TITLE 19.EDUCATION

Part 2. TEXAS EDUCATION AGENCY

Chapter 33. STATEMENT OF INVESTMENT OBJECTIVES, POLICIES, AND GUIDELINES OF THE TEXAS PERMANENT SCHOOL FUND

19 TAC §33.65

The State Board of Education (SBOE) proposes an amendment to §33.65, concerning the Texas Permanent School Fund (PSF). Section 33.65 establishes provisions for the administration of the guarantee program for school district bonds. The proposed amendment would modify the administration of the PSF bond guarantee program. Changes to this rule are prompted by the need to clarify the current application process and to act on advice of legal counsel regarding the amount of capacity to be held in reserve.

Texas Education Code (TEC), §7.102(c)(33), authorizes the SBOE to adopt rules for the implementation of the guaranteed bond program as authorized in TEC, Chapter 45, School District Funds, Subchapter C, Guaranteed Bonds. Section 33.65 is the rule the SBOE adopted to implement the program. TEC, §45.053, limits the amount of bonds that can be guaranteed and requires an annual report to determine whether the amount of bonds guaranteed is within the limit. In November 2004, the SBOE adopted changes to the rule because the capacity of the fund to guarantee bonds was at its limit, prompting the need to limit access to the program. These rule changes took effect December 5, 2004. Further revisions are necessary to clarify the administration of the program and to increase the amount of capacity held in reserve based on the advice of legal counsel.

19 TAC §33.65 establishes the administration of the guarantee bond program, definitions applicable to the program, data sources used for the purposes of prioritization, and provisions related to application processing, including refunding issues, estimates of available capacity, and capacity reserved for emergencies; school district applications for guarantees, including commissioner review of applicants; limitations on the total amounts of bonds that may be guaranteed under the program; allocation of specific holdings of the PSF; defeasement of bonds; issuance of bonds; payments; guarantee restrictions; and transition for certain applications.

The proposed amendment to 19 TAC §33.65 would clarify the treatment of applications for the guarantee for which there remains insufficient capacity to guarantee fully and increase the amount of capacity that is held in reserve. In addition, the proposed amendment would clarify the types of bonds which are eligible for a guarantee. Specifically, the proposed amendment includes the following changes.

Language regarding the deadline by which to receive applications would be added to subsection (b)(3). New language would be added to subsection (b) to define new money and refunding issues and to clarify the eligibility of combination issues. In subsection (d)(3), language would be added to subparagraph (B) to clarify refunding issues eligible for guarantee and new subparagraph (E) would be added regarding refunding transactions. Language regarding the amount of capacity to be held in reserve for emergencies and the treatment of applications for which capacity is insufficient to fully guarantee the proposed bond issue would be added to subsection (d)(5). Language regarding the time period to receive bond approval from the Office of the Attorney General would be added to subsection (d)(7). Subsection (d)(7) would also be modified by adding new subparagraph (D) to specify the requirement that bonds not be represented as guaranteed until the date of the letter granting approval. New paragraph (3) would be added to subsection (f) to specify that the eligibility of bonds to receive guarantee is limited to new money, refunding, and combination issues. Subsection (n) regarding transitional provisions would be deleted.

Joe Wisnoski, deputy associate commissioner for school finance and fiscal analysis, has determined that for the first five-year period the amendment is in effect there are no anticipated costs to the state as a result of enforcing or administering the amendment related to these modifications of policies that guide the administration of the guaranteed bond program. In terms of impact on local governments, the need to create a larger reserve would reduce the capacity of the fund to guarantee bond issues, which may create a greater need on the part of districts to purchase private bond insurance. Potential costs to districts related to the purchase of private bond insurance is impossible to estimate because the costs are driven by the unique circumstances of school districts that propose to issue bonds, including the market's assessment of the district's financial condition and the proposed bond issue.

Mr. Wisnoski has determined that for each year of the first five years the amendment is in effect the public benefit anticipated as a result of enforcing the amendment is that the PSF bond guarantee program provides low-cost bond insurance to school districts in Texas and ensures that the bonds issued by school districts under this program will be rated AAA in the bond market. This superior bond rating allows districts to market their bonds at the lowest possible interest rates and thus reduces the long-term costs of the bonds for school districts and taxpayers. There will be no effect on small businesses. There is no anticipated economic cost to persons who are required to comply with the amendment.

Comments on the proposal may be submitted to Cristina De La Fuente-Valadez, Policy Coordination Division, Texas Education Agency, 1701 North Congress Avenue, Austin, Texas 78701, (512) 475-1497. Comments may also be submitted electronically to rules@tea.state.tx.us or faxed to (512) 463-0028. All requests for a public hearing on the proposed amendment submitted under the Administrative Procedure Act must be received by the commissioner of education not more than 15 calendar days after notice of the proposal has been published in the Texas Register .

The amendment is proposed under the Texas Education Code, §7.102(c)(33), which authorizes the SBOE to adopt rules as necessary for the administration of the guaranteed bond program as provided under TEC, Chapter 45, Subchapter C.

The amendment implements the Texas Education Code, §7.102(c)(33), and §45.053.

§33.65.Guarantee Program for School District Bonds.

(a) Statutory provision. The commissioner of education shall administer the guarantee program for school district bonds according to the provisions of the Texas Education Code (TEC), Chapter 45, Subchapter C.

(b) Definitions. The following definitions apply to the guarantee program for school district bonds.

(1) Annual debt service--Payments of principal and interest on outstanding bonded debt scheduled to occur between September 1 and August 31 during the fiscal year in which the guarantee is sought as set forth in the final official statement or the final bond order for the bonds most recently issued by the district, if the district has outstanding bonded indebtedness.

(A) The annual debt service does not include the amount of debt service to be paid on the bonds for which the reservation is sought.

(B) The debt service amounts used in this calculation for variable rate bonds will be that which is published in the final official statement.

(2) Bond order--The order adopted by the governing body of a school district that authorizes the issuance of bonds.

(3) Application deadline--The last business day of the month in which an application for a guarantee is filed. Applications must be received by the Texas Education Agency division responsible for state funding by 5:00 p.m. on the last business day of the month in order to be considered in that month's application processing.

(4) New money issue--An issuance of bonds for the purposes of constructing, renovating, acquiring, and equipping school buildings; the purchase of property; or the purchase of school buses. Eligibility for the guarantee for new money issues is limited to the issuance of bonds authorized under TEC, §45.003. A new money issue does not include the issuance of bonds to purchase a facility from a public facility corporation created by the school district or to purchase any property that is currently under a lease-purchase contract under Local Government Code, Chapter 271, Subchapter A. A new money issue does not include an issuance of bonds to refinance any type maintenance of tax-supported debt. Maintenance tax-supported debt includes, but is not limited to:

(A) time warrants or loans entered under TEC, Chapter 45, Subchapter E; or

(B) any other type of loan or warrant that is not supported by bond taxes as defined by TEC, §45.003.

(5) Refunding issue--An issuance of bonds for the purpose of refunding bonds that are supported by bond taxes as defined by TEC, §45.003. Eligibility for the guarantee for refunding issues is limited to refunding issues that refund bonds that were authorized by a bond election under TEC, §45.003.

(6) [ (4) ] Combination issue--An issuance of bonds for which an application is filed for a guarantee that includes both a new money portion and a refunding portion, as permitted by Texas Government Code, Chapter 1207. The eligibility of combination issues for the guarantee is limited by the eligibility of the new money and refunding portions as defined in this subsection.

(7) [ (5) ] Average daily attendance (ADA)--Total refined average daily attendance as defined by §129.1025 of this title (relating to Adoption By Reference: Student Attendance Accounting Handbook).

(8) [ (6) ] Enrollment growth--Growth in student enrollment that has occurred over the previous five years.

(c) Data sources.

(1) The following data sources shall be used for purposes of prioritization:

(A) projected ADA as adopted by the legislature for appropriations purposes;

(B) final property values certified by the comptroller of public accounts for the tax year preceding the year in which the bonds will be issued. If final property values are unavailable, the most recent projection of property values by the comptroller shall be used;

(C) annual debt service, as defined in subsection (b)(1) of this section, due during the fiscal year in which the proposed debt will be issued. The amount of debt service on the proposed bond issue will not be included in the calculation of annual debt service; and

(D) enrollment increases over the previous five years shall be determined using Public Education Information Management System (PEIMS) submission data available at the time of application.

(2) The commissioner may consider adjustments to data values determined to be erroneous prior to the deadline for receipt of applications for that application cycle.

(d) Application processing. To facilitate prioritization of applications for the guarantee, all applications received during a calendar month will be held until the tenth business day of the subsequent month. On the tenth business day of each month, the commissioner of education will announce the results of the prioritization described in paragraph (5) of this subsection and process applications for the guarantee up to the available capacity, subject to the requirements of this subsection.

(1) The school district may not submit an application for a guarantee prior to the successful passage of an authorizing proposition.

(2) The actual guarantee of the bonds is subject to the approval process prescribed in subsection (e) of this section.

(3) Refunding issues must comply with the following requirements in order to retain eligibility for the guarantee for the refunding bonds.

(A) The district must be accredited.

(B) The bonds to be refunded must have been previously guaranteed by the Permanent School Fund (PSF). Only refunding issues as defined in subsection (b)(5) of this section are eligible for the guarantee.

(C) The district must demonstrate that issuing the refunding bond(s) will result in a present value savings to the district and must not have a maturity date later than the final maturity date of the bonds being refunded. Present value savings is determined by computing the net present value of the difference between each scheduled payment on the original bonds and each scheduled payment on the refunding bonds. Present value savings shall be computed at the true interest cost of the refunding bonds.

