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14.23: Section 5.6 Answers - Mathematics


1. (y = 1 − 2x + c_{1}e^{−x} + c_{2}xe^{x} ;: {e^{−x}, xe^{x} })

2. (y=frac{4}{3x^{2}}+c_{1}x+frac{c_{2}}{x};: {x, 1/x})

3. (y=frac{x(ln |x|)^{2}}{2}+c_{1}x+c_{2}xln |x|;: {x, xln |x|})

4. (y = (e^{2x} + e^{x} ) ln(1 + e^{−x}) + c_{1}e^{2x} + c_{2}e^{x};: {e^{2x}, e^{x}})

5. (y=e^{x}left(frac{4}{5}x^{7/2}+c_{1}+c_{2}x ight);: {e^{x}, xe^{x}})

6. (y = e^{x} (2x^{3/2} + x^{1/2} ln x + c_{1}x^{1/2} + c_{2}x^{−1/2} );: {x^{1/2} e^{x}, x^{−1/2}e^{−x}})

7. (y = e^{x} (x sin x + cos x ln | cos x| + c_{1} cos x + c_{2} sin x);: {e^{x} cos x, e^{x} sin x})

8. (y = e ^{−x ^{2}} (2e^{−2x} + c_{1} + c_{2}x);: {e^{ −x^{2}}, xe^{−x ^{2}} })

9. (y = 2x + 1 + c_{1}x^{2} + frac{c_{2}}{x^{2}};: {x^{2}, 1/x^{2} })

10. (y=frac{xe^{2x}}{9}+xe^{-x}(c_{1}+c_{2}x);: {xe^{-x}, x^{2}e^{-x}})

11. (y=xe^{x}left(frac{x}{3}+c_{1}+frac{c_{2}}{x^{2}} ight);: {xe^{x}, e^{x}/x})

12. (y=-frac{(2x-1)^{2}e^{x}}{8}+c_{1}e^{x}+c_{2}xe^{-x};: {e^{x}, xe^{-x}})

13. (y=x^{4}+c_{1}x^{2}+c_{2}x^{2}ln |x|; {x^{2}, x^{2}ln |x|})

14. (y = e^{−x} (x^{3/2} + c_{1} + c_{2}x ^{1/2} );: {e^{−x}, x^{1/2} e^{−x} })

15. (y = e^{x} (x+c_{1}+c_{2}x^{2});: {e^{x}, x^{2}e^{x} })

16. (y=x^{1/2}left(frac{e^{2x}}{2}+c_{1}+c_{2}e^{x} ight);:{x^{1/2}, x^{1/2}e^{x}})

17. (y = −2x^{2} ln x + c_{1}x^{2} + c_{2}x^{4} ;: {x^{2}, x^{4} })

18. ({e^{x}, e^{x}/x})

19. ({x^{2}, x^{3}})

20. ({ln |x|, xln |x|})

21. ({sinsqrt{x},cossqrt{x}})

22. ({e^{x}, x^{3}e^{x}})

23. ({x^{a}, x^{a}ln |x|})

24. ({xsin x,xcos x})

25. ({e^{2x}, x^{2}e^{2x}})

26. ({x^{1/2}, x^{1/2}cos x})

27. ({x^{1/2}e^{2x}, x^{1/2}e^{-2x}})

28. ({1/x, e^{2x}})

29. ({e^{x}, x^{2}})

30. ({e^{2x}, x^{2}e^{2x}})

31. (y=x^{4}=6x^{2}-8x^{2}ln |x|)

32. (y=2e^{2x}-xe^{-x})

33. (y=frac{(x+1)}{4}[-e^{x}(3-2x)+7e^{-x}])

34. (y=frac{x^{2}}{4}+x)

35. (y=frac{(x+2)^{2}}{6(x-2)}+frac{2x}{x^{2}-4})

38.

  1. (y=frac{-kc_{1}sin kx+kc_{2}cos kx}{c_{1}cos kx+c_{2}sin kx})
  2. (y=frac{c_{1}+2c_{2}e^{x}}{c_{1}+c_{2}e^{x}})
  3. (y=frac{-6c_{1}+c_{2}e^{7x}}{c_{1}+c_{2}e^{7x}})
  4. (y=-frac{7c_{1}+c_{2}e^{6x}}{c_{1}+c_{2}e^{6x}})
  5. (y=-frac{(7c_{1}-c_{2})cos x+(c_{1}+7c_{2})sin x}{c_{1}cos x+c_{2}sin x})
  6. (y=frac{-2c_{1}+3c_{2}e^{5x/6}}{6(c_{1}+c_{2}e^{5x/6})})
  7. (y=frac{c_{1}+c_{2}(x+6)}{6(c_{1}+c_{2}x)})

39.

  1. (y=frac{c_{1}+c_{2}e^{x}(1+x)}{x(c_{1}+c_{2}e^{x})})
  2. (y=frac{-2c_{1}x+c_{2}(1-2x^{2})}{c_{1}+c_{2}x})
  3. (y=frac{-c_{1}+c_{2}e^{2x}(x+1)}{c_{1}+c_{2}xe^{2x}})
  4. (y=frac{2c_{1}+c_{2}e^{-3x}(1-x)}{c_{1}+c_{2}xe^{-3x}})
  5. (y=frac{(2c_{2}x-c_{1})cos x-(2c_{1}x+c_{2})sin x}{2x(c_{1}cos x+c_{2}sin x)})
  6. (y=frac{c_{1}+7c_{2}x^{6}}{x(c_{1}+c_{2}x^{6})})

Math 120: Groups and Rings

This class will cover groups, fields, rings, and ideals. More explicitly: Groups acting on sets, examples of finite groups, Sylow theorems, solvable and simple groups. Fields, rings, and ideals polynomial rings over a field PID and non-PID. Unique factorization domains.

Math 120 will be a fast-moving, high-workload class. This is also a Writing in the Major class. Most students interested in this material will find Math 109 (offered in spring quarter) more appropriate.

Professor: Ravi Vakil, 383-Q, vakil-at-math.stanford.edu.

Course assistant: Francois Greer, 381-A, fgreer-at-math.stanford.edu.

Office hours: Tuesday 8-10 am and 11 am-noon (Francois Greer), Wednesday 9-10:30 am (Ravi Vakil), Thursday 8-10 am and 11 am-noon (Francois Greer), Friday 10:30 am - noon (Ravi Vakil). (In the week of the final exam, these office hours are replaced by 12 hours of bonus office hours shown below.)

