Name Period Date POLYNOMIALS Student Packet 4: Polynomial Arithmetic Applications POLY4.1 Hundred Chart Patterns Gather empirical data to form conjectures about number patterns. Write algebraic expressions. Practice polynomial arithmetic. Use algebraic expressions to prove (or disprove) conjectures. View algebra as a useful mathematical tool. POLY4. Picture Frames Use mathematical reasoning to create polynomial expressions that generalize patterns. Practice polynomial arithmetic. POLY4.3 Number Tricks Use algebraic expressions to generalize patterns. Practice polynomial arithmetic Write verbal expressions as algebraic expressions. POLY4 STUDENT PAGES POLY4.4 Vocabulary, Skill Builders, and Review 16 1 7 11 POLY4 SP
WORD BANK (POLY4) Word Definition or Explanation Example or Picture conjecture deductive reasoning empirical evidence generalization inductive reasoning proof POLY4 SP0
4.1 Hundred Chart Patterns HUNDRED CHART PATTERNS Ready (Summary) We will investigate patterns on the hundred chart. We will write and use algebraic expressions to prove conjectures based on the patterns. Pick any four consecutive numbers on the hundred chart. Go (Warmup) Set (Goals) Gather empirical data to form conjectures about number patterns. Write algebraic expressions. Practice polynomial arithmetic. Use algebraic expressions to prove (or disprove) conjectures. View algebra as a useful mathematical tool. Find the inner product. Find the outer product. Example: 5 6 7 8 6 7 = 4 5 8 = 40 1. Compare the products: Try this for at least four other groups of four consecutive numbers.. 0 1 3 1 = = 3. 4. 5. 6. Write a conjecture about this pattern as a complete sentence. POLY4 SP1
4.1 Hundred Chart Patterns HUNDRED CHART 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 1 3 4 5 6 7 8 9 30 31 3 33 34 35 36 37 38 39 40 41 4 43 44 45 46 47 48 49 50 51 5 53 54 55 56 57 58 59 60 61 6 63 64 65 66 67 68 69 70 71 7 73 74 75 76 77 78 79 80 81 8 83 84 85 86 87 88 89 90 91 9 93 94 95 96 97 98 99 100 POLY4 SP
4.1 Hundred Chart Patterns 1. Conjecture that the class is going to prove: PROVING A CONJECTURE. Prove your conjecture by using algebra to label each of the four consecutive numbers and then by multiplying the inner and outer products. Inner Product ( )( ) 3. Why does this prove the conjecture? Outer Product ( )( ) POLY4 SP3
4.1 Hundred Chart Patterns A SECOND CONJECTURE Now try a different experiment from the hundred chart with four numbers that form a square. Pick any four numbers on the hundred chart that form a x square. 1. 5 6 15 16. Compare them: Find the product of the diagonal that begins in the upper left corner Find the product of the diagonal that begins in the upper right corner 5 16 = 6 = Try this for at least three other groups of four numbers that form a square. 3. 4. 5. 3 3 33 33 = = 6. What do you notice in all of these examples? Make a conjecture using a complete sentence. POLY4 SP4
4.1 Hundred Chart Patterns A SECOND CONJECTURE (continued) 1. Conjecture that the class is going to prove:. Prove your conjecture by using algebra to label each of the four numbers and then by multiplying each diagonal. Find the product of the diagonal that begins in the upper left corner. ( )( ) 3. Why does this prove the conjecture? Find the product of the diagonal that begins in the upper right corner. ( )( ) POLY4 SP5
4.1 Hundred Chart Patterns A THIRD CONJECTURE Experiment with patterns like these four numbers on a 3 x 3 square. Multiply the vertical numbers. Multiply the horizontal numbers. Do this for at least three more 3 x 3 squares. 5 14 16 5 Horizontal Product Vertical Product 1.. 3. 4. 14 16 = 4 5 5 = 5. Write your conjecture in words. 6. Prove your conjecture algebraically: 7. Does your conjecture hold? Horizontal Product Vertical Product POLY4 SP6