(D) In the event that a district files an application for a combination issue, the application will be treated as a single issue for the purposes of eligibility for the guarantee. A guarantee for the combination issue will be awarded only if both the new money portion and the refunding portion meet all of the eligibility requirements described in this subsection. The district making the application must present data to the commissioner that demonstrates compliance for both the new money portion of the issue and the refunding portion of the issue.

(E) The refunding transaction must comply with the provisions of paragraph (7)(A) and (C) of this subsection.

(4) The commissioner of education will estimate the available capacity of the Texas Permanent School Fund (PSF) on a monthly basis.

(5) The State Board of Education (SBOE) shall establish an amount of capacity to be held in reserve of no less than 5.0% of the fund's capacity. Guarantees will be awarded each month beginning with the districts with the lowest property wealth per ADA until the PSF reaches [ 98% of ] its net capacity to guarantee bonds , as determined by subtracting the amount to be held in reserve from the total available capacity . The reserved [ remaining ] capacity can be used to award guarantees [ is to be held in reserve ] for districts that experience unforeseen catastrophes or emergencies that require the renovation or replacement of school facilities as described in the TEC, §44.031(h).

(A) The amount to be held in reserve may be increased by a majority vote of the SBOE based on changes in the asset allocation and risk in the portfolio and unrealized gains in the portfolio.

(B) Guarantees will be awarded to applicants based on the fund's capacity to fully guarantee the bond issue for which the guarantee is sought. Applications for bond issues that cannot be fully guaranteed will not receive an award. The amount of bond issue for which the guarantee was requested may not be modified after the monthly application deadline for the purposes of securing the guarantee during the award process.

(6) An application received after the application deadline shall be considered a valid application for the subsequent month, unless withdrawn by the submitting district before the end of the subsequent month.

(7) Each district that submits a valid application shall be notified of the application status within ten business days of the end of the month following the application deadline.

(A) If a district is awarded a guarantee, the bonds must be approved by the Office of the Attorney General within 120 [ 180 ] days of the date of the letter granting the approval of the guarantee [ application deadline ].

(B) If a district does not receive a guarantee or for any reason does not receive approval of the bonds from the Office of the Attorney General within the specified time period, the district may re-apply in a subsequent month. Applications that were denied a guarantee will not be retained for consideration in subsequent months.

(C) If the bonds are not approved by the Office of the Attorney General within 120 [ 180 ] days of the date of the letter granting the approval of the guarantee [ application deadline ], the commissioner shall consider the application withdrawn and the district must re-apply for a guarantee.

(D) Districts may not represent the bonds as guaranteed for the purposes of pricing or marketing the bonds prior to the date of the letter granting approval of the guarantee.

(e) Application for the guarantee.

(1) Districts shall apply to the commissioner of education for the guarantee of eligible bonds. The district shall submit, in a form specified by the commissioner, the information required under the TEC, §45.055(b), and this section and any additional information the commissioner may require. The application and all additional information required by the commissioner must be received before the application will be processed. The application shall be accompanied by a fee to be set by the commissioner and approved by the SBOE [ State Board of Education (SBOE) ].

(2) Under the TEC, §45.056, the commissioner shall investigate the applicant school district's accreditation status and financial status. A district must be accredited and financially sound to be eligible for approval by the commissioner. The commissioner's review shall include the following:

(A) the purpose of the bond issue;

(B) the district's accreditation status and compliance with statutes and rules of the Texas Education Agency; and

(C) the district's financial status and stability, including approval of the bonds by the attorney general under the provisions of the TEC, §45.0031 and §45.005.

(f) Limitations on access to the guarantee.

(1) The following limitations apply to bonds for which the election authorizing the issuance of bonds was called after July 15, 2004.

(2) The commissioner shall limit approval of the guarantee to a district with less than $1,250 of annual debt service per student in ADA at the time of the application for a guarantee. The limitation shall not apply to school districts that have enrollment that is 25% higher than the enrollment reported five years earlier, based on PEIMS data available at the time of application. The annual debt service amount is the amount defined by subsection (b)(1) of this section.

(3) The eligibility of bonds to receive the guarantee is limited to those new money, refunding, and combination issues as defined in subsection (b)(4)-(6) of this section.

(g) Allocation of specific holdings. If necessary to successfully operate the guarantee program, the commissioner may allocate specific holdings of the PSF to specific bond issues guaranteed under this section. This allocation shall not prejudice the right of the SBOE to dispose of the holdings according to law and requirements applicable to the fund; however, the SBOE shall ensure that holdings of the PSF are available for a substitute allocation sufficient to meet the purposes of the initial allocation. This allocation shall not affect any rights of the bond holders under law.

(h) Defeasement. The guarantee shall be completely removed when bonds guaranteed by this program are defeased, and such a provision shall be specifically stated in the bond resolution. If bonds guaranteed by this program are defeased, the district shall notify the commissioner in writing within ten calendar days of the action.

(i) Bonds issued before August 15, 1993. For bonds issued before August 15, 1993, a school district seeking the guarantee of eligible bonds shall certify that, on the date of issuance of any bond, no funds received by the district from the Available School Fund (ASF) are reasonably expected to be used directly or indirectly to pay the principal or interest on, or the tender or retirement price of, any bond of the political subdivision or to fund a reserve or placement fund for any such bond.

(j) Bonds guaranteed before December 31, 1993. For bonds guaranteed before December 1, 1993, if a school district cannot pay the maturing or matured principal or interest on a guaranteed bond, the commissioner shall cause the amount needed to pay the principal or interest to be transferred to the district's paying agent solely from the PSF and not from the ASF. The commissioner also shall direct the comptroller of public accounts to withhold the amount paid, plus interest, from the first state money payable to the district, excluding payments from the ASF.

(k) Bonds issued after August 15, 1993, and guaranteed on or after December 1, 1993. If a school district cannot pay the maturing or matured principal or interest on a guaranteed bond, the commissioner shall cause the amount needed to pay the principal or interest to be transferred to the district's paying agent from the PSF. The commissioner also shall direct the comptroller of public accounts to withhold the amount paid, plus interest, from the first state money payable to the district, regardless of source, including the ASF.

(l) Payments. For purposes of the provisions of the TEC, Chapter 45, Subchapter C, matured principal and interest payments are limited to amounts due on guaranteed bonds at scheduled maturity, at scheduled interest payment dates, and at dates when bonds are subject to mandatory redemption, including extraordinary mandatory redemption, in accordance with their terms. All such payment dates, including mandatory redemption dates, must be specified in the order or other document pursuant to which the bonds initially are issued. Without limiting the provisions of this subsection, payments attributable to an optional redemption or a right granted to a bondholder to demand payment upon a tender of such bonds in accordance with the terms of the bonds do not constitute matured principal and interest payments.

(m) Guarantee restrictions. The guarantee provided for eligible bonds in accordance with the provisions of the TEC, Chapter 45, Subchapter C, is restricted to matured bond principal and interest. The guarantee does not extend to any obligation of a district under any agreement with a third party relating to bonds that is defined or described in state law as a "bond enhancement agreement" or a "credit agreement," unless the right to payment of such third party is directly as a result of such third party being a bondholder.

[(n) Transitional provisions. Applications with a sale date beyond December 31, 2004, or received after October 8, 2004, will be subject to the provisions of this section as amended to be effective December 5, 2004. This subsection expires effective September 1, 2005.]

This agency hereby certifies that the proposal has been reviewed by legal counsel and found to be within the agency's legal authority to adopt.

Filed with the Office of the Secretary of State on July 25, 2005.

TRD-200503035

Cristina De La Fuente-Valadez

Director, Policy Coordination

Texas Education Agency

Earliest possible date of adoption: September 4, 2005

For further information, please call: (512) 475-1497


Chapter 74. CURRICULUM REQUIREMENTS

Subchapter C. OTHER PROVISIONS

19 TAC §74.25

The State Board of Education (SBOE) proposes an amendment to §74.25, concerning curriculum requirements. The section establishes provisions relating to high school credit for college courses. The proposed amendment would change the process through which students receive high school graduation credit for college courses.

Adopted to be effective September 1, 1996, 19 TAC §74.25 currently allows a school district board of trustees to adopt a policy that allows a student to be awarded credit toward high school graduation for completion of a college-level course. The rule requires that the course must be provided only by an institution of higher education that is accredited by one of several regional accrediting associations.

The rule currently establishes that to be eligible to enroll and be awarded credit toward state graduation requirements, a student must have the approval of the high school principal or other school official designated by the school district. The rule specifies that the course for which credit is awarded must provide advanced academic instruction beyond, or in greater depth than, the essential knowledge and skills for the equivalent high school course.

The proposed amendment would change the process for awarding high school graduation credit for completion of college courses. Language referring to a policy adopted by the school district board of trustees would be removed from subsection (a). In addition, language would be added to subsection (b) to require the acceptance and transferability of credit earned from an accredited institution of higher education toward state high school graduation requirements. Subsection (b) would also be revised to remove the requirement for prior approval of the high school principal or other designated school official.

Susan Barnes, Associate Commissioner for Standards and Programs, has determined that for the first five-year period the amendment is in effect there will be no fiscal implications for state or local government as a result of enforcing or administering the amendment.

Dr. Barnes has determined that for each year of the first five years the amendment is in effect the public benefit anticipated as a result of enforcing the amendment would be clarification of the provision providing students with additional options for completing high school graduation requirements. There will be no effect on small businesses. There is no anticipated economic cost to persons who are required to comply with the amendment.