Text: Dummit and Foote's Abstract Algebra, third edition (be careful to get the right edition!).

Grading scheme:
Short online assignments 5%
Problem sets 20%
Writing in the Major Assignment 15%
Midterm 20%
Final exam 40%

Short online assignments. There will be weekly short online assignments on Coursework. You needn't answer more than one or two sentences per question. This is intended to be low-stress: you will get full marks for any good-faith answer. This is intended to give us feedback on how the reading has been going. Because I will have to review your feedback in a limited time period on Sunday afternoon, the deadlines will be on Sunday at noon sharp.

Email lists and study groups. There is a list of email addresses and dorm locations of people willing to be contacted to discuss homework or to form study groups. If you are interested in being on the list and seeing the list, just let me know.

Problem sets. There will be weekly homework assignments, posted here. You are encouraged to work together to solve problems. But you must write up your solutions individually, and give credit for ideas that others had. You should give complete proofs. Because the graders will need to process a huge volume of homework in a small amount of time: Please staple your homework, and write your name on each page.

Homework due dates: Thursday at 3 pm outside my office. No lates will be allowed (so the graders can just grade one problem set at a time, and hence have a better chance of getting them back promptly). But to give everyone a chance to get sick, or have busy periods, the lowest problem set will be dropped.

  • Problem set 1 is due on Thursday October 9. Do 12 of the following problems, including at least one from each section. Section 1.1: 1, 23, 26. Section 1.2: 5. Section 1.3: 2 (answers only). Section 1.4: 3, 10. Section 1.6: 6, 18, 20. Section 1.7: 3, 4, 17, 23. Section 2.1: 12, 13. Section 2.2: 3, 10. Section 2.3: 3, 21. Section 2.4: 7, 13.
  • Problem set 2 is due on Thursday October 16. Do 12 of the following problems, including at least one from each section in chapter 3. Section 2.3: 9, 23. Section 3.1: 3, 6, 9, 16, 22, 24, 25, 27, 41, 42. Section 3.2: 4, 8, 10, 11, 17. Section 3.3: 1, 3, 4, 7, 9. Section 3.4: 4, 5, 6.
  • Problem set 3 is due on Thursday October 23. Do 12 of the following problems, including at least one from each section. Section 3.5: 9, 10, 12. Section 4.1: 1, 4, 9, 10. Section 4.2: 1, 6, 8, 10, 11, 12. Section 4.3: 2a, 4, 5.
  • Problem set 4 is due on Thursday October 30. Do 11 of the following, including at least one from each section. Section 4.3: 17, 22, 25, 26, 29, 33. Section 4.4: 1, 2, 3, 8, 16, 18 (worth 2 problems), 19 (worth 2 problems). Section 4.5: 3, 7, 30.
  • Problem set 5 is due on Thursday November 20. Do 11 of the following, including at least one from each section. Section 4.6: 1, 6. Section 5.4: 2, 5, 15. Section 5.5: 2, 8. Section 7.1: 14, 23, 26. Section 7.2: 3, 12. Section 7.3: 10, 13, 18, 19, 22, 30, 34.
  • Problem set 6 is due on Thursday December 4. Do 10 of the following problems, including at least one from each section excpt 7.5. Section 7.4: 8, 9, 10, 15. Section 7.5: 5 (worth 2). Section 7.6: 1, 5 (worth 2), 7. Section 8.1: 1(d)+2(b) (together worth 1), 3, 6, 7, 12. 8.2: 3, 5. Also, for each of the 6 problem sets, pick one problem you particularly liked, and explain why in a sentence or two.

Writing in the major assignment. Clear writing is essential to mathematical communication, as you probably realize from reading better and worse mathematical texts. Clarity in writing may be more important in mathematics than in any other science, for a number of reasons. Good exposition is an acquired and important skill. Throughout this class, you'll received feedback on solutions to your problem sets, and you should use this to refine your ability to communicate your ideas clearly and effectively. This writing project will give you an opportunity to focus on your exposition, as opposed to absorbing new mathematical content. This course will emphasize both exposition in communciating mathematics and the structure of proofs. Part of your grade on each assignment and on the exams will be on your exposition of your solutions to problems. Information about the Writing in the Major assignment is here. Deadlines to be aware of: Fri. Oct. 31, Friday Nov. 7, Fri. Nov. 21.

Miscellaneous: Josh Meisel (who took 120 in 2009) found this link about a Futurama episode whose plot involved group theory. To my home page.


§ 23.573 Damage tolerance and fatigue evaluation of structure.

(a) Composite airframe structure. Composite airframe structure must be evaluated under this paragraph instead of §§23.571 and 23.572. The applicant must evaluate the composite airframe structure, the failure of which would result in catastrophic loss of the airplane, in each wing (including canards, tandem wings, and winglets), empennage, their carrythrough and attaching structure, moveable control surfaces and their attaching structure fuselage, and pressure cabin using the damage-tolerance criteria prescribed in paragraphs (a)(1) through (a)(4) of this section unless shown to be impractical. If the applicant establishes that damage-tolerance criteria is impractical for a particular structure, the structure must be evaluated in accordance with paragraphs (a)(1) and (a)(6) of this section. Where bonded joints are used, the structure must also be evaluated in accordance with paragraph (a)(5) of this section. The effects of material variability and environmental conditions on the strength and durability properties of the composite materials must be accounted for in the evaluations required by this section.

(1) It must be demonstrated by tests, or by analysis supported by tests, that the structure is capable of carrying ultimate load with damage up to the threshold of detectability considering the inspection procedures employed.

(2) The growth rate or no-growth of damage that may occur from fatigue, corrosion, manufacturing flaws or impact damage, under repeated loads expected in service, must be established by tests or analysis supported by tests.

(3) The structure must be shown by residual strength tests, or analysis supported by residual strength tests, to be able to withstand critical limit flight loads, considered as ultimate loads, with the extent of detectable damage consistent with the results of the damage tolerance evaluations. For pressurized cabins, the following loads must be withstood:

(i) Critical limit flight loads with the combined effects of normal operating pressure and expected external aerodynamic pressures.

(ii) The expected external aerodynamic pressures in 1g flight combined with a cabin differential pressure equal to 1.1 times the normal operating differential pressure without any other load.

(4) The damage growth, between initial detectability and the value selected for residual strength demonstrations, factored to obtain inspection intervals, must allow development of an inspection program suitable for application by operation and maintenance personnel.

