# 12.2E: Exercises - Mathematics

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### Practice Makes Perfect

Exercise (PageIndex{19}) Find and Evaluate Composite Functions

In the following exercises, find

1. ((f circ g)(x))
2. ((g circ f)(x))
3. ((f cdot g)(x))
1. (f(x)=4 x+3) and (g(x)=2 x+5)
2. (f(x)=3 x-1) and (g(x)=5 x-3)
3. (f(x)=6 x-5) and (g(x)=4 x+1)
4. (f(x)=2 x+7) and (g(x)=3 x-4)
5. (f(x)=3 x) and (g(x)=2 x^{2}-3 x)
6. (f(x)=2 x) and (g(x)=3 x^{2}-1)
7. (f(x)=2 x-1) and (g(x)=x^{2}+2)
8. (f(x)=4 x+3) and (g(x)=x^{2}-4)

1.

1. (8x+23)
2. (8x+11)
3. (8 x^{2}+26 x+15)

3.

1. (24x+1)
2. (24x-19)
3. (24x^{2}+19x-5)

5.

1. (6 x^{2}-9 x)
2. (18 x^{2}-9 x)
3. (6 x^{3}-9 x^{2})

7.

1. (2 x^{2}+3)
2. (4 x^{2}-4 x+3)
3. (2 x^{3}-x^{2}+4 x-2)

Exercise (PageIndex{20}) Find and Evaluate Composite Functions

In the following exercises, find the values described.

1. For functions (f(x)=2 x^{2}+3) and (g(x)=5x-1), find
1. ((f circ g)(-2))
2. ((g circ f)(-3))
3. ((f circ f)(-1))
2. For functions (f(x)=5 x^{2}-1) and (g(x)=4x−1), find
1. ((f circ g)(1))
2. ((g circ f)(-1))
3. ((f circ f)(2))
3. For functions (f(x)=2x^{3}) and (g(x)=3x^{2}+2), find
1. ((f circ g)(-1))
2. ((g circ f)(1))
3. ((g circ g)(1))
4. For functions (f(x)=3 x^{3}+1) and (g(x)=2 x^{2}=3), find
1. ((f circ g)(-2))
2. ((g circ f)(-1))
3. ((g circ g)(1))

1.

1. (245)
2. (104)
3. (53)

3.

1. (250)
2. (14)
3. (77)

Exercise (PageIndex{21}) Determine Whether a Function is One-to-One

In the following exercises, determine if the set of ordered pairs represents a function and if so, is the function one-to-one.

1. (egin{array}{l}{{(-3,9),(-2,4),(-1,1),(0,0)}, {(1,1),(2,4),(3,9) }}end{array})
2. (egin{array}{l}{{(9,-3),(4,-2),(1,-1),(0,0)}, {(1,1),(4,2),(9,3) }}end{array})
3. (egin{array}{l}{{(-3,-5),(-2,-3),(-1,-1)}, {(0,1),(1,3),(2,5),(3,7) }}end{array})
4. (egin{array}{l}{{(5,3),(4,2),(3,1),(2,0)}, {(1,-1),(0,-2),(-1,-3) }}end{array})

1. Function; not one-to-one

3. One-to-one function

Exercise (PageIndex{22}) Determine Whether a Function is One-to-One

In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.

1.

1. Figure 10.1.65

2. Figure 10.1.66

2.

1. Figure 10.1.67

2. Figure 10.1.68

3.

1. Figure 10.1.69

2. Figure 10.1.70

4.

1. Figure 10.1.71

2. Figure 10.1.72

1.

1. Not a function
2. Function; not one-to-one

3.

1. One-to-one function
2. Function; not one-to-one

Exercise (PageIndex{23}) Determine Whether a Function is One-to-One

In the following exercises, find the inverse of each function. Determine the domain and range of the inverse function.

1. ({(2,1),(4,2),(6,3),(8,4)})
2. ({(6,2),(9,5),(12,8),(15,11)})
3. ({(0,-2),(1,3),(2,7),(3,12)})
4. ({(0,0),(1,1),(2,4),(3,9)})
5. ({(-2,-3),(-1,-1),(0,1),(1,3)})
6. ({(5,3),(4,2),(3,1),(2,0)})

1. (egin{array}{l}{ ext { Inverse function: }{(1,2),(2,4),(3,6),(4,8)} . ext { Domain: }{1,2,3,4} . ext { Range: }} {{2,4,6,8} .}end{array})

3. (egin{array}{l}{ ext { Inverse function: }{(-2,0),(3,1),(7,2),(12,3)} . ext { Domain: }{-2,3,7,12} ext { . }} { ext { Range: }{0,1,2,3}}end{array})

5. (egin{array}{l}{ ext { Inverse function: }{(-3,-2),(-1,-1),(1,0),(3,1)} . ext { Domain: }} {{-3,-1,1,3} . ext { Range: }{-2,-1,0,1}}end{array})

Exercise (PageIndex{24}) Determine Whether a Function is One-to-One

In the following exercises, graph, on the same coordinate system, the inverse of the one-to-one function shown.

1. Figure 10.1.73

2. Figure 10.1.74

3. Figure 10.1.75

4. Figure 10.1.76

1.

3.

Exercise (PageIndex{25}) Determine Whether the given functions are inverses

In the following exercises, determine whether or not the given functions are inverses.

1. (f(x)=x+8) and (g(x)=x-8)
2. (f(x)=x-9) and (g(x)=x+9)
3. (f(x)=7 x) and (g(x)=frac{x}{7})
4. (f(x)=frac{x}{11}) and (g(x)=11 x)
5. (f(x)=7 x+3) and (g(x)=frac{x-3}{7})
6. (f(x)=5 x-4) and (g(x)=frac{x-4}{5})
7. (f(x)=sqrt{x+2}) and (g(x)=x^{2}-2)
8. (f(x)=sqrt[3]{x-4}) and (g(x)=x^{3}+4)

1. (g(f(x))=x,) and (f(g(x))=x,) so they are inverses.

3. (g(f(x))=x,) and (f(g(x))=x,) so they are inverses.

5. (g(f(x))=x,) and (f(g(x))=x,) so they are inverses.

7. (g(f(x))=x,) and (f(g(x))=x,) so they are inverses (for nonnegative (x ))

Exercise (PageIndex{26}) Determine the inverse of a function

In the following exercises, find the inverse of each function.