4. Picture Frames Ready (Summary) We will create polynomial expressions that generalize a geometric pattern, and simplify the expressions. We will use our understanding of the construction of each pattern to verify the accuracy of the polynomial we created. PICTURE FRAMES Go (Warmup) 1. Write the formula for finding the area of each: a. A square with side length s. b. A rectangle with a base, b, and a height, h. c. A circle with radius r.. For this square-inside-of-a-square picture: a. Write in words the steps you could take to find the area of the shaded region. Set (Goals) Use mathematical reasoning to create polynomial expressions that generalize patterns Practice polynomial arithmetic b. Find the area of the shaded region if the side length if the larger square is 8 inches and the side length of the smaller square is 5 inches. 3. Find the area of a circle with a radius of 7 cm. Leave the answer in terms of π. POLY4 SP7
4. Picture Frames A SQUARE PICTURE FRAME PATTERN 1. Study the given square picture frame pattern, and fill in the first 3 rows of the table.. Sketch the picture for square n. 3. Fill in the last row of the table to find a generalized expression for the shaded area. Square pattern number Side length of outer square Side length of inner square 8 8 5 9 9 6 10 n square 8 square 9 square 10 square n Un-simplified expression for shaded area (picture frame) 8 5 = 64 5 Simplified expression for shaded area 4. Substitute the values from squares 8, 9, and 10 into the simplified expression for the th shaded area of the n figure as a check. 39 POLY4 SP8
4. Picture Frames A CIRCULAR PICTURE FRAME PATTERN 1. Study the given circle pattern and fill in the first 3 rows of the table. Leave the answers in terms of π.. Sketch circle n. 3. Fill in the last row of the table to find a generalized expression for the shaded area of this circle pattern. circle 7 circle 8 circle 9 circle n Circle pattern number Outer circle radius Inner circle radius 7 7 3 8 8 4 9 n Shaded area expression (un-simplified) Shaded area (simplified) 4. Substitute the values from circles 7, 8, and 9 into the simplified expression for the shaded th area of the n figure as a check. POLY4 SP9
4. Picture Frames A RECTANGULAR PICTURE FRAME PATTERN 1. Study the given rectangle pattern and fill in the first 3 rows of the table.. Sketch rectangle n. 3. Fill in the last row of the table to find a generalized expression for the shaded area of this rectangle pattern. rectangle 4 rectangle 5 rectangle 6 rectangle n Rectangle pattern number Outer rectangle dimensions Inner rectangle dimensions 4 4 7 4 5 5 8 3 5 6 n Shaded area expression (un-simplified) Shaded area (simplified) 4. Substitute the values from rectangles 4, 5, and 6 into the simplified expression for the th shaded area of the n figure as a check. POLY4 SP10
4.3 Number Tricks Ready (Summary) We will perform mathematical number tricks and use algebraic expressions to show how they work. 1. Perform the number trick below. NUMBER TRICKS Go (Warmup) Set (Goals) Use algebraic expressions to generalize patterns. Practice polynomial arithmetic Write verbal expressions as algebraic expressions. Step Directions Numbers Algebraic Process 1 Choose a natural number between 1 and 10. Multiply your number by. n 3 Add 8 to your answer. 4 Divide your answer by. 5 Subtract your original number from your answer. 6 What number did you end with?. What is the number trick? 3. Explain why the algebraic process supports that this trick will work for all numbers? n POLY4 SP11
4.3 Number Tricks 1. Perform the number trick below. NUMBER TRICK 1 Step Words Numbers Algebraic Process 1 Choose a number. n Multiply your number by one more than the original number. 3 Add your original number. 4 Add 1. 5 Divide by 1 more than your original number 6 Subtract 1 7 What number do you have now?. What is the number trick? n(n + 1) = 3. Explain why the algebraic process supports that this trick will work for all numbers? POLY4 SP1