(5) For any bonded joint, the failure of which would result in catastrophic loss of the airplane, the limit load capacity must be substantiated by one of the following methods&mdash

(i) The maximum disbonds of each bonded joint consistent with the capability to withstand the loads in paragraph (a)(3) of this section must be determined by analysis, tests, or both. Disbonds of each bonded joint greater than this must be prevented by design features or

(ii) Proof testing must be conducted on each production article that will apply the critical limit design load to each critical bonded joint or

(iii) Repeatable and reliable non-destructive inspection techniques must be established that ensure the strength of each joint.

(6) Structural components for which the damage tolerance method is shown to be impractical must be shown by component fatigue tests, or analysis supported by tests, to be able to withstand the repeated loads of variable magnitude expected in service. Sufficient component, subcomponent, element, or coupon tests must be done to establish the fatigue scatter factor and the environmental effects. Damage up to the threshold of detectability and ultimate load residual strength capability must be considered in the demonstration.

(b) Metallic airframe structure. If the applicant elects to use §23.571(c) or §23.572(a)(3), then the damage tolerance evaluation must include a determination of the probable locations and modes of damage due to fatigue, corrosion, or accidental damage. Damage at multiple sites due to fatigue must be included where the design is such that this type of damage can be expected to occur. The evaluation must incorporate repeated load and static analyses supported by test evidence. The extent of damage for residual strength evaluation at any time within the operational life of the airplane must be consistent with the initial detectability and subsequent growth under repeated loads. The residual strength evaluation must show that the remaining structure is able to withstand critical limit flight loads, considered as ultimate, with the extent of detectable damage consistent with the results of the damage tolerance evaluations. For pressurized cabins, the following load must be withstood:

(1) The normal operating differential pressure combined with the expected external aerodynamic pressures applied simultaneously with the flight loading conditions specified in this part, and

(2) The expected external aerodynamic pressures in 1g flight combined with a cabin differential pressure equal to 1.1 times the normal operating differential pressure without any other load.

[Doc. No. 26269, 58 FR 42163, Aug. 6, 1993 58 FR 51970, Oct. 5, 1993, as amended by Amdt. 23-48, 61 FR 5147, Feb. 9, 1996 73 FR 19746, Apr. 11, 2008]

[Doc. No. 26269, 58 FR 42163, Aug. 6, 1993 58 FR 51970, Oct. 5, 1993, as amended by Amdt. 23-48, 61 FR 5147, Feb. 9, 1996 73 FR 19746, Apr. 11, 2008


2013 New Jersey Revised StatutesTitle 23 - FISH AND GAME, WILD BIRDS AND ANIMALSSection 23:2B-14 - "Act" defined, penalties enforcement.

73.For purposes of this section, the "act" means and includes all the new sections and amended sections contained herein, all the remaining sections of Title 50 of the Revised Statutes, sections 23:3-41, 23:3-46, 23:3-47, 23:3-48, 23:3-51, 23:3-52, 23:5-9, 23:5-16, 23:9-114, 23:9-115 and 23:9-120 of Title 23 of the Revised Statutes, sections 1, 2, and 3 of P.L.1941, c.211 (C.23:5-24.1 to 23:5-24.3), and sections 4 through 17 of P.L.2013, c.74 (C.23:3-51.2 through C.23:3-51.15).

The commissioner may utilize any or all of the following remedies for any violation of this act:

a. (1) A person who violates the provisions of this act or of any rule, regulation, license or permit adopted or issued pursuant to this act shall be liable to a penalty of not less than $300 or more than $3,000 for the first offense and not less than $500 or more than $5,000 for any subsequent offense, unless the commissioner has established an alternate penalty for a specific offense pursuant to paragraph (2) of this subsection.

(2)The Commissioner of Environmental Protection, with the approval of the Marine Fisheries Council, may, by regulation, establish a penalty schedule for any specific violation of this act or of any rule or regulation adopted pursuant to this act. No such penalty may be less than $30 or more than $100 for the first offense or less than $50 or more than $200 for any subsequent offense. Any penalty provided for by this act or by the fee schedule adopted by the commissioner shall be collected in a civil action by a summary proceeding pursuant to the "Penalty Enforcement Law of 1999," P.L.1999, c.274 (C.2A:58-10 et seq.). The Superior Court or any municipal court shall have jurisdiction to enforce the "Penalty Enforcement Law of 1999." If the violation is of a continuing nature, each day during which it continues shall constitute an additional, separate and distinct offense.

b. (1) A person who violates the provisions of this act or any rule or regulation or any license or permit adopted or issued pursuant to this act shall be liable to the revocation or suspension of any license or permit held by the violator pursuant to this act for such period of time as the court may choose, except when the rule or regulation violated includes a mandatory revocation or suspension schedule in which case that schedule shall determine the period of time of the revocation or suspension.

(2)In the case of a person who knowingly violates the provisions of R.S.50:2-11,R.S.50:3-13, R.S.50:4-2 or R.S.50:4-3, or any rule or regulation or any license or permit adopted or issued pursuant thereto, the violator shall be subject to a mandatory revocation or suspension of the license or permit or privilege for the taking of oysters, clams or other shellfish held by the violator, or in the case of a violator without the necessary license or permit the loss of the privilege to take oysters, clams or other shellfish, for a period of three years for a first offense, five years for a second offense, and 10 years for the third and any subsequent offense. For purposes of this paragraph, a knowing violator shall include, but need not be limited to: (a) a person who is the holder of a commercial shellfish license or permit (b) a person who is the holder of a recreational shellfish license or permit who is in possession of more than the daily recreational shellfish limit (c) a person without any shellfish license or permit who is in possession of more than the daily recreational shellfish limit (d) a person who is the holder of a recreational shellfish license or permit who is engaging in shellfish activities with the holder of a commercial shellfish license or permit or (e) a person without the necessary shellfish license or permit who is engaging in shellfish activities with the holder of a commercial shellfish license or permit.

c.If a person violates any of the provisions of this act, or any rule or regulation or any license or permit adopted or issued pursuant to this act, the department may institute a civil action in a court of competent jurisdiction for injunctive relief to prohibit and prevent the violation or violations and the court may proceed in the action in a summary manner.

