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Hamilton Secondary Numeracy Project Shining Term 2 Homework Answers Name ___________________________ Decimals and Fractions 1 1. Use a calculator to divide the numerator by the denominator to find the decimal equivalents of 1/9, 1/99, 1/999… Can you predict the next decimal? 1/9 1÷9= 0.1111111 1/99 1 ÷ 99 = 0.0101010101 1/999 1 ÷ 999 = 0.001001001 1/9999 1 ÷ 9999 = 0.0001000100 1/99999 1 ÷ 99999 = 0.00001000010 1/999999 1 ÷ 999999 = 0.000001000001 2. Use a 0-9 dice or 1-9 cards (or remove the 10s, Jacks, Queens, Kings and Jokers from a set of cards and use Aces as 1s, or write 1-9 on pieces of paper). Roll dice twice (if you roll 0 roll again) or choose two cards to make a fraction less than 1. Multiply it by 2 /3. For example, if you roll a 5 and a 2, multiply 2/5 by 2/3. How many can you do in two minutes? 2 × 2 = 4 5 3 15 E.g. 1 x 2 = 2 = 1 E.g. 3 x 2 = 6 = 2 6 3 18 9 5 3 15 5 E.g. 4 x 7 E.g. 1 x 2 8 3 E.g. 8 x 2 = 16 9 3 27 E.g. 3 x 2 = 6 = 1 8 3 24 4 2= 8 3 21 =2=1 24 12 HSNP © Hamilton 2013 E.g. 2 x 2 = 4 7 3 21 E.g. 5 x 2 = 10 7 3 21 Shining Term 2 E.g. 1 x 2 = 2 9 3 27 E.g. 2 x 2 = 4 = 2 6 3 18 9 E.g. 5 x 2 = 10= 5 6 3 18 9 Page 1 Decimals and Fractions 1 If you have internet access, try this Play Fractone at bit.ly/13xZBVl. Choose ‘Pretty good’ or ‘I’m going for it’! Click on pairs of fractions with a total of 1 as quickly as you can. Sometimes you will need to click on pairs with different denominators, for example, 4/8 and 1/2. This is a computer game. My time was __________ just right How did you find these problems? too hard too easy Page 2 Addition 1 Remove the 10s, Jacks, Queens, Kings and Jokers from a set of cards and use Aces as 1s (or use 1-9 cards or write 1-9 on pieces of paper). Take two cards to make a fraction. Take two more to make another fraction. Find the total of the two fractions using ‘smile and kiss’. Only record the additions with a total of between 1 and 2. How many can you do in five minutes with totals of between 1 and 2? 8+ 7 8 9 X 9 x 8 = 72 64/72 + 63/72 = 127/72 or 1 55/72 5+4 9 6 9 x 6 = 54 30/54 + 36/54 = 66/54 or 1 12/54 HSNP © Hamilton 2013 7+5 9 6 9 x 6 = 54 42/54 + 45/54 = 87/54 or 1 33/54 7+5 8 7 8 x 7 = 56 49/56 + 40/56 = 89/56 or 1 33/56 7+3 9 5 9 x 5 = 45 35/45 + 27/45 = 62/45 or 1 17/45 3+5 4 8 4 x 8 = 32 24/32 + 20/32 = 44/32 or 1 12/32 2+5 5 6 5 x 6 = 30 12/30 + 25/30 = 37/30 or 1 7/30 6+3 7 6 7 x 6 = 42 36/42 + 21/42 = 57/42 or 1 15/42 Shining Term 2 Page 3 Addition 1 If you have internet access, try this Play Fruit Shoot Fraction Addition at bit.ly/11fmvhA. Choose ‘Level 3’ and ‘Relaxed mode’. Add the pair of fractions shown and click on the fruit with the answer. Record the additions you complete below. What was your score out of 10? This is a computer game. Level 3a Score = just right Level 3b __ 10 Score = __ 10 How did you find these problems? too hard too easy Page 4 Subtraction 1 1. Roll a dice twice to make a fraction less than one. Use ‘smile and kiss’ to subtract this fraction from 8/9. Do three of these. 8 4 − 9X 6 9 x 6 = 54 - 36/54 = 12/54 or 6/27 48/54 E.g. 8/9 - 3/5 9 x 5 = 45 8 x 5 = 40 3 x 9 = 27 40/45 - 27/45 = 13/45 8/9 - 3/5 = 13/45 E.g. 8/9 - 2/5 9 x 5 = 45 8 x 5 = 40 2 x 9 = 18 40/45 - 18/45 = 22/40 8/9 - 2/5 = 22/40 or 11/20 E.g. 8/9 - 3/6 9 x 6 = 54 8 x 6 = 48 3 x 9 = 27 48/54 - 27/54 = 21/54 8/9 - 3/6 = 27/54 or 1/2 2. Solve the following by thinking about how many of each fraction are in each number. For example, how many halves are in 20? 20 ÷ 1/2 6 