4.3 Number Tricks 1. Perform the number trick below. NUMBER TRICK Step Words Numbers Algebraic Process 1 Choose a number. n Add 4. n + 4 3 Multiply by your original number. 4 Multiply by 4. 5 Divide by your original number. 6 Subtract 16. 7 Divide by 4. 8 What number do you have now?. What is the number trick? nn+ ( 4) = 3. Explain why the algebraic process supports that this trick will work for all numbers? POLY4 SP13
4.3 Number Tricks 1. Perform the number trick below. MORE NUMBER TRICKS Step Words Numbers Algebraic Process 1 Choose a number. n Square your number. n 3 4 Add three more than four times your original number Divide by 1 more than your original number. 5 Subtract the original number. 6 What is the result?. Perform the number trick below. n + 4n+ 3 Step Words Numbers Algebraic Process 1 Choose a number. n Square it. 3 Subtract 4. 4 Divide by less than your original number. 5 Subtract your original number 6 What is the result? POLY4 SP14
4.3 Number Tricks Use this page to create your own number trick. NUMBER TRICK TEMPLATE Step Words Numbers Algebraic Process POLY4 SP15
4.4 Vocabulary, Skill Builders, and Review Here is a number trick. FOCUS ON VOCABULARY (POLY 4) Choose a number. Add 4. Multiply by. Subtract 8. Divide by your original number. The result is Felicity did some work to verify this trick. Match her work with the word or words that describe what she did. You may use a word more than once. 1. First she tried it with numbers.. 3. 4. 5 9 18 10 1 16 3 4 Then she said, I think the result is always going to be. Then she drew these pictures. Then she said, This shows that the result is always going to be. A. conjecture B. deductive reasoning C. empirical evidence D. generalization E. inductive reasoning F. Proof POLY4 SP16
4.4 Vocabulary, Skill Builders, and Review 1. SKILL BUILDER 1 Use an area model to factor. First find the GCF of the terms. 5x 15x =. When factoring a binomial: First look for the GCF of all the terms Then look for the difference of two squares pattern 4x + 1x 0 Factor completely. If it cannot be factored, write not factorable. 3. x + 8 4. 6. GCF 4x 16x 5. 1x 7 When factoring a trinomial: First look for the GCF of all the terms Then look for a quadratic trinomial where coefficient of the square term is 1 Then look for a quadratic trinomial where the coefficient of the square term is not 1 Factor completely. If it cannot be factored, write not factorable. 5x + 15x 0 7. 3x + 1x 9 8. 4x + 7x 5 POLY4 SP17
4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER Rewrite each expression as a power of. 1.. 3. (4 ) (4 ) 3 ( ) 4. 5. 6. 7 7 8 4 3 4 5 8 4 Evaluate when a = 1, b = -, c = - 7. 8. 9. -b 4 b 4ac a a 10. 11. -b + b 4ac a 4 1-8 16 4 (8 ) -b b 4ac a 3 3 POLY4 SP18
4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 3 Find equations of lines in different forms. Use the information given. 1. Given: (graph). Given (table): slope-intercept form point-slope form standard form Evaluate each expression if x = - 3 x y 0-6 1-3 0 3 3 4 6 slope-intercept form point-slope form standard form 3. x 4 4. x 3 5. x 6. x 1 7. x 0 8. x -1 9. x - 10. x -3 11. Examine your answers to problems 5-1. Under what conditions is the result positive? negative? a fraction between 0 and 1? POLY4 SP19
4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 4 A radical expression containing square roots is simplified when there are: no perfect squares under the radical no fractions under the radical no radicals in the denominator. Simplify each expression. 1.. 3. 44 9 48 75 4. 5. 6. 3 1 9 13 49 10 11 16 4 1 8 7. 8. 9. 4 ( + ) ( + ) ( + )( ) 10. 11. 1. 40 16 4 5 + 5 4()(3) + 3 5 9 6 () POLY4 SP0
