Cluster Note: This cluster has many possibilities for Make sense of optional enrichment, such as relating the example in A.APR.4 to the problems and solution of the system u2+v2=1, v = t(u+1), relating the Pascal persevere in solving triangle property of binomial coefficients to (x+y)n+1 = (x+y)(x+y)n, them. deriving explicit formulas for the coefficients, or proving the binomial theorem by induction. Reason abstractly and quantitatively. A.APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity Construct viable (x2 + y2)2 = (x2 y2)2 + (2xy)2 can be used to generate Pythagorean – arguments and triples. critique the Essential Skills and Knowledge reasoning of others. Knowledge of the process for proving identities Ability to see, use and manipulate the structure in an Model with expression as needed to an identity mathematics. Use appropriate Use polynomial A.APR.5 (+) Know and apply the Binomial Theorem for the expansion identities to tools of (x + y)n in powers of x and y for a positive integer n, where x and y solve problems. strategically. are any numbers, with coefficients determined for example by Pascal’s Triangle. Attend to precision. Note: The Binomial Theorem can be proved by mathematical induction or by combinatorial argument. Look for and make use of structure. Essential Skills and Knowledge Ability to replicate Pascal’s triangle Look for and express regularity in repeated reasoning.
Use polynomial identities to solve problems. Cluster Note: This cluster has many possibilities for optional enrichment, such as relating the example in A.APR.4 to the solution of the system u2+v2=1, v = t(u+1), relating the Pascal triangle property of binomial coefficients to (x+y)n+1 = (x+y)(x+y)n, deriving explicit formulas for the coefficients, or proving the binomial theorem by induction. A.APR.4 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. Essential Skills and Knowledge Knowledge of the process for proving identities Ability to see, use and manipulate the structure in an expression as needed to an identity A.APR.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in 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. Note: The Binomial Theorem can be proved by mathematical induction or by combinatorial argument. Essential Skills and Knowledge Ability to replicate Pascal’s triangle Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
AlgebraIIFramework.docx