performance task?One method the participants used was to use color-coding to ensure a connection. First, the group members color-coded each standard. Second, they looked at each learning strand and checked off, using the color system, where elements of each standard were contained in the strand. Last, they looked at the performance task, and highlighted or checked, using the color system, where elements of each standard were contained in the task. (These key elements were translated into actionable skills accessed in the rubric. See #5)If connections were not achieved, group members made a decision to reorganize, omit, add, condense or adjust as needed. 3. How do you ensure that the learning strands and activities within the activities are sequenced correctly for your students?Several resources were used, such as the CCLS Skills Progression at a Glance, Wisconsin Early Learning Skills, Equals chapter/skills sequencing, etc. (Note: please remember that the use of available resources such as language skills progressions, other content curricular models from various states, reading skills checklists, etc. should be referenced when revising other content area modules)4. How do you ensure that the learning activities are appropriate for each level (B, C, and D)?Participants referred back to Piaget’s Cognitive Levels of Development, their own students IEPs, as well as, keeping the individual needs of the learners in alternate assessment classes at the forefront of their minds When developing the learning activities for all levels.5. What should you consider for creating a rubric against the performance task for Level B, C, and D?Isolated key skills were identified in the standards and translated to actionable learning targets for the students when developing the Level C and B rubrics. Content expectations played a significant role in establishing the rubrics. Aspects of the rubric quantified skills for the B and C level learners and included a simple rating system (4-1, 3-1, etc.).

D 75 Alternate Assessment Curriculum Grade 8 Math Module 1: Expressions and Equations

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It was determined by the revision group that a specific rubric that could be used across the modules for the level D student would provide teachers with the ability to track skills related to engagement. This was determined to be the best approach to tracking progress for student who are cognitively young and require mastery of those skills related to engagement before any further content knowledge acquisition could be expected.


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District 75 Alternate Assessment Curriculum Framework Grade 8 MATH Module 1

Expressions and Equations

CONTEXT

UNIT TOPIC: Mathematical Practices

Mathematics is a language for translating real-life situations into numerical models. Expressions and equations are two ways we can translate situations into the language of mathematics and in grades 6-8 students are exposed and taught this. Students are to be exposed to and taught these mathematical concepts through hands-on instruction that emphasizes concrete manipuliatives, examples, and application to real world problems.

In 6th grade students learn to use variables to represent an unknown number in a mathematical sentence or phrase, solve simple one-step addition and multiplication sentences, and be able to write inequalities based on real-world situations. In 7th grade students extend what they learned in 6th grade by using variables in inequalities, to expand their expressions based on the all of the basic operations, and solve two-step problems, including problems with the distributive property in the 8th grade, students will primarily work with graphing and solving equations and inequalities, in addition work more with exponents.

The number system is reviewed in this module, which will focus on expanding students’ understanding of how numbers work with one-another based on various types of operations. In grades 6-8, students work with the order of operations, fractions, and they will start using positive and negative numbers. Students are to be exposed to and taught these mathematical concepts through hands-on instruction that emphasizes concrete manipuliatives, examples, and application to real world problems.

What happens when we manipulate shapes? In the 8th grade students spend a lot of time working on different types of geometric transformations and congruency. These concepts are presented visually in order to make transformations that are not static and allow students to see them in action. Students will start out the module looking for and creating congruent shapes—meaning the shapes are exactly the same in every way. After that, the first set of transformations the students will learn about is reflection, rotation, and transformation, which could also be called flipping, spinning, and sliding.


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The sample activities outlined are designed to elicit performances of mathematical thinking and behaviors, but also provide opportunities for students to get a concrete understanding of we use mathematical language to describe situations in the real-world. Teachers should emphasize concrete examples and repeated regular practice using manipuliatives and visualizations.

The activities in this module should be reinforced with regularly vocabulary review and simple equations throughout the day. Simple rate formulas should be used regularly during this module, in order to prepare students for rates, ratios, and percent later in later modules. Also, these are real-life examples that provide functional math skills for reasoning.

ASSESSMENTFORMATIVE ASSESSMENT EVIDENCE:

Pictures of students participating in various classroom lessons and activities

Data collection

Student work samples, as appropriate

STANDARDS

MATH COMMON CORE ANCHOR STANDARDS:8. EE.5 Graph proportional relationships, 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.