4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 5 Write each polynomial as a sum of terms in decreasing order of powers. 1. 5 + (x + 3)(x 3 ). 6 (x )(x 3) 3. x (x + 3)(x ) Factor completely. If it cannot be factored, write not factorable. 4. x 7x + 10 5. x + 10x + 5 6. x - 64 7. -50x + 30 8. x + 15x + 6 9. x + 8x + 3 Use >, <, or = to make each statement true. Show work. 10. 11. 1. 1 4 3 4 4 3 3 7 7 7 7 3 1 1 1 (9 16) 9 16 Determine which numbers are in scientific notation. If NOT, write it in scientific notation. 13. 8.1 x 10 5 14. 0.13 x 10-4 POLY4 SP1
4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 6 1. Between which two consecutive integers is 59? and Simplify each radical expression.. 3. 4. 49 6 4 14 6 4 5. 6. 7. 7 49 3 50 8. 9. 10. 3 1 36 17 81 ( + 5 )( 5 ) 18 4 30 3 4 Solve and graph: 11. 1. -x + 7 5-3 33x + 8 POLY4 SP
4.4 Vocabulary, Skill Builders, and Review 1. Solve for x. Check by substitution. Given graph: 1 6 1 = -36 3 x + x SKILL BUILDER 7. 0.5x + 4 = -x 6 4. Write the equation in slope-intercept form. 5. Write the equation in standard form. 6. State the x- and y- intercepts. Compute: 7. (-3) 8. 5 3 9. -3 10. -1(-6 + 4) 11. 36 1 (-6 + 4) 1. 36 1(-6 + 4) POLY4 SP3
4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 8 Simplify each expression by combining like terms. Write each polynomial as a sum. 1. x 5y + 6x 8y. -13 x + 5y 7 x 3. 4. 6x 9 ( x 7) ( xy 4 x ) (3x 5 xy) 5. -5( x y) 3( y x) 6. x( x + y) + y( x + y) 7. ( x 6)( x + 6) 8. ( x 7)( x 10) 8. ( x 3)(-x + 8) 10. (5 x)( x + 9) 11. Write a variable expression for the area of a square whose side is x + 8. POLY4 SP4
4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 9 Write each statement using symbols. If the variable is on the right side, change it to the left side using appropriate properties. Then graph each. 1. x is less than -. x is greater or equal to than - 3. the opposite of x is less than or equal to -3 4. 3 is less than x 5. - is less than or equal to the opposite of x. Graph each inequality. Be sure they are in slope-intercept form first. 6. 7. y > x+ 1 - y+ x 10 3 8. Describe the differences between the graph of an inequality in one variable and the graph of an inequality in two variables. POLY4 SP5
4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 10 Factor using any method. Be sure to factor each polynomial completely. If it cannot be factored write not factorable. 1. 8x + 40x. 4. x + 10x + 9 5. 7. x 15x + 6 8. 10. 10x + 80x + 160 11. 13. x 9 14. 16. 3x 18x 17. 15x 35x 3. 7y 49y x x 4x 77 6. x + 9x 36 30x + 00 9. x + x + 6 x 4x 30 1. 3x + 18x 3 x 50 15. x 10 x + 4x + 144 18. x 4x 144 POLY4 SP6
4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 11 Graph each system of inequalities. Test several points to verify correct shading. 1. y -3x + 5 y > 1 - x 3 Complete the table. Decimal Ex. 0.009 1. 0.008. Fraction 9 1,000 4.76 100. y 3 < - 5 x x + y -4 Product of a number between 1 and 10, and a multiple of 10 1 3. 3.5 7 10 Scientific notation 9 0.001 or 9 1 9 10-3 10 3 4. 4. 10 3 POLY4 SP7
4.4 Vocabulary, Skill Builders, and Review Solve each system using algebra. 1. 6x y = -16 4x + y = 1 SKILL BUILDER POLY 1. 4x = 6 + y x 1 y = 3 Arnon wants to record the 1-hour opera marathon on the radio. He has 90-minute discs and 60-minute discs. If he uses 9 discs, how many of each type will he use? 3. Solve the algebraically problem using 4. Solve the problem algebraically using one variable. two variables (a system of equations). 5. Could you solve this problem without algebra? Explain. 6. Think of a number. Subtract 7. Multiply by 3. Add 30. Divide by 3. Subtract the original number. The result is always 3. Use polynomials to illustrate this number trick. POLY4 SP8
