Kansas State Department of Educationaugust 2006

Kansas

extended

mAthematics

standards

Kansas State Department of EducationAugust 2006

Contents

Table of Contents...... i

Kansas Extended Standards in Mathematics Committee Members...... ii

General Introduction...... vi

Purpose...... vi

Individuals with Disabilities Education Act (IDEA) Requirements and Background...... vii

Title 1 – No Child Left Behind Requirements...... viii

Definitions...... viii

Clarifying Examples...... ix

Responses & Communication ...... ix

Use of this Document ...... x

Kansas Extended Curricular Standards – Mathematics...... 1

Standard 1 – Number and Computation...... 3

Standard 2 – Algebra...... 7

Standard 3 – Geometry...... 11

Standard 4 – Data...... 14

Kansas General and Extended Curricular Standards – Grade Level Alignment...... 17

3rd GradeExtended Standards...... 19

Extended Standard 1 – Numbers and Computation...... 21

Extended Standard 2 – Algebra...... 25

Extended Standard 3 – Geometry...... 26

Extended Standard 4 – Data...... 27

4th Grade Extended Standard ...... 29

Extended Standard 1 – Numbers and Computation...... 31

Extended Standard 2 – Algebra...... 35

Extended Standard 3 – Geometry...... 37

Extended Standard 4 – Data...... 39

5th Grade Extended Standards...... 41

Extended Standard 1 – Numbers and Computation...... 43

Extended Standard 2 – Algebra...... 47

Extended Standard 3 – Geometry...... 48

Extended Standard 4 – Data...... 50

6th Grade Extended Standards...... 53

Extended Standard 1 – Numbers and Computation...... 55

Extended Standard 2 – Algebra...... 58

Extended Standard 3 – Geometry...... 59

Extended Standard 4 – Data...... 61

7th Grade Extended Standards...... 63

Extended Standard 1 – Numbers and Computation...... 65

Extended Standard 2 – Algebra...... 67

Extended Standard 3 – Geometry...... 69

Extended Standard 4 – Data...... 71

8th Grade Extendeds...... 73

Extended Standard 1 – Numbers and Computation...... 75

Extended Standard 2 – Algebra...... 78

Extended Standard 3 – Geometry...... 79

Extended Standard 4 – Data...... 80

High School Extended Standards...... 81

High School Extended Standard 1 – Numbers and Computation...... 83

High School Extended Standard 2 – Algebra...... 86

High School Extended Standard 3 – Geometry...... 88

High School Extended Standard 4 – Data...... 89

Kansas General and Extended Curricular Standards with clarifying examples...... 91

Extended Standard 1 – Numbers and Computation...... 93

Extended Standard 2 – Algebra...... 137

Extended Standard 3 – Geometry...... 165

Extended Standard 4 – Data...... 193

- 1 -

Kansas Extended Standards Mathematics

Committee Members

George Abel...... Co-Chair General Education Mathematics Committee

EmporiaPublic Schools, USD 253

Robert Aman...... Teacher, EmporiaHigh School, USD. 253

Kerri Anderson...... Teacher, SMD class, South High, USD 259

Shirley Armentrout...... Parent Representative, Topeka

Susan M. Bashinski...... University of Kansas, BeechCenter on Disability

Penny Carrithers...... Teacher, LevySpecialEducationCenter, Wichita, USD 259

Tammy Esposito...... Principal, LevySpecialEducationCenter, Wichita, USD 259

Pauline Hintz...... Principal, LakemaryCenter, Paola

Cheryl Murra...... LeavenworthPublic Schools, USD 453

Jo Musselwhite...... SalinaPublic Schools, USD 305

Glenda Parma...... Special Education Teacher, SharonLearningCenter Sharon,

South Central Kansas Coop

Linda Sullivan...... School Psychologist, ChisholmLifeCenter, Wichita, USD 259

Tracy Newell...... Mathematics Consultant, Kansas State Department of Education

Betsy Wiens...... Auburn Washburn Public Schools (Topeka), USD 437

Lynnett Wright...... Extended Standards and Alternate Assessment Lead Consultant, Kansas State Department of Education