1. (f(x)=x-12)
2. (f(x)=x+17)
3. (f(x)=9 x)
4. (f(x)=8 x)
5. (f(x)=frac{x}{6})
6. (f(x)=frac{x}{4})
7. (f(x)=6 x-7)
8. (f(x)=7 x-1)
9. (f(x)=-2 x+5)
10. (f(x)=-5 x-4)
11. (f(x)=x^{2}+6, x geq 0)
12. (f(x)=x^{2}-9, x geq 0)
13. (f(x)=x^{3}-4)
14. (f(x)=x^{3}+6)
15. (f(x)=frac{1}{x+2})
16. (f(x)=frac{1}{x-6})
17. (f(x)=sqrt{x-2}, x geq 2)
18. (f(x)=sqrt{x+8}, x geq-8)
19. (f(x)=sqrt[3]{x-3})
20. (f(x)=sqrt[3]{x+5})
21. (f(x)=sqrt[4]{9 x-5}, x geq frac{5}{9})
22. (f(x)=sqrt[4]{8 x-3}, x geq frac{3}{8})
23. (f(x)=sqrt[5]{-3 x+5})
24. (f(x)=sqrt[5]{-4 x-3})

1. (f^{-1}(x)=x+12)

3. (f^{-1}(x)=frac{x}{9})

5. (f^{-1}(x)=6 x)

7. (f^{-1}(x)=frac{x+7}{6})

9. (f^{-1}(x)=frac{x-5}{-2})

11. (f^{-1}(x)=sqrt{x-6})

13. (f^{-1}(x)=sqrt[3]{x+4})

15. (f^{-1}(x)=frac{1}{x}-2)

17. (f^{-1}(x)=x^{2}+2, x geq 0)

19. (f^{-1}(x)=x^{3}+3)

21. (f^{-1}(x)=frac{x^{4}+5}{9}, x geq 0)

23. (f^{-1}(x)=frac{x^{5}-5}{-3})

Exercise (PageIndex{27}) Writing Exercises

1. Explain how the graph of the inverse of a function is related to the graph of the function.
2. Explain how to find the inverse of a function from its equation. Use an example to demonstrate the steps.

## Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

b. If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

For the linear equation (y = a + b ext), (b =) slope and (a = y)-intercept. From algebra recall that the slope is a number that describes the steepness of a line, and the (y)-intercept is the (y) coordinate of the point ((0, a)) where the line crosses the (y)-axis.

0 and so the line slopes upward to the right. For the second, b = 0 and the graph of the equation is a horizontal line. In the third graph, (c), b < 0 and the line slopes downward to the right." src="http://cnx.org/resources/917c2e46d01. ch12_03_01.jpg" style="width: 725px height: 170px"> Figure (PageIndex<3>):​​​​​​. Three possible graphs of (y = a + b ext) (a) If (b > 0), the line slopes upward to the right. (b) If (b = 0), the line is horizontal. (c) If (b < 0), the line slopes downward to the right.

Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of $25 plus$15 per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is (y = 25 + 15 ext).

What are the independent and dependent variables? What is the (y)-intercept and what is the slope? Interpret them using complete sentences.

The independent variable ((x)) is the number of hours Svetlana tutors each session. The dependent variable ((y)) is the amount, in dollars, Svetlana earns for each session.

The (y)-intercept is 25 ((a = 25)). At the start of the tutoring session, Svetlana charges a one-time fee of $25 (this is when (x = 0)). The slope is 15 ((b = 15)). For each session, Svetlana earns$15 for each hour she tutors.

Ethan repairs household appliances like dishwashers and refrigerators. For each visit, he charges $25 plus$20 per hour of work. A linear equation that expresses the total amount of money Ethan earns per visit is (y = 25 + 20 ext).

What are the independent and dependent variables? What is the (y)-intercept and what is the slope? Interpret them using complete sentences.

The independent variable ((x)) is the number of hours Ethan works each visit. The dependent variable ((y)) is the amount, in dollars, Ethan earns for each visit.

The y-intercept is 25 ((a = 25)). At the start of a visit, Ethan charges a one-time fee of $25 (this is when (x = 0)). The slope is 20 ((b = 20)). For each visit, Ethan earns$20 for each hour he works.

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## Impossible Polyhedra

Imagine making a polyhedron by taking polygons and fixing them together along their edges. We need four triangles to make a tetrahedron, six squares to make a cube and so on. There are five regular polyhedra and each of these is made entirely out of triangular, square or pentagonal faces. But what about irregular polyhedra? Is it possible, for example, to make an irregular polyhedron using only polygons of, say, six, seven and eight sides? The answer (rather surprisingly) is 'no', but how do we prove a statement like this?

We will need Euler's formula for the polyhedron this says that if the polyhedron has $F$ faces, $E$ edges, and $V$ vertices, then $F-E+V=2$. We shall also need to count the number of triangular faces, say $F_3$, the number of faces with exactly four edges, say $F_4$, and so on. Obviously, eginF= F_3 + F_4 + ldots, quad (1) end and also as every edge lies on the edge of exactly two faces, we have egin2E = 3F_3 + 4F_4 + 5F_5 + ldots . quad (2) end This shows that $2E geq 3F$ because egin2E = 3F_3 + 4F_4 + 5F_5 + ldots geq 3F_3 + 3F_4 + 3F_5 + ldots = 3F. end Next, let $V_m$ be the number of vertices which have exactly $m$ edges ending at that vertex. Then eginV= V_3 + V_4 + ldots,quad (3) end and also as every edge has two vertices, we have egin2E = 3V_3+4V_4+5V_5+ldots . quad (4) end This gives the inequality $2E geq 3V$ because egin2E = 3V_3+4V_4+5V_5+ldots geq 3V_3+3V_4+3V_5+ldots = 3V.end We are going to show that eginF_3+F_4+F_5 geq 4, quad F_3+V_3 geq 8. quad (5) end This tells us, for example, that not all of $F_3$, $F_4$ and $F_5$ can be zero, so there must be at least one face with exactly three, four of five sides. Thus it is indeed impossible to make a polyhedron with each face having at least six sides. We know that eginF-E+V = 2, quad 2E geq 3F, quad 2E geq 3V,end so that egin2+E = F+V leq F + 2E/3.end This shows that $2+E/3 leq F$ and hence that $12+2E leq 6F$. If we substitute for $F$ and for $E$ their expressions in terms of the $F_k$ given in (1) and (2), we get egin12 + 3F_3 + 4F_4 + 5F_5 + ldots leq 6F_3 + 6F_4 + 6F_5 + ldots.end This gives egin12+F_7+2F_8+3F_9+ldots leq 3F_3+2F_4+F_5end so that, finally, $12 leq 3F_3 + 2F_4 + F_5$. This is the first inequality in (5).

To prove the second inequality in (5), we start with Euler's formula $F-E+V=2$ and write this as $(F-E/2)+(V-E/2) = 2$. Multiplying throughout by $4$, we now get $(4F-2E)+(4V-2E) = 8$. If we now replace $4F-2E$ by their expressions in terms of the $F_i$ in (1) and (2), and $4V-2E$ by the expressions in (3) and (4), we get egin4(F_3 + F_4+ldots) -(3F_3+4F_4+ldots) +4(V_3 + V_4+ldots) -(3V_3+4V_4+ldots) =8.end

Simplifying this we see that $F_3+V_3 geq 8$ which is the second inequality in (5).