The department is hereby authorized and empowered to compromise and settle any claim for a penalty under this section in such amount in the discretion of the department as may appear appropriate and equitable under all of the circumstances.

d.In addition to the penalties prescribed by this section, a person who violates the provisions of R.S.50:2-11, R.S.50:3-13, R.S.50:4-2 or R.S.50:4-3, or any rule or regulation or any license or permit adopted or issued pursuant thereto, shall be subject to the forfeiture of any vessel, vehicle, or equipment used in the commission of the violation. A designated conservation officer of the Department of Environmental Protection, a member of the State Police, or any other law enforcement officer may seize and secure any vessel, vehicle, or equipment used in the commission of such a violation. Upon the seizure of the vessel, vehicle, or equipment, the conservation officer, member of the State Police, or other law enforcement officer shall immediately thereafter institute a civil action to determine if the forfeiture is warranted in the court in which the penalty action was filed pursuant to this section, which court shall have jurisdiction to adjudicate the forfeiture action. The owner or any person having a security interest in the vessel, vehicle, or equipment may secure its release by depositing with the clerk of the court in which the action is pending a bond with good and sufficient sureties in an amount to be fixed by the court, conditioned upon the return of the vessel, vehicle, or equipment to the Department of Environmental Protection upon demand after completion of the court proceeding. The court may proceed in a summary manner and may direct the confiscation of the vessel, vehicle, or equipment by the department for its use or for disposal by sale or public auction. Moneys collected by the department through the sale or public auction of the vessel, vehicle, or equipment shall be used by the Division of Fish and Wildlife for the enforcement of the provisions of this act.

L.1979, c.199, s.73 amended 1983, c.219, s.1 1991, c.91, s.279 2003, c.8, s.1 2013, c.74, s.1.

Disclaimer: These codes may not be the most recent version. New Jersey may have more current or accurate information. We make no warranties or guarantees about the accuracy, completeness, or adequacy of the information contained on this site or the information linked to on the state site. Please check official sources.



Problem A
(Grade 3/4)
Problem B
(Grade 5/6)
Problem C
(Grade 7/8)
Problem D
(Grade 9/10)
Problem E
(Grade 11/12)
Week 30Problem PDF | HTML PDF | HTML PDF | HTML PDF | HTML PDF | HTML
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Week 11Problem PDF | HTML PDF | HTML PDF | HTML PDF | HTML PDF | HTML
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Week 10Problem PDF | HTML PDF | HTML PDF | HTML PDF | HTML PDF | HTML
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Week 08Problem PDF | HTML PDF | HTML PDF | HTML PDF | HTML PDF | HTML
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Week 06Problem PDF | HTML PDF | HTML PDF | HTML PDF | HTML PDF | HTML
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Week 05Problem PDF | HTML PDF | HTML PDF | HTML PDF | HTML PDF | HTML
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Week 04Problem PDF | HTML PDF | HTML PDF | HTML PDF | HTML PDF | HTML
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Week 03Problem PDF | HTML PDF | HTML PDF | HTML PDF | HTML PDF | HTML
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Week 02Problem PDF | HTML PDF | HTML PDF | HTML PDF | HTML PDF | HTML
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Week 01Problem PDF | HTML PDF | HTML PDF | HTML PDF | HTML PDF | HTML
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Each of the following links is to a booklet containing all the problems from particular years with solutions. The problems are organized into strands, grouping problems into various areas of the curriculum. A problem often appears in multiple strands.


News & Updates

Monday March 29th, 2021 Spelling Distance Learning Assignments Hey all,
So today something that has been in the works for a while and I think it's (finally) ready for the public.
Now on the Spelling Worksheet Maker you'll see an option for 'Distance Learning'.
Just click on that and instead of outputing worksheets it'll generate distance learning assignments.
And as a bonus, any previously created spelling list is compatible with the distance learning option. Huzzah!

I've been testing and tweaking it for a few months now, so I'm absolutely 100% positive there are no bugs or glitches.
That being said. when you find any bug or glitches please let me know in the comments. :P

Easter Coloring Sheets Also just in time for easter there are a few coloring sheets available here:
Coloring Sheets

Robert Smith (Admin)
[email protected]


RECORDINGS

Welcome to this webinar about coding in the mathematics curriculum. We will be taking you through the pedagogy and social emotional learning skills that coding brings to your classroom as well as the lessons that have been created as supports for Ontario teachers in the primary grades.
NOTE: This recording does not yet contain the Q&A section - those will be added soon from all the recordings with captioning.

Welcome to this webinar about coding in the mathematics curriculum. We will be taking you through the pedagogy and social emotional learning skills that coding brings to your classroom as well as the lessons that have been created as supports for Ontario teachers in the Junior grades.
NOTE: This recording does not yet contain the Q&A section - those will be added soon from all the recordings with captioning.

Welcome to this webinar about teaching the new coding expectations for the intermediate grades of the revised mathematics curriculum. We will be taking you through the lessons that have been created as supports for Ontario educators as well as the pedagogy and social emotional learning skills that coding brings to your classroom.
NOTE: This recording does not yet contain the Q&A section - those will be added soon from all the recordings with captioning.

Welcome to this webinar about teaching the Number expectations for the primary grades of the revised mathematics curriculum. In this webinar, we take you through the lessons that have been created by Ontario educators to support Ontario educators. These Number lessons have a focus on developing fraction concepts and skills. These lessons benefit from the latest research about learning fractions, including an emphasis on high-yield visual representations.

Welcome to this webinar about teaching the Number expectations for the junior grades of the revised mathematics curriculum. In this webinar, we take you through the lessons that have been created by Ontario educators to support Ontario educators. These Number lessons have a focus on developing fraction concepts and skills. These lessons benefit from the latest research about learning fractions, including an emphasis on high-yield visual representations.

Welcome to this webinar about teaching the Number expectations for the intermediate grades of the revised mathematics curriculum. In this webinar, we take you through the lessons that have been created by Ontario educators to support Ontario educators. These Number lessons have a focus on developing fraction concepts and skills. These lessons benefit from the latest research about learning fractions, including an emphasis on high-yield visual representations.

Welcome to this webinar about teaching the new Financial Literacy expectations for the primary grades of the revised mathematics curriculum. In this webinar, we take you through the lessons that have been created by Ontario educators to support Ontario educators. In developing the Financial Literacy lessons for this project, the intentional decision to use scenarios-based tasks was made. The goal of the lessons were to provide real-life contexts for students to develop their understanding of money concepts, as an individual and a community.

Welcome to this webinar about teaching the new Financial Literacy expectations for the junior grades of the revised mathematics curriculum. In this webinar, we take you through the lessons that have been created by Ontario educators to support Ontario educators. In developing the Financial Literacy lessons for this project, the intentional decision to use scenarios was made. Scenarios give the opportunity to have a discussion about financial choices and the inequities of access to those choices. The goal was to provide students with a chance to learn and understand the individual skills needed that impact personal financial choices and also the economic literacy needed to understand the impact of financial systems as a whole.