Comments on the proposal may be submitted to Cristina De La Fuente-Valadez, Policy Coordination Division, Texas Education Agency, 1701 North Congress Avenue, Austin, Texas 78701, (512) 475-1497. Comments may also be submitted electronically to rules@tea.state.tx.us or faxed to (512) 463-0028. All requests for a public hearing on the proposed amendment submitted under the Administrative Procedure Act must be received by the commissioner of education not more than 15 calendar days after notice of the proposal has been published in the Texas Register .

The amendment is proposed under the Texas Education Code, §7.102, which authorizes the SBOE to establish curriculum and graduation requirements; §28.002, which authorizes the SBOE to by rule identify the essential knowledge and skills of each subject of the required curriculum that all students should be able to demonstrate and that will be used in evaluating textbooks and addressed on the assessment instruments; and §28.025, which authorizes the SBOE to by rule determine curriculum requirements for the minimum, recommended, and advanced high school programs that are consistent with the required curriculum.

The amendment implements the Texas Education Code, §§7.102, 28.002, and 28.025.

§74.25.High School Credit for College Courses.

(a) Credit [ A school district board of trustees may adopt a policy that allows a student to be awarded credit ] toward high school graduation for completion of [ completing ] a college-level course shall be granted under this section only when the [ . The ] course is [ must be ] provided [ only ] by an institution of higher education that is accredited by one of the following regional accrediting associations:

(1) - (6) (No change.)

(b) Credit earned toward state graduation requirements by a student in an accredited institution of higher education within Texas shall be transferable and must be accepted by a school district in the state. [ To be eligible to enroll and be awarded credit toward state graduation requirements, a student must have the approval of the high school principal or other school official designated by the school district. ] The course for which credit is awarded must provide advanced academic instruction beyond, or in greater depth than, the essential knowledge and skills for the equivalent high school course.

This agency hereby certifies that the proposal has been reviewed by legal counsel and found to be within the agency's legal authority to adopt.

Filed with the Office of the Secretary of State on July 25, 2005.

TRD-200503036

Cristina De La Fuente-Valadez

Director, Policy Coordination

Texas Education Agency

Earliest possible date of adoption: September 4, 2005

For further information, please call: (512) 475-1497


Chapter 101. ASSESSMENT

Subchapter B. DEVELOPMENT AND ADMINISTRATION OF TESTS

19 TAC §101.23

The State Board of Education (SBOE) proposes an amendment to §101.23, concerning student assessment. The section sets forth the SBOE-determined level of performance considered to be satisfactory on assessment instruments. The proposed amendment would set the performance standards for the Grade 8 science assessment. The Texas Education Code (TEC), §39.024(a), authorizes the SBOE to set the standard for satisfactory performance on the Texas Assessment of Knowledge and Skills (TAKS).

The 2001 federal No Child Left Behind Act requires that science be assessed in each of the following grade spans: Grades 3 - 5, 6 - 9, and 10 - 12 by the 2007 - 2008 school year. At the state level, Senate Bill 1108 and House Bill 411, passed by the 78th Texas Legislature in 2003, mandated the development of a Grade 8 science assessment to be administered to students no later than the 2006 - 2007 school year. TEC, §39.023(a)(6), was amended by this legislation to add the Grade 8 science assessment and requires that results from the assessment be included in the accountability system no later than the 2008 - 2009 school year. Currently, TAKS measures the statewide curriculum in science at Grades 5, 10, and 11. Development activities were undertaken for the new assessment, and the Grade 8 science assessment was field-tested both in a paper-and-pencil format and in an online format in April 2005.

When the TAKS program was first developed, a national Technical Advisory Committee (TAC) was assembled to advise the SBOE on standard-setting activities. This committee was composed of prominent educational testing experts with experience in standard setting in other major testing programs across the country. The current TAC met in February 2005 to discuss standard setting for the Grade 8 science assessment. At this meeting the TAC discussed the plan for conducting standard setting, a summary of the methods for standard setting, impact data, and ways to examine recommended standards in comparison with standards in other Grade 8 subjects and other grade level science assessments. At the April 2005 SBOE meeting, the SBOE approved the proposed standard-setting plan for the TAKS Grade 8 science assessment. As set forth in the standard-setting plan approved by the SBOE at its April 2005 meeting, a panel of experts was convened to evaluate data for the Grade 8 science assessment and develop recommendations on the performance standards for that assessment. During its July 2005 meeting, the SBOE reviewed and considered panel recommendations on performance standards for the TAKS Grade 8 science assessment. Impact data as well as other relevant information from the spring 2005 field test was also presented at the July 2005 meeting to assist the SBOE with determining the performance standards.

The SBOE took action that would establish a two-year phase-in period for the "met standard" level using the standard error of measurement (SEM) statistic to determine the standards during the phase-in period. For spring 2006, the passing standard would be set at 2 SEM below the panel recommendation, moving up to 1 SEM below the next year, and then to the panel recommendation in spring 2008.

The proposed amendment to §101.23 would add a new subsection (b), including a new figure, identifying the performance standards established by the SBOE for the TAKS Grade 8 science assessment. This figure reflects the TAKS scale scores required to achieve the "met standard" and "commended performance" at the standards equivalent to the panel recommendations, as well as those scale score standards at 1 SEM and 2 SEM below the panel recommendation for the "met standard" level. This is in accordance with the phase-in schedule established by the SBOE for full implementation of the TAKS Grade 8 science assessment performance standards. Language is also included in the proposed new subsection (b) and figure to maintain equivalent standards in future test forms.

Susan Barnes, Associate Commissioner for Standards and Programs, has determined that for the first five-year period the amendment is in effect there will be no fiscal implications for state or local government as a result of enforcing or administering the amendment.

Dr. Barnes has determined that for each year of the first five years the amendment is in effect the public benefit anticipated as a result of enforcing the amendment would be the addition of the middle school science assessment augmenting the Texas student assessment program's capacity to provide Texas students, schools, and the public with an accurate gauge of students' academic progress in learning the key components of the state-mandated curriculum. The middle school science test, which will assess Texas Essential Knowledge and Skills student expectations taught at Grades 6, 7, and 8, will also serve as a bridge between the elementary Grade 5 TAKS science test and the Grade 10 and exit-level TAKS science tests. There will be no effect on small businesses. There is no anticipated economic cost to persons who are required to comply with the amendment.

Comments on the proposal may be submitted to Cristina De La Fuente-Valadez, Policy Coordination Division, Texas Education Agency, 1701 North Congress Avenue, Austin, Texas 78701, (512) 475-1497. Comments may also be submitted electronically to rules@tea.state.tx.us or faxed to (512) 463-0028. All requests for a public hearing on the proposed amendment submitted under the Administrative Procedure Act must be received by the commissioner of education not more than 15 calendar days after notice of the proposal has been published in the Texas Register .

The amendment is proposed under the Texas Education Code, Chapter 39, Subchapter B, which authorizes the State Board of Education to adopt rules to create and implement a statewide assessment program, and, specifically, §39.024(a), which authorizes the State Board of Education to set the standard for satisfactory performance on the TAKS.

The amendment implements the TEC, Chapter 39, Subchapter B.

§101.23.Performance Standards.

(a) (No change.)

(b) As established in subsection (a) of this section, the SBOE shall determine the level of performance considered satisfactory on assessment instruments. The table in this subsection identifies the performance standards established by the SBOE for the TAKS Grade 8 science assessment. The "commended" and "met" standards are based on the spring 2006 operational test form. Future forms of the test will be equated by the Texas Education Agency to the 2006 assessment in order to ensure that equivalent standards are maintained.

Figure: 19 TAC §101.23(b) (.pdf)

(c) [ (b) ] The alternative assessment of academic skills will measure annual growth based on appropriate expectations for each student receiving special education services, as determined by the student's admission, review, and dismissal (ARD) committee in accordance with criteria established by the commissioner of education as required by the TEC, §39.024(a).

This agency hereby certifies that the proposal has been reviewed by legal counsel and found to be within the agency's legal authority to adopt.

Filed with the Office of the Secretary of State on July 25, 2005.

TRD-200503037

Cristina De La Fuente-Valadez

Director, Policy Coordination

Texas Education Agency

Earliest possible date of adoption: September 4, 2005

For further information, please call: (512) 475-1497


Chapter 111. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS FOR MATHEMATICS

Subchapter A. ELEMENTARY

19 TAC §§111.11 - 111.17

The State Board of Education (SBOE) proposes amendments to §§111.11-111.17, concerning the Texas Essential Knowledge and Skills (TEKS) for mathematics. The sections establish the curriculum standards for elementary mathematics, Kindergarten-Grade 5. The proposed amendments would refine and align elementary mathematics TEKS, for implementation beginning with the 2006-2007 school year.

Following a November 2003 directive from the SBOE to provide a schedule for reviewing the TEKS, Texas Education Agency (TEA) staff prepared a proposed 2004-2005 TEKS review calendar. The TEKS review process is designed to follow the same timeline as the textbook adoption process.

TEA staff has begun the review process for the elementary mathematics TEKS. A work group of teachers, central office staff, and university personnel was assembled to review these TEKS. After the work group refined and aligned the elementary mathematics TEKS, the draft revisions were placed on the TEA web site in the form of a survey to collect feedback from the public for 30 days beginning in mid-December 2004. A summary of the survey results was provided to the SBOE at the April meeting. The SBOE was also provided with an explanation of the changes for alignment and refinement of the TEKS during the April 2005 meeting.