For the cube, we have $F_3 = 0$ and $V_3=8$, so that $F_3+V_3=8$. As $F_3+V_3geq 8$ in all cases, $8$ is the smallest possible value of $F_3+V_3$. This leads to the question of finding other polyhedra that give other values of $F_3$ and $V_3$ with $F_3+V_3=8$ (that is, its smallest value). We leave this as an exercise for the reader, and suggest that you try the regular polyhedra first.

The inequalities (5) give us a lot of information about regular polyhedra. Suppose that we have a regular polyhedron. Because $F_3+F_4+F_5 geq 4$, there must be at least four faces which are triangular, quadrilateral, or pentagonal. As the polyhedron is regular, all faces have the same number of sides, and so all are triangles, all are squares, or all are pentagons.

In the case, of squares and pentagons, $F_3=0$ so that $V_3geq 8$. So, by regularity, every vertex has exactly three edges leaving it. These polyhedra are the cube and the dodecahedron.

It remains to consider the cases when every face is trianglular. Suppose that there are $q$ edges that leave each vertex. Then $F-E+V=2$, $2E = 3F$ and $2E =qV$. If we eliminate $F$ and $V$ from these equations we get $(6-q)E= 6q$ so that $q = 3,4,5$ giving the tetrahedron, octahedron and icosahedron, respectively. We have now listed all possible regular polyhedra.

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## 12.2E: Exercises - Mathematics

Showing results 1 to 20 of 27
1 2

in Retos (2007), 12(2e semestre), 5-12

Forty-six PE teachers participated in the study. Among them, 27 had an experience of more than five years in teaching (12 women and 15 men) and 19 beginning teachers (12 women and 7 men). Teachers were . [more ▼]

Forty-six PE teachers participated in the study. Among them, 27 had an experience of more than five years in teaching (12 women and 15 men) and 19 beginning teachers (12 women and 7 men). Teachers were presented five scenarios of instructional situations. They were asked to read each scenario and write a response detailing how they would react if the situation occurred during their teaching. Four hundred thirty-four proposals of actions were gathered and analysed since an answer could have many different proposals. The actions proposed were analysed from audiotapes of teachers’ interviews. They were classified according to an inductive technique. The opinions of experienced teachers, men and women, differed than those of their beginning colleagues, as indicated by the number and content of categories. Four categories of decisions dealt with the content of education: «adaptation of the exercises to the student», «creation of levels groups», «gradation of the difficulty», and «same activity for all.» The teachers had strategies immediately available to deal with the students’ individual differences. According to their personal characteristics, experienced teachers displayed a greater variety of application of sound principles of teaching. [less ▲]

Detailed reference viewed: 90 (7 ULiège)

in Carlier, Ghislain (Ed.) Si l’on parlait du plaisir d’enseigner l’éducation physique (2004)

Detailed reference viewed: 265 (11 ULiège)

Detailed reference viewed: 241 (7 ULiège)

in Point sur la Recherche en Education (2003), 27

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in eJournal de la Recherche sur l'Intervention en Éducation Physique et Sport (2002), 1

Le sport-études a été créé afin de permettre aux jeunes ayant de bonnes dispositions sportives de les développer dans le cadre d’un programme intégré à leur formation scolaire. Si les modalités pratiques . [more ▼]

Le sport-études a été créé afin de permettre aux jeunes ayant de bonnes dispositions sportives de les développer dans le cadre d’un programme intégré à leur formation scolaire. Si les modalités pratiques diffèrent largement d’une structure à l’autre selon la législation en vigueur, les sportifs sont inévitablement confrontés à des problèmes communs. Bizarrement, peu d’études semblent s’y être intéressées. Par ailleurs, les élèves qui partagent la formation scolaire des sportifs constituent un groupe auquel il n’est jamais fait référence. [less ▲]

Detailed reference viewed: 1911 (20 ULiège)

In physical education settings, the development of intrinsic motivation in students is a central concern for every teacher. However, most studies have focused on isolated variables using quantitative . [more ▼]

In physical education settings, the development of intrinsic motivation in students is a central concern for every teacher. However, most studies have focused on isolated variables using quantitative approaches. The purpose of this study was to underline the existence of relationships between students’ intrinsic motivation (assessed by dispositional factors) and individual/collective situational factors. A secondary school boys’ class and its teacher were the subjects of the study. Questionnaires and interviews were used. Data were initially processed from the class point of view. Based on a cluster analysis determined by intrinsic motivation level, two groups of students were compared. Qualitative analysis was based on the answers of the two most and the two least motivated students of the class. Extracurricular practice (individual situational variable), teacher-students’ interaction and mastery climate (collective situational variables) were identified among the factors related to students’ intrinsic motivation. Inter-individual diversity of influencing variables was pointed out. [less ▲]

Detailed reference viewed: 140 (4 ULiège)

in International Journal of Physical Education (2002), 39(1), 12-20

In French, the term "intervention" is used since a few years. The paper will: (1) deal with the topic of instruction limites to teachers' decisions and interventions during the interactive phase in . [more ▼]

In French, the term "intervention" is used since a few years. The paper will: (1) deal with the topic of instruction limites to teachers' decisions and interventions during the interactive phase in teaching during this phase, activities planned in the pre-interactive phase are implemented: the interactive phase concerns the task's presentation, the pupils' activity and the control of this activity in terms of behavioural management and of the quality of the eprformance (feedback) (2) refer to the main research methods used in gathering data, and (3) develop the main findings in the domain of instruction: teacher thinking and behaviour, pupils' thoughts and behaviour, management of the class or in other words control of pupils' behaviour. It is evident that thougts and behaviours are closely related and are part of the relationship between teachers and pupils. The paper focused only on texts appearing in French, Portuguese and Spanish published since 1997 in Physical eeducation journals and Proceeddings of international congresses. [less ▲]

Detailed reference viewed: 72 (2 ULiège)

in Chin, K. N. Jwo, H. (Eds.) AIESEP Taiwan 2001 International Conference Proceedings: The Exchange and Development of Sport Culture in East and West (2001)

There is no doubt that motivaton of schoolchildren is a concern for physical education teachers. Research data shows that usually physical education is pretty well considered by students. An European . [more ▼]

There is no doubt that motivaton of schoolchildren is a concern for physical education teachers. Research data shows that usually physical education is pretty well considered by students. An European study of youth lifestyle showed that in many countries, 12 and 15 year old students like physical education (Piéron et al., 1996). Percentage over 70% of these youth answered that they liked physical education lessons . [less ▲]

Detailed reference viewed: 105 (1 ULiège)

in Torralbo Lanza, R. Mazón Cobo, V. Sarabia Lavín, D. (Eds.) et al La enseñanza de la educación física y el deporte escolar. (2001)

Detailed reference viewed: 98 (1 ULiège)

The goals of the study were : (1) to determine if team sport players are able to recall verbal interventions provided by the coach during the match (2) to analyze the characteristics of those that are . [more ▼]

The goals of the study were : (1) to determine if team sport players are able to recall verbal interventions provided by the coach during the match (2) to analyze the characteristics of those that are well recalled. We followed one volleyball team and one basketball team during the 1998-1999 season. Both teams competed in the second Belgian division and belonged to the best clubs of the region of Liege. Data were collected during four matches. Players were interviewed after each match in order to collect their opinions about the interventions emitted by the coach. [less ▲]

