1. The document provides a sample test item specification for a mathematics assessment. It includes details of the test such as the competencies being measured, number of items, and item content details.
2. Sample test items are provided measuring concepts such as volume, applying laws of exponents, and dividing polynomials. Correct answers are categorized by depth of knowledge.
3. The specification outlines the test plan, including objectives being measured, cognitive level of items, and item content details for multiple choice questions.
I am Simon M. I am a Stochastic Processes Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Stochastic Processes, from Texas, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
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Math 235 - Summer 2015Homework 2Due Monday June 8 in cla.docxandreecapon
Math 235 - Summer 2015
Homework 2
Due Monday June 8 in class
Remember: In this course, you must always show reasoning for your answers. You can use any result we have
proved in class, in textbook reading, or in a previous homework.
Problem 1 For each of the following problems, you must justify your answer by finding the general solution
to the corresponding system of linear equations, or by showing that no solution exists.
(a) In the vector space P3(R), can −2x3 − 11x2 + 3x+ 2 be written as a linear combination of vectors in
{x3 − 2x2 + 3x− 1, 2x3 + x2 + 3x− 2}?
(b) In the vector space M2×2(R), can
(
1 0
0 1
)
be written as a linear combination of vectors
in
{(
1 0
−1 0
)
,
(
0 1
0 1
)
,
(
1 1
0 0
)}
?
Problem 2 Show that a subset W of a vector space V (over a field F ) is a subspace of V if and only if
span(W ) = W .
Problem 3 You are given a subset S of a vector space V . Determine whether S is linearly dependent or
linearly independent using exclusively methods developed in this course, and justify your answers.
(a) V = R3 and S = {(1, 2,−1), (2,−3, 1), (2, 3,−5)}.
(b) V = P3(R) and S = {1, 1 + 2t+ t2, 1− 2t+ t3, t2 + t3}.
(c) V = F(R,R) and S = {t, et, sin(t)}.
Problem 4 Prove that a subset S of a vector space V is linearly dependent if and only if there exists a
proper subset S′ ( S with the same span as S.
Problem 5 Exercise 1.6.13 from the textbook.
Problem 6 You are given a subspace S of M2×2(F ), the vector space of 2 × 2 matrices with entries in a
field F . You are required to find a basis for this subspace, and to find the dimension of this subspace.
For each problem, you DO NOT need to prove that S is a subspace, but you DO need to prove that your
conjectured basis is, in fact, a basis (that is, you need to show it is a linearly independent generating set for
S).
(a) S is the subspace of all diagonal 2× 2 matrices with entries in F .
(b) S is the subspace of all symmetric 2× 2 matrices with entries in F .
(c) S is the subspace of all skew-symmetric 2× 2 matrices with entries in F .
Problem 7 Let W1 and W2 be subspaces of a finite-dimensional vector space V . Prove that dim(W1∩W2) ≤
min{dim(W1),dim(W2)} and dim(W1 +W2) ≥ max{dim(W1),dim(W2)}.
Problem 8 Each of the maps below goes from one vector space to another (where both vectors spaces are
over the same field). For each map: prove that it is linear, determine whether it is one-to-one or not (prove
your answer), and determine whether it is onto or not (prove your answer).
(a) T : P3(R)→M2×2(R) defined by T (p) =
(
p(0) p′(0)
p′′(0) p′′′(0)
)
.
(b) T : M2×2(F ) → F defined by T (A) = tr(A), where F is a field. (Recall that for an n × n matrix,
tr(A) =
∑n
i=1Aii.)
1
(c) T : R2 → R3 defined by T ((a, b)) = (a, b, a+ b).
(Hint: You may find an analysis of rank and nullity useful here.)
Problem 9 Suppose that T : R2 → R2 is linear and that T ((1, 2)) = (3, 4) and T ((1, 3)) = (0, 1). Find
T ((1, 0)). Is T one-to-one? Justify your answer.
Problem 10 Let ...
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
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Similar to 2nd Quarter CBEA-Math 7 - Week 5 ot 8.docx
I am Simon M. I am a Stochastic Processes Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Stochastic Processes, from Texas, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
I am Kennedy, G. I am a Stochastic Processes Assignment Expert at excelhomeworkhelp.com. I hold a Ph.D. in Stochastic Processes, from Indiana, USA. I have been helping students with their homework for the past 7 years. I solve assignments related to Stochastic Processes. Visit excelhomeworkhelp.com or email info@excelhomeworkhelp.com. You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
Math 235 - Summer 2015Homework 2Due Monday June 8 in cla.docxandreecapon
Math 235 - Summer 2015
Homework 2
Due Monday June 8 in class
Remember: In this course, you must always show reasoning for your answers. You can use any result we have
proved in class, in textbook reading, or in a previous homework.