Welcome to this webinar about teaching the new Financial Literacy expectations for the intermediate grades of the revised mathematics curriculum. In this webinar, we take you through the lessons that have been created by Ontario educators to support Ontario educators. In developing the Financial Literacy lessons for this project, the intentional decision to use scenarios was made. Scenarios give the opportunity to have a discussion about financial choices and the inequities of access to those choices. The goal was to provide students with a chance to learn and understand the individual skills needed that impact personal financial choices and also the economic literacy needed to understand the impact of financial systems as a whole.

Welcome to this webinar about teaching the new Mathematical Modelling expectations for the primary grades of the revised mathematics curriculum. In this webinar, we take you through the lessons that have been created by Ontario educators to support Ontario educators. These Mathematical Modelling lessons have a focus on developing students&rsquo appreciation for the role mathematics plays in real &ldquoreal life&rdquo while providing students with opportunities to experience mathematics in a more personal and relevant way.

Welcome to this webinar about teaching the new Mathematical Modelling expectations for the junior grades of the revised mathematics curriculum. In this webinar, we take you through the lessons that have been created by Ontario educators to support Ontario educators. These Mathematical Modelling lessons have a focus on developing students&rsquo appreciation for the role mathematics plays in real &ldquoreal life&rdquo while providing students with opportunities to experience mathematics in a more personal and relevant way.

Welcome to this webinar about teaching the new Mathematical Modelling expectations for the intermediate grades of the revised mathematics curriculum. In this webinar, we take you through the lessons that have been created by Ontario educators to support Ontario educators. These Mathematical Modelling lessons have a focus on developing students&rsquo appreciation for the role mathematics plays in real &ldquoreal life&rdquo while providing students with opportunities to experience mathematics in a more personal and relevant way.

This hour-long webinar will outline the vision, goals and strands of the revised Ontario Elementary Mathematics Curriculum. In this webinar, we will unpack these very important sections of the Curriculum Context of the revised curriculum, providing educators with an introduction as to how this "front matter" has expanded and what these changes mean to educators as they plan their math programs. We will also give an overview of the changes in the curriculum, particularly as they relate to the strands.

Also available:
* the slides for this presentation, in Google Slides format.
* links and resources document, "Resources for Curriculum Context Webinar 1" , comprising links to research, videos, audio, and websites which may be of use to you as you continue to explore the Vision, Goals, and Strands in the Revised Elementary Math Curriculum (2020). [Google Doc]

This hour-long webinar will focus on program planning for the revised Elementary Mathematics Curriculum. In this webinar, we will explore the contents of the Program Planning section of the &ldquoCurriculum Context&rdquo, and provide educators with an introduction to what these changes mean to educators as they plan their math programs.

Also available:
* the slides for this presentation, in Google Slides format.
* links and resources document, "Resources for Curriculum Context Webinar 2" , comprising links to research, videos, audio, and websites which may be of use to you as you continue to learn more about the considerations for program planning found in the Revised Elementary Math Curriculum (2020). [Google Doc]

This hour-long webinar will focus on Mathematical Modelling. In this webinar, we will explore in detail what the mathematical modelling process looks like and how it is truly new to the curriculum. We hope to deepen educators&rsquo understanding of what Mathematical Modelling is and how it can be woven into a math program. Sample tasks will be discussed.

Also available:
* the slides for this presentation, in Google Slides format.
* links and resources document, "Resources for Curriculum Context Webinar 3", comprising links to research, videos, audio, and websites which may be of use to you as you continue to learn more about the Mathematical Modelling process.

This hour-long webinar focuses on Social-Emotional Learning Skills in Mathematics & the Mathematical Processes. Examine this strand in more detail, and look at connecting SEL Skills, Mathematical Processes, and assessment and instruction of the remaining five strands.

Also available:
* link and resources document, "Resources for Curriculum Context Webinar 4: Strand A - Social-Emotional Learning Skills in Mathematics and the Mathematical Processes", you will find links to research, videos, audio, and websites which may be of use to you as you continue to learn more about SES and the Mathematical Processes.

This hour-long webinar focuses on Assessment and Evaluation. Revisit assessment and evaluation of student learning in light of the revised elementary math curriculum and its expanded Curriculum Context.

Also available:
* link and resources document, "Resources for Curriculum Context Webinar 5: Assessment and Evaluation", you will find links to research, videos, audio, and websites which may be of use to you as you continue to learn more about this topic.


Official FAQs on DBE Program Regulations (49 CFR 23)

The General Counsel of the Department of Transportation has reviewed these questions and answers and approved them as consistent with the language and intent of 49 CFR Part 23. These questions and answers therefore represent the institutional position of the Department of Transportation.

These questions and answers provide guidance and information for compliance with the provisions under 49 CFR part 23, pertaining to the implementation of the Department's disadvantaged business enterprise program. Like all guidance material, these questions and answers are not, in themselves, legally binding or mandatory, and do not constitute regulations. They are issued to provide an acceptable means, but not the only means, of compliance with Part 23. While these questions and answers are not mandatory, they are derived from extensive DOT, recipient, and contractor experience and input concerning the determination of compliance with Part 23.

Section 23.75 - Can recipients enter into long-term, exclusive agreements with concessionaires?

The purpose of this guidance on Long-Term, Exclusive (LTE) agreements for concessions is to provide information and direction to airport sponsors, ACDBE program staff, stakeholders, and all other interested parties on how to determine whether an agreement is considered a LTE agreement subject to the prohibition against such agreements in the ACDBE program rules.

Section 23.31 27.67(b)(2) - Personal Net Worth

Question: If the owner of a DBE or ACDBE certified firm or applicant firm has a personal net worth of less than $1.32 million, does that necessarily mean that the recipient must regard the owner as being economically disadvantaged?

Answer: No. A person cannot be regarded as economically disadvantaged if he or she exceeds the $1.32 million personal net worth (PNW) cap. However, there may be some cases in which an individual whose PNW is less than $1.32 million may properly be regarded as not being economically disadvantaged.

