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Factorizing Quadratics

High school

• • • • • • • • • • • • • • • • • • • • • Objective • • • • • • • • • • • • • • • • • • • • •

Evaluate formulas and expressions.
Manipulate algebraic expressions.

• • • • • • • • • • • • • • • Explanation of the activity • • • • • • • • • • • • • •

Confirm the equation

+ 5

+6 = (

+ 2) (

+ 3) is always true by plugging in values for

While working on this activity, students will be learning about the factorization of quadratic
expressions. Some students may also study the graphs of factorized quadratic functions.

• • • • • • • • • • • • • • • • • Using the calculator • • • • • • • • • • • • • • • • •

Calculator functions used: Addition, subtraction, multiplication, division, squaring,

memory calculation, Multi-line Playback

Example:

Confirm the equation

+ 5

+6 = (

+ 2) (

+ 3) is always true.

(

) Store any number,

, into your calculator.

(2) Use your calculator to evaluate x

+ 5x + 6

(3) Now use your calculator to evaluate (x + 2) (x + 3)

Store

Calculate

+ 5

+ 6.

Calculate (

+ 2)(

+ 3).

Press the following buttons and then start operation.
(Variable

printed in blue under one of keys)

(X+2)(X+3)=

DEG

+5X+6=

DEG

Repeat for other values of

until you are sure

the equation always holds true.

(X+2)(X+3)=

DEG

+5X+6=

DEG

Check with Multi-Line Playback

Summary of Contents for EL-531RH

Page 1: ...SCIENTIFIC CALCULATOR TEACHER S GUIDE JULY 1999 EL 531RH ...

Page 2: ...n page 39 marked with We encourage you to put this guide to good use Introduction p 2 How to Operate p 3 Number of Bowling p 4 Down to One p 6 Reverse the Order p 8 Different Products p 10 Sums and Products p 12 Target 100 p 14 Ordering Fractions p 16 Addting Fractions p 18 Halfway Between p 20 Near Integers p 22 Reshaping Cuboids p 24 Function Tables p 26 Palindromes p 28 Trial and Improvement p ...

Page 3: ...the calculator as an effective learning tool in the classroom This Teacher s Guide presents several classroom activities that make use of Sharp scientific calculators The purpose of these activities is not to introduce the calcula tor as a device to relieve the burden of performing difficult calculations but rather to develop the students understanding of mathematical concepts and explore areas of...

Page 4: ...normal arithmetic and function calculations Mode 1 STAT 1 mode for performing 1 variable statistical calculations Mode 2 STAT 2 mode for performing 2 variable statistical calculations If the calculator fails to operate normally press the reset switch on the back to reinitialise the unit The display format and calculation mode will return to their initial settings RESET 2 RESET SWITCH Reset switch ...

Page 5: ... 3 digit number and enter it into your calculator Pretend each digit is a bowling pin Knock down each pin one at a time so that your calculator display shows 0 A Using subtraction B Using addition 1 Enter a 3 digit number 2 Knock down one digit or pin i e change the last digit to a 0 3 Knock down the next pin i e change the tens column digit to 0 4 Knock down the pin of the hundreds column A Using...

Page 6: ... the calculator and the other student has to knock each digit or pin down Example 638 8 630 630 30 600 600 600 0 Points for students to discuss It is important for students to talk about what they are doing and use the appropriate language for example six hundred and thirty minus thirty equals six hundred Students should be challenged to vary the starting point i e sometimes starting with the hund...

Page 7: ... number 1 Players can use any of the numbers 1 9 together with any of the keys below You cannot put numbers together to make 2 or 3 digit numbers You can use each number only once The first player to get his her calculator display to show 1 scores five points If after an agreed time limit no player has reached 1 the player who is closest scores two points While working on this activity students sh...

Page 8: ... set to fixed decimal place mode by pressing the 2ndF key once and then pressing the key which has FSE written in yellow above it until FIX is displayed at the top of the calculator screen And press 2ndF TAB and 0 keys Doing this will round answers to 0 decimal places The starting number can then be generated by multiplying a random number by 100 Further Ideas Play the game using decimal starting ...

