Sail into Summer with Math!

Transcription

1 Sail ito Summer with Math! For Studets Eterig Hoors Geometry This summer math booklet was developed to provide studets i kidergarte through the eighth grade a opportuity to review grade level math objectives ad to improve math performace.

2 Hoors Geometry Summer Packet Table of Cotets Page Objective Suggested Completio Date 1 Fractio Operatios Jue 24 th 2 Order of Operatios July 1 st 3 Proportios July 8 th 4 Squares, Square Roots, ad the Laws of Epoets..... July 15 th 5 Solvig Equatios I July 26 th 6 Solvig Equatios II July 29 th 7 Iequalities August 5 th 8 Irregular Areas August 9 th 9 Factorig Quadratic Equatios August 12 th 10 Solvig Systems of Equatios August 16 th Graphig Liear Equatios August 19 th 13 Simplifyig Radicals (Preview) August 25 th

3 Fractio Operatios Eercises: Perform the idicated operatio No Calculators! SHOW ALL WORK. Use a separate sheet of paper (if eeded) ad staple to this page Hoors Geometry Page 1 Summer

4 Order of Operatios Evaluate each epressio. SHOW ALL WORK. No calculators [16 (3 + 7) 5] (2) 9 5. [1 + 3(9 + 12)] (4 2) (3 5 27) (17 11) (9 + 7) 2 4 (2 3) Hoors Geometry Page 2 Summer

5 Proportios Solve each proportio for the missig term. SHOW YOUR WORK. No Calculators = = y = = = 7 a = = = 8 9. Seth ears $152 i 4 days. At that rate, how may days will it take to ear $532? 10. Laette drove 248 miles i 4 hours. At that rate, how log will it take her to drive a additioal 93 miles? 11. A blueprit for a house states 2.5 iches equals 10 feet. If the legth of the wall is 12 feet, how log is the wall i the blueprit? Hoors Geometry Page 3 Summer

6 Hits/Guide: Squares, Square Roots, ad the Laws of Epoets Epoets are a way to represet repeated multiplicatio, so that 3 4 meas 3 multiplied four times, or , which equals 81. I this eample, 3 is the base ad 4 is the power. Roots are the base umbers that correspod to a give power, so the square (referrig to the power of 2) root of 81 is 9 because 9 9 = 81 ad the fourth root of 81 is 3 because is 81., where is the root ide ad is the radicad There are certai rules whe dealig with epoets that we ca use to simplify problems. They m m are: Addig powers a a a Eercises: Evaluate: m m Multiplyig powers a a Subtractig powers a a m a Negative powers 1 a a To the zero power a (8 4)² = 2. (4 2)² (5 8)³ = 3. 5 (8 3)² = m (9 125) 6. ( 8 4)(1 3) Simplify the followig problems usig epoets (Do ot multiply out): = = = = = = Hoors Geometry Page 4 Summer

7 Hits/Guide: Solvig Equatios I The key i equatio solvig is to isolate the variable, to get the letter by itself. I two-step equatios, we must udo additio ad subtractio first, the multiplicatio ad divisio. Remember the golde rule of equatio solvig: If we do somethig to oe side of the equatio, we must do the eact same thig to the other side. Eamples: = = = -2 Solve: 4 (-2) - 6 = -14 = = -14 Solve: (24/-6) - 4 = = = -8-8 = -8 Whe solvig equatios that iclude basic mathematical operatios, we must simplify the mathematics first, the solve the equatios. For eample: 5 (4-3) + 7 = 4 (9-6) 5 (1) + 7 = 4 (3) Check: 5 (4-3) + 7 (1) = 4 (9-6) = = 4 (3) = 12 7 = = 1 Eercises: Solve the followig equatios usig the rules listed o the previous pages: SHOW ALL WORK. Use a separate sheet of paper (if ecessary) ad staple to this page. -6 m 1. -4t + 3t - 8 = r + 5-6r = ( 7) 6 3 y 5. 6g + (-3) = ( 4) t 9 5(4 3) t t = 8 (7 (-2)) 9. 7(6 ( 8)) (3-6) = 6 (4 + t) 11. 4r + 5r - 8r = (7 + ) = 5(7 - (-4)) 13. Eplai i words how to solve a two step equatio, similar to ay of the equatios i problems 2 through 6 above. Hoors Geometry Page 5 Summer

8 Hits/Guide: Solvig Equatios II As we kow, the key i equatio solvig is to isolate the variable. I equatios with variables o each side of the equatio, we must combie the variables first by addig or subtractig the amout of oe variable o each side of the equatio to have a variable term o oe side of the equatio. The, we must udo the additio ad subtractio, the multiplicatio ad divisio. Remember the golde rule of equatio solvig. Eamples: 8-6 = t = t t + 6t 4-6 = 5 5 = t = = 10t = Eercises: Solve the followig problems: No Calculators! SHOW ALL WORK. Use a separate sheet of paper (if ecessary) ad staple to this page r - 7 = 6r r t = 5t = 3-3 t 4. 6y + 5 = 6y = p - 8 = -6p Rowboat Retals: $5.00 per hour plus a $ deposit. Deposit will be refuded if the boat is retured udamaged. Which equatio represets the total cost for retig ad returig a row-boat udamaged? Let c be the total cost i dollars ad t be the time i hours. a. c = 5t b. c = 500t c. c = 100t + 5 d. c = 5t 8. Ted wats to buy a $ bike. He has two optios for paymet. Optio Oe: Ted ca borrow the $ from his father ad repay him $40.00 a moth for a year. Optio Two: The bike shop will fiace the bike for oe year at a 15% aual iterest rate. The formula for the total amout paid (a) is: a = p + prt, where p i the amout borrowed, r is the rate of iterest, ad t is the time i years. Which optio would cost Ted the least amout of moey? Eplai how you determied your aswer. Use words, symbols, or both i your eplaatio. Hoors Geometry Page 6 Summer