Background:
The extended standards writing committee studied the Kansas Mathematics Standards and invited a range of educational professionals and content experts to participate in aligning the Extended Standards with the General Standards. The focus of the standards alignment work was on aligning to the general education grade level academic standards to the extended standards.
The work of the group ALWAYS started with the intent of the grade-level standard before considering the extended standard
Teachers should use this document to:

Serve as a guide for instruction
Serve as a guide for developing IEP goals
Select indicators to be assessed on the Kansas Alternate Assessment
Review what is assessed in each grade level on the Kansas State Assessments

Codes:
General
SampleM.3.1.1.K2a
M.3.1.1.K2a
Mathematics,3rd grade, standard 1, benchmark 1, knowledge indicator 2a
Extended
Sample EM.1.1.6
EM.1.1.6
Extended Mathematics, standard 1, benchmark 1, indicator 6

General Introduction

Purpose

This document replaces all Extended Standards in Mathematics developed prior to August 2006. These extended standards are an extension of the Kansas Mathematics Standards. These extended standards, benchmarks, indicators, and examples are intended to be used in developing curricular materials for students with significant cognitive disabilities who are eligible for the alternate assessment. In addition, the extended standards will be helpful in developing IEP goals, benchmarks, or objectives for students with significant cognitive disabilities and who are eligible to take the Kansas Alternate Assessment. These standards are designed for students who require substantial adjustments to the general curriculum, yet allow that curriculum to maintain a clear connection to the general education standards.

The extended benchmarks, indicators, and examples have been developed to provide high expectations for students.

With the expertise and assistance of the members of the General Mathematics Curricular standards committee, these new extended mathematics standards have been aligned to the general grade level indicators that are assessed on the grades 3 - 8 and high school. Many of the extended standards can appropriately be repeated in several general indicators. It is critical to remember that these extended indicators are taught in the appropriate context including grade and age appropriateness.

When the extended benchmarks and the indicators are listed separately they have been organized in a hierarchical fashion (where possible), reflecting the wide variation in performance expectations for these students.

Because of the extremely wide range of abilities of students there remain very few students who are eligible to take the alternate assessment, whose level of skill acquisition is not possible to establish grade-specific alignment and expectations. These indicators are listed on each page and are to be taught in the appropriate context, including grade and age appropriateness.

Individuals with Disabilities Education Act (IDEA) Requirements and Background

The Individuals with Disabilities Act of 1997 (IDEA 97) resulted in a significant clarification of the educational expectations for students with disabilities. The reauthorization of this act in 2004 strengthens those requirements.

Section 614(d)(VI)(bb): “if the IEP Team determines that the child shall take an alternate assessment on a particular State or district wide assessment of student achievement, a statement of why – (AA) the child cannot participate in the regular assessment; and (BB) the particular alternate assessment selected is appropriate for the child;”

Accordingly, the Kansas Extended Standards have been developed to be consistent with the general standards, thus ensuring that the education of all students, including those with the most significant disabilities, is consistent with goals and standards for students as established by the Kansas State Board of Education. Further, the state is required to develop an alternate assessment for students with disabilities who are unable to participate in general state and district assessments:

Section 611(1)(16)(A): In general – All children with disabilities are included in all general State and district-wide assessment programs, including assessments described under section 111 of the Elementary and Secondary Education Act of 1965, with appropriate accommodations and alternate assessments where necessary and as indicted in their respective individualized education programs.

(C)(i) In general – The state (or, in the case of a district wide assessment, the local education agency) has developed and implemented guidelines for the participation of children with disabilities in alternate assessments for those children who cannot participate in regular assessments.

(ii) Requirements for Alternate Assessments – The guidelines under this clause (i) shall provide for alternate assessments that – (I) are aligned with the State’s challenging academic content standards and challenging student academic achievement standards;

(iii) Conduct of alternate Assessments – The state conducts the alternate assessments described in the subparagraph.

In keeping with this requirement, the extended standards serve as the basis for the development of the Kansas Alternate Assessment.