Page 9: ...hile working on this activity students should develop their skills of mental mathematics They should also be interpreting and generalizing their answers Example A To reverse the order of 58 Solution Add 27 to 58 to get 85 Now try using a 3 digit number Example B Enter 432 into the calculator Solution Add 198 to 234 to get 432 58 27 432 234 234 198 Press the following buttons and then start operati...

Page 10: ...scuss After trying an example the students can talk about the operations and numbers that they used This discussion should lead to the generalization that one way to reverse the order of the digits is to add or subtract a multiple of 9 More able students could be asked to try and prove this gener alization 10a b N 10b a N 10b a 10a b N 9b 9a 9 b a Further Ideas Try using the activity with 3 digit ...

Page 11: ... among the possible answers will be While working on this activity students should practice their skills of mental estimation They should also be interpreting and generalizing their answers What is the largest number you can make by pressing the keys and once and only once Example Can you make a larger number Using algebra for any four digits a b c d where a b c d the largest product is given by 1...

Page 12: ... can explore other sets of four numbers generalizing the rule to find the largest product using words or symbols After generalizing explain the rule that for any four digits a b c d where a b c d the largest product is given by 10d a x 10c b If the investigation is extended to the five digits 1 2 3 4 5 then the largest product is given by 431 x 52 22412 For some students it may be appropriate to b...

Page 13: ... 10 Find out which number pair gives the largest possible product This activity helps to reinforce students understanding of the mathematical terms sum and prod uct and develops trial and improvement methods Using the calculator Calculator functions used Addition multiplication subtraction parentheses Try to find the largest product of any two numbers whose sum is 10 Example 2 8 2 8 You can also c...

Page 14: ...couraged to devise similar problems to give to each other involving numbers with different sums Further Ideas Investigate products of 3 4 5 numbers which have the same sum This could be explored graphically Generally for two numbers whose sum is n the largest product is given by n 2 2 for three numbers whose sum is n the largest product is given by n 3 3 The nearest integer to n e where e 2 718 is...

Page 15: ...the answer is as close as possible to 100 Players score points according to how close they are to 100 within 10 1 point within 5 2 points within 1 5 points exactly 100 10 points Player 2 then enters a number and the game continues The first player to score 20 points wins While working on this activity students will be extending their understanding of decimals and improving their estimation skills ...

Page 16: ...t is obtained from the activity when students are playing together in small teams discussing their choices of a number to multiply by Points for students to discuss At the end of the activity students strategies should be discussed and compared Further Ideas Play the game with different target numbers For example students could multiply or divide a random number to reach a target of 1 The first pl...

Page 17: ...ir understanding of the relative sizes of common fractions The activity suggests an approach to teaching equivalence of common fractions Using the calculator Calculator functions used Addition division fractional calculation You will need a 0 2 number line Estimate where the following common fractions should be placed on the number line and then record estimates Use a calculator and a ruler to che...

Page 18: ... the calculator It is important that students are aware of the general method of converting common fractions into decimal notation dividing the numerator by the denominator as well as the use of the fraction key on the calculator Points for students to discuss It will be discovered that some of the fractions are equivalent to each other and this leads into the second part of the activity When the ...

Page 19: ... This activity suggests an approach to teaching addition of common fractions Using the calculator Calculator functions used Addition fractional calculation Example Using fractional calculation find the sum of and 1 2 1 3 on the calculator display means Convert to decimal notation 0 83333 on the calculator display means Find the sums of other common fractions 1 5 1 7 Junior high school Elementary s...

Page 20: ...nd explain what is happening It should be noted that the pattern may appear to break down when fractions with a common denominator are added Points for students to discuss Students can then explore what happens when other common fractions are added For some students it may be appropriate to begin by considering a pair of fractions that includes one unit fraction It is important that students are e...