Detailed reference viewed: 170 (3 ULiège)

in Revue de l'Education Physique (2000), 40(1), 35-43

Dans le cadre des activités de recherche du service de Pédagogie des activités physiques et sportives de l’Université de Liège, les auteurs ont effectué une large étude portant sur le traitement . [more ▼]

Dans le cadre des activités de recherche du service de Pédagogie des activités physiques et sportives de l’Université de Liège, les auteurs ont effectué une large étude portant sur le traitement différencié des élèves par les enseignements en éducation physique. Cet article présente un aspect des résultats qui n’avaient pas encore été abordés. Il fut préparé à partir d’une étude intitulée «Analyse de la prise en considération des caractéristiques individuelles des élèves dans les décisions et les comportements d'enseignants experts et débutants» subventionnée par la Communauté française de Belgique, Administration de l’Enseignement et de la Recherche Scientifique. [less ▲]

Detailed reference viewed: 721 (11 ULiège)

in Revue de l'Education Physique (2000), 40(2), 61-72

Les auteurs font partie du service de Pédagogie des activités physiques et sportives de l’Université de Liège. Ils ont effectué une étude portant sur le traitement différencié des élèves par les . [more ▼]

Les auteurs font partie du service de Pédagogie des activités physiques et sportives de l’Université de Liège. Ils ont effectué une étude portant sur le traitement différencié des élèves par les enseignements en éducation physique, subventionnée par la Communauté française de Belgique, Administration de l’Enseignement et de la Recherche Scientifique. [less ▲]

Detailed reference viewed: 169 (5 ULiège)

in Revue de l'Education Physique (2000), 40(4), 173-180

L'étude fait partie du projet de recherche 157/1997 de la Communauté française, projet confié au service de Pédagogie des activités physiques et sportives de l'Université de Liège. D'autres aspects de . [more ▼]

L'étude fait partie du projet de recherche 157/1997 de la Communauté française, projet confié au service de Pédagogie des activités physiques et sportives de l'Université de Liège. D'autres aspects de cette recherche furent présentés dans les numéros 1, 2 et 3 du volume 40 de la Revue de l'Education Physique. [less ▲]

Detailed reference viewed: 497 (13 ULiège)

in Revue de l'Education Physique (2000), 40(3), 119-129

S'intéresser aux avis des acteurs de la relation pédagogique sur ce qu'ils vivent en classe représente une démarche particulièrement utile pour mieux apprécier la signification des comportements qu'ils . [more ▼]

S'intéresser aux avis des acteurs de la relation pédagogique sur ce qu'ils vivent en classe représente une démarche particulièrement utile pour mieux apprécier la signification des comportements qu'ils ont adoptés. Il est clair qu'à l'issue d'une leçon d'éducation physique, il est hautement souhaitable que les élèves éprouvent autant de satisfaction que possible. Une estimation globale de satisfaction doit être complétée de perceptions plus spécifiques relatives à l'intensité de la leçon, aux sentiments de compétence en relation avec ce qui fut réalisé, aux corrections reçues, au progrès réalisé, au sentiment de s'être donné à fond, de s'être amusé et d'avoir prêté attention aux interventions de l'enseignant. [less ▲]

Detailed reference viewed: 209 (5 ULiège)

in Gréhaigne, J. F. Mahut, N. Marchal, D. (Eds.) Qu'apprennent les élèves en faisant des activités physiques et sportives?/What do people learn from physical activity program? Actes sur CD Rom du Congrès International de l'AIESEP (1999)

Detailed reference viewed: 45 (5 ULiège)

in Revue de l'Education Physique (1999), 39(1), 25-36

Lorsqu'on tente de comprendre les comportements observés en classe, aborder la réflexion, les décisions et les valeurs des intervenants permet de mieux interpréter ce que l'on a observé. Cette démarche . [more ▼]

Lorsqu'on tente de comprendre les comportements observés en classe, aborder la réflexion, les décisions et les valeurs des intervenants permet de mieux interpréter ce que l'on a observé. Cette démarche est évidemment délicate puisqu'il s'agit d'entrer dans un domaine sujet à des problèmes de crédibilité. Il est crucial de distinguer les valeurs et croyances, d'une part, et les prises de décisions, d'autre part. Celles-ci concernent deux moment de la relation pédagogique, . [less ▲]

Detailed reference viewed: 77 (2 ULiège)

in Informations Pédagogiques : le point sur la Recherche en Education (1998), 6

Dans le cadre d'une Ecole de la Réussite, il importe que l'action pédagogique repose sur une démarche individualisée (Ministère de l'Education, 1996). En effet, les caractéristiques d'un apprenant . [more ▼]

Dans le cadre d'une Ecole de la Réussite, il importe que l'action pédagogique repose sur une démarche individualisée (Ministère de l'Education, 1996). En effet, les caractéristiques d'un apprenant interviennent sur l'efficacité des stimuli pédagogiques. Le niveau d'habileté figure parmi les variables le plus souvent prises en considération à ce propos dans l'enseignement des activités physiques et sportives. En outre, on peut s'attendre à ce que les enseignants en éducation physique soient amenés à tenir compte de l'écolution des processus de motivation dans l'enseignement. L'article traite des résultats spécifiques au niveau de l'enseignement primaire. Ils ont été fournis par un projet de recherche qui concernait également le secondaire (Piéron, Cloes, Luts, Ledent, Pirottin & Delfosse, 1998). [less ▲]

Detailed reference viewed: 144 (7 ULiège)

in Revue de l'Education Physique (1998), 38(1), 11-24

Afin de recueillir une image valide tu traitement différencié des élèves dans des leçons d'éducation physique, il est primordial de proposer une approche multidimensionnelle, prenant comme cible les . [more ▼]

Afin de recueillir une image valide tu traitement différencié des élèves dans des leçons d'éducation physique, il est primordial de proposer une approche multidimensionnelle, prenant comme cible les différents acteurs du processus édagogique, à la fois les enseignants et les élèves. Par ailleurs, deux types d'informations devraient être récoltés afin de rendre compte de la réalité de la relation pédagogique: celles qui concernent les comportements et actions concrètes et celles relevant des valeurs, attitudes, perceptions et mécanismes de prise de décision. Les premières reposent sur l'observation alors que les secondes exigent l'interrogation des sujets. [less ▲]

Detailed reference viewed: 327 (7 ULiège)

in Revue de l'Education Physique (1998), 38(3), 109-125

Tout service universitaire compte parmi ses missions celle de développer la connaissance du domaine au moyen d'une recherche structurée. On sait qu'enseignement et recherche se nourrissent mutuellement . [more ▼]

Tout service universitaire compte parmi ses missions celle de développer la connaissance du domaine au moyen d'une recherche structurée. On sait qu'enseignement et recherche se nourrissent mutuellement. La pédagogie des activités physiques et sportives ne fait évidemment pas exception, même si elle ne possède aucun caractère prioritaire dans les organismes qui financent la recherche. . [less ▲]