8. NS.2 Use rational approximations of irrational numbers to compare the size of

irrational numbers, locate them approximately on a number line diagram, and estimate

the value of expressions (e.g., π2).

8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations

8.G.A.2 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.


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http://www.corestandards.org/Math/Content/8/G/A/2


8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-

dimensional figures using coordinates.

8.G.A.4 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.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use

them to solve real-world and mathematical problems.

8.SP.A.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.

8.SP.A.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.

8.SP.A.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.

CAREER DEVELOPMENT AND OCCUPATIONAL STANDARDS

2.1: Integrated learning encourages students to use essential academic concepts, facts,

and procedures in applications related to life skills and the world of work. This approach

allows students to see the usefulness of the concepts that they are being asked to learn

and to understand their potential application in the world of work.

3a Understand how a linear equation can supply information (rate of pay) that can

be used in developing a career plan. Understand how interest is calculated.


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http://www.corestandards.org/Math/Content/8/G/C/9



3a Integrated Learning: Use knowledge of linear relationships to solve problems

(estimating job costs).

3a.1 Universal Foundational Skills: Basic skills include the ability to read, write, listen,

and speak as well as perform arithmetical and mathematical functions.

Vocabulary

Solution Variable Unknown Equation Parts (Pieces) Whole Line Equations Line Graph Coordinate Plane Points Line Formula (y = mx+b) Slope Rate Proportional Relationships Proportion Chart 2-dimensional shape square rectangle triangle circle hexagon Bar Category Chart Collect Compare Data Display

Add Subtract Multiply Divide Plus Minus Times Skip-Count Integer Operation Positive Negative Chart Number Line 3-dimensional shape Cube rectangular prism triangular prism pyramid sphere graph plot record relationship set interpret Findings gather

ESSENTIAL QUESTIONS

1. How does application of expressions and equations help us make sense of real-

world problems?


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2. What are the properties of 2D and 3D geometric shapes?

3. What is volume and how is measured?

4. How do graphs help in understanding trends, making comparisons, and identifying

relationships and how can graphs and data be used to predict outcomes of future

events?

LESSON STRANDS OVERVIEW

REVIEW OF STRAND 1: Solve one or more equations when the variable is by itself (e.g., 3 +1= x).

Solve equations when the variable is within one expression (e.g., 3 + x = 5).

Identify a line in a coordinate plane as the representation of a linear equation and

recognize the meaning of its’ slope.

Identify and/or interpret proportional relationships by graphing and/or making

tables using unit rates (e.g., time & distance; amount of bags & dollar cost; large

food portions & increased weight or other familiar units to the students).

REVIEW OF STRAND 2: Identify positive and negative numbers on a number line and/or in real-world

situations.

Add and/or subtract positive and negative numbers with or without decimals

and apply to real-world situations using a number line diagram.

Use multiplication and/or division to describe real-world problems.

REVIEW OF STRAND 3: Identify congruent geometric figures.

Demonstrate a flip (reflection), a turn (rotation), or a slide (translation) using

manipulatives.


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Determine if a translated figure is congruent to the original figure previously

presented.

Describe the change to an image/or object when it is transformed. e.g. labels image

or object as having been flipped/reflected.

Know/use the formula for volume.

Review of Strand 4: Collect and record one set, or two sets of related data in a simple chart/graph.

Collect and display data on a two-way table showing data that pertains to two

different categories from one survey group.

Generate a statement to express relationships between two sets of data.

Display two sets of data in a scatter plot.

LEARNING PLANS AND ACTIVITIES

NOTE: Preferred Mode of Communication (PMC) should be considered

for all students in all activities across all levels.


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Lesson Strand 1: Solve one or more equations when the variable is by itself

(e.g., 3 + 1 = x). Solve equations when the variable is within one expression. Identify a line in

a coordinate plane and how to graph a line. Recognize that a line is the representation of a

linear equation and the meaning of its slope is a proportional relationship by graphing

and/or making tables using unit rates.

LEARNING PLANS AND ACTIVITIES LEVEL D: Have students observe how to place objects on a mat (see below) to “solve for

x”.

Have students attend to equations with a variable by activating a voice output

device that says, ‘3 + 1 = x’ while teacher or other students arrange number and

symbol cards to make the equation.