Page 21: ...o develop an increasing feel for the relative sizes of fractions Using the calculator Calculator functions used Addition division multiplication fraction calculation Example A Find the fraction that is halfway between and Using fractional calculation obtain the sum of and 1 2 1 3 Half of this fraction is the number you are looking for so divide this fraction by 2 2 Or after Display 1 multiply by 1...

Page 22: ...ts can be asked to give fractions that lie between 1 2 and 1 3 and identify the common fraction that is halfway between them At this stage it may be necessary to discuss methods for finding a number that is halfway between two numbers Students can then use their calculators to identify fractions that are halfway between other unit fractions This can be extended to non unit fractions It is importan...

Page 23: ...e calculator to find fractions that are near integers in decimal form While working on this activity students will be developing their understanding of decimals particularly their relationship with fractions Using the calculator Calculator functions used Multiplication division The 35th multiple of 0 314 is 10 99 35 x 0 314 10 99 35 0 314 10 99 is a near integer it is nearly 11 Using fractional ca...

Page 24: ... investigate the multiples of some decimals Points for students to discuss The teacher could ask the students to generate the multiples of 0 314 challenging them to find a multiple that is nearly an integer Students can then begin to investigate the situation further Further Ideas Use this idea to investigate different approximations for π For example 22 7 3 142857 whereas π 3 141593 However 179 5...

Page 25: ...each having a volume of 12 cm3 Find the equivalent equations for each of the cuboids for example 1 x 1 x 12 12 2 x 2 x 3 12 etc Using the calculator Calculator functions used Multiplication Multi line Playback Introduce students to the calculator s Multi line playback feature which will be useful to display sets of solutions for each volume number 1 1 12 2 2 3 1 3 4 etc Press the following buttons...

Page 26: ...nd improvement to find the side of a cube having a volume of 180 cm3 Move into four or more dimensions as a means of finding the factors of a number For example 6006 77 x 78 7 x 11 x 6 x 13 2 x 3 x 7 x 11 x 13 All stages can be displayed using the replay function 1 x 1 x 12 12 1 x 2 x 6 12 2 x 2 x 3 12 1 x 3 x 4 12 Find the five calculations that represent cuboids that each have a volume of 30 cm3...

Page 27: ...Multiplication editing Multi line Playback Enter the y values for the function y 4x 5 using the values from 5 to 5 for x Use the calculator s playback function to calculate the functions efficiently After calcu lating the values use them to plot the graph of y 4x 5 4 5 5 4 Press the following buttons and then start operation Each press of a key takes you one calculation step forward or backward x ...

Page 28: ... calculate function values can be applied to a wide range of equations including polynomials trigonometric functions etc Students can do calculations in one sequence and then use the playback function to go back through the answers and record or plot them all at once Further Ideas Investigations on graphs can be done more quickly if the playback function is used so each function does not have to b...

Page 29: ...ool Objective Understand and use the concept of place value in whole numbers Explore a variety of situations that lead to the expression of relationships Construct and interpret formulas and expressions Manipulate algebraic expressions Explanation of the activity A word that reads the same forwards and backwards such as mom and level is called a palin drome A palindromic number is exactly the same...

Page 30: ...so it may be necessary to give them the following hint Try writing the number in its long form for example 2332 2 x 1000 3 x 100 3 x 10 2 Further Ideas Try to explain the problem using algebra You could start off by giving the class just the first line of the calculation below and let them work on the rest individually or in groups a b b a a x 1000 b x 100 b x 10 a a x 1000 a b x 100 b x 10 a 1000...

Page 31: ... gives students the opportunity to enhance their understanding of decimals and im prove their skills in estimation Using the calculator Calculator functions used Multiplication FSE TAB Set the calculator to fixed point notation with a TAB value of 0 Doing this will display answers to the nearest whole number Adjust the TAB setting to 1 and then continue to improve the accuracy of the answer 1 4 4 ...

Page 32: ...acy are appropriate in the case of cubic centimeters of ice cream possibly only to one decimal place Further Ideas Find the side of a cubical carton whose volume is 1 2 liter It may be necessary to remind students of the equivalence of 500 ml fluid measure and 500 cm3 solid measure Find the dimensions of a fruit juice carton whose sides are in the proportion 1 2 3 and whose capacity is 1 liter Fin...