Detailed reference viewed: 466 (8 ULiège)

in Revue des Maladies Respiratoires (1997), 14(5), 379-85

The data of the literature concerning bronchial reactivity in diabetic patients are controversial. Therefore, we studied the influence of the presence of a diabetic cardiac autonomic neuropathy (CAN) on . [more ▼]

The data of the literature concerning bronchial reactivity in diabetic patients are controversial. Therefore, we studied the influence of the presence of a diabetic cardiac autonomic neuropathy (CAN) on the ventilatory parameters measured during a methacholine-induced bronchoconstriction test. Ten insulin-dependent diabetic patients without CAN, ten insulin-dependent diabetic patients with CAN and ten healthy volunteers, all non-smokers and free of respiratory symptoms, have undergone a functional respiratory check-up before the methacholine test. The presence of CAN was classically studied by the decrease in heart rate changes during three standardized tests (deep breathing at 6 cycles/min, Valsalva manoeuver, orthostatism) which all mainly explore the parasympathetic function. The bronchial response to methacholine was similar in the healthy subjects and in the diabetic patients without CAN. However, the fall in forced expiratory volume in 1 second induced by the highest dose of methacholine was significantly less marked in the diabetic subjects with CAN than in the two other groups. These results suggest that the diabetic autonomic neuropathy also involves the vagal innervation of the respiratory tract. [less ▲]

You asked for an algebraic answer to 1, and it is given below. However, a non-algebraic answer would be much better, so that is given first.

The usual technique used to run linear regressions is called Ordinary Least Squares (OLS). The formula you give above is the formula for the OLS estimator. However, the formula you give above is not the definition of the OLS estimator. The definition of the OLS estimator is "that coefficient vector, $hat<eta>$, which minimizes the sum of squared residuals." First, verbally . . .

Define the number of observations as $N$, so that each of the $delta$ and $alpha$ models have $N$ observations, and the $eta$ model has $2N$. Consider the $delta$ and $alpha$ which solve the first two OLS problems you pose above, and ask, do they (transformed as you have transformed them) solve the third, pooled, $eta$, OLS problem? Well, think about that third problem's objective function $sum_^ <2N>(Y_i-hat_i)^2$. We know that $hat$ minimizes the first $N$ terms (definition of OLS) and has no effect on the last $N$ terms (by inspection). We know that $hat$ minimizes the last $N$ terms (definition of OLS) and has no effect on the first $N$ terms (by inspection). Last, there are "enough" $eta$ such that the $alpha$ and $delta$ may be adjusted independently in the third problem. That is essentially a proof that the OLS estimators have to be exactly the same (again up to your transformation). The proof is not completely formal and explicit, but it's not hard from here to make it so.

To your question 2, yes this result is true much more generally. For any estimator defined by an optimization problem (any "M Estimator," like maximum likelihood for example), there is a result like this. The key requirements are that the parameter spaces of the pooled and separate models are "the same" up to an invertible transformation (like the one you gave to transform $alpha$ and $delta$ to $eta$) and that the objective function of the pooled model can be decomposed (linearly or multiplicatively) into non-interacting parts corresponding (up to an increasing transformation) to the objective functions of the separate problems. This is a lot of models.

The kind of argument I give above is incredibly useful when you are using any kind of M-estimator. For example, it is a famous result for OLS that if you re-scale (change the units of) one of the $X$ variables, that its coefficient will be rescaled by OLS in an exactly offsetting way and that nothing else about the OLS estimator will change. This can be proved by a fairly tedious algebra exercise or by a very brief optimization argument like the one I gave above. The result is not just true for OLS, though. Any M estimator which has the $X$ multiplied against a coefficient will have this magic re-scaling property as well.

Now, to the algebraic demonstration. Changing your notation a little for clarity, start by comparing estimating these two equations by OLS: egin Y_1 &= X_1delta + epsilon_1 Y_2 &= X_1alpha + epsilon_2 end With estimating this equation by OLS: egin left(egin Y_1 Y_2 end ight) &= left[egin X_1 & 0 X_1 & X_1 end ight] left(egin gamma_1 gamma_2 end ight) + left(egin epsilon_1 epsilon_2 end ight) strut Y &= Xgamma + epsilon end My $gamma_1$ is your $eta_0$ and $eta_1$ stacked up and etc. egin hat &= (X'X)^<-1>X'Y strut &= left[egin X_1'X_1 & 0 2X_1'X_1 & X_1'X_1 end ight]^ <-1>left(egin X_1'Y_1 X_1'Y_1 + X_1'Y_2 end ight) strut &= left[egin (X_1'X_1)^ <-1>& 0 -2(X_1'X_1)^ <-1>& (X_1'X_1)^ <-1>end ight] left(egin X_1'Y_1 X_1'Y_1 + X_1'Y_2 end ight) strut left( egin hat_1 hat_2 end ight) &= left( egin (X_1'X_1)^<-1>X_1'Y_1 -2(X_1'X_1)^<-1>X_1'Y_1+(X_1'X_1)^<-1>X_1'Y_1 + (X_1'X_1)^<-1>X_1'Y_2 end ight) strut left( egin hat_1 hat_2 end ight) &= left( egin hat_1 hat_1 - hat_1 end ight) end

## How 2e Schools Transition to Online Learning

As the coronavirus impacted schools across the world, 2e News reached out to 2e-friendly programs to learn how they were continuing to support their students. Below are stories from program leaders who transitioned their school, clinic, or program to a virtual environment. Are you part of a 2e school or program and would like to share its story? Share your experiences here.

##### Chris Wiebe, Ed.D., High School Director — Bridges Academy, Studio City, CA

The Bridges Academy high school program was already online with Google Classroom (our learning management system) and Alma (our student information system) however, our use of these platforms was not intended to accommodate an exclusively virtual environment. The biggest initial challenge was moving the broad community toward an online mindset. To aid in this process, I built a website that explains the logistics and components of our interim online learning program. The site includes basic information about our schedule, daily routines, and expectations for students, as well as a parent primer about our tech platforms and a framework for how our teachers assess students’ online engagement.

Despite proficiency with online tools, teachers and students at brick-and-mortar schools are not necessarily attuned to the unique opportunities of online learning. Some teachers will use an LMS like a bulletin board, posting materials that students retrieve, complete, and then return to an inbox. Whether that inbox is in the cloud, or a plastic tray at the corner of a desk, the instructional strategy is exactly the same and technology is not being leveraged to enhance the student experience.

That is why I am using the SAMR model to anchor ongoing faculty discussions about high-quality instruction in the virtual space. This model categorizes different levels of technology integration to help teachers make creative and informed instructional choices. For instance, substitution (“S”) occurs when technology acts as only a direct replacement for a learning tool with no functional change (e.g. using a keyboard instead of a pencil). Augmentation (“A”) means that technology is still a replacement for a non-digital tool, but there is also some enhancement (e.g., incorporating audio/video into a verbal presentation). Modification (“M”) occurs when the actual task/assignment changes as a result of the affordances of the chosen technology (e.g., using a cloud doc to enable students to color code a document while the teacher offers real-time feedback.) Finally, redefinition (“R”) occurs when technology affords the creation of a new task, previously inconceivable in a non-virtual environment (see example below).