Find the missing number and/or part: discuss that you have 3 objects, but want

to have 5 objects altogether, how many more objects do you need?

Engage in collecting classroom temperature data (or other kinds of data) over

the course of a day, generating a line graph on a coordinate plane, and using

large arrow symbols to label up and down (showing increase or decrease).

Engage students to click and color in number patterns on number charts. Color

all numbers one color, than start creating an ABAB pattern by coloring every

other number a different color.

LEARNING STUDENTS AND ACTIVITIES LEVEL C: Students use manipulatives and laminated mats (see below) to learn that x and

other letters represent a numerical value.

Use mats and manipulatives to complete an addition and/or subtraction problem

with 2 or more variables that ask to “solve for x” (e.g. 3 + 1 = x 3 – 1 = x). Write

equation below the mats.


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Use manipulatives and a mat (see resources) labeled with 3 + x = 5 and x = ___.

Students solve for the missing number by breaking the sum into its parts.

Collect classroom temperature data over the course of a day, then work as a

class to create a line graph on a coordinate plane and explain if the temperature

was increasing or decreasing by analyzing the line and choosing picture symbols

to answer.

Give students a set of data with x and y columns, then model using a coordinate

plane to graph the Identify the letters x and y and the numbers on the lines.

LEARNING PLANS AND ACTIVITIES LEVEL B: Solve written vertical equations with 2-digit numbers that equal x

(e.g., 12 + 7= x).

Use simple word problems such as, ‘Katlin has 4 cookies and Adul has 3 cookies,

how many cookies are there altogether?’ Student write’s the equation and/or

written equation is given to student and solves for x.

Students create word problems given simple equations individually or pairs or as

a class.

Collect classroom temperature data over the course of a day and graph the data,

with or without guidance, given a template of a coordinate plane with the x and

y-axis labeled.

After graphing several proportional lines as a class, students will be given a

proportion chart and asked to graph the line that demonstrates a proportional

relationship.

Lesson Strand 2: Review of: Identify positive and negative numbers on a number line

and/or in real-world situations. Add and/or subtract positive and negative numbers with or

without decimals and apply to real-world situations using a number line diagram. Use

multiplication and/or division to describe real-world problems.


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LEARNING PLANS AND ACTIVITIES LEVEL D: Engage in real-life experiences involving negative numbers (money/debt, using

an elevator, temperature, football) by participating in these experiences with

support as needed

Activate an AAC device indicating ‘plus’ and/or ‘minus’.

Engage with a number line with positive and/or negative numbers using

manipulatives.

Engage in real world division problems by equally distributing materials (plates

for snacks, notebooks, cookies etc.) among classmates.

LEARNING PLANS AND ACTIVITIES LEVEL C: Participate in real-life experiences involving integers (money and debt, using an

elevator, temperature, football) by identifying positive and negative numbers in

these scenarios.

Use a number line and/or manipulatives to solve real world subtraction

problems with positive and/or negative numbers.

Participate in adding and/or subtracting number card games using a number line

and/or manipulatives to determine whose card has the higher value or the sum

and/or the difference of their cards.

Participate in real world division problems by independently distributing

materials to classmates using JARS routines and scripts with mathematical

language.

LEARNING PLANS AND ACTIVITIES LEVEL B: Have students generated real life examples where positive and negative

numbers are used then, have the other students identify the positive and

negative numbers.


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Use a number line or other tools strategically, to solve real world addition and

subtraction problems involving rational numbers in any form (positive, negative,

0, fraction, decimals etc.).

Identify and complete equations that represent real-life multiplication and/or

division problems and have them create multiplication and/or division problems.

Lesson Strand 3: Review of: Identify congruent geometric figures. Demonstrate a flip

(reflection), a turn (rotation), or a slide (translation) using manipulatives. Determine if a

translated or transformed figure is congruent to the original figure previously presented.

Describe the change to an image/or object when it is transformed. Know/use the formula

for volume.

LEARNING PLANS AND ACTIVITIES LEVEL D: Student engages with shape blocks by placing them over drawn shapes in order

to identify congruent shapes.

Student engages in looking at shapes with measurements next to them to

determine if the shape is congruent or not (scaffold depending on students

ability).

Student engages with materials to complete an art activity where half the image is on

one side of the paper and the student must complete the other side by making a

reflection of it.