9 Iequalities Hits/Guide: I solvig iequalities, the solutio process is very similar to solvig equalities. The goal is still to isolate the variable, to get the letter by itself. However, the oe differece betwee equatios ad iequalities is that whe solvig iequalities, whe we multiply or divide by a egative umber, we must chage the directio of the iequality. Also, sice a iequality as may solutios, we ca represet the solutio of a iequality by a set of umbers or by the umbers o a umber lie. Iequality - a statemet cotaiig oe of the followig symbols: < is less tha > is greater tha < is less tha or equal to > is greater tha or equal to _ is ot equal to Eamples: 1. Itegers betwee -4 ad All umbers betwee -4 ad The positive umbers So, to solve the iequality -4 < -8 becomes -4 < ad therefore > 2 is the solutio (this is because wheever we multiply or divide a iequality by a egative umber, the directio of the iequality must chage) ad ca be represeted as: Eercises: Solve the followig problems: 1. 4 > 9 No Calculators! 2. -5t > Hoors Geometry Page 7 Summer

10 Hits/Guide: Irregular Area To solve problems ivolvig irregular area, use either a additive or a subtractive method. I a additive area problem, break the object dow ito kow shapes ad the add the areas together. I a subtractive area problem, subtract the area of kow shapes from a larger whole. Eercises: 1. The bakig sheet show holds 12 cookies. Each cookie has a diameter of 3 iches. What is the area of the uused part of the bakig sheet? Roud your aswer to the earest square ich. 2. Fid the area of the shaded regios. r = 6 m a. b. 20 m 9 ft 4 ft 15 ft c. d. 6 m 10 m 2 m 6 m 10 m 4 cm 4 cm 14 m Hoors Geometry Page 8 Summer

11 Hits/Guide: Factorig Quadratic Equatios Factorig a polyomial ca make a problem easier to solve or allow oe to easily fid the roots of a equatio. Factorig ca be thought of as the opposite of distributio because terms are epaded, usually from a triomial (three term) equatio to a equatio which is the product of two biomial (two) terms. Eamples: ² = ( + 2)( + 3) 2² = (2 + 1)( - 2) If these equatios are set to zero, the we ca solve for the roots of the equatio by settig each biomial term to zero. Eample: 2² = 0 (2 + 1)( - 2) = 0 which meas that = 0 or - 2 = 0 because if the product is zero, the oe of the factors must be zero. therefore, = -0.5 or = 2. Eercises: Fid the roots of each equatio. 1. a² + a - 30 = 0 2. b² + 7b + 12 = 0 3. m² - 14m + 40 = 0 4. s² + 3s = a² + 22a + 3 = ² = ² = y + 2y² = m + 6m² = ² - 2 = ² + = ² - 4 = 4 Hoors Geometry Page 9 Summer

12 Hits/Guide: Solvig Systems of Equatios A system of equatios occurs whe a situatio ivolves multiple compoets that ca idividually be described usig equatios. Where these equatios itersect, their ad y values are the same. Graphically, this appears as a itersectio of lies. Algebraically, the ad y values that solve simultaeous equatios are the same. The three primary methods of solvig systems of equatios are graphically, by substitutio, ad by liear combiatio. Eercises: Solve each system of equatios usig ay method y = = 5 + 2y 6 + 8y = = 3 4y y = y = y = y = 8 5. Haz ad Mario wet to a sale at a music store where all CDs were oe price ad all cassettes were aother price. Haz bought 2 CDs ad 2 cassettes for $40.00, while Mario bought 1 CD ad 4 cassettes for $ The equatios below represet these purchases, where is the cost of a CD ad y is the cost of a cassette. Haz 2 + 2y = 40 Mario + 4y = 44 What are the costs of a sigle CD ad a sigle cassette? Solve the system of equatios by either costructig a graph o a sheet of graph paper or by usig a algebraic process. Eplai how you determied the costs. Use words, symbols, or both i your eplaatio. 6. A eam will have 20 questios ad be worth a total of 100 poits. There will be a true/false sectio where the questios are worth 3 poits each ad a short essay sectio where the questios will be worth 11 poits each. How may essay questios will there be o the test? Hoors Geometry Page 10 Summer

13 Graphig Liear Equatios Referece Sheet Hits/Guide: Hoors Geometry Page 11 Summer

14 Graphig Liear Equatios Practice Idetify the slope ad y-itercept of the followig liear equatios ad the graph them o the coordiate plae. 3. Put the followig equatios ito slope-itercept form (solve for y) ad the graph them o the coordiate plae Hoors Geometry Page 12 Summer

15 Simplifyig Radicals PREVIEW (**This iformatio will be eplicitly taught i Hoors Geometry) Hits/Guide: To simplify radicals, first factor the radicad as much as possible, the "pull out" square terms usig the followig rules: 2 a a a a ab a b as log as b 0 b b A radical is i simplest form whe: there is o iteger uder the radical sig with a perfect square factor, there are o fractios uder the radical sig, ad there are o radicals i the deomiator. Eercises: Simplify each epressio Nia says that caot be simplified. George says that is ca be simplified to Who is correct? Eplai how you kow. Hoors Geometry Page 13 Summer