At an initial planning meeting, one of my faculty members was exploring how to archive a class discussion held via chat to scaffold the writing process. The assignment would go something like this: The archived transcript (“brainstorming”) would provide material for a subsequent assignment that asked students to collectively complete a shared digital graphic organizer (“organizing”), transferring main ideas and details from the text chat onto the document. Then students would draw from the graphic organizer to create outlines for distinct, individual essays on this common topic (“drafting”). Finally, students in pairs would take turns sharing each other’s screens to provide one another with feedback through verbal comments via video conference and written feedback via the comment function on Google Docs (“revision”).

While strategies like this can bolster opportunities for collaboration and deep learning during this difficult time, only a gifted and committed teacher can create supportive and successful learning environments for their 2e students, built on strong relationships. At Bridges we are lucky to have dozens of just those types of teachers.

##### Melanie Hayes, Ph.D., Founder — Big Minds Unschool, Pinole, CA

Like everyone else, we had to act fast to adapt to this new way of being. We had one short spring break week to convert our entire school and recreate Big Minds online. Thanks to a wonderful faculty and administrative team, we have been able to provide consistency and comfort through offering nearly the same schedule, activities, and teaching that we did in the real world. Our students now can enjoy their morning free time together on a private Big Minds Minecraft server, then they attend their math/literacy coaching and daily classes as an interactive group online. Our site director holds our regular lunchtime conversations via Zoom, where she eats lunch with our students and they enjoy one another’s company.

We are also offering a parent support time for a “virtual drop-off” to give parents a place to chat with the director and each other. Our parent support coordinator offers a homeschooling support session each week to ensure parents are not feeling overwhelmed by this new reality. We are in the early stages, but so far so good. It has been really sweet to connect to our students virtually and to know that our community bond is holding strong!

##### Jacqueline Byrne, Founder — FlexSchool — Berkeley Heights, NJ & Bronxville, NY

FlexSchool and FlexStudents seem to be adapting quite well to the new normal. Fortunately, FlexSchool already offered an online school in the form of its cloud classroom. Several weeks ago, we began putting a plan in place to move all of our classes to the cloud classroom while the coronavirus outbreak would prevent us from being on our campuses. Teachers spent several hours in training, and on Monday, March 16, online classes began without missing a single school day. In many ways, it has been reassuring for our students to be able to stick to their normal class schedule during the week.

One challenge of working completely online is that it can be difficult for students to spend so much time sitting in front of the screen. Students therefore are encouraged to spend 30 minutes per day doing some form of physical activity, and we have asked them to send in pictures to document how they spent their time. Many of the teachers have also built breaks into the class period. As always, students who need to move around while they are learning are still encouraged to do so.

Providing the students with their regular schedule not only keeps up with the rigor of the curriculum but also allows students to have real-time interaction with their classmates. Class discussions are still the most integral part of learning at Flex. Because the classes are live and online, our students are experiencing human connection and are not receiving packets of rote work. Our teachers are still able to provide academic and emotional support. We realize how important that connection is, especially in this time of isolation. In addition to normal classes, we have created several online after-school clubs. The students really enjoy the opportunity to play and engage in a non-academic setting.

One student said, “I would be going crazy if I didn’t have classes.” Another said that he is grateful to have something interesting to do. This new normal is, of course, a work in progress, and we are constantly reviewing feedback to find areas that could be improved. This has been an adjustment for everyone, and we are very proud of our staff and students who made a smooth transition to this new way of learning.

##### Callie Turk, co-founder — Resilience and Engagement for Every Learner (REEL), Palo Alto, CA

This past Wednesday marked two weeks since my three children enjoyed communities of learning and friendship in person, which is hard to wrap my head around. The private schools they attend already make use of online systems and their teachers made a nimble transition to distance learning.

At home, it took a few days for us to settle into a routine. No, Mom isn’t the only person who is going to pick up the house. Yes, we are all going to break for lunch at noon. For sure, everyone is going to move their bodies for at least 30 minutes each day. And, definitely, work blocks and schoolwork will happen before video games or episodes of The Office.

I have found that distance learning has some upsides. For instance, one of my eighth-grade daughters is in a self-paced math class. In the past, she received the attention she needed in geometry, but the teacher had limited time in a one-hour class. Now the teacher schedules at least 15 minutes per day of one-on-one time with my twice-exceptional daughter, who noted, “Linda is such a great teacher and person. I’m glad I get to work with her.”

Only a gifted and committed teacher can create supportive and successful learning environments for their 2e students, built on strong relationships.

But there are also downsides. It’s tough on teens to be away from their peers. My most extroverted child quickly learned to navigate Zoom and now spends hours online with her friends, the way I spent hours on the telephone with friends during my teen years. But my more introverted kids, who soak up positive social energy through proximity to others, aren’t having their needs met. I’m uncertain what the long-term social-emotional implications are for all three of them.

All things considered, we’ve been blessed with teachers who are putting their hearts into maintaining valuable learning experiences during this difficult time. But we will be grateful to get our kids back to a face-to-face community of learning and growth. Two weeks have gone by faster and more successfully than I ever thought they would, and I am confident that all of us — parents, teachers, students — will learn new ways of doing things that we might want to carry forward into the future.

##### Max Melby, School Director — Arete Academy, St. Louis Park, MN

Arete Academy started working with community partners in the first week of March to respond to the potential for school closure and distance learning, but it’s hard to feel adequately prepared when the task is to transform your entire school program. Our teachers, students, and parents have absolutely risen to the occasion and I’ve never been more glad to be a part of the Arete school community.

We started distance learning on March 19 and we are hoping to resume on-campus classes on May 3, barring worsened conditions. We are using our Google Apps for Education as our learning platform and, with what little data we have, Google Classroom and Google Meet hold a lot of promise for sustainable success.

Teachers are keeping their virtual classrooms current with assignments and providing ongoing feedback, as well as holding one-on-one meetings and group classes. Students follow recommended schedules that we put together or work in whatever ways fit into their home routine.

Our program is predominantly asynchronous right now, but we are actively working on ways to shift the balance to provide opportunities for synchronous learning. One of my favorite examples of synchronous learning right now isn’t necessarily academic: Each day, our whole school is invited to a standing Google Meeting to eat lunch together. There’s something really special about the nonsensical hypothetical questions and laughter our students share over lunch and I’m grateful for our teachers who are making that work and the students who consistently sign on for it

I have visited a handful of very special schools (2e schools included, of course) and each school has its own special “lightning in a jar” quality. I think that one of the big challenges that all of us face is how to capture that lightning in a jar and recreate it over the internet. How much work is reasonable from afar? How are we supporting our students’ social connections? How can we support our parents? We all have a lot of learning to do and I look forward to hearing more about how the broader 2e community is responding.

