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Medium 9781576336984

Addition and Subtraction: Whole Numbers: SSAT-ISEE Arithmetic

Ace Academics Ace Academics ePub
Medium 9781576336991

Quadratic Equations and Radicals: SSAT-ISEE Algebra

Ace Academics Ace Academics ePub
Medium 9781576336397

Variables: GED Algebra

Ace Academics Ace Academics ePub
Medium 9781576336984

Simplify Fractions: SSAT-ISEE Arithmetic

Ace Academics Ace Academics ePub
Medium 9781936763313

Chapter 4 Instructional Practices for Application-Based Mathematical Learning

Chris Weber Solution Tree Press ePub

Teaching mathematics is arguably one of the most complex elements of an elementary teacher’s profession. The nature of mathematical learning is such that students must master specific skills with a fluid understanding that allows them to apply the learning in a variety of contexts and in conjunction with other skills and understandings. In addition, students must master the requisite language and tools in order to be able to communicate and model mathematics. To design and implement instruction that ensures such rich learning, teachers must be able to weave evidence-based instructional strategies with mathematical practices and apply those elements strategically within engaging contexts.

Effective instruction includes those instructional decisions that positively impact student learning and engagement. The NMAP (2008) identifies these practices as follows:

• Maintenance of the balance between student-centered and teacher-directed instruction

• Explicit instruction for students having mathematics difficulties

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Medium 9781576336724

Units of Measurement: SAT Arithmetic

Ace Academics Ace Academics ePub
Medium 9781576336281

Equations with Several Variables: COOP-HSPT Algebra

Ace Academics Ace Academics ePub
Medium 9781576336922

Addition: Fractions: PSAT Arithmetic

Ace Academics Ace Academics ePub
Medium 9789380298818

6

Parmananda Gupta Laxmi Publications PDF

62

DIFFERENTIAL GEOMETRY AND CALCULUS OF VARIATIONS

Dividing (3) by (4), we get dθ

− a sin θ

κ (n . a) ds

= dφ

− τ (n . a)

− a sin φ ds

κ sin θ dθ

=

.

τ sin φ dφ

Example 14. For the curve r = r(s), if

dt

= w × t, ds

Sol. Given equations are

dn db

= w × n and

= w × b, find the vector w. ds ds

dt

=w×t ds

...(1)

dn

= w × n ...(2) ds

db

=w×b ds

By Frenet formula,

dt

= κn . ds

dt

= 0 + κn = τ(t × t) + κ(b × t) = (τt + κb) × t ds

dt

= (τt + κb) × t ds

By Frenet formula,

dn

= – κt + τb. ds

By

If

...(3)

...(4)

dn

= – κ(n × b) + τ(t × n) = κ(b × n) + τ(t × n) ds

= (κb + τt) × n = (τt + κb) × n dn

= (τt + κb) × n ds db

Frenet formula,

= – τn. ds db

= – τ(b × t) + 0 = τ(t × b) + κ( b × b ) = (τt + κb) × b ds db

= (τt + κb) × b ds w = τt + κb, then given equations (1), (2) and (3) are satisfied.

...(5)

...(6)

Note. The vector w = τt + κb is called the Darboux vector for the curve r = r(s).

Example 15. Using Serret-Frenet formula, find the direction cosines of the unit principal normal vector and the unit binormal vector at the point ‘s’ for the curve r = r(s).

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Medium 9781576336946

Circles - Regular Polygons: PSAT Geometry

Ace Academics Ace Academics ePub
Medium 9781576336922

Real Numbers: PSAT Arithmetic

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Medium 9781576336281

Signed Numbers: COOP-HSPT Algebra

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Medium 9781576336939

Like Terms: PSAT Algebra

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Medium 9781576336830

Properties of Numbers: Praxis I Arithmetic

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Medium 9781576336281

Concepts: COOP-HSPT Algebra

Ace Academics Ace Academics ePub

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