  • The legal and policy rationale behind the PNW provision of the rule is that a program designed to assist socially and economically disadvantaged individuals should not include people who can reasonably be regarded as having accumulated wealth too substantial to need the program’s assistance.
  • Consequently, in determining whether an individual is economically disadvantaged, a recipient is entitled to look not only at the individual’s PNW but also at his or her overall economic situation to make a reasonable determination of whether the individual is fairly regarded as being economically disadvantaged.
  • Consistent with Small Business Administration practice in the 8(a) program, it is appropriate for recipients to review the total fair market value of the individual’s assets and determine if that level appears to be substantial and indicates an ability to accumulate substantial wealth.
  • For example, an individual with very high assets and significant liabilities may, in accounting terms, have a PNW of less than $750,000. However, the person’s assets (e.g., a very expensive house, a yacht, extensive real or personal property holdings) may lead to a conclusion that he or she is not economically disadvantaged. The recipient can rebut the individual’s presumption of economic disadvantage under these circumstances, as provided in sec. 26.67(b)(2).
  • This guidance applies to determinations of economic disadvantage under both 49 CFR Part 23 and 49 CFR Part 26.

Section 23.37 12/9/2011

Question: May a recipient or UCP require a firm certified as a DBE under Part 26 to complete an entire new application to be certified as an ACDBE under part 23?

Answer: No. Section 23.37 provides that as a recipient or UCP, you are required to presume that a firm certified in your State as a DBE under Part 26 meets size, disadvantage, ownership, and control requirements for certification as an ACDBE under Part 23.

  • A recipient or UCP must check to make sure a Part 23 applicant meets the Part 23 PNW cap.
  • There is only one additional determination that you need to make in order to certify such a firm as an ACDBE: whether the disadvantaged owners of the Part 26-certified firm can control the activities of the firm with respect to its participation in the ACDBE program.
  • You are not required to certify a DBE firm as an ACDBE if the firm does not do work relevant to the airport concessions program (e.g., operating a concession or providing goods and services to concessions).
  • In summary, when you receive an ACDBE application that has a current Part 26 DBE certification in your State, you should only seek information on two subjects: the firm owner’s PNW and the disadvantaged owner’s ability to control the firm with respect to airport concession activities. You may not require the firm to file a complete new application.

Section 23.55 06/12/2015

Question: How do recipients count ACDBE participation that involves the sale of advertising displays or messages to the public on the airport?

Answer: The primary function of an advertising concession is selling advertising displays or messages at an airport pursuant to a written agreement with the airport owner, with another concessionaire, or with the owner or lessor of a terminal (if other than the recipient).


14.23: Section 5.6 Answers - Mathematics

(a) General rule.--Subject to subsection (a.1), the court may grant any protection order or approve any consent agreement to bring about a cessation of abuse of the plaintiff or minor children. The order or agreement may include:

(1) Directing the defendant to refrain from abusing the plaintiff or minor children.

(2) Granting possession to the plaintiff of the residence or household to the exclusion of the defendant by evicting the defendant or restoring possession to the plaintiff if the residence or household is jointly owned or leased by the parties, is owned or leased by the entireties or is owned or leased solely by the plaintiff.

(3) If the defendant has a duty to support the plaintiff or minor children living in the residence or household and the defendant is the sole owner or lessee, granting possession to the plaintiff of the residence or household to the exclusion of the defendant by evicting the defendant or restoring possession to the plaintiff or, with the consent of the plaintiff, ordering the defendant to provide suitable alternate housing.

(4) Awarding temporary custody of or establishing temporary visitation rights with regard to minor children. In determining whether to award temporary custody or establish temporary visitation rights pursuant to this paragraph, the court shall consider any risk posed by the defendant to the children as well as risk to the plaintiff. The following shall apply:

(i) A defendant shall not be granted custody, partial custody or unsupervised visitation where it is alleged in the petition, and the court finds after a hearing under this chapter, that the defendant:

(A) abused the minor children of the parties or poses a risk of abuse toward the minor children of the parties or

(B) has been convicted of violating 18 Pa.C.S. § 2904 (relating to interference with custody of children) within two calendar years prior to the filing of the petition for protection order or that the defendant poses a risk of violating 18 Pa.C.S. § 2904.

(ii) Where the court finds after a hearing under this chapter that the defendant has inflicted abuse upon the plaintiff or a child, the court may require supervised custodial access by a third party. The third party must agree to be accountable to the court for supervision and execute an affidavit of accountability.

(iii) Where the court finds after a hearing under this chapter that the defendant has inflicted serious abuse upon the plaintiff or a child or poses a risk of abuse toward the plaintiff or a child, the court may:

(A) award supervised visitation in a secure visitation facility or

(B) deny the defendant custodial access to a child.

(iv) If a plaintiff petitions for a temporary order under section 6107(b) (relating to hearings) and the defendant has partial, shared or full custody of the minor children of the parties by order of court or written agreement of the parties, the custody shall not be disturbed or changed unless the court finds that the defendant is likely to inflict abuse upon the children or to remove the children from the jurisdiction of the court prior to the hearing under section 6107(a). Where the defendant has forcibly or fraudulently removed any minor child from the care and custody of a plaintiff, the court shall order the return of the child to the plaintiff unless the child would be endangered by restoration to the plaintiff.

(v) Nothing in this paragraph shall bar either party from filing a petition for custody under Chapter 53 (relating to custody) or under the Pennsylvania Rules of Civil Procedure.

(vi) In order to prevent further abuse during periods of access to the plaintiff and child during the exercise of custodial rights, the court shall consider, and may impose on a custody award, conditions necessary to assure the safety of the plaintiff and minor children from abuse.

(5) After a hearing in accordance with section 6107(a), directing the defendant to pay financial support to those persons the defendant has a duty to support, requiring the defendant, under sections 4324 (relating to inclusion of medical support) and 4326 (relating to mandatory inclusion of child medical support), to provide health coverage for the minor child and spouse, directing the defendant to pay all of the unreimbursed medical expenses of a spouse or minor child of the defendant to the provider or to the plaintiff when he or she has paid for the medical treatment, and directing the defendant to make or continue to make rent or mortgage payments on the residence of the plaintiff to the extent that the defendant has a duty to support the plaintiff or other dependent household members. The support order shall be temporary, and any beneficiary of the order must file a complaint for support under the provisions of Chapters 43 (relating to support matters generally) and 45 (relating to reciprocal enforcement of support orders) within two weeks of the date of the issuance of the protection order. If a complaint for support is not filed, that portion of the protection order requiring the defendant to pay support is void. When there is a subsequent ruling on a complaint for support, the portion of the protection order requiring the defendant to pay support expires.