Student engages with transparency paper or physical manipulatives in order to identify

if a reflected shape is congruent to the original shape.

Student engages in identifying if a shape has been rotated, dilated, or reflected by

activating a pre-programmed switch.

Student engages with materials to fill cereal boxes, tissues boxes, etc. with cubed blocks

to determine the volume.


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LEARNING PLANS AND ACTIVITIES LEVEL C: Student participates on matching shapes on a paper with their congruent shape

printed out on a transparency paper. (Have 2-4 shapes on each sheet)

Student participates in looking at shapes with measurements next to them to

determine if the shape is congruent or not (scaffold depending on students

ability)

Student participates in art activity where half the image is on one side of the paper and

the student must complete the other side by making a reflection of it.

Student participates in identifying if various shapes remain congruent to their original

depending on what type of translation or transformation the teacher does.

Student participates in identifying if a shape has been rotated, dilated, or reflected.

Student participates in filling cereal boxes, tissues boxes, etc. with cubed blocks to

determine the volume.

LEARNING PLANS AND ACTIVITIES LEVEL B: Student matches shapes on a paper with their congruent shape printed out on a

transparency paper. (Have 3-6 shapes on each sheet)

Student determines if a shape is congruent based on measurements. (If the

student is at a higher level, have some measurements missing, but enough to

determine the answer.)

Student completes an art activity where half the image is on one side of the paper and

the student must complete the other side by making a reflection of it.

Student identifies if various shapes remain congruent to their original depending on

what type of translation or transformation the teacher does.

Student identifies if a shape has been rotated, dilated, or reflected.

Student fills cereal boxes, tissues boxes, etc. with cubed blocks to determine the

volume.


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Lesson Strand 4: Review of: Collect and record one set, or two sets of related data in a

simple chart/graph. Collect and display data on a two-way table showing data that pertains

to two different categories from one sample group. Generate a statement to express

relationships between two sets of data. Display two sets of data in a scatter plot.

LEARNING PLANS AND ACTIVITIES LEVEL D: Engages with asking the questions to collect the data for survey questions using

a voice output device

Engages with a survey by collecting/holding the pictures that represent the

responses from others

Engages with representations of two categories of related information that will be focus

of data collection by others.

Students engage with the numerical representations that will placed into the sections of

a two-way table

Engage with rulers and a yardstick when exploring the relationship that is being

represented between 3 one foot rulers and a yard

Engages with representations of scatter plots ( visual or tactile).

LEARNING PLANS AND ACTIVITIES LEVEL C: Draws a bar graph that represents data already collected

Collects simple data from classmates for one data question by gathering a

picture representation of each students’ response related to the question, and

place into a pre-created format.

Collects data on favorite food and favorite meal by giving choices to peers and using a

tally sheet to collect data

Collects and records data from two classmates using tally marks into a two way table

format


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Selects the appropriate statement that expresses a relationship between two sets of

data give two-three options

Reads the scatter plot about date of birth and month of birth by identifying the number

of birthdays charted in the same month but different days for students in the class from

a set of choices

LEARNING PLANS AND ACTIVITIES LEVEL B:

Explains the purposes of different types of graphs. For example, bar graphs are

used to compare data, line graphs are used to show change over time, circle

graphs are used to show parts of a whole.

Create a survey question then ask classmates for their responses by tallying

under the possible choices

Collects data from 11 classmates using data collection activity similar to one described in

level D with chart above, then transfers results in a two way table format using

numerical values.

Creates a two-way table survey activity- writing the steps for doing the survey for others

to follow

Writes a statement about a relationship that is represented between 3 rulers

and yardstick.

Plots dates and months of birth for classmates on a scatterplot graph after they have

gathered it from the class

Create a graph and plot results of two sets of data they collected to show a scatter plot