Page 33: ...ation Enter the following equations into the calculator 3 7 13 7 3 17 Find other last digit combinations that give answers ending in 1 Which of the numbers in the following set can be made from the product of two numbers excluding equations using 1 multiplied by the number itself 21 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201 Which of the numbers can be made in more than one way ...

Page 34: ...ents to discuss After investigating patterns on their own students should share their discoveries with the rest of the class Further Ideas Examine the first 20 prime numbers Except for the number 2 they all end with an odd last digit Repeat the procedure for the last digits of 3 7 and 9 Find all the prime numbers between 1 and 201 Find the last digits to 1 the answers to the multiplication tables ...

Page 35: ...t rates While working on this activity students will develop their understanding of percentages within the context of compound interest situations Using the calculator Calculator functions used calculation multiplication Press the following buttons and then start operation If you invest money at a certain level of interest by how much will your money grow Example 100 is invested at 0 1 annual inte...

Page 36: ...st rates that would double their money for different numbers of years These interest rates could be plotted on a graph 5 years 14 9 10 years 7 2 15 years 4 7 20 years 3 6 25 years 2 8 Points for students to discuss It may be useful to show students how to generate sequences on the calculator Further Ideas Investigate interest rates that would triple an investment or From 1970 to 1980 prices triple...

Page 37: ...years is 100 so n 100 The interest is 0 1 so x 1 001 100 1 001 100 You have 110 50 after 100 years How many years would it take for the money to double Let s make an equation The money invested multiplies each year by the amount x After n years the money doubles so axn 2a Divide both sides by a xn 2 Calculates the log of both sides log xn log2 nlogx log2 n log2 logx If a is the money deposited the...

Page 38: ... to prove generaliza tions Using the calculator Calculator functions used Addition A game for two players The first player enters any number into his her calculator without showing it to the other player For example 298 The second player then enters a number into his her calculator without showing it to the other player For example 55 The players then show each other their numbers and add them If ...

Page 39: ...asons for their conjectures Odd Even Odd Even Odd Odd Odd Odd Even Even Even Even Points for students to discuss More able students could try to formally prove their conjectures The idea can be extended by students thinking about the conditions for obtaining even or odd answers when three numbers are added or four numbers or Further Ideas More able students could try to formally prove their conjec...

Page 40: ...ng algebraic expressions While working on this activity students will be developing skills of generalization and refining their methods of expressing mathematical rules Using the calculator Calculator functions used Addition subtraction multiplication division last answer memory Let s look at a sequence find a rule to generalize it and check it with our calculators Example 3 5 7 9 11 The above seq...

Page 41: ...sequences and discuss the different rules Below are some possible solutions It is important to emphasize that there may be different solutions ANS 4 ANS 5 2 ANS ANS 3 ANS x 3 1 ANS2 1 Points for students to discuss It may be useful to show students how to generate sequences on the calculator Further Ideas Find different sequences that begin 1 2 5 Is it possible to generate the multiples of any num...

Page 42: ...imes of two trains together with the distance of each station from Norwich How long does it take to complete the various stages of the journey Norwich Diss 17 75 Diss Stowmarket 13 5 Stowmarket Ipswich 10 25 Ipswich Colchester 15 25 Colchester Chelmsford 19 5 Chelmsford London 26 Example Try calculating the average speed of a train traveling from Norwich to London The train leaves Norwich at 11 30...

Page 43: ... M S to enter and compute various times The actual purpose of the key should be discussed with students however Further Ideas Compare other train lines and other forms of transport What factors affect the speed of a train Investigate the statement Longer journeys are quicker Let s find the average speed at which the train travels between Norwich and London Speed distance time 115 miles 1 52 is cal...

Page 44: ...tion fixed number calculation Let s use our calculators to simulate rolling dice Example To simulate rolling dice we want the numbers from 1 to 6 so the formula will be 1 RANDOM 5 However this will also generate decimals So change to the FIX mode by pressing TAB and 0 until appears FIX on a screen Now only integers from 1 to 6 will be generated SetTAB to zero so that values to the right of the dec...