Problem 1 For each of the following problems, you must justify your answer by finding the general solution
to the corresponding system of linear equations, or by showing that no solution exists.
(a) In the vector space P3(R), can −2x3 − 11x2 + 3x+ 2 be written as a linear combination of vectors in
{x3 − 2x2 + 3x− 1, 2x3 + x2 + 3x− 2}?
(b) In the vector space M2×2(R), can
(
1 0
0 1
)
be written as a linear combination of vectors
in
{(
1 0
−1 0
)
,
(
0 1
0 1
)
,
(
1 1
0 0
)}
?
Problem 2 Show that a subset W of a vector space V (over a field F ) is a subspace of V if and only if
span(W ) = W .
Problem 3 You are given a subset S of a vector space V . Determine whether S is linearly dependent or
linearly independent using exclusively methods developed in this course, and justify your answers.
(a) V = R3 and S = {(1, 2,−1), (2,−3, 1), (2, 3,−5)}.
(b) V = P3(R) and S = {1, 1 + 2t+ t2, 1− 2t+ t3, t2 + t3}.
(c) V = F(R,R) and S = {t, et, sin(t)}.
Problem 4 Prove that a subset S of a vector space V is linearly dependent if and only if there exists a
proper subset S′ ( S with the same span as S.
Problem 5 Exercise 1.6.13 from the textbook.
Problem 6 You are given a subspace S of M2×2(F ), the vector space of 2 × 2 matrices with entries in a
field F . You are required to find a basis for this subspace, and to find the dimension of this subspace.
For each problem, you DO NOT need to prove that S is a subspace, but you DO need to prove that your
conjectured basis is, in fact, a basis (that is, you need to show it is a linearly independent generating set for
S).
(a) S is the subspace of all diagonal 2× 2 matrices with entries in F .
(b) S is the subspace of all symmetric 2× 2 matrices with entries in F .
(c) S is the subspace of all skew-symmetric 2× 2 matrices with entries in F .
Problem 7 Let W1 and W2 be subspaces of a finite-dimensional vector space V . Prove that dim(W1∩W2) ≤
min{dim(W1),dim(W2)} and dim(W1 +W2) ≥ max{dim(W1),dim(W2)}.
Problem 8 Each of the maps below goes from one vector space to another (where both vectors spaces are
over the same field). For each map: prove that it is linear, determine whether it is one-to-one or not (prove
your answer), and determine whether it is onto or not (prove your answer).
(a) T : P3(R)→M2×2(R) defined by T (p) =
(
p(0) p′(0)
p′′(0) p′′′(0)
)
.
(b) T : M2×2(F ) → F defined by T (A) = tr(A), where F is a field. (Recall that for an n × n matrix,
tr(A) =
∑n
i=1Aii.)
1
(c) T : R2 → R3 defined by T ((a, b)) = (a, b, a+ b).
(Hint: You may find an analysis of rank and nullity useful here.)
Problem 9 Suppose that T : R2 → R2 is linear and that T ((1, 2)) = (3, 4) and T ((1, 3)) = (0, 1). Find
T ((1, 0)). Is T one-to-one? Justify your answer.
Problem 10 Let ...
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
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1. CBEA-like Test Items – SOLO Framework
Name: ANNA LIZETTE C. DE GUZMAN Learning Area: Mathematics 7
TABLE OF SPECIFICATION
I. Quarter: 2 Week: Week 5 No. of Days/Hours: 4
Competency Level in the RBT
II. MELC/s:The learners derives the laws of exponent. Multiplies and divides
polynomials.
Apply
III. Enabling Competency: Apply
IV. Test Item Specifications Item No.1 Item No. 2 Item No. 3
Number of Test
Items
Level in the
RBT
Apply Apply Remembering
Knowledge
Dimension
Conceptual Conceptual. Factual
Item Content
Stimuli(Personal/Local/National/Global):
1. Your mom ordered coloring materials from Shopee Apps for your Arts project. The dimensions of the shipping box containing your Mom’s purchases
was shown below. You remember that formula for solving the volume of rectangular prism is V= L X W X H
3a
2a2 3a
Prime:
What is the volume of the box written as a monomial?
POSSIBLE ANSWERS
Pre-Structural Uni-Structural Multi-Structural Relational Extended Abstract
2. No Mastery Least Mastered Nearly Mastered Mastered Highly Mastered
2a2
2a2
+ 6a 18a2
18a4
The answer is far from the rules of
multiplying polynomials(Laws of
exponent).
The answer used Perimeter
formula
The numerical coefficient
were performed correctly but
the exponent were not.
The Product law of exponent
were properly applied.