MATERIALS/ RESOURCES http://www.mathsisfun.com

Explanation of Linear Equations

o http://www.mathsisfun.com/algebra/linear-equations.html

100’s chart that you can customize with colors

www.coolmath.com


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http://www.coolmath.com/

http://www.mathsisfun.com/algebra/linear-equations.html

http://www.sheppardsoftware.com/math.htm

www.aaamath.com

illuminations.nctm.org/activitysearch.aspx

funschool.kaboose.com/arcade/math/index.html

www.learningplanet.com/stu/index.asp

www.helpkidzlearn.com/early.html

www.northerngrid.org/ngflwebsite/sen/NetSwitch/index.htm

www.superteacherworksheets.com - Worksheets and Games

http://www.mathsisfun.com/number-line.html

http://www.echalk.co.uk/Maths/dfes_numeracy/Assets/number_line_flash.swf

http://www.crickweb.co.uk/ks2numeracy-tools.html#Toolkit%20index2a

http://mathwire.com/algebra/integers.html

http://www.math-salamanders.com/image-files/3rd-grade-math-games-pit-of-

doom-minus-20-to-5.gif

http://mdk12.org/instruction/curriculum/hsa/geometry/math_reference_sheet.html

http://www.mangahigh.com/en-us/games/transtar

http://www.teachingmaths.net/Two%20way%20tables.pdf

http://www.stattrek.com/statistics/two-way-table.aspx

MANIPULATIVES Various objects to count add and subtract

Laminated Number Lines

Calculators

A laminated addition worksheet with boxes

Sorry or Trouble Board Game – Counting, 1:1 Correspondence, Forwards and

Backwards

cereal box

plastic container (rectangular prism) to fill with water


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http://www.stattrek.com/statistics/two-way-table.aspx

http://www.teachingmaths.net/Two%20way%20tables.pdf

http://www.mangahigh.com/en-us/games/transtar

http://mdk12.org/instruction/curriculum/hsa/geometry/math_reference_sheet.html

http://www.math-salamanders.com/image-files/3rd-grade-math-games-pit-of-doom-minus-20-to-5.gif

http://www.math-salamanders.com/image-files/3rd-grade-math-games-pit-of-doom-minus-20-to-5.gif

http://mathwire.com/algebra/integers.html

http://www.superteacherworksheets.com/

http://www.helpkidzlearn.com/early.html

http://www.learningplanet.com/stu/index.asp

http://www.aaamath.com/

plastic container (cylinder) to fill with water

small cube manipulatives (interlocking cubes, unifix cubes, foam cubes)

popsicle sticks

BOOKS

The Missing Piece , by Shel Silverstein

VIDEOs Brainpop.com – Math and Data Analysis Section

- Coordinate Plane

- Graphing Linear Equations

http://teachertube.com/viewVideo.php?

video_id=119155&title=Integers_Rap&vpkey=4c7340bdfb (Integer Rap)

BrainPop.com Video – Math – Numbers and Operations – Adding and

Subtracting Integers

http://www.onlinemathlearning.com/bar-graph.html

http://www.onlinemathlearning.com/scatter-plots.html

https://learnzillion.com/lessons/1179-construct-a-scatter-plot


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https://learnzillion.com/lessons/1179-construct-a-scatter-plot

http://www.onlinemathlearning.com/scatter-plots.html

http://www.onlinemathlearning.com/bar-graph.html

Essential Thinking Skills and Behaviors: Definitions and Explanatory Notes

EngagementEngagement is a behavior involving the focusing of the mental process upon someone or something. It is commonly demonstrated by a voluntary and sustained or repeated attention to stimuli. Engagement may be expressed through a wide variety of sensory, motor and/or speech, communication and language forms. Student’s physical, emotional, cognitive, social and cultural development impact significantly on the nature of the attention they are able, or choose, to demonstrate. Therefore, individual modes of student engagement need to be identified, taught, developed, refined, and/or expanded upon. These modes may include, but not limited to: exploration through touching, listening, looking, smelling, and/or tasting; and increase/decrease or initiation/cessation of body movement; and vocalizations/verbalizations. Without engagement, additional information processing cannot take place.

Explanatory Notes: When providing students with opportunities for engagement it is critical that the

same opportunities be presented daily over time. Variation in the means of story presentation, along with increased familiarity with expectations, should serve to sustain student motivation and interest. In addition, the presentation of materials should be supplemented with ongoing, direct instruction to facilitate targeted skills and behaviors specific to the content area.

Emphasis should be placed on relating meaningful activities/materials to student’s prior knowledge and experience.

Extensive efforts should be placed on involving, to the greatest extent possible, a student’s family in providing opportunities for student engagement. Such efforts might include: planning instructional materials; inviting family members to read stories in class; planning family related fairs; encourage family members to learn about and visit public and other community resources; and responding to educational needs as expressed by a student’s family.