4.4 Vocabulary, Skill Builders, and Review TEST PREPARATION (POLY4) Show your work on a separate sheet of paper and choose the best answer. 1. Below is an excerpt from a hundreds chart. If 40 = n, which expression(s) represent(s) 54? 40 41 4 43 44 45 50 51 5 53 54 55 60 61 6 63 64 65 A. n + 10 + 4 B. n + 5 C. n + 14 D. n + 54. Below is a 3 3 square taken from a hundreds chart. What expression (in terms of n) should go in the square with the question mark? n A. n + 1 B. 3n + 1 C. n + 1 D. n + 1 3. Below is a square taken from a hundreds chart. What expression (in terms of n) represents ab. a b n A. ( n+ 1)( n+ 10) B. ( n 1)( n 10) C. ( n+ 1)( n 10) D. ( n 1)( n+ 10) 4. If n is a number, what is an expression for four more than four times a number. A. 4n + 4 B. n(4 + 4) C. (4 + 4) n D. 4+ 4+ n 5. The radius of the smaller circle is given. If the radius of the larger circle is twice the radius of the smaller circle, what is the area of the shaded part? A. 8π B. 36π C. 64π D. 48π? 4 POLY4 SP9
4.4 Vocabulary, Skill Builders, and Review 4.1 Hundred Chart Proofs KNOWLEDGE CHECK (POLY4) 1. Below is an excerpt from a hundreds chart. If 5 = n, what does 6 represent? Give the most simplified answer.. 40 41 4 43 44 45 50 51 5 53 54 55 60 61 6 63 64 65 1 4 36 39 74 77 11 14 46 49 84 87 1 14 = 14 11 14 = 44 36 49 = 1,764 46 39 = 1,794 74 87 = 6,438 84 77 = 6,468 If the following table follows the pattern above, write a conjecture for the relationship between ad and bc. 4. Picture Frames Use the following picture to answer questions 3 and 4. 3. Write an expression for the area of the smaller rectangle. 4. Write an expression for the area of the shaded part. 4.3 Proving Number Tricks a c 5. If n is a number, write an expression for the product of a number and one more than the number. b d 4 n n+3 7 POLY4 SP30
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HOME-SCHOOL CONNECTION (POLY4) For problems 1 and, consider the 3 3square below taken from a hundreds chart. a n b 1. What expression (in terms of n) should go in the square with the a and what expression (in terms of n) should go in the square with the b?. Multiply ab so that it is a sum of terms 3. If the length of the smaller square is n and the length of the larger square is 1, what is an expression for the area of the shaded part? Signature Date POLY4 SP33
COMMON CORE STATE STANDARDS MATHEMATICS A-SSE-1a A-SSE-1b A-SSE- CA Addition (CA.A.8a) CA Addition (CA.A.8b) A-APR-1 A-APR-4 MP1 MP MP3 MP4 MP5 MP6 MP7 MP8 STANDARDS FOR MATHEMATICAL CONTENT Interpret expressions that represent a quantity in terms of its context: Interpret parts of an expression, such as terms, factors, and coefficients. Interpret expressions that represent a quantity in terms of its context: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r) n as the product of P and a factor not depending on P. Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x ) (y ), thus recognizing it as a difference of squares that can be factored as (x y )(x + y ). Use the distributive property to express a sum of terms with a common factor as a multiple of a sum of terms with no common factor. For example, express xy + x y as xy (y + x). Use the properties of operations to express a product of a sum of terms as a sum of products. For example, use the properties of operations to express (x + 5)(3 - x + c) as -x + cx - x + 5c + 15. 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, and divide polynomials by monomials. Solve problems in and out of context. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x + y ) = (x y ) + (xy) can be used to generate Pythagorean triples. STANDARDS FOR MATHEMATICAL PRACTICE 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. First Printing DO NOT DUPLICATE 01 POLY4 SP34