(6) Prohibiting the defendant from having any contact with the plaintiff or minor children, including, but not limited to, restraining the defendant from entering the place of employment or business or school of the plaintiff or minor children and from harassing the plaintiff or plaintiff's relatives or minor children.

(7) Prohibiting the defendant from acquiring or possessing any firearm for the duration of the order, ordering the defendant to temporarily relinquish to the sheriff or the appropriate law enforcement agency any firearms under the defendant's possession or control, and requiring the defendant to relinquish to the sheriff or the appropriate law enforcement agency any firearm license issued under section 6108.3 (relating to relinquishment to third party for safekeeping) or 18 Pa.C.S. § 6106 (relating to firearms not to be carried without a license) or 6109 (relating to licenses) the defendant may possess. The court may also order the defendant to relinquish the defendant's other weapons or ammunition that have been used or been threatened to be used in an incident of abuse against the plaintiff or the minor children. A copy of the court's order shall be transmitted to the chief or head of the appropriate law enforcement agency and to the sheriff of the county of which the defendant is a resident. When relinquishment is ordered, the following shall apply:

(i) (A) The court's order shall require the defendant to relinquish such firearms, other weapons, ammunition and any firearm license pursuant to the provisions of this chapter within 24 hours of service of a temporary order or the entry of a final order or the close of the next business day as necessary by closure of the sheriffs' offices, except for cause shown at the hearing, in which case the court shall specify the time for relinquishment of any or all of the defendant's firearms.

(B) A defendant subject to a temporary order requiring the relinquishment of firearms, other weapons or ammunition shall, in lieu of relinquishing specific firearms, other weapons or ammunition which cannot reasonably be retrieved within the time for relinquishment in clause (A) due to their current location, provide the sheriff or the appropriate law enforcement agency with an affidavit listing the firearms, other weapons or ammunition and their current location. If the defendant, within the time for relinquishment in clause (A), fails to provide the affidavit or fails to relinquish, pursuant to this chapter, any firearms, other weapons or ammunition ordered to be relinquished which are not specified in the affidavit, the sheriff or the appropriate law enforcement agency shall, at a minimum, provide immediate notice to the court, the plaintiff and appropriate law enforcement authorities. The defendant shall not possess any firearms, other weapons or ammunition specifically listed in the affidavit provided to the sheriff or the appropriate law enforcement agency pursuant to this clause for the duration of the temporary order.

(C) As used in this subparagraph, the term "cause" shall be limited to facts relating to the inability of the defendant to retrieve a specific firearm within 24 hours due to the current location of the firearm.

(ii) The court's order shall contain a list of any firearm, other weapon or ammunition ordered relinquished. Upon the entry of a final order, the defendant shall inform the court in what manner the defendant is going to relinquish any firearm, other weapon or ammunition ordered relinquished. Relinquishment may occur pursuant to section 6108.2 (relating to relinquishment for consignment sale, lawful transfer or safekeeping) or 6108.3 or to the sheriff or the appropriate law enforcement agency pursuant to this paragraph. Where the sheriff or the appropriate law enforcement agency is designated, the sheriff or the appropriate law enforcement agency shall secure custody of the defendant's firearms, other weapons or ammunition and any firearm license listed in the court's order for the duration of the order or until otherwise directed by court order. In securing custody of the defendant's relinquished firearms, the sheriff or the appropriate law enforcement agency shall comply with 18 Pa.C.S. § 6105(f)(4) (relating to persons not to possess, use, manufacture, control, sell or transfer firearms). In securing custody of the defendant's other weapons and ammunition, the sheriff or the appropriate law enforcement agency shall provide the defendant with a signed and dated written receipt which shall include a detailed description of the other weapon or ammunition and its condition. The court shall inform the defendant that firearms, other weapons or ammunition shall be deemed abandoned when the conditions under 18 Pa.C.S. § 6128(a) (relating to abandonment of firearms, weapons or ammunition) are satisfied and may then be disposed of in accordance with 18 Pa.C.S. § 6128.

(iii) The sheriff or the appropriate law enforcement agency shall provide the plaintiff with the name of the person to which any firearm, other weapon or ammunition was relinquished.

(iv) Unless the defendant has complied with subparagraph (i)(B) or section 6108.2 or 6108.3, if the defendant fails to relinquish any firearm, other weapon, ammunition or firearm license within 24 hours or upon the close of the next business day due to closure of sheriffs' or appropriate law enforcement agencies' offices or within the time ordered by the court upon cause being shown at the hearing, the sheriff or the appropriate law enforcement agency shall, at a minimum, provide immediate notice to the court, the plaintiff and appropriate law enforcement agencies, as appropriate.

(v) Any portion of any order or any petition or other paper which includes a list of any firearm, other weapon or ammunition ordered relinquished shall be kept in the files of the court as a permanent record thereof and withheld from public inspection except:

(A) upon an order of the court granted upon cause shown

(B) as necessary, by law enforcement and court personnel or

(C) after redaction of information listing any firearm, other weapon or ammunition.

(vi) As used in this paragraph, the term "defendant's firearms" shall, if the defendant is a licensed firearms dealer, only include firearms in the defendant's personal firearms collection pursuant to 27 CFR § 478.125a (relating to personal firearms collection).

(7.1) If the defendant is a licensed firearms dealer, ordering the defendant to follow such restrictions as the court may require concerning the conduct of his business, which may include ordering the defendant to relinquish any Federal or State license for the sale, manufacture or importation of firearms as well as firearms in the defendant's business inventory. In restricting the defendant pursuant to this paragraph, the court shall make a reasonable effort to preserve the financial assets of the defendant's business while fulfilling the goals of this chapter.

(8) Directing the defendant to pay the plaintiff for reasonable losses suffered as a result of the abuse, including medical, dental, relocation and moving expenses counseling loss of earnings or support costs of repair or replacement of real or personal property damaged, destroyed or taken by the defendant or at the direction of the defendant and other out-of-pocket losses for injuries sustained. In addition to out-of-pocket losses, the court may direct the defendant to pay reasonable attorney fees. An award under this chapter shall not constitute a bar to litigation for civil damages for injuries sustained from the acts of abuse giving rise to the award or a finding of contempt under this chapter.

(9) Directing the defendant to refrain from stalking or harassing the plaintiff and other designated persons as defined in 18 Pa.C.S. §§ 2709 (relating to harassment) and 2709.1 (relating to stalking).

(10) Granting any other appropriate relief sought by the plaintiff.