The draft revisions have been provided to a review panel consisting of highly regarded mathematics experts. Feedback from mathematics experts and from SBOE members at the April meeting have been incorporated into the proposed amendments to the elementary mathematics TEKS. Examples of the proposed amendments include revisions for precision in language, mathematical correctness, developmental appropriateness, vertical alignment, and parallel language from Kindergarten-Grade 5.

The SBOE recently concluded the review process for the secondary mathematics TEKS in the areas of mathematics, Grades 6-8 (including Grade 6 Spanish mathematics); Algebra I and II; Geometry; Precalculus; and Mathematical Models with Applications. During the February 2005 meeting, the SBOE adopted amendments that refine and align the TEKS for secondary mathematics and specified that implementation begin with the 2006-2007 school year. The proposed implementation for the refined and aligned elementary mathematics TEKS would begin with school year 2006-2007 to coincide with the effective date of the amended secondary mathematics TEKS.

Susan Barnes, associate commissioner for standards and programs, has determined that for the first five-year period the amendments are in effect there will be no significant fiscal implications for state or local government as a result of enforcing or administering the amendments. Normal business costs would be associated with the TEKS updating process for the Texas Education Agency, including staff travel, meeting accommodations, and production and dissemination of documents.

Dr. Barnes has determined that for each year of the first five years the amendments are in effect the public benefit anticipated as a result of enforcing the amendments would include better alignment of the TEKS and coordination of the TEKS revision with the textbook adoption process. There will be no effect on small businesses. There is no anticipated economic cost to persons who are required to comply with the amendments.

Comments on the proposal may be submitted to Cristina De La Fuente-Valadez, Policy Coordination Division, Texas Education Agency, 1701 North Congress Avenue, Austin, Texas 78701, (512) 475-1497. Comments may also be submitted electronically to rules@tea.state.tx.us or faxed to (512) 463-0028. All requests for a public hearing on the proposed amendments submitted under the Administrative Procedure Act must be received by the commissioner of education not more than 15 calendar days after notice of the proposal has been published in the Texas Register .

The amendments are proposed under the Texas Education Code, §7.102, which authorizes the SBOE to establish curriculum and graduation requirements and §28.002, which authorizes the SBOE to by rule identify the essential knowledge and skills of each subject of the required curriculum that all students should be able to demonstrate and that will be used in evaluating textbooks and addressed on the assessment instruments.

The amendments implement the Texas Education Code, §7.102, and §28.002.

§111.11.Implementation of Texas Essential Knowledge and Skills for Mathematics, Grades K-5.

The provisions of this subchapter shall be implemented by school districts beginning with the 2006-2007 school year. [ September 1, 1998, and at that time shall supersede §75.27(a)-(f) of this title (relating to Mathematics). ]

§111.12.Mathematics, Kindergarten.

(a) Introduction.

(1) Within a well-balanced mathematics curriculum, the primary focal points at Kindergarten are developing whole-number concepts and using patterns and sorting to explore number, data, and shape.

(2) Throughout mathematics in Kindergarten-Grade 2, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use numbers in ordering, labeling, and expressing quantities and relationships to solve problems and translate informal language into mathematical language and symbols. Students use [ patterns to describe ] objects to create and identify patterns and use those patterns to [ , ] express relationships, make predictions, and solve problems as they build an understanding of number, operation, shape, and space. Students progress from informal to formal [ use informal ] language [ and observation of geometric properties ] to describe two- and three-dimensional geometric figures [ shapes, solids, ] and likenesses [ locations ] in the physical world . Students [ and ] begin to develop measurement concepts as they identify and compare attributes of objects and situations. Students collect, organize, and display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.

(3) Throughout mathematics in Kindergarten-Grade 2, students develop numerical fluency with conceptual understanding and computational accuracy. Students in Kindergarten-Grade 2 use basic number sense to compose and decompose numbers in order to solve problems requiring precision, estimation, and reasonableness. By the end of Grade 2, students know basic addition and subtraction facts and are using them to work flexibly, efficiently, and accurately with numbers during addition and subtraction computation.

(4) [ (3) ] Problem solving, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Throughout mathematics in Kindergarten-Grade 2, students use these processes together with technology and other mathematical tools such as manipulative materials to develop conceptual understanding and solve meaningful problems as they do mathematics.

(b) Knowledge and skills.

(1) Number, operation, and quantitative reasoning. The student uses numbers to name quantities. The student is expected to:

(A) use one-to-one correspondence and language such as more than, same number as, or two less than to describe relative sizes of sets of concrete objects;

(B) use sets of concrete objects to represent quantities given in verbal or written form (through 20 [ 9 ]); and

(C) use numbers to describe how many objects are in a set (through 20) using verbal and symbolic descriptions .

(2) Number, operation, and quantitative reasoning. The student describes order of events or objects. The student is expected to:

(A) use language such as before or after to describe relative position in a sequence of events or objects; and

(B) name the ordinal positions in a sequence such as first, second, third, etc.

(3) Number, operation, and quantitative reasoning. The student recognizes that there are quantities less than a whole. The student is expected to:

(A) share a whole by separating it into two equal parts; and

(B) explain why a given part is half of the whole.

(4) Number, operation, and quantitative reasoning. The student models addition (joining) and subtraction (separating) . The student is expected to model and create addition and subtraction problems in real situations with concrete objects.

(5) Patterns, relationships, and algebraic thinking. The student identifies, extends, and creates patterns. The student is expected to identify, extend, and create patterns of sounds, physical movement, and concrete objects.

(6) Patterns, relationships, and algebraic thinking. The student uses patterns to make predictions. The student is expected to:

(A) use patterns to predict what comes next, including cause-and-effect relationships; and

(B) count by ones to 100.

(7) Geometry and spatial reasoning. The student describes the relative positions of objects. The student is expected to:

(A) describe one object in relation to another using informal language such as over, under, above, and below; and

(B) place an object in a specified position.

(8) Geometry and spatial reasoning. The student uses attributes to determine how objects are alike and different. The student is expected to:

(A) describe and identify an object by its attributes using informal language;

(B) compare two objects based on their attributes; and

(C) sort a variety of objects including two- and three-dimensional geometric figures according to their attributes and describe how the objects are sorted [ those groups are formed ].

(9) Geometry and spatial reasoning. The student recognizes attributes [ characteristics ] of two- and three-dimensional geometric figures [ shapes and solids ]. The student is expected to:

(A) describe and compare the attributes of real-life objects such as balls, boxes, cans, and cones or models of three-dimensional geometric figures [ solids ];

(B) recognize shapes in real-life three-dimensional geometric figures [ objects ] or models of three-dimensional geometric figures [ solids ]; and

(C) describe, identify, and compare circles, triangles, [ and ] rectangles , and [ including ] squares (a special type of rectangle) .

(10) Measurement. The student directly compares the [ uses ] attributes of [ such as ] length, area, weight/mass, [ weight, or ] capacity , and/or relative temperature [ to compare and order objects ]. The student uses comparative language to solve problems and answer questions. The student is expected to:

(A) compare and order two or three concrete objects according to length (longer/shorter than, or the same); [ (shorter or longer), capacity (holds more or holds less), or weight (lighter or heavier); and ]

(B) compare the areas of two flat surfaces of two-dimensional figures (covers more, covers less, or covers the same);

[ (B) find concrete objects that are about the same as, less than, or greater than a given object according to length, capacity, or weight.]

(C) compare two containers according to capacity (holds more, holds less, or holds the same);

(D) compare two objects according to weight/mass (heavier than, lighter than or equal to); and

(E) compare situations or objects according to relative temperature (hotter/colder than, or the same as).

(11) Measurement. The student uses time [ and temperature ] to describe, compare , and order events [ , ] and situations [ , and/or objects ]. The student is expected to:

[ (A) compare situations or objects according to temperature such as hotter or colder;]

(A) [ (B) ] compare events according to duration such as more time than or less time than;

(B) [ (C) ] sequence events (up to three) ; and

(C) [ (D) ] read a calendar using days, weeks, and months.

(12) Probability and statistics. The student constructs and uses graphs of real objects or pictures to answer questions. The student is expected to:

(A) construct graphs using real objects or pictures in order to answer questions; and

(B) use information from a graph of real objects or pictures in order to answer questions.

(13) Underlying processes and mathematical tools. The student applies Kindergarten mathematics to solve problems connected to everyday experiences and activities in and outside of school. The student is expected to:

(A) identify mathematics in everyday situations;

(B) solve problems [ use a problem-solving model, ] with guidance [ , ] that incorporates the processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;

(C) select or develop an appropriate problem-solving strategy including drawing a picture, looking for a pattern, systematic guessing and checking, or acting it out in order to solve a problem; and

(D) use tools such as real objects, manipulatives, and technology to solve problems.

(14) Underlying processes and mathematical tools. The student communicates about Kindergarten mathematics using informal language. The student is expected to:

(A) communicate mathematical ideas [ explain and record observations ] using objects, words, pictures, numbers, and technology; and

(B) relate everyday language to mathematical language and symbols.

(15) Underlying processes and mathematical tools. The student uses logical reasoning [ to make sense of his or her world ]. The student is expected to justify [ reason and support ] his or her thinking using objects, words, pictures, numbers, and technology.

§111.13.Mathematics, Grade 1.

(a) Introduction.

(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 1 are building number sense through number relationships, adding and subtracting whole numbers , [ and ] organizing and analyzing data , and working with two- and three-dimensional geometric figures .