Title 1 – No Child Left Behind Requirements

States are required to hold all students to the same standards except that these regulations permit States to measure the achievement of students with the most significant cognitive disabilities based on alternate achievement standards. (These are called “The Kansas Extended Standards.”)
These regulations clarify that a State is permitted to use alternate achievement standards to evaluate the performance of students with the most significant cognitive disabilities.
Alternate achievement standards are acceptable only for the small number of students with the most significant cognitive disabilities.
The alternate achievement standards embody challenging academic expectations appropriate for those students who are assessed against them.
The use of ‘‘highest learning standards possible’’ is intended to reflect that the alternate achievement standards should be no less challenging for students with the most significant cognitive disabilities.

Definitions:

The following definitions clarify the four levels of this extended standards document. These definitions are very closely aligned with the definitions that are used in the Kansas Curriculum Standards in Mathematics.

Standard:

A curricular standard is a general statement of what a student should know and be able to do in academic subjects.

Example of a standard:

Numbers and computation - The student uses numerical and computational concepts and procedures in a variety of situations.

Benchmarks:

A specific statement of what a student should know and be able to do. Benchmarks are used to measure a student’s progress towards meeting a standard. Benchmarks are listed in hierarchical order under a standard.

Example of a benchmark:

The learner demonstrates number sense in a variety of situations.

Indicators:

A statement of the knowledge or skills that a student demonstrates in order to meet a benchmark. Indicators are critical to understanding the benchmarks and standards. Where possible, the indicators are listed in hierarchical order under a benchmark, from lower-level indicators to indicators of higher performance. This design allows every student from within the population covered by these extended standards to demonstrate a level of performance for the standards.

Example of an indicator:

The learner counts by rote.

Clarifying Examples:

Examples show how a student might demonstrate an indicator, using practical, real-world examples. Clarifying examples are NOT listed in hierarchical order. These examples are taken from the domains of school, vocational/career, community, recreation/leisure, and home. These domains are those in which students receive instruction in order to practice and maintain skills. The clarifying examples should provide a clear connection between the standards and instructional practice.

Clarifying Example:

School – Counts math manipulatives

Vocational/Career – Counts numbers embedded in steps of a task

Community – Counts to self to calm anxiety or maintain temper

Recreation/Leisure – Joins in count down the seconds remaining on a game clock at a sporting event

Home – Counts numbers in sequence while handling own money

Blank example pages are provided at the end of each benchmark. Teachers may use these pages to generate additional clarifying examples for their individual students.

Responses & Communication:

The demonstration of mathematical competencies may be mediated through any of the following:

Manipulatives / Assistive technology
Paper and pencil / Mental mathematics
Calculators

The extended standards are written to address the wide variety of communication methods used by students who qualify for the alternate assessment. These methods are determined by the IEP team and may include, but are not limited to:

Speech / Computer access
Large print / Pictures (picture board, notebook, etc.)
Braille / Vocal responses (cries, utterances, etc.)
Augmentative communication / Written responses (handwritten, computer generated, etc.)
Sign language / Eye gaze
Touch and object cues / Body movements

Use of this Document:

This document may be used for a variety of purposes. First, as with the general education standards, the document will assist Kansas teachers in planning local curriculum and assessments for students with disabilities. Although the document is intended to provide a curricular focus, it is not a state mandated curriculum. In addition, the document provides a resource that can and should be used in developing the IEP, yet it is not intended that the document contain everything a student may need regarding mathematics or that may appropriately appear on a student’s IEP.

It is also not expected that districts will develop curriculum to include every indicator; instead, the document has been developed to provide information to support a broad range of different local curricular emphases.

It should be noted:

If students are taught only the items that are assessed, a comprehensive individualized instructional plan has not been made available and the student does NOT have a quality educational program.

The entire standards document should be used for instruction. These are only indicators to be assessed on the

KansasState Assessment.

Kansas

Extended Mathematics Standards

Standard 1 – Numbers and Computation

Standard 1: Number and Computation -

The student uses numerical and computational concepts and procedures in a variety of situations.

Extended Benchmark 1: Number Sense -

The student demonstrates number sense in a variety of situations.

