Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Polynomials
|
2 - Understand (polynomials are closed under addition)
2 - Understand (polynomials are closed under subtraction)
2 - Understand (polynomials are closed under multiplication)
2 - Add (polynomials)
2 - Subtract (polynomials)
2 - Multiply (polynomials)
|
2 - I can understand polynomials are closed under addition.
2 - I can understand polynomials are closed under subtraction.
2 - I can understand polynomials are closed under multiplication.
2 - I can add polynomials.
2 - I can subtract polynomials.
2 - I can multiply polynomials.
|
A. Polynomials can be manipulated arithmetically just as integers can.
B. Polynomials are closed for the operations of addition, subtraction, and multiplication.
|
A.1 When do I need to perform operations on polynomials?
B. 1 What does it mean to say that polynomials are closed for the operations of addition, subtraction, and multiplication?
|
Understand the relationship between zeros and factors of polynomials.
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Polynomials
|
1 - Know (the Remainder Theorem for polynomials)
3 - Apply (the Remainder Theorem for polynomials)
|
1 - I can know the Remainder Theorem for polynomials.
3 - I can apply the Remainder Theorem for polynomials.
|
A. One can can evaluate a polynomial function for a specific value by using the Remainder Theorem.
B. Applying the Remainder Theorem, one can determine if the binomial x - a is a factor of the polynomial p(x).
|
A.1. How can the Remainder Theorem for polynomials be used to determine the remainder when dividing a polynomial p(x) and a binomial x - a?
B. 1 What does it mean when the Remainder Theorem reveals a remainder other than zero after polynomial division?
|
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Polynomials
|
1 - Identify (zeros of a polynomial by factorization)
3 - Use (zeros to roughly graph polynomial function)
|
1 - I can identify zeros of a polynomial function by factorization.
3 - I can use zeros to roughly graph a polynomial function.
|
A. Zeros of a polynomial function, along with an understanding of the function's behavior, contribute to graph sketching.
|
A.1 Why do I need to be able to find the zeros of a polynomial?
A.2 How would the zeros of a polynomial function enable me to construct a rough graph?
|
Prove polynomial identities and use them to describe numerical relationships.
For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Polynomials
|
3 - Prove (polynomial identities)
3 - Use (identities to describe numerical relationships)
|
3 - I can prove polynomial identities.
3 - I can use identities to describe numerical relationships.
|
A. Polynomial identities can be used to generate other numerical relationships.
|
A.1 Why do I need to know polynomial identities?
A.2 How are polynomial identities used?
|
(+) Know and apply the Binomial Theorem for the expansion of (x + y)n powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Binomial Theorem
|
1 - Know (Binomial Theorem)
3 - Apply (Binomial Theorem)
3 - Determine (coefficients of expansion from Pascal's Triangle)
|
1 - I can know the Binomial Theorem
3 - I can apply the Binomial Theorem.
3 - I can expand [x+y]^n.
3 - I can determine coefficients of expansions by using Pascal's Triangle.
|
A. Binomial Theorem makes the expansion of (x+y)^n easier by using patterns.
B. Pascal's triangle is a very useful tool for determining coefficients of expansions.
|
A.1 What is the Binomial Theorem?
A. 2 Why should I know and understand the Binomial Theorem?
A.3 How do I use the Binomial Theorem?
B.1 What is Pascal's triangle, and where did it come from?
B.2 How many number patterns are there in Pascal's triangle?
B. 3 How can Pascal's triangle be used to determine the coefficients of an algebraic expansion?
|
Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Rational expressions
|
2 - Rewrite (rational expressions in different forms)
3 - Use (inspection)
3 - Use (long division)
3 - Use (computer algebra system)
|
2 - I can rewrite rational expressions in different forms.
2 - I can write a(x)/b(x) in the form q(x)+r(x)/b(x).
3 - I can use inspection.
3 - I can use long division.
3 - I can use a computer algebra system.
|
A. Rational expressions can be re-written into simpler forms.
|
A.1 How can I re-write the rational expression a(x)/b(x) into a more simple form of q(x) + r(x)/b(x)?
A.2 Why should I be able to re-write a rational expression using inspection, long division, or a computer algebra system?
|
(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
|
Rational expressions
|
2 - Understand (rational expressions form a system analogous to rational numbers)
2 - Understand (rational expressions are closed under addition)
2 - Understand (rational expressions are closed under subtraction)
2 - Understand (rational expressions are closed under multiplication)
2 - Understand (rational expressions are closed under division by a nonzero expression)
2 - Add (rational expressions)
2 - Subtract (rational expressions)
2 - Multiply (rational expressions)
2 - Divide (rational expressions)
|
2 - I can understand (rational expressions form a system analogous to rational numbers)
2 - I can understand rational expressions are closed under addition.
2 - I can understand rational expressions are closed under subtraction.
2 - I can understand rational expressions are closed under multiplication.
2 - I can understand rational expressions are closed under division by a nonzero expression.
2 - I can add rational expressions.
2 - I can subtract rational expressions.
2 - I can multiply rational expressions.
2 - I can divide rational expressions.
|
A. One can work with rational expressions just as with rational numbers.
B. Rational Expressions form a system that is closed under addition, subtraction, multiplication, and division by a nonzero rational experssion.
|
A.1 How can I add rational expressions?
A. 2 How can I subract rational expressions?
A. 3 How can I multiply rational expressions?
A. 4 How can I divide rational expressions?
B. 1 What does it mean to say the rational expressions are closed under addition, subtraction, multiplication, and division by a nonzero rational expression?
|
Be the first to comment below.
Please enter a Registration Key to continue.