(2) Throughout mathematics in Kindergarten-Grade 2, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use numbers in ordering, labeling, and expressing quantities and relationships to solve problems and translate informal language into mathematical language and symbols. Students use [ patterns to describe ] objects to create and identify patterns and use those patterns to [ , ] express relationships, make predictions, and solve problems as they build an understanding of number, operation, shape, and space. Students progress from informal to formal [ use informal ] language [ and observation of geometric properties ] to describe two- and three-dimensional geometric figures [ shapes, solids, ] and likenesses [ locations ] in the physical world . Students [ and ] begin to develop measurement concepts as they identify and compare attributes of objects and situations. Students collect, organize, and display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.

(3) Throughout mathematics in Kindergarten-Grade 2, students develop numerical fluency with conceptual understanding and computational accuracy. Students in Kindergarten-Grade 2 use basic number sense to compose and decompose numbers in order to solve problems requiring precision, estimation, and reasonableness. By the end of Grade 2, students know basic addition and subtraction facts and are using them to work flexibly, efficiently, and accurately with numbers during addition and subtraction computation.

(4) [ (3) ] Problem solving, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Throughout mathematics in Kindergarten-Grade 2, students use these processes together with technology and other mathematical tools such as manipulative materials to develop conceptual understanding and solve meaningful problems as they do mathematics.

(b) Knowledge and skills.

(1) Number, operation, and quantitative reasoning. The student uses whole numbers to describe and compare quantities. The student is expected to:

(A) compare and order whole numbers up to 99 (less than, greater than, or equal to) using sets of concrete objects and pictorial models;

(B) create sets of tens and ones using concrete objects to describe, compare, and order whole numbers;

(C) identify individual coins by name and value and describe relationships among them; and

[ (C) use words and numbers to describe the values of individual coins such as penny, nickel, dime, and quarter and their relationships; and]

(D) read and write numbers to 99 to describe sets of concrete objects.

(2) Number, operation, and quantitative reasoning. The student uses pairs of whole numbers to describe fractional parts of whole objects or sets of objects. The student is expected to:

(A) separate [ share ] a whole [ by separating it ] into two, three, or four equal parts and use appropriate language to describe the parts such as three out of four equal parts; and

(B) use appropriate language to describe part of a set such as three out of the eight crayons are red.

(3) Number, operation, and quantitative reasoning. The student recognizes and solves problems in addition and subtraction situations. The student is expected to:

(A) model and create addition and subtraction problem situations with concrete objects and write corresponding number sentences; and

(B) use concrete and pictorial models to [ learn and ] apply basic addition and subtraction facts (up to 9 + 9 = 18 and 18 - 9 = 9) [ (sums to 18) using concrete models ].

(4) Patterns, relationships, and algebraic thinking. The student uses repeating patterns and additive patterns to make predictions. The student is expected to identify, describe, and extend concrete and pictorial patterns in order to make predictions and solve problems. [ : ]

[ (A) identify, describe, and extend concrete and pictorial patterns in order to make predictions and solve problems; and]

[ (B) use patterns to skip count by twos, fives, and tens.]

(5) Patterns, relationships, and algebraic thinking. The student recognizes patterns in numbers and operations. The student is expected to:

(A) use patterns to skip count by twos, fives, and tens;

(B) [ (A) ] find patterns in numbers, including odd and even;

(C) [ (B) ] compare and order whole numbers using place value; [ and ]

(D) use patterns to develop strategies to solve basic addition and basic subtraction problems; and

(E) [ (C) ] identify patterns in related addition and subtraction sentences (fact families for sums to 18) such as 2 + 3 = 5, 3 + 2 = 5, 5 - 2 = 3, and 5 - 3 = 2.

(6) Geometry and spatial reasoning. The student uses attributes to identify two- and three-dimensional geometric figures. The student compares and contrasts two- and three-dimensional geometric figures or both [ , compare, and contrast shapes and solids ]. The student is expected to:

[ (A) describe and identify objects in order to sort them according to a given attribute using informal language;]

(A) [ (B) ] describe and identify two-dimensional geometric figures, including circles, triangles, [ and ] rectangles, and [ including ] squares (a special type of rectangle); [ , and describe the shape of balls, boxes, cans, and cones; and ]

(B) describe and identify three-dimensional geometric figures, including spheres, rectangular prisms (including cubes), cylinders, and cones;

(C) describe and identify two- and three-dimensional geometric figures in order to sort them according to a given attribute using informal and formal language; and

(D) [ (C) ] use concrete models to combine two-dimensional geometric figures [ shapes ] to make new geometric figures [ shapes using concrete models ].

(7) Measurement. The student directly compares the attributes of length, area, weight/mass, capacity, and temperature. The student uses comparative language to solve problems and answer questions. The student selects and uses nonstandard units to describe length [ , weight, and capacity ]. The student is expected to:

(A) estimate and measure length [ , capacity, and weight of objects ] using nonstandard units such as paper clips or sides of color tiles ; [ and ]

(B) compare and order two or more concrete objects according to length (from longest to shortest);

(C) [ (B) ] describe the relationship between the size of the unit and the number of units needed to measure the length of an object; [ in a measurement. ]

(D) compare and order the area of two or more two-dimensional surfaces (from covers the most to covers the least);

(E) compare and order two or more containers according to capacity (from holds the most to holds the least);

(F) compare and order two or more objects according to weight/mass (from heaviest to lightest); and

(G) compare and order two or more objects according to relative temperature (from hottest to coldest).

(8) Measurement. The student understands that time [ and temperature ] can be measured. The student uses time to describe and compare situations. The student is expected to:

[ (A) recognize temperatures such as a hot day or a cold day;]

[ (B) describe time on a clock using hours and half hours; and]

(A) [ (C) ] order three or more events according to duration; and [ by how much time they take. ]

(B) read time to the hour and half-hour using analog and digital clocks.

(9) Probability and statistics. The student displays data in an organized form. The student is expected to:

(A) collect and sort data; and

(B) use organized data to construct real-object graphs, picture graphs, and bar-type graphs.

(10) Probability and statistics. The student uses information from organized data. The student is expected to:

(A) draw conclusions and answer questions using information organized in real-object graphs, picture graphs, and bar-type graphs; and

(B) identify events as certain or impossible such as drawing a red crayon from a bag of green crayons.

(11) Underlying processes and mathematical tools. The student applies Grade 1 mathematics to solve problems connected to everyday experiences and activities in and outside of school. The student is expected to:

(A) identify mathematics in everyday situations;

(B) solve problems [ use a problem-solving model, ] with guidance [ as needed, ] that incorporates the processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;

(C) select or develop an appropriate problem-solving plan or strategy including drawing a picture, looking for a pattern, systematic guessing and checking, or acting it out in order to solve a problem; and

(D) use tools such as real objects, manipulatives, and technology to solve problems.

(12) Underlying processes and mathematical tools. The student communicates about Grade 1 mathematics using informal language. The student is expected to:

(A) explain and record observations using objects, words, pictures, numbers, and technology; and

(B) relate informal language to mathematical language and symbols.

(13) Underlying processes and mathematical tools. The student uses logical reasoning [ to make sense of his or her world ]. The student is expected to justify [ reason and support ] his or her thinking using objects, words, pictures, numbers, and technology.

§111.14.Mathematics, Grade 2.

(a) Introduction.

(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 2 are developing an understanding of the base-ten place value system, comparing and ordering whole numbers, applying addition and subtraction, and using measurement processes.

(2) Throughout mathematics in Kindergarten-Grade 2, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use numbers in ordering, labeling, and expressing quantities and relationships to solve problems and translate informal language into mathematical language and symbols. Students use [ patterns to describe ] objects to create and identify patterns and use those patterns to [ , ] express relationships, make predictions, and solve problems as they build an understanding of number, operation, shape, and space. Students progress from [ use ] informal to formal language [ and observation of geometric properties ] to describe two- and three-dimensional geometric figures [ shapes, solids, ] and likenesses [ locations ] in the physical world . Students [ and ] begin to develop measurement concepts as they identify and compare attributes of objects and situations. Students collect, organize, and display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.

(3) Throughout mathematics in Kindergarten-Grade 2, students develop numerical fluency with conceptual understanding and computational accuracy. Students in Kindergarten-Grade 2 use basic number sense to compose and decompose numbers in order to solve problems requiring precision, estimation, and reasonableness. By the end of Grade 2, students know basic addition and subtraction facts and are using them to work flexibly, efficiently, and accurately with numbers during addition and subtraction computation.

(4) [ (3) ] Problem solving, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Throughout mathematics in Kindergarten-Grade 2, students use these processes together with technology and other mathematical tools such as manipulative materials to develop conceptual understanding and solve meaningful problems as they do mathematics.

(b) Knowledge and skills.

(1) Number, operation, and quantitative reasoning. The student understands how place value is used to represent whole numbers. The student is expected to : [ use concrete models to represent, compare, and order whole numbers (through 999), read the numbers, and record the comparisons using numbers and symbols (>, <,=). ]

(A) use concrete models of hundreds, tens, and ones to represent a given whole number (up to 999) in various ways;

(B) use place value to read, write, and describe the value of whole numbers to 999; and

(C) use place value to compare and order whole numbers to 999 and record the comparisons using numbers and symbols (<,=, >).

(2) Number, operation, and quantitative reasoning. The student describes how fractions are used [ uses fraction words ] to name parts of whole objects or sets of objects. The student is expected to:

(A) use concrete models to represent and name fractional parts of a whole object (with denominators of 12 or less); [ (not to exceed twelfths) when given a concrete representation; and ]

(B) use concrete models to represent and name fractional parts of a set of objects (with denominators of 12 or less); and [ (not to exceed twelfths) when given a concrete representation. ]

(C) use concrete models to determine if a fractional part of a whole is closer to 0, 1/2, or 1.