(a.1) Final order or agreement.--The following apply:

(1) Any final order must direct the defendant to refrain from abusing, harassing, stalking, threatening or attempting or threatening to use physical force against the plaintiff or minor children and must order that the defendant is subject to the firearms, other weapons or ammunition and firearms license prohibition relinquishment provisions under subsection (a)(7).

(2) A final agreement may direct the defendant to refrain from abusing, harassing, stalking, threatening or attempting or threatening to use physical force against the plaintiff or minor children and may order that the defendant is subject to the firearms, other weapons or ammunition and firearms license prohibition and relinquishment provisions under subsection (a)(7).

(b) Identifying information.--Any order issued under this section shall, where furnished by either party, specify the Social Security number and date of birth of the defendant.

(c) Mutual orders of protection.--Mutual orders of protection shall not be awarded unless both parties have filed timely written petitions, complied with service requirements under section 6106 (relating to commencement of proceedings) and are eligible for protection under this chapter. The court shall make separate findings and, where issuing orders on behalf of both petitioners, enter separate orders.

(d) Duration and amendment of order or agreement.--A protection order or approved consent agreement shall be for a fixed period of time not to exceed three years. The court may amend its order or agreement at any time upon subsequent petition filed by either party.

(e) Extension of protection orders.--

(1) An extension of a protection order may be granted:

(i) Where the court finds, after a duly filed petition, notice to the defendant and a hearing, in accordance with the procedures set forth in sections 6106 and 6107, that the defendant committed one or more acts of abuse subsequent to the entry of the final order or that the defendant engaged in a pattern or practice that indicates continued risk of harm to the plaintiff or minor child.

(ii) When a contempt petition or charge has been filed with the court or with a hearing officer in Philadelphia County, but the hearing has not occurred before the expiration of the protection order, the order shall be extended, at a minimum, until the disposition of the contempt petition and may be extended for another term beyond the disposition of the contempt petition.

(iii) If the plaintiff files a petition for an extension of the order and the defendant is or was incarcerated and will be released from custody in the next 90 days or has been released from custody within the past 90 days. The plaintiff does not need to show that the defendant committed one or more acts of abuse subsequent to the entry of the order or that the defendant engaged in a pattern or practice that indicates continued risk of harm to the plaintiff or minor children as set forth in subparagraph (i).

(2) Service of an extended order shall be made in accordance with section 6109 (relating to service of orders).

(3) There shall be no limitation on the number of extensions that may be granted.

(f) Support procedure.--The domestic relations section shall enforce any support award in a protection order where the plaintiff files a complaint for support under subsection (a)(5).

(g) Notice.--Notice shall be given to the defendant, in orders issued under this section, stating that violations of an order will subject the defendant to arrest under section 6113 (relating to arrest for violation of order) or contempt of court under section 6114 (relating to contempt for violation of order or agreement). Resumption of coresidency on the part of the plaintiff and defendant shall not nullify the provisions of the court order.

(h) Title to real property unaffected.--No order or agreement under this chapter shall in any manner affect title to any real property.

(i) Third parties and affidavits.--A court requiring relinquishment of firearms under this section shall provide for the hearing of petitions by third parties who request the return of a firearm relinquished by the defendant under subsection (a)(7). The following apply:

(1) A third party claiming to be the lawful owner of a firearm relinquished by the defendant under subsection (a)(7) may request the return of the firearm by providing proof of ownership and a sworn affidavit.

(2) The affidavit under paragraph (1) must affirm all of the following:

(i) The third party who is the lawful owner will not intentionally or knowingly return to the defendant the firearm or allow access to the firearm by the defendant.

(ii) The third party who is the lawful owner understands that violating subparagraph (i) constitutes a misdemeanor of the second degree under 18 Pa.C.S. Ch. 61 (relating to firearms and other dangerous articles).

(iii) If the third party who is the lawful owner is a family or household member of the defendant, any firearm returned under this section must be stored in a gun safe to which the defendant does not have access and will not be permitted to access, or stored in a location outside the third party's home to which the defendant does not have access.

(3) If the court orders the return of a firearm under this section, prior to the return of the firearm, the sheriff shall independently confirm that the person seeking relief under this section is legally eligible to possess firearms under Federal and State law. The sheriff shall conduct the background check as soon as practicable after the court enters an order under this section.

(June 23, 1993, P.L.124, No.28, eff. imd. Oct. 6, 1994, P.L.574, No.85, eff. 60 days Dec. 16, 1997, P.L.549, No.58, eff. 60 days May 10, 2000, P.L.35, No.10, eff. imd. Dec. 9, 2002, P.L.1759, No.218, eff. 60 days Nov. 10, 2005, P.L.335, No.66, eff. 180 days Oct. 12, 2018, P.L.519, No.79, eff. 180 days)

2018 Amendment. Act 79 amended subsec. (a) intro. par. and (7) and added subsecs. (a.1), (e)(1)(iii) and (i). Act 79 shall apply to orders issued pursuant to 23 Pa.C.S. § 6108 on or after the effective date of section 11 of Act 79.

References in Text. Chapter 45, referred to in subsec. (a)(5), was repealed by the act of April 4, 1996, P.L.58, No.20. The subject matter is now contained in Parts VIII and VIII-A of this title.

Former Chapter 53 (Custody), referred to in this section, is repealed. The subject matter is now contained in Chapter 53 (Child Custody).

Cross References. Section 6108 is referred to in sections 6105, 6107, 6108.2, 6108.3, 6108.5, 6108.7, 6110, 6113, 6117, 6120, 6303, 6711 of this title sections 2709, 2709.1, 6105, 6106, 6128 of Title 18 (Crimes and Offenses).


Ratios & Proportions & Problem Solving is a is a grade 5/6 math workbook that covers ratios and proportions.

Some of the topics covered are:

  • ratios and fractions
  • ratios and rates
  • aspect ratios
  • equivalent rates
  • cross-multiplying
  • scaling figures
  • proportional relationships
  • significant digits
  • problem solving with bar models
  • ratio problems and bar models.

This math workbook is divided into 24 sections. Each section begins with a bite-sized introduction to a topic with an example, followed by practice exercises including word problems. Answers are in the back. The format is ideal for independent or parent-guided study . The Math Mammoth series of workbooks is highly recommended by K5 Learning!


Watch the video: ΜΑΘΗΜΑΤΙΚΑ ΛΥΣΕΙΣ ΠΑΝΕΛΛΗΝΙΩΝ 2021 ΘΕΜΑ Γ (November 2021).