Item Content
Stimuli(Personal/Local/National/Global):
2. Your teacher’s assignment is to simplify the given polynomials: ( b2) (3c2) (2c)3
Prime:
What laws of exponents will you apply to get the answer?
POSSIBLE ANSWERS
Pre-Structural Uni-Structural Multi-Structural Relational Extended Abstract
No Mastery Least Mastered Nearly Mastered Mastered Highly Mastered
Law of Zero Exponent Power of a Power rule
and Law of Negative
exponent
Product Law of
Exponents
Product Law of
Exponents and Power of
a Product law
There is no Zero exponent on the given
polynomials.
There is no negative
exponent on the given
polynomials but Power to
power rule will be applied.
Product law of exponent will
be applied to simplify ( b2)
(3c2
) but can’t be applied
to simplify (2c)3
( b2
) (3c2
) = 3b2
c2
Product law of
exponents
(2c)3
= 23
c3
= 8c3
Power of a product law
(3b2c2) (8c3) = 24b2c5
Product law of
exponents
Item Content
Stimuli(Personal/Local/National/Global):
3. 3. Your classmate is stuck on a problem while working on dividing polynomials, (14x4
- 70x3
+ 21x2
- 63x ) ÷ (7x) . You helped your classmate since you
know the rules in dividing polynomials?
Prime: What will be the quotient of the given polynomials?
POSSIBLE ANSWERS
Pre-Structural Uni-Structural Multi-Structural Relational Extended
Abstract
No Mastery Least Mastered Nearly Mastered Mastered Highly
Mastered
2x5
- 10x4
+ 3x3
- 9x2
2x4
+ 10x3
+ 3x2
+ 9x 2x3
+ 10x2
+ 3x + 9 2x3
- 10x2
+ 3x - 9
Explanation of Answers
In dividing polynomials exponents of the
same bases should not be added.
In dividing polynomials
exponents of the same bases
should not be multiplied.
The answer is almost correct
because laws for dividing
polynomials were applied but
the rules on sign numbers
were not.
All laws were properly applied to this answer. Quotient law
of exponent and rules in dividing signs.
Name: Learning Area: Mathematics
TABLE OF SPECIFICATION
I. Quarter: 2 Week: Week 6 No. of Days/Hours: 4
Competency Level in the RBT
4. MELC/s: Uses models and algebraic methods to find the: (a) product of two binomials;
(b) product of the sum and difference of two terms; (c) square of a binomial; (d) cube of
a binomial; (e) product of a binomial and a trinomial. M7AL-IIe-g-1
Apply
II. Enabling Competency: Apply
IV. Test Item Specifications Item No.1 Item No. 2 Item No. 3
Number of Test
Items
Level in the
RBT
Apply Remembering
Knowledge
Dimension
Conceptual Factual .
Item Content
Stimuli(Personal/Local/National/Global):
1. Student A wants to know the area of the school quadrangle. You learned from your math class that the formula in finding area of
rectangle is A= L x W. If the school quadrangle has a length of (x + 5) meters and width of (x + 3) meters.
Prime:
What is the area of the school quadrangle?
POSSIBLE ANSWERS
Pre-Structural Uni-Structural Multi-Structural Relational Extended Abstract
No Mastery Least Mastered Nearly Mastered Mastered Highly Mastered
x+5 2x+8 x2
+ 15 x2
+ 8x + 15
It’s just the first binomial no operation
was performed.
The two binomials were
added. It should be
multiplied.
Only first terms and last
terms were multiplied.
Laws for multiplying
polynomials were fully
applied.
F - (x) (x) = x2
O - (x) (3)= 3x
I - (5) (x) = 5x
L - (5)(3) = 15
= x2
+ 8x + 15
Item Content
Stimuli(Personal/Local/National/Global):
2. Teacher A asked his students to find the product of these two binomials, (2y + 5) and (3y - 5) using FOIL method.
5. Prime:
What is the product of the two binomials?
POSSIBLE ANSWERS
Pre-Structural Uni-Structural Multi-Structural Relational Extended Abstract
No Mastery Least Mastered Nearly Mastered Mastered Highly Mastered
Name: Learning Area: Mathematics 7
TABLE OF SPECIFICATION
III. Quarter: 1 Week:Week 7 No. of Days/Hours: 4
Competency Level in the RBT
IV. MELC/s: solves problems involving algebraic expressions Apply
V. Enabling Competency: Illustrates well-defined sets, subsets, universal sets, null set,
cardinality of sets, union and intersection of sets and the different of two sets.