Page 45: ...equally likely outcomes Other ways of using the calculator to simulate dice can also be explored For example 0 5 RANDOM 6 Students could then explore the reliability of other dice simulations such as adding or multiplying the scores on two dice Points for students to discuss It is probably best to explain orally to students how to use the calculator to simulate dice before starting the activity Fu...

Page 46: ...he importance of experimentation and simulation Using the calculator Calculator functions used Random numbers addition Let s use our calculators to find out the mean score if we roll a die 10 times The mean is calculated by dividing the total score by the number of rolls of the die Calculate 4 4 1 5 2 3 6 5 4 6 10 4 4 1 6 10 Let s see what happens to the mean score if we roll the die 20 times 50 t...

Page 47: ...r to simulate dice before starting the activity If the calculator is set to STAT mode it is possible to store the rolls of the die and use the calculator to find the mean score It is important to note that the calculator should be set back to floating decimal place mode when calculating the mean score Alternatively students could work in pairs using one calculator to simulate the die and another c...

Page 48: ...s can also be used to introduce an iterative method for solving quadratic equations Using the calculator Calculator functions used Addition subtraction multiplication division last answer memory Let s see what happens when we repeat the same equation over and over Think of a number Divide it by 2 and add 1 Example Try the number 6 6 2 1 Divide the answer by 2 and add 1 2 1 Keep on repeating this 2...

Page 49: ... 2 2x 2 2x x 2 x Students can investigate different rules such as 3 1 or 2 5 generalizing which rules con verge to an integer limit Generally the sequence a b converges to ab a 1 where a 1 and a 0 Points for students to discuss It may be useful to show students how to generate sequences on the calculator Further Ideas Students use their calculators to generate different convergent sequences They t...

Page 50: ...sion last answer memory Multi line Playback The Italian mathematician Fibonacci discovered this sequence 1 1 2 3 5 8 13 21 34 55 89 144 To generate a similar sequence begin with two numbers and add them to get the next Continue adding the last two numbers to get the next term Let s try making a regular series similar to the Fibonacci series above First select two suitable integers Press the follow...

Page 51: ...students to discuss Explain orally to students how to generate the Fibonacci sequence on the calculator Further Ideas Ask students for two starting values and then generate the first ten terms of the sequence Predict the sum of these of ten numbers and then ask the students to confirm your answer The trick is that the sum of these numbers will always be the sixth number multiplied by 11 2 6 8 14 2...

Page 52: ...lculator functions used Addition subtraction multiplication division squaring memory calculation Multi line Playback Example Confirm the equation x2 5x 6 x 2 x 3 is always true 1 Store any number x into your calculator 2 Use your calculator to evaluate x2 5x 6 3 Now use your calculator to evaluate x 2 x 3 Store 14 14 Calculate x2 5x 6 5 6 Calculate x 2 x 3 2 3 Press the following buttons and then ...

Page 53: ...n investigation into factorizing quadratic expressions The teacher could do an example on the board and then other expressions could be given to the students to factorize using trial and improvement methods on their calculator It is important that students are encouraged to reflect on the factorized expressions and look for strategies to help them factorize quadratics These strategies should be co...

Page 54: ... Calculator functions used Coordinate conversion Example using the EL 531RH Try to find the length of the hypotenuse the side opposite the right angle of a right triangle whose other two sides are 3 cm and 4 cm long Input the lengths of the two shorter sides 3 4 Find the length of the hypotenuse We find that the length of the hypotenuse is 5 cm Extra Try to find the angle of the corner between the...

Page 55: ...s of right angles triangles e g 3 4 5 and 6 8 10 etc and concepts of similarity should be discussed It is important that students generalizations are collated and discussed Points for students to discuss To generate a Pythagorean triple square any odd number and split the answer into two consecu tive integers The original number and the two consecutive integers will form a Pythagorean triple For e...

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