Each student should possess a public library card, and be a member of other community organizations when appropriate and feasible.


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Environmental Differentiation

Environmental Differentiation is the recognition of differences in the attributes of things/places with which, and individuals with whom, one comes in contact and includes recognition of self as a distinct entity. It is usually demonstrated by distinct patterns of exploration or reaction to different stimuli and may be evidenced through various modes of student response. Environmental Differentiation may, but does not necessarily, include knowledge of the names/functions of the materials/places/individuals involved.

Explanatory Notes: The purpose for having students learn to differentiate is to help them develop a

basis from which they will be able to use materials functionally, make informed choices and develop concepts related to materials. However, instruction related to Environmental Differentiation should not preclude instruction toward other essential skills or behaviors (e.g. Functional Use of Objects; Self Regulation).

When various content area materials are being functionally used by a student, the student is already demonstrating environmental differentiation.

For a student with a limited response repertoire (i.e. a student with additional significant physical/sensory impairments), differentiation may be evidenced through the engagement with different stimuli. For example, a student might demonstrate differentiation simply by focusing on or maintaining hand contact with one stimulus for a significantly longer period of time than another stimulus.

For a student who is not environmentally differentiating, an implication for instruction is that the student may need to be provided with increased opportunities for sensory exploration of/interaction with the materials and for using the materials functionally. In providing these increased opportunities, it is essential to insure that a student’s safety and dignity are maintained, especially with regard to social context and age appropriateness.


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Conceptualization

Conceptualization is the formation of mental representations or ideas for categorizing information or mental connections to prior experiences. As children develop, new concepts about objects, people, places and the relationship between them are continually being learned. Conceptualization may be demonstrated through a range of initiated utterances/actions or responses to questions, comments, or directions. Individual communication modes may vary, and need to be identified, taught, developed, refined and/or expanded upon.

Explanatory Notes: In identifying a concept that a student is expected to learn, it is important to make

known to instructors and students the intended definition of that concept.

It is important that incidental displays of knowledge of identified concepts/meanings are noted/documented as they occur throughout the day.

In order for a student to demonstrate the knowledge of a concept/meaning, it is necessary for the student to exhibit a behavior that is intentional. For instance, a student who might typically sit without movement would not be considered to demonstrate knowledge of “wait” by remaining in a motionless position. Rather, the student would need to initiate a movement at the proper turn-taking time in order to have displayed knowledge of what “waiting” means.

Learning environments should be picture cue/object cue/print rich, so as to facilitate the learning of the concepts.

In expecting demonstration of knowledge of specific concepts, it is important that the other concepts/meanings used contextually by the instructor are known by the student or made clear (e.g. through demonstration) to the student. This is especially important with regards to concepts/meanings that define an expected mode of performance (e.g. touch, press, look).

Beyond the concepts/meanings that are found in this curriculum frameworks, which is based on the ELA and Math Common Core Learning Standards and Science and Social Studies NYS/NYC Scope and Sequence for grade level instructional content, there are other NYS standards based concepts that may be important to explicitly address in relation to each content area. For example, in Career Development and Occupational Studies, these may include: work; start/begin; end/finish; put away/put back; more/enough; and no. In Health, these may include; privacy, danger, emergency, clean, stranger, helper, friend, “feeling


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uncomfortable”, sick/hurt, exercise, medicine, and choice. These other concepts can identified by referring to New York State’s Learning Standards for Family and Consumer Sciences, Health, Phys. Ed., Career Development and Occupational Studies, The Arts, as well as, the NYSAA Alternate Grade Level Indicators for Science and Social Studies, and the grade level Extensions for English Language Arts and Math.

In addition to basic key concepts related to a content area, it is critical that students learn concepts needed for them to use their individual system of communication during assessment and instructional situations (e.g. point, touch, look, press, pick-up, give, tell, me/say).


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Functional Use of Objects

Functional Use of Objects is the appropriate utilization of materials in alignment with the purpose(s) for which they exist in a given culture. It may be applied to the use of an object that has undergone modifications. Students unable to utilize materials functionally due to a physical impairment may achieve this standard by communicating the purpose of the materials.