##### Dan Peters, Ph.D., Executive Director — Summit Center, Walnut Creek & Torrance, CA

At Summit Center we spend lots of time working with parents of gifted and 2e children and adolescents, helping to provide strategies to manage screen time and technology use. During this time of “shelter in place,” we are increasingly grateful for technology and the online platforms that help us stay connected to each other and maintain as many of Summit’s services as possible.

We are using Google Meet for virtual counseling and consultation sessions with parents, adolescents, and children. This tool has also helped us continue to offer our clients educational therapy. While we have temporarily cancelled psychological evaluations, our team is researching options for testing virtually. As telehealth becomes more commonplace, some studies are showing positive results. Providing testing questions through a shared screen as well as audio recordings of test questions are some of the ways that providers are exploring assessment in the digital space.

We are also starting online support groups and offering our library of webinars to our community for free. These resources include Taming the Worry Monster, Psychosocial development of Gifted Children, Time Management, and more. I continue to record podcasts (Parent Footprint with Dr. Dan) and livestream on Facebook to maintain community among our members, urging them to stay present in these uncertain times.

I am hopeful that the creativity and inventiveness required by the current circumstances will provide us with perspectives and information that boost our capacity to provide families with our services over the long term. Times like these require problem solving and thinking outside of the box. Twice-exceptional people are needed now more than ever.

##### Kim Busi, M.D., Founder — Quad Preparatory School, New York, NY

The Quad Preparatory School, which meets the needs of K-12 2e students in New York City, moved to an online continuous learning environment on March 12. In conversations with other school leaders, the transition to distance learning at Quad Prep seemed easier and less complicated than it was for our peer independent schools in NYC.

Everyone’s staff has worked incredibly hard, but we believe that our shift was eased because our fundamental approach lends itself much more easily to a continuous learning/online environment. Over the past weeks, our 2e pedagogy has guided the opportunities we’ve leveraged — maybe this can exemplify how 2e models of education are models for all education!

Opportunity 1: Strengths are more important than ever. As the novelty of the online classroom recedes, engagement in and through strengths and talents is central. The ingenuity and creativity of our educators and providers has been nothing short of magical to witness. Lead with strengths — and lead early — to prevent disengagement online.

Opportunity 2: Personalized learning is easy to convert. This is a good time to rethink the necessity of group cohorting all day and all the time. Consider a schedule that allows for full group, small (or just-right sized) group, and 1:1 instruction to alleviate screen group burnout and frustrations.

Opportunity 3: Seize the enhanced opportunity (yes, enhanced!) for integrated social, emotional, and executive functioning learning (SEL & EF). Who could have imagined that an online platform could offer such unique SEL and EF skill-development opportunities?! Using breakout groups and leveraging the fact that kids can see themselves and others online, teachers and providers (in our case, psychosocial teachers that already exist in every class) have a unique, visual opportunity for in-the-moment coaching.

Opportunity 4: Take care of your community. Parents are now co-teachers, so have a strength-based approach to parents navigating the needs of a 2e child in the family. Solicit feedback early and often, and provide responsive mechanisms for the range of support they’ll need, including technology assistance, scheduling, and other guidance as they navigate this new landscape. And don’t underestimate the need to support their own community interaction. Virtual “coffees” and “cocktail hours,” workshops, and robust resource lists are excellent ways to take care of your parent community. And, don’t forget the staff for every place that I wrote parents, please substitute staff.

Opportunity 5: Structure is the monarch, but flexibility rules the day! Start early: The transition is very similar to the excitement and ramp-up during the first weeks of school, and you can’t start planning too early. Set the frame, in writing, to all parties. Outline expectations, structure, remaining work in progress, and note that you anticipate changes. Expectation setting and previewing is more important than ever. If you haven’t done so, create technology protocols and guidelines for staff, families, and students. Just because you’re online doesn’t mean that expectations diminish.

And then, be willing to change! Our Early Childhood program is of course not the same as our Upper School’s, and we have empowered our divisional leadership to adapt and adapt again. Deploy staff as needed, whether it’s in support of additional personalized learning, after-school programming, or additional technology or scheduling support. Listen to your teachers, students, and families, and make changes in real time as needed. At Quad Prep, one of our stated values is “we never give up on each other and are driven by the ethos of “not if, but how.” This has been a tremendous opportunity to put that value into action — while different, our goal is to make this time as rich and supportive a time as possible for our students, families, faculty, and staff.

##### Danny Boyer, 7th/8th Grade Science Teacher, Helios School, Sunnyvale, CA

As a teacher, one of our main responsibilities is managing our students’ physical and mental well-being during school. Much of this is for the simple reason we want our students to be happy, productive, and focused when we are teaching them. This requires providing diverse modes of learning, frequent movement breaks, and chances to talk with classmates as well as quiet work time. We also recognize the importance of allowing students to blow off steam at recess, to eat healthy snacks and lunch, and to drink water. We need to be vigilant when computers are being used to help students stay on task and avoid the oh-so-tempting distractions available on the web. We know which students need us to look over their shoulder or check in with them regularly to make sure they are staying on task. We know our students and what they need.

When COVID-19 caused schools to close and we had to suddenly shift to remote learning, I realized students and families were going to need to manage their work days themselves. I imagined our students sitting at a computer all day, knowing some of them would have a very difficult time drawing boundaries between work time, social time, and play time. If they did not create good habits from the beginning, this could be a disaster for students without the executive functioning to create structure on their own. It also is difficult for parents to jump into this role and, for many, they are not aware of the schedule management that is built into a school day.

For this reason, I decided to create a study plan as one of our first assignments. I saw it as a teachable moment for both students and parents — an opportunity to have families think about healthy habits when it comes to screen time, social media, school work, exercise, diet, etc. And this is just a starting point. As we all settle in, teachers can meet with individual students or parents to discuss tools, strategies, and solutions for improving how we do this thing we call “remote learning.” Because one thing is always true — there is no one-size-fits-all solution. Each student and family is different and will require their own study plan, which will evolve as we adapt to our new learning environments.

##### Debbie Steinberg Kuntz, LMFT, founder of brightandquirky.com

Bright & Quirky hosted a webinar providing multiple perspectives on supporting 2e kids and their families in this environment. “Bright, Quirky & Coping: Living and learning at home during the pandemic,” focused on how to deal with anxiety, stress, and worry creating daily family structure making the most of temporary homeschooling for 2e kids how to create community and more.

The panel featured Debbie Steinberg Kuntz, Melanie Hayes, neuropsychologist and author Robin McEvoy, school counselor and 2e specialist Lauren Hutchinson, and Michelle Bronwell, homeschooler of 2e kids.

##### Lisa Reid, Ed.D., Principal — STEM 3 Academy, Irvine, CA

Stem 3 Academy has navigated numerous challenging circumstances during these uncertain times. Abrupt departures from routine and the need to support learning from home are challenging for all families, but especially to those within our population. Given that, what has been most important is daily personal communication with our families.