(3) Number, operation, and quantitative reasoning. The student adds and subtracts whole numbers to solve problems. The student is expected to:

(A) recall and apply basic addition and subtraction facts ([ sums ] to 18);

(B) model addition and subtraction of two-digit numbers with objects, pictures, words, and numbers;

(C) [ (B) ] select addition or subtraction to [ and ] solve problems using two-digit numbers, whether or not regrouping is necessary; [ and ]

(D) [ (C) ] determine the value of a collection of coins up to [ less than ] one dollar ; and [ . ]

(E) describe how the cent symbol, dollar symbol, and the decimal point are used to name the value of a collection of coins.

(4) Number, operation, and quantitative reasoning. The student models multiplication and division. The student is expected to:

(A) model, create, and describe multiplication situations in which equivalent sets of concrete objects are joined; and

(B) model, create, and describe division situations in which a set of concrete objects is separated into equivalent sets.

(5) Patterns, relationships, and algebraic thinking. The student uses patterns in numbers and operations. The student is expected to:

(A) find patterns in numbers such as in a 100s chart;

(B) use patterns in place value to compare and order whole numbers through 999; and

(C) use patterns and relationships to develop strategies to remember basic addition and subtraction facts . Determine patterns in related addition and subtraction number sentences (including fact families) such as 8 + 9 = 17, 9 + 8 = 17, 17 - 8 = 9, and 17 - 9 = 8. [ ; and ]

[ (D) solve subtraction problems related to addition facts (fact families) such as 8 + 9 = 17, 9 + 8 = 17, 17 - 8 = 9, and 17 - 9 = 8.]

(6) Patterns, relationships, and algebraic thinking. The student uses patterns to describe relationships and make predictions. The student is expected to:

(A) generate a list of paired numbers based on a real-life situation such as number of tricycles related to number of wheels;

(B) identify patterns in a list of related number pairs based on a real-life situation and extend the list; and

(C) identify, describe, and extend repeating and additive patterns to make predictions and solve problems.

(7) Geometry and spatial reasoning. The student uses attributes to identify two- and three-dimensional geometric figures. The student compares and contrasts two- and three-dimensional geometric figures or both [ , compare, and contrast shapes and solids ]. The student is expected to:

(A) describe [ identify ] attributes (the number of vertices, faces, edges, sides) of two- and three-dimensional geometric figures such as circles, polygons, spheres, cones, cylinders, prisms, and pyramids, etc. [ any shape or solid ];

(B) use attributes to describe how 2 [ two ] two-dimensional figures or 2 three-dimensional geometric figures [ shapes or two solids ] are alike or different; and

(C) cut two-dimensional geometric figures [ shapes ] apart and identify the new geometric figures formed [ shapes made ].

(8) Geometry and spatial reasoning. The student recognizes that [ numbers can be represented by points on ] a line can be used to represent a set of numbers and its properties . The student is expected to use whole numbers to locate and name points on a number line.

(9) Measurement. The student directly compares the attributes of length, area, weight/mass, and capacity, and uses comparative language to solve problems and answer questions. The student selects and uses nonstandard units to describe length, area, capacity, and weight/mass. The student recognizes and uses models that approximate standard units ( from both SI, also known as metric , and customary systems ) of length, weight/mass [ weight ], capacity, and time. The student is expected to:

(A) identify concrete models that approximate standard units of length and use them to measure length [ , capacity, and weight ];

(B) select a non-standard unit of measure such as square tiles to determine the area of a two-dimensional surface;

[ (B) measure length, capacity, and weight using concrete models that approximate standard units; and]

(C) select a non-standard unit of measure such as a bathroom cup or a jar to determine the capacity of a given container; and

[ (C) describe activities that take approximately one second, one minute, and one hour.]

(D) select a non-standard unit of measure such as beans or marbles to determine the weight/mass of a given object.

(10) Measurement. The student uses standard tools to estimate and measure time and temperature (in degrees Fahrenheit) . The student is expected to:

(A) read a thermometer to gather data; [ and ]

(B) read and write times shown [ describe time ] on analog and digital clocks using five-minute increments; and [ a clock using hours and minutes. ]

(C) describe activities that take approximately one second, one minute, and one hour.

(11) Probability and statistics. The student organizes data to make it useful for interpreting information. The student is expected to:

(A) construct picture graphs and bar-type graphs;

(B) draw conclusions and answer questions based on picture graphs and bar-type graphs; and

(C) use data to describe events as more likely or less likely such as drawing a certain color crayon from a bag of seven red crayons and three green crayons.

(12) Underlying processes and mathematical tools. The student applies Grade 2 mathematics to solve problems connected to everyday experiences and activities in and outside of school. The student is expected to:

(A) identify the mathematics in everyday situations;

(B) solve problems with guidance [ use a problem-solving model ] that incorporates the processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;

(C) select or develop an appropriate problem-solving plan or strategy including drawing a picture, looking for a pattern, systematic guessing and checking, or acting it out in order to solve a problem; and

(D) use tools such as real objects, manipulatives, and technology to solve problems.

(13) Underlying processes and mathematical tools. The student communicates about Grade 2 mathematics using informal language. The student is expected to:

(A) explain and record observations using objects, words, pictures, numbers, and technology; and

(B) relate informal language to mathematical language and symbols.

(14) Underlying processes and mathematical tools. The student uses logical reasoning [ to make sense of his or her world ]. The student is expected to justify [ reason and support ] his or her thinking using objects, words, pictures, numbers, and technology.

§111.15.Mathematics, Grade 3.

(a) Introduction.

(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 3 are multiplying and dividing whole numbers, connecting fraction symbols to fractional quantities, and standardizing language and procedures in geometry and measurement.

(2) Throughout mathematics in Grades 3-5, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use algorithms for addition, subtraction, multiplication, and division as generalizations connected to concrete experiences; and they concretely develop basic concepts of fractions and decimals. Students use appropriate language and organizational structures such as tables and charts to represent and communicate relationships, make predictions, and solve problems. Students select and use formal language to describe their reasoning as they identify, compare, and classify two- or three-dimensional geometric figures [ shapes and solids ]; and they use numbers, standard units, and measurement tools to describe and compare objects, make estimates, and solve application problems. Students organize data, choose an appropriate method to display the data, and interpret the data to make decisions and predictions and solve problems.

(3) Throughout mathematics in Grades 3-5, students develop numerical fluency with conceptual understanding and computational accuracy. Students in Grades 3-5 use knowledge of the base-ten place value system to compose and decompose numbers in order to solve problems requiring precision, estimation, and reasonableness. By the end of Grade 5, students know basic addition, subtraction, multiplication, and division facts and are using them to work flexibly, efficiently, and accurately with numbers during addition, subtraction, multiplication, and division computation.

(4) [ (3) ] Problem solving, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Throughout mathematics in Grades 3-5, students use these processes together with technology and other mathematical tools such as manipulative materials to develop conceptual understanding and solve meaningful problems as they do mathematics.

(b) Knowledge and skills.

(1) Number, operation, and quantitative reasoning. The student uses place value to communicate about increasingly large whole numbers in verbal and written form, including money. The student is expected to:

(A) use place value to read, write (in symbols and words), and describe the value of whole numbers through 999,999;

(B) use place value to compare and order whole numbers through 9,999; and

(C) determine the value of a collection of coins and bills.

(2) Number, operation, and quantitative reasoning. The student uses fraction names and symbols (with denominators of 12 or less) to describe fractional parts of whole objects or sets of objects. The student is expected to:

(A) construct concrete models of fractions;

(B) compare fractional parts of whole objects or sets of objects in a problem situation using concrete models;

(C) use fraction names and symbols to describe fractional parts of whole objects or sets of objects [ with denominators of 12 or less ]; and

(D) construct concrete models of equivalent fractions for fractional parts of whole objects.

(3) Number, operation, and quantitative reasoning. The student adds and subtracts to solve meaningful problems involving whole numbers. The student is expected to:

(A) model addition and subtraction using pictures, words, and numbers; and

(B) select addition or subtraction and use the operation to solve problems involving whole numbers through 999.

(4) Number, operation, and quantitative reasoning. The student recognizes and solves problems in multiplication and division situations. The student is expected to:

(A) learn and apply multiplication facts through 12 by 12 [ the tens ] using concrete models and objects ;

(B) solve and record multiplication problems (up to two digits times one digit) [ (one-digit multiplier) ]; and

(C) use models to solve division problems and use number sentences to record the solutions.

(5) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected to:

(A) round whole numbers [ two-digit numbers ] to the nearest ten or hundred to approximate reasonable results in problem situations [ and three-digit numbers to the nearest hundred ]; and

(B) use strategies including rounding and compatible numbers to estimate solutions to addition and subtraction problems.

[ (B) estimate sums and differences beyond basic facts.]

(6) Patterns, relationships, and algebraic thinking. The student uses patterns to solve problems. The student is expected to:

(A) identify and extend whole-number and geometric patterns to make predictions and solve problems;

(B) identify patterns in multiplication facts using concrete objects, pictorial models, or technology; and

(C) identify patterns in related multiplication and division sentences (fact families) such as 2 x 3 = 6, 3 x 2 = 6, 6 ÷ 2 = 3, 6 ÷ 3 = 2.

(7) Patterns, relationships, and algebraic thinking. The student uses lists, tables, and charts to express patterns and relationships. The student is expected to:

(A) generate a table of paired numbers based on a real-life situation such as insects and legs; and

(B) identify and describe patterns in a table of related number pairs based on a meaningful problem [ real-life situation ] and extend the table.