Apply
IV. Test Item Specifications Item No.1 Item No. 2 Item No. 3
Number of Test
Items
Level in the
RBT
Apply Apply Remembering
Knowledge
Dimension
Conceptual Conceptual Factual
6. Item Content
Stimuli(Personal/Local/National/Global):
1. Student B need P1000 to buy a scientific calculator to be used in math class. Her neighbor will pay her P100 per hour as tutoring fee and her father gave
her P300 for cleaning the house.
Prime:
Which inequality represents the minimum amount of hours she will need to tutor in order for her to buy her calculator?
POSSIBLE ANSWERS
Pre-Structural Uni-Structural Multi-Structural Relational Extended
Abstract
No Mastery Least Mastered Nearly Mastered Mastered Highly
Mastered
1000x + 100 ≤ 300 300x + 100 ≥ 1000 100x - 300 ≥ 1000 100x +300 ≥ 1000
Explanation of Answers
It is not the answer because the total
money, Student B needed is 1000.
Item Content
Stimuli(Personal/Local/National/Global):
2. A rectangular badminton play area has a length that is seven inches longer than its width (w). The perimeter of this badminton play area
is at most 30 inches.
Prime:
Which of the following inequalities gives the possible values of "w"?
POSSIBLE ANSWERS
Pre-Structural Uni-Structural Multi-Structural Relational Extended Abstract
7. No Mastery Least Mastered Nearly Mastered Mastered Highly Mastered
w ≤ 8 w 11 w ≥4 w ≤4
Explanation of Answers
It is from the measure of the width. It is the measure of the
length of the rectangle.
The symbol used in the
answer is at least instead
of at most which was
given in the problem.
From the solution below:
2L + 2W ≤ P
2(W+4) + 2W ≤ 30
2W+8+ 2W ≤ 30
4W ≤ 30 - 18
4W ≤ 16
W ≤ 16/4
W ≤ 4
Item Content
Stimuli(Personal/Local/National/Global):
3. A family went to an amusement park and plan to take different rides. The amusement park charges P40.00 per ride. If the family have
P360.00,
Prime:
How many ride can they take?
8. POSSIBLE ANSWERS
Pre-Structural Uni-Structural Multi-Structural Relational Extended
Abstract
No Mastery Least Mastered Nearly Mastered Mastered Highly Mastered
y= 5 y= 200-40 y = 40/200 y= 200/40
Explanation of Answers
It shows the answer on the
equation being asked.
The operation used was is
incorrect.
The operation used was
right but the given number
was placed incorrectly.
The operation and placing of the given are correct. Linear
equation written correctly thus, represents the
relationships of the given problems.
Name: Learning Area: Mathematics
TABLE OF SPECIFICATION
VI. Quarter: 2 Week: Week 8 No. of Days/Hours: 4
Competency Level in the RBT
MELC/s: differentiates algebraic expressions, equations and inequalities.
illustrates linear equation and inequality in one variable.
Apply
VII. Enabling Competency: Apply
IV. Test Item Specifications Item No.1 Item No. 2 Item No. 3
Number of Test
Items
Level in the
RBT
Apply Remembering
Knowledge
Dimension
Conceptual Factual .
Item Content
Stimuli(Personal/Local/National/Global):
1. A number is 4 times another. If their sum is 70. Find the larger number.
9. Prime:
What is the larger number?
POSSIBLE ANSWERS
Pre-Structural Uni-Structural Multi-Structural Relational Extended Abstract
No Mastery Least Mastered Nearly Mastered Mastered Highly Mastered
4x x+4x=70 14 56
This answer is just one of the given facts. The answer is the equation or
number sentence of the
problem.
This answer is the smaller
number you’ll get in solving
the equation.
By the solution below:
x + 4x = 70
5x= 70
x=70/5
x=14 - the smaller
number
4(x)
4(14)
56 - the larger number
Item Content
Stimuli(Personal/Local/National/Global):
2.A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree
equation.It uses equal sign that indicates equality.
Prime:
Which of the following algebraic expressions are linear equations?
i.2x+ 5 = 3
ii. 3b2 ≤ 20
iii. ab + a + 5 = 10
iv.x ≥ 6
v. 3x - 6 = 2
10. POSSIBLE ANSWERS
Pre-Structural Uni-Structural Multi-Structural Relational Extended
Abstract
No Mastery Least Mastered Nearly Mastered Mastered Highly Mastered
Only b and d are linear equations. a, b, c are linear equations. Only a nd c are linear
equations.
a, c, e are linear equations.
Explanation of Answers
b and d are linear inequalities.
The symbol used expresses
inequality.
Only a and c are linear
equation while b is an
inequality also, it is q
quadratic equation.
a and c are linear
equations but e expression
is linear equation too.
These are all linear equations because the expressions’
highest exponent of the variable is one. And uses equal
sign that indicates equality.