Work obligations, access to technology, family dynamics, and differing learning needs — as well as social-emotional variables — call for us to differentiate our support, as we always do. It has been heartening to see our staff’s unwavering support and every home supporting their children’s continued progress in any way they can. And families are not only thinking of their own situations, but supporting us too.

We do not typically send much work home, so remote learning tools, such as video conferencing, have meant a steep learning curve for our families and teachers. It was initially a bit scary, but everyone stepped up to the challenge. As a result, aspects of the student-teacher-family relationship have evolved in interesting ways: teachers may have their toddlers crawling through learning sessions, and children do live-streamed show-and-tell with their pets while at home with their parents and guardians. Teachers and so many people around the world have shared ideas about making the best of this temporary learning situation, to make it as supportive and effective as possible. This means something different for every child, family, and teacher.

We make sure our families have the technology they need to access curriculum, where instruction and opportunities for social interaction are provided through Google Classroom and Zoom. When needed, we mail home hard copies of assignments and materials. Students are able to access daily, differentiated work based upon their personal situation and ability to engage, given their individual circumstances and accommodations outlined within their Individual Education Programs (IEP).

Work obligations, access to technology, family dynamics, and differing learning needs — as well as social-emotional variables — call for us to differentiate our support, as we always do.

Our service providers are coordinating sessions remotely, and I’ve been pleased to see their continuing support — especially through counseling and social skills groups, which have helped students feel connected.

Our world took a big hit a few weeks ago, and all of the changes that everyone needed to make in response created a huge amount of work, learning, and flexibility. Those three words represent variables that, when seen as obstacles, can make educational progress seem impossible at times for our population. It would have been easy for anyone involved to throw their hands in the air and quit, but they haven’t. Everyone has stepped up in their own way and has shown not only continued dedication to growth, but also to one another as a caring community.

I’m grateful and proud of the way in which our students, families, teachers, organization, and schools have come together. Even physical separation cannot break the commitment to kids that we’ve all made together. Below are the elements of our distance learning model, which is constantly evolving with the changing needs of our students.

• Teachers are using Google Classroom and Google Sites to post class schedules and teacher schedules. Interaction and participation is managed on Zoom, telephone, email, and in the cloud.

• Students check in with each of their teachers during the day and work on assignments individually or collaboratively. Some work is done on Google Docs, which allows for real-time editing. At the end of the day, students have the opportunity to engage with their teachers for questions, clarification, or additional help.

• Related services, such as speech and counseling, are provided on Zoom. In addition, given the reason for the school closures, social and emotional support by way of Zoom chats and Google Meet provide opportunities for students to relate to one another while being moderated by a teacher.

• We have also recently issued a resource newsletter to parents outlining a number of activities across grade levels, which students can engage in at home. These are both plugged and unplugged activities to ensure that students of all levels of sophistication are engaged, whether or not they have access to electronic devices.

## Use of logarithms in economics

Why do economists always want to take the natural logarithm of everything? Here’s the answer,if you don’t mind looking at a few equations and graphs.

Let me begin with a quick review of compound interest rates. If you invest an amount for one year at an r% interest rate, at the end of the year you’ll have an amount . For example, with a 4% return, Sometimes your interest might be compounded, for example, you get your money back plus 2% at the end of six months, and then can earn 2% interest on both your original principal as well as on the first six month’s interest: . With an interest rate of r = 4%, compounded twice annually like this your money would actually have grown 4.04% at the end of the year. Compounded quarterly, or about 4.06% at the end of the year. It turns out the formula converges to a particular function as the frequency of compounding ngets arbitrarily large, this limit being wheree is a special number (approximately equal to 2.72) that possesses this and a number of other amazing properties. Thus if you earn r% interest that is compounded continuously, at the end of the year your money will have grown by . For example, if r = 4%, continuous compounding would actually give you 4.08% at the end of the year, a little more than the 4.06% from quarterly compounding or 4.0% with no compounding.

Taking natural logarithms is just the inverse of the above operation: , or since the log of a ratio is the difference of the logs, In other words, taking the difference between the log of a stock price in year 2 and the log of the price in year 1 is just calculating a rate of return on the holding, quoted in terms of a continuously compounded rate.

For low values of r, the continuously compounded return is almost the same as the noncompounded return, so that the log difference is almost the same number as the percentage change. For the example just given, the percentage change is whereas the log change is So any time that you see a graph that is measured in logs, an increase of 0.01 on that scale corresponds very closely to a 1% increase. A graph that is a straight line over time when plotted in logs corresponds to growth at a constant percentage rate each year.

Using logs, or summarizing changes in terms of continuous compounding, has a number of advantages over looking at simple percent changes. For example, if your portfolio goes up by 50% (say from $100 to$150) and then declines by 50% (say from $150 to$75), you’re not back where you started. If you calculate your average percentage return (in this case, 0%), that’s not a particularly useful summary of the fact that you actually ended up 25% below where you started. By contrast, if your portfolio goes up in logarithmic terms by 0.5, and then falls in logarithmic terms by 0.5, you are exactly back where you started. The average log return on your portfolio is exactly the same number as the change in log price between the time you bought it and the time you sold it, divided by the number of years that you held it.

Logarithms are often a much more useful way to look at economic data. For example, here is a graph of an overall U.S. stock price index going back to 1871. Plotted on this scale, one can see nothing in the first century, whereas the most recent decade appears insanely volatile.

On the other hand, if you plot these same data on a log scale, a vertical move of 0.01 corresponds to a 1% change at any point in the figure. Plotted this way, it’s clear that, in percentage terms, the recent volatility of stock prices is actually modest relative to what happened in the Great Depression in the 1930’s.

There are a couple of other changes that can make the graph even more meaningful and easy to read. For one thing, we probably want to take out the effect of inflation. The graph below plots the natural log of the real instead of the nominal stock price, and subtracts off the log from 1871, so that the graph starts at zero and reports cumulative log returns since then. The numbers have also been multiplied by 100, so that if the value goes up by 1 unit on this graph, it corresponds to a 1% increase in the real value of stocks.

Natural logs are also very convenient for describing relations between economic variables. For example, here’s a scatter plot of the relation between the log of U.S. real GDP on the horizontal axis (multiplied by 100 and normalized to begin at 0) and the log of U.S. oil consumption on the vertical axis. The relation between these series follows a straight line with slope of 1 up until about 1970, corresponding to an income elasticity of oil demand of around 1– for every 1% increase in real GDP, U.S. oil consumption also rose by 1%. By contrast, since 1970, the slope of the graph is about 0.2– U.S. real GDP has grown at a (continuously compounded) annual rate since then of 2.9%, whereas oil consumption has only grown 0.6%. You can read those numbers directly off the graph below by taking the difference between any two points and dividing by the number of years separating them.

So next time somebody wants to summarize data in terms of natural logs, don’t complain– it’s usually a much more meaningful and robust way to see what’s going on.