(8) Geometry and spatial reasoning. The student uses formal geometric vocabulary. The student is expected to identify, classify, and [ name, ] describe two- and three-dimensional geometric figures by their attributes. The student compares two-dimensional figures, three-dimensional figures, or both by their attributes [ , and compare shapes and solids ] using formal geometry [ geometric ] vocabulary.

(9) Geometry and spatial reasoning. The student recognizes congruence and symmetry. The student is expected to:

(A) identify congruent two-dimensional figures [ shapes ];

(B) create two-dimensional figures [ shapes ] with lines of symmetry using concrete models and technology; and

(C) identify lines of symmetry in two-dimensional geometric figures [ shapes ].

(10) Geometry and spatial reasoning. The student recognizes that [ numbers can be represented by points on ] a line can be used to represent numbers and fractions and their properties and relationships . The student is expected to locate and name points on a number line using whole numbers and fractions , including [ such as ] halves and fourths .

(11) Measurement. The student directly compares the attributes of length, area, weight/mass, and capacity, and uses comparative language to solve problems and answer questions. The student selects and uses standard units to describe length, area, capacity/volume, and weight/mass.

[ (11) Measurement. The student selects and uses appropriate units and procedures to measure length and area. ] The student is expected to:

(A) use linear measurement tools to estimate and measure lengths using standard units [ such as inch, foot, yard, centimeter, decimeter, and meter ];

(B) use standard units [ linear measure ] to find the perimeter of a shape; [ and ]

(C) use concrete and pictorial models of square units to determine the area of two-dimensional surfaces; [ shapes. ]

(D) identify concrete models that approximate standard units of weight/mass and use them to measure weight/mass;

(E) identify concrete models that approximate standard units for capacity and use them to measure capacity; and

(F) use concrete models that approximate cubic units to determine the volume of a given container or other three-dimensional geometric figure.

(12) Measurement. The student reads and writes time and measures [ time and ] temperature in degrees Fahrenheit to solve problems . The student is expected to:

(A) use a thermometer to measure temperature; and

(B) [ (A) ] tell and write time shown on analog [ traditional ] and digital clocks . [ ; and ]

[ (B) use a thermometer to measure temperature.]

[ (13) Measurement. The student applies measurement concepts. The student is expected to measure to solve problems involving length, area, temperature, and time.]

(13) [ (14) ] Probability and statistics. The student solves problems by collecting, organizing, displaying, and interpreting sets of data. The student is expected to:

(A) collect, organize, record, and display data in pictographs and bar graphs where each picture or cell might represent more than one piece of data;

(B) interpret information from pictographs and bar graphs; and

(C) use data to describe events as more likely than , less likely than , or equally likely as .

(14) [ (15) ] Underlying processes and mathematical tools. The student applies Grade 3 mathematics to solve problems connected to everyday experiences and activities in and outside of school. The student is expected to:

(A) identify the mathematics in everyday situations;

(B) solve problems [ use a problem-solving model ] that incorporate [ incorporates ] understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;

(C) select or develop an appropriate problem-solving plan or strategy, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem; and

(D) use tools such as real objects, manipulatives, and technology to solve problems.

(15) [ (16) ] Underlying processes and mathematical tools. The student communicates about Grade 3 mathematics using informal language. The student is expected to:

(A) explain and record observations using objects, words, pictures, numbers, and technology; and

(B) relate informal language to mathematical language and symbols.

(16) [ (17) ] Underlying processes and mathematical tools. The student uses logical reasoning [ to make sense of his or her world ]. The student is expected to:

(A) make generalizations from patterns or sets of examples and nonexamples; and

(B) justify why an answer is reasonable and explain the solution process.

§111.16.Mathematics, Grade 4.

(a) Introduction.

(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 4 are comparing and ordering fractions and decimals, applying multiplication and division, and developing ideas related to congruence and symmetry.

(2) Throughout mathematics in Grades 3-5, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use algorithms for addition, subtraction, multiplication, and division as generalizations connected to concrete experiences; and they concretely develop basic concepts of fractions and decimals. Students use appropriate language and organizational structures such as tables and charts to represent and communicate relationships, make predictions, and solve problems. Students select and use formal language to describe their reasoning as they identify, compare, and classify two- or three-dimensional geometric figures [ shapes and solids ]; and they use numbers, standard units, and measurement tools to describe and compare objects, make estimates, and solve application problems. Students organize data, choose an appropriate method to display the data, and interpret the data to make decisions and predictions and solve problems.

(3) Throughout mathematics in Grades 3-5, students develop numerical fluency with conceptual understanding and computational accuracy. Students in Grades 3-5 use knowledge of the base-ten place value system to compose and decompose numbers in order to solve problems requiring precision, estimation, and reasonableness. By the end of Grade 5, students know basic addition, subtraction, multiplication, and division facts and are using them to work flexibly, efficiently, and accurately with numbers during addition, subtraction, multiplication, and division computation.

(4) [ (3) ] Problem solving, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Throughout mathematics in Grades 3-5, students use these processes together with technology and other mathematical tools such as manipulative materials to develop conceptual understanding and solve meaningful problems as they do mathematics.

(b) Knowledge and skills.

(1) Number, operation, and quantitative reasoning. The student uses place value to represent whole numbers and decimals. The student is expected to:

(A) use place value to read, write, compare, and order whole numbers through the millions place; and

(B) use place value to read, write, compare, and order decimals involving tenths and hundredths, including money, using concrete objects and pictorial models.

(2) Number, operation, and quantitative reasoning. The student describes and compares fractional parts of whole objects or sets of objects. The student is expected to:

(A) use concrete objects and pictorial models to generate equivalent fractions [ using concrete and pictorial models ];

(B) model fraction quantities greater than one using concrete objects and pictorial models [ materials and pictures ];

(C) compare and order fractions using concrete objects and pictorial models; and

(D) relate decimals to fractions that name tenths and hundredths using concrete objects and pictorial models.

(3) Number, operation, and quantitative reasoning. The student adds and subtracts to solve meaningful problems involving whole numbers and decimals. The student is expected to:

(A) use addition and subtraction to solve problems involving whole numbers; and

(B) add and subtract decimals to the hundredths place using concrete objects and pictorial models.

(4) Number, operation, and quantitative reasoning. The student multiplies and divides to solve meaningful problems involving whole numbers. The student is expected to:

(A) model factors and products using arrays and area models;

(B) represent multiplication and division situations in picture, word, and number form;

(C) recall and apply multiplication facts through 12 x 12;

(D) use multiplication to solve problems (no more than two digits times two digits without technology) [ involving two-digit numbers ]; and

(E) use division to solve problems (no more than one-digit divisors and three-digit dividends without technology) [ involving one-digit divisors ].

(5) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected to:

(A) round whole numbers to the nearest ten, hundred, or thousand to approximate reasonable results in problem situations; and

(B) use strategies including rounding and compatible numbers to estimate solutions to multiplication and division problems.

[ (B) estimate a product or quotient beyond basic facts.]

(6) Patterns, relationships, and algebraic thinking. The student uses patterns in multiplication and division. The student is expected to:

(A) use patterns and relationships to develop strategies to remember basic multiplication and division facts (such as the patterns in related multiplication and division number sentences (fact families) such as 9 x 9 = 81 and 81 ÷ 9 = 9) ; and

[ (B) solve division problems related to multiplication facts (fact families) such as 9 x 9 = 81 and 81 ÷ 9 = 9; and]

(B) [ (C) ] use patterns to multiply by 10 and 100.

(7) Patterns, relationships, and algebraic thinking. The student uses organizational structures to analyze and describe patterns and relationships. The student is expected to describe the relationship between two sets of related data such as ordered pairs in a table.

(8) Geometry and spatial reasoning. The student identifies and describes attributes of geometric figures [ lines, shapes, and solids ] using formal geometric language. The student is expected to:

(A) identify and describe right, acute, and obtuse angles;

(B) identify and describe [ models of ] parallel and intersecting (including perpendicular ) lines using concrete objects and pictorial models ; and

(C) use essential attributes to define two- and three-dimensional geometric figures.

[ (C) describe shapes and solids in terms of vertices, edges, and faces.]

(9) Geometry and spatial reasoning. The student connects transformations to congruence and symmetry. The student is expected to:

(A) demonstrate translations, reflections, and rotations using concrete models;

(B) use translations, reflections, and rotations to verify that two shapes are congruent; and

(C) use reflections to verify that a shape has symmetry.

(10) Geometry and spatial reasoning. The student recognizes the connection between numbers and their properties and points on a [ number ] line. The student is expected to locate and name points on a number line using whole numbers, fractions such as halves and fourths, and decimals such as tenths.

(11) Measurement. The student applies measurement concepts. The student is expected to estimate and measure to solve problems involving length (including perimeter) and area. The student uses measurement tools to measure capacity/volume and weight/mass. The student is expected to:

(A) estimate and use measurement tools to determine length (including perimeter), area, capacity and weight/mass using standard units SI (metric) and customary;

(B) perform simple conversions between different units of length, between different units of capacity, and between different units of weight within the customary measurement system;

(C) use concrete models of standard cubic units to measure volume;

(D) estimate volume in cubic units; and

(E) explain the difference between weight and mass.

[ (11) Measurement. The student selects and uses appropriate units and procedures to measure weight and capacity. The student is expected to:]

[ (A) estimate and measure weight using standard units including ounces, pounds, grams, and kilograms; and]

[ (B) estimate and measure capacity using standard units including milliliters, liters, cups, pints, quarts, and gallons.]