Explanatory Notes: Emphasis should be placed on involving family members in encouraging a

student to use content related materials during functional daily activities. For example, in the area of English Language Arts/Native Language Arts, some activities might include: giving a greeting card to a relative or friend; bringing a shopping list, with accompanying tangible symbols, to the supermarket; marking important dates on a calendar; labeling household items; and engaging with books and magazines.


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Problem SolvingProblem solving is the directing of one’s actions towards achieving a goal that presents uncertainty or difficulty. It presupposes an awareness of the existence of a problem. It generally involves taking into account factors related to a problem, and trying or considering more than one way to solve a problem. Resolution of a problem may be unattainable even though problem solving behaviors have been applied. Explanatory Notes:

When considering problem solving, an emphasis should be placed on a student’s involvement in the process of solving a problem rather than on a student’s resolution of a problem.

A student’s performance of Problem Solving may take the form of a variety of actions/response modes.

An implication for instruction is a recognition of the need to provide students with adequate time and opportunities “to try” or consider more than one way of solving a problem before intervening in the process.

Problem Solving may be accomplished through the completion of tasks formulated with the intent of providing opportunities for students to demonstrate specific problem solving behaviors. It may be accomplished, however, within a broader framework of general content area assignments, which naturally include a variety of problem solving situations.

A distinction involves the student’s completion of the task that the student has previously demonstrated an ability to do readily, while problem solving involves an element of uncertainly or difficulty for the student.

When a student secures needed help, instructors should not simply complete an action for the student. Rather, the student should be guided through the problem solving process, with help provided only to the extent actually needed by the student. In this way, a student hopefully will begin to approach future problem solving situations by trying another way before securing help.


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Self-Regulation

Self-regulation is an ongoing monitoring of ones’ own sensory/physical/social/cognitive conditions, and an adjusting of these conditions to maintain a desired and comfortable internal state. Self-regulation involves knowing and applying a repertoire of behaviors to diverse settings, making informed choices, and acting upon or indicating a desire or need for change.

Explanatory Notes: (Self-Regulation, General) The following conditions may necessitate self-regulation

o Sensory, including sensitivities to light, sound texture taste, smell and surrounding physical space.

o Physical, including pain, pleasure, hunger, thirst, discomfort, fatigue, hyperactivity, illness, and a need to use the bathroom.

o Emotional, including distress, loneliness, need for solitude, anger, aggressiveness, withdrawal, sadness, frustration, disappointment, elation, fear, anxiety, and stress.

o Social, including segregation, lack of privacy, and numbers/appearance/behaviors of individuals in the environment

o Cognitive, including level of subject content (either too high or too low), nature of subject matter presentation, and lack of appropriate means for accessing/expressing information.

Students may exhibit behaviors that are self-regulatory in nature but fail to meet the standard for self-regulation (as they are not desired behaviors). These include:

o Behaviors which are unsafe (e.g. abuse to self or others; object destruction)o Behaviors which interfere with one’s own learning or the learning of others

(e.g. replacing attention to task with stereotypic response; continuous noise production)

o Behaviors which interfere with positive social interactions (e.g. grabbing belongings of others; public disrobing).

Recognition should be given to the fact that most individuals engage in some common mannerisms or behaviors (e.g. finger-tapping; shaking of a glass with ice cubes; nail biting) through which they express their internal state. These behaviors, for the most part, are accepted by other individuals and do not seem to interfere in the development and maintenance of social relationships. Although the behavior of


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a student may differ in nature from these more common expressions, there is an expectation that such student behaviors, if exhibited in a safe and healthy manner, should be understood and accepted by others as an inherent part of “who” the student is. In fact, it may be precisely through such a particular behavior that a student is self-regulating.

In order to maintain internal control for self-regulating, students may need to be provided with positive behavioral support systems, including attention to communication and/or sensory needs and abilities.

Explanatory Notes: (Self-Regulation, Informed Choice-Making)

An informed choice refers to a student’s selection (within a single activity) of one of two (or possibly more) objects, activities, or environments for which opportunities for exploration/acquisition of knowledge have been provided. The informed nature of the choice may be demonstrated through a consistent response to an initial presentation (e.g. verbal; tangible; pictorial) and then to a second presentation with order/position altered**. If any doubt about a student’s selection still exists, a final presentation in either order/position can be made. Informed choice may be demonstrated in a different manner by a student who clearly has a demonstrated knowledge of the concept “yes” or “no”. Such a student needs only to reaffirm his/her choice by responding “yes” or “no” when asked if this choice is what he/she wants. Informed choice may also be demonstrated through independent indication of a choice different from the objects, activities, or environments offered.