Extended Knowledge Base Indicators

Extended Clarifying Examples

The student:

This indicator is no longer assessed due to federal guidance.
understands the concept of more
understands the concept of one

counts by rote
establishes one to one correspondence
recognizes (receptive) and/or labels (expressive) numerals

generates whole numerals
understands concepts of numbers greater than one
recognizes a whole and parts of a whole

recognizes coins and/or currency

identifies symbols for dollar and cent notations
understands ordinal numbers
understands the values of coins and/or currency
understands the equivalencies of coins and/or currency

/ The student:

N/A
Responds appropriately when requested to provide more of an item
selects one item from container of many (milk carton from milk cooler, book from library shelf)
counts aloud or signs when requested by teacher.
distributes one paper or piece of equipment to each member in class
Presses numeral keys on a calculator to coincide with numbers named by peers or paraeducator during a lesson
keys in numerals on voice output device, to correctly answer a question
double clicks (i.e., two times) with computer mouse to open desired file
explains that two halves of an object are equal to one another and combine to make a whole
identifies pennies, nickels, dimes, and quarters in school cafeteria
correctly identifies the dollars and cents signs as indicating money amounts
goes to the start of the line when told by the teacher to, “Please, go first.”
states the value of penny, nickel, dime, or quarter
demonstrates different ways to combine coins to equal $1.00

Special Notes

A variety of clarifying examples that illustrate the range of application possibilities is included in chapter 3.
The extended standards are written to address a wide variety of response and communication modalities or methods used by students who qualify for the alternate assessment. These are individually determined by the IEP team.
Demonstration of mathematical competencies may be mediated through any of the following: manipulatives, paper and pencil, calculators, assistive technology, and/or mental mathematics.

Standard 1 – Numbers and Computation

Standard 1 - Number and Computation:

The student uses numerical and computational concepts and procedures in a variety of situations.

Extended Benchmark 2 - Number Systems and their properties:

The student demonstrates an understanding of number systems and their properties in a variety of situations.

Extended Knowledge Base Indicators

Extended Clarifying Examples

The student:

matches like numerals

counts manipulatives

establishes number/numerical correspondence

identifies sub-sets

identifies place value

/ The student:

matches bus number on ID card to numerals on side of school bus parked in school loading area
determines the number of steps s/he must take from the classroom to reach another designated location in the school.
counts school newsletters according to number of students in classroom

sort recyclable items from trash collected during lunch period

lines up numerals vertically in a problem when copying a problem

Special Notes

A variety of clarifying examples that illustrate the range of application possibilities is included in chapter 3.
The extended standards are written to address a wide variety of response and communication modalities or methods used by students who qualify for the alternate assessment. These are individually determined by the IEP team.
Demonstration of mathematical competencies may be mediated through any of the following: manipulatives, paper and pencil, calculators, assistive technology, and/or mental mathematics.

Standard 1 – Numbers and Computation

Standard 1 - Number and Computation:

The student uses numerical and computational concepts and procedures in a variety of situations.

Extended Benchmark 3 - Estimation:

The student uses numerical estimation in a variety of situations.

Extended Knowledge Base Indicators

Extended Clarifying Examples

The student:

identifies (receptive) and/or estimates (expressive) a little more, a little less, or about the same

rounds whole numbers

estimates amount of purchase

estimates quantities and checks whether or not results are reasonable

/ The student:
1.selects appropriately from a one digit, two digit, and three digit number (3,30,300) to estimate the amount of objects in a set
2.rounds a decimal number to the nearest whole number
3.chooses items for lunch that are less than the amount of money s/he has available to spend
4.estimates the cost of a full school lunch will be about $5, and all items can not purchased with $1
Special Notes

A variety of clarifying examples that illustrate the range of application possibilities is included in chapter 3.
The extended standards are written to address a wide variety of response and communication modalities or methods used by students who qualify for the alternate assessment. These are individually determined by the IEP team.
Demonstration of mathematical competencies may be mediated through any of the following: manipulatives, paper and pencil, calculators, assistive technology, and/or mental mathematics.

Standard 1 – Numbers and Computation