(12) Measurement. The student applies measurement concepts. The student measures time and temperature (in degrees Fahrenheit and Celsius). The student is expected to : [ measure to solve problems involving length, including perimeter, time, temperature, and area. ]

(A) use a thermometer to measure temperature and changes in temperature; and

(B) use tools such as a clock with gears or a stopwatch to solve problems involving elapsed time.

(13) Probability and statistics. The student solves problems by collecting, organizing, displaying, and interpreting sets of data. The student is expected to:

(A) use concrete objects or pictures to make generalizations about determining all possible combinations of a given set of data or of objects in a problem situation; and

[ (A) list all possible outcomes of a probability experiment such as tossing a coin;]

[ (B) use a pair of numbers to compare favorable outcomes to all possible outcomes such as four heads out of six tosses of a coin; and]

(B) [ (C) ] interpret bar graphs.

(14) Underlying processes and mathematical tools. The student applies Grade 4 mathematics to solve problems connected to everyday experiences and activities in and outside of school. The student is expected to:

(A) identify the mathematics in everyday situations;

(B) solve problems [ use a problem-solving model ] that incorporate [ incorporates ] understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;

(C) select or develop an appropriate problem-solving plan or strategy, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem; and

(D) use tools such as real objects, manipulatives, and technology to solve problems.

(15) Underlying processes and mathematical tools. The student communicates about Grade 4 mathematics using informal language. The student is expected to:

(A) explain and record observations using objects, words, pictures, numbers, and technology; and

(B) relate informal language to mathematical language and symbols.

(16) Underlying processes and mathematical tools. The student uses logical reasoning [ to make sense of his or her world ]. The student is expected to:

(A) make generalizations from patterns or sets of examples and nonexamples; and

(B) justify why an answer is reasonable and explain the solution process.

§111.17.Mathematics, Grade 5.

(a) Introduction.

(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 5 are comparing and contrasting lengths, areas [ area ], and volumes [ volume ] of two- or three-dimensional geometric figures [ geometric shapes and solids ]; representing and interpreting data in graphs, charts, and tables; and applying whole number operations in a variety of contexts.

(2) Throughout mathematics in Grades 3-5, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use algorithms for addition, subtraction, multiplication, and division as generalizations connected to concrete experiences; and they concretely develop basic concepts of fractions and decimals. Students use appropriate language and organizational structures such as tables and charts to represent and communicate relationships, make predictions, and solve problems. Students select and use formal language to describe their reasoning as they identify, compare, and classify two- or three-dimensional geometric figures [ shapes and solids ]; and they use numbers, standard units, and measurement tools to describe and compare objects, make estimates, and solve application problems. Students organize data, choose an appropriate method to display the data, and interpret the data to make decisions and predictions and solve problems.

(3) Throughout mathematics in Grades 3-5, students develop numerical fluency with conceptual understanding and computational accuracy. Students in Grades 3-5 use knowledge of the base-ten place value system to compose and decompose numbers in order to solve problems requiring precision, estimation, and reasonableness. By the end of Grade 5, students know basic addition, subtraction, multiplication, and division facts and are using them to work flexibly, efficiently, and accurately with numbers during addition, subtraction, multiplication, and division computation.

(4) [ (3) ] Problem solving, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Throughout mathematics in Grades 3-5, students use these processes together with technology and other mathematical tools such as manipulative materials to develop conceptual understanding and solve meaningful problems as they do mathematics.

(b) Knowledge and skills.

(1) Number, operation, and quantitative reasoning. The student uses place value to represent whole numbers and decimals. The student is expected to:

(A) use place value to read, write, compare, and order whole numbers through the billions place; and

(B) use place value to read, write, compare, and order decimals through the thousandths place.

(2) Number, operation, and quantitative reasoning. The student uses fractions in problem-solving situations. The student is expected to:

(A) generate a fraction equivalent to a given fraction such as 1/2 and 3/6 or 4/12 and 1/3;

[ (A) generate equivalent fractions;]

(B) generate a mixed number equivalent to a given improper fraction or generate an improper fraction equivalent to a given mixed number;

(C) [ (B) ] compare two fractional quantities in problem-solving situations using a variety of methods, including common denominators; and

(D) [ (C) ] use models to relate decimals to fractions that name tenths, hundredths, and thousandths.

(3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems. The student is expected to:

(A) use addition and subtraction to solve problems involving whole numbers and decimals;

(B) use multiplication to solve problems involving whole numbers (no more than three digits times two digits without technology);

(C) use division to solve problems involving whole numbers (no more than two-digit divisors and three-digit dividends without technology) , including interpreting the remainder within a given context ;

(D) identify [ prime factors of a whole number and ] common factors of a set of whole numbers; and

(E) model situations using [ and record ] addition and/or [ and ] subtraction involving [ of ] fractions with like denominators using concrete objects, pictures, words, and numbers [ in problem-solving situations ].

(4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected to use strategies, including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division problems. [ : ]

[ (A) round whole numbers and decimals through tenths to approximate reasonable results in problem situations; and]

[ (B) estimate to solve problems where exact answers are not required.]

(5) Patterns, relationships, and algebraic thinking. The student makes generalizations based on observed patterns and relationships. The student is expected to:

[ (A) use concrete objects or pictures to make generalizations about determining all possible combinations;]

(A) [ (B) ] describe the relationship between sets of data in graphic organizers such as [ use ] lists, tables, charts, and diagrams [ to find patterns and make generalizations such as a procedure for determining equivalent fractions ]; and

(B) [ (C) ] identify prime and composite numbers using concrete objects, pictorial models , and patterns in factor pairs.

(6) Patterns, relationships, and algebraic thinking. The student describes relationships mathematically. The student is expected to select from and use diagrams and equations such as y = 5 + 3 [ number sentences ] to represent meaningful problem [ real-life ] situations.

(7) Geometry and spatial reasoning. The student generates geometric definitions using critical attributes. The student is expected to identify essential attributes including parallel, perpendicular, and congruent parts of two- and three-dimensional geometric figures. [ : ]

[ (A) identify critical attributes including parallel, perpendicular, and congruent parts of geometric shapes and solids; and]

[ (B) use critical attributes to define geometric shapes or solids.]

(8) Geometry and spatial reasoning. The student models transformations. The student is expected to:

(A) sketch the results of translations, rotations, and reflections on a Quadrant I coordinate grid ; and

(B) identify [ describe ] the transformation that generates one figure from the other when given two congruent figures on a Quadrant I coordinate grid .

(9) Geometry and spatial reasoning. The student recognizes the connection between ordered pairs of numbers and locations of points on a plane. The student is expected to locate and name points on a coordinate grid using ordered pairs of whole numbers.

(10) Measurement. The student applies measurement concepts involving length (including perimeter), area, capacity/volume, and weight/mass to solve problems [ selects and uses appropriate units and procedures to measure volume ]. The student is expected to:

(A) perform simple conversions within the same measurement system (SI (metric) or customary);

[ (A) measure volume using concrete models of cubic units; and]

(B) connect models for perimeter, area, and volume with their respective formulas; and

[ (B) estimate volume in cubic units.]

(C) select and use appropriate units and formulas to measure length, perimeter, area, and volume.

(11) Measurement. The student applies measurement concepts. The student measures time and temperature (in degrees Fahrenheit and Celsius). The student is expected to:

(A) solve problems involving changes in temperature; and

[ (A) measure to solve problems involving length (including perimeter), weight, capacity, time, temperature, and area; and]

(B) solve problems involving elapsed time.

[ (B) describe numerical relationships between units of measure within the same measurement system such as an inch is one-twelfth of a foot.]

(12) Probability and statistics. The student describes and predicts the results of a probability experiment. The student is expected to:

(A) use fractions to describe the results of an experiment; [ and ]

(B) use experimental results to make predictions ; and [ . ]

(C) list all possible outcomes of a probability experiment such as tossing a coin.

(13) Probability and statistics. The student solves problems by collecting, organizing, displaying, and interpreting sets of data. The student is expected to:

(A) use tables of related number pairs to make line graphs;

(B) describe characteristics of data presented in tables and graphs including median, mode, and range [ the shape and spread of the data and the middle number ]; and

(C) graph a given set of data using an appropriate graphical representation such as a picture or line graph .

(14) Underlying processes and mathematical tools. The student applies Grade 5 mathematics to solve problems connected to everyday experiences and activities in and outside of school. The student is expected to:

(A) identify the mathematics in everyday situations;

(B) solve problems [ use a problem-solving model ] that incorporate [ incorporates ] understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;

(C) select or develop an appropriate problem-solving plan or strategy, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem; and

(D) use tools such as real objects, manipulatives, and technology to solve problems.

(15) Underlying processes and mathematical tools. The student communicates about Grade 5 mathematics using informal language. The student is expected to:

(A) explain and record observations using objects, words, pictures, numbers, and technology; and

(B) relate informal language to mathematical language and symbols.

(16) Underlying processes and mathematical tools. The student uses logical reasoning [ to make sense of his or her world ]. The student is expected to:

(A) make generalizations from patterns or sets of examples and nonexamples; and

(B) justify why an answer is reasonable and explain the solution process.

This agency hereby certifies that the proposal has been reviewed by legal counsel and found to be within the agency's legal authority to adopt.

Filed with the Office of the Secretary of State on July 25, 2005.

TRD-200503038

Cristina De La Fuente-Valadez

Director, Policy Coordination

Texas Education Agency

Earliest possible date of adoption: September 4, 2005

For further information, please call: (512) 475-1497