An informed choice also assumes that a student possesses an equal opportunity to choose either of the sections available. This is especially important to consider when the student has limited motor and/or sensory abilities.

Given the concept of informed choice, various implications for instruction are evident, and include consideration of the placement of materials, the communicative means utilized by students to make choices, and steps taken to familiarize students with materials/activities/ environments available as choices.

Instructional efforts to increase a student’s opportunities to make informed choices will increase the probability of a student’s demonstration of general self-regulatory behavior, decision-making and awareness of the consequences of


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one’s decisions. Therefore, instructional provision for facilitating informed choice-making should be ongoing throughout a students’ day.

**It is recognized that repeatedly presenting choices in a different order/position may result in frustration on the part of students. Therefore, this type of procedure for insuring informed choice is designed primarily for the purpose of occasional assessment rather than for the purpose of ongoing instruction.


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Social Interaction

Social Interaction is reciprocal in nature and involves the use of communication for a variety of purposes. These may include having one’s desires or needs realized, or becoming involved in personal relationships. Such relationships may vary and may include being a one-time partner on a project, a member of a frequently meeting group, a helper, or a friend. Social interaction presupposes self-recognition, that is, the perception of self as a separate being, distinct form people/objects in the surrounding world. Explanatory Notes:

In general, communication refers to a process through which individuals receive from, transmit to, or exchange with others information, feelings or thoughts.

In order to help a student to learn how to socially interact, it is imperative that a student be assessed in a comprehensive and ongoing manner to determine which modes of communication are most appropriate for that student. Individual communication modes may vary and need to be identified, taught, refined, and /or expanded upon. Some students may even need to have meaning assigned to some of their naturally occurring behaviors (e.g. movements; facial expressions; vocalizations) so that they might begin intentionally to use these behaviors to communicate. Such a process should result in a student having ongoing access to and use of an effective system of communication.

In interactions with a student, it is critical to be aware of and respond immediately and consistently to any form of communication exhibited by the student, especially one of a subtle nature. In so doing, one is helping the student understand and come to expect that a communication causes others to act or respond. If such student communications are not attended to, the student most likely will discontinue communication since his/her communicative intent is not being realized.

It is beneficial to use a variety of communicative means (e.g. pictures, speech, gestures) when the student is engaged in receptive communication, even if some of these means appear to be of a nature that is beyond a student’s present cognitive level. However, a student should be taught and then have access to a means of communicating expressively that is consistent with that student’s present cognitive level.

It is critical that a student’s requests/directives and rejections/protests be addressed. Even if it is determined that the student’s attempt to control the environment cannot be accommodated, the attempt should at least be acknowledged.


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To maximize a student’s social interactions, emphasis needs to be placed on providing a student with an opportunity to communicate in the context of authentic situations and environments.

A student’s alternative/augmentative communication system (e.g. a device, board, and/or set of tangible symbols) needs to be accessible to the student throughout the day - at home, at school, and in community settings.

Significant emphasis should be placed on encouraging a student’s communication partners to accept and respond to alternate/augmentative forms of communication.

In order to interpret a student’s utterance or other communication as a request, it is subsequently necessary for the student to accept/interact with the referred to object/action/person. Otherwise, it may be that the student is merely recognizing the existence of an object/action/person.

To the greatest extent possible, and certainly to the degree mandated by a student’s IEP and by applicable educational regulations, a student should be learning to socially interact with students receiving general education services.

Certainly there is value in social interactions that occur between students and adults. Adults are able to provide appropriate models of communication and to respond readily to student initiations of communications. However, a significant emphasis also needs to be placed on providing opportunities for students to interact with peers (those receiving general and special education services).

When teaching a student to use a communication system expressively, it is critical that an instructor consistently model the use of the system in communications with the student.

The District 75 Office of Technology Solutions provides resources to students, staff, administrators, and parents in the areas of instructional, informational, and assistive technologies.


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Documents

ps37r.altervista.orgps37r.altervista.org/nysaacc/Math M5 8.docx · Web viewD 75 Alternate Assessment Curriculum Grade 8 Math Module 5: Mathematical Practices. Page