UNIT FIVE RATIONAL EXPRESSIONS 18 HOURS MATH 521B

Transcription

1 UNIT FIVE RATIONAL EXPRESSIONS 8 HOURS MATH 5B Revised Nov4, 00 07

2 SCO: By the end of grade students will be epected to: B model (with concrete materials and pictorial representations) and epress the relationship between arithmetic operations and operations on algebraic epressions and equations Elaborations - Instructional Strategies/Suggestions Note to Teachers: Students may need a day or two of review on factoring What is a Rational Epression? Encourage students to create a definition for a rational number. Common fraction definition a A rational number is a number of the form where a, b 0 I and b 0. b Decimal fraction definition A rational number is any terminating or non-terminating repeating decimal fraction. E A rational algebraic epression is a fraction where the numerator and/or denominator are polynomials. Simplifying Rational Epressions Challenge students to simplify epressions like y + 3+ ) ) 3) 7 0 y 6 After a short discussion with students on how a rational epression is reduced, students should be able to develop rules for arithmetic or algebraic rational epressions. Find the largest possible factor common to the numerator and denominator. For algebraic epressions this will involve: < factor the numerator and denominator separately < state the restrictions < cancel out the common factors E: ( + )( + ) ( 3)( + ) +, 3, 3 Rational epressions give people who use statistics fleibility in using formulas in their fields. 08

3 Worthwhile Tasks for Instruction and/or Assessment Suggested resources What is a Rational Epression? Group Activity Generate a list of as many different appearing rational numbers as possible. Discussion For what values are the following rational epressions undefined: 4 ) 3) ) 4) What is a Rational Epression? Math Power 0 p.58 # 5 Algebra, Structure & Method Book p.9 # -8 See Worksheet at the end of the unit on restriction implications. Problem Solving p.46 #-3 Journal Evaluate each rational epression for 3 and notice a pattern: 4 ) ) 3) Simplifying Rational Epressions Performance a) For pattern epress the number of stars in the nth diagram in terms of n. b) Repeat for pattern. c) write rational epression that represents the ratio of pattern to pattern. Simplify the epression. d) for what values is the rational epression undefined? Pattern Simplifying Rational Epressions Choose sparingly from Math power 0 p.58 #9-47 odd p.59 # 53,54,56 Algebra, Structure & Method Book p.8 #-5 odd 9-35 odd continued on p. 5 09

4 SCO: By the end of grade students will be epected to: B model (with concrete materials and pictorial representations) and epress the relationship between arithmetic operations and operations on algebraic epressions and equations Elaborations - Instructional Strategies/Suggestions Multiplying or Dividing Rational Epressions Challenge student groups to simplify rational arithmetic and algebraic epressions. They should appreciate that the process is the same for both. Students must state any restrictions on the variable for rational algebraic epressions. E. ) ) B8 understand the relationships that eist between operations on integers, fractions and decimals, and operations on algebraic epressions 3) 4) a a a a 4 + a3 B37 demonstrate number and operation sense in solving inequality relationships and operating on rational epressions Worthwhile Tasks for Instruction and/or Assessment Suggested resources 0

5 Simplifying Rational Epressions continued from p. 3 Simplifying Rational Epressions Journal Why must restrictions be considered for rational epressions? Multiplying or Dividing Rational Epressions Multiplying or Dividing Rational Epressions Math Power 0 p.63 #7-7 odd #37-47 odd p.64 #53-59 Algebra, Structure & Method Book p.34 # 3- odd Group Activity Write two rational epressions with binomial denominators 8 that have a product of 6

6 SCO: By the end of grade students will be epected to: B model (with concrete materials and pictorial representations) and epress the relationship between arithmetic operations and operations on algebraic epressions and equations Elaborations - Instructional Strategies/Suggestions +,! Rational Epressions Use and investigate a range of eamples to allow student groups to generate algorithms for adding and subtracting rational epressions. Begin with eamples that have numeric denominators, progressing to eamples containing algebraic denominators. Hopefully, student groups will be able to generate the algorithms and see the common method no matter what the denominator. Good eamples to challenge the groups with might be: a) B8 understand the relationships that eist between operations on integers, fractions and decimals, and operations on algebraic epressions b) c) a a a d) B37 demonstrate number and operation sense in solving inequality relationships and operating on rational epressions

7 Worthwhile Tasks for Instruction and/or Assessment Suggested resources Multiplying or Dividing Rational Epressions Pencil/Paper/Discussion Write, but don t simplify, epressions for ªABC and ªDBC. Then write and simplify an epression for the ratio of the area of ªABC to the area of ªDBC. Multiplying or Dividing Rational Epressions +,! Rational Epressions Pencil/Paper Find the perimeter of this triangle: Performance Write two rational epressions with binomial denominators 4 5 that have a sum of. ( + )( ) +,! Rational Epressions Choose sparing from: Math Power 0 p.67 #9,,3 p.68 #9-3 p.7 #7-3 odd #43-53 odd # 64,66 p.73 # 67 Algebra, Structure & Method Book p.37 #,3,7,8,7, #33-36 Problem Solving p.03 #3,6,7 3

8 SCO: By the end of grade students will be epected to: Elaborations - Instructional Strategies/Suggestions Rational Equations Allow time for student groups to investigate a range of rational equations including: A6 apply properties of numbers when operating upon and epressing equations D4 apply the Pythagorean Theorem < equations with numerical denominators < equations with algebraic denominators < applications < geometric applications Remind groups that restrictions to the denominator are important here also. D5 apply and determine formulae for perimeter, area, surface area and volume 4

9 Worthwhile Tasks for Instruction and/or Assessment Suggested resources Rational Equations Research/Presentation Find the length of the bases and the altitude of the trapezoid if it has an area of m. Rational Equations Algebra, Structure & Method Book p.45 #,3,,7,9 p.46 #5,6,9 p.49 #3-3 odd p.50 #5,6,9, Pencil/Paper Find the dimensions of the rectangle below which has a shaded area of 64 cm. Pencil/Paper Solve each of the following: a) b) c) y 0 y y + 3 y + y 6 5

10 Rational Restrictions Worksheet Pencil/Paper/Technology Graph each of the following and show the discontinuity (applicable restrictions): a) y + b) y + c) y d) y + 4 e) y + 6 f) y

11 Note to Teacher: TI-83 Solution If you press zoom 4: decimal it will show the hole in the graph. You may have to set the window dimensions to multiples of the basic to get some of the problems to fit on the TI-83 screen. basic dimensions the dimensions I used press graph if you press trace and cursor along the graph to where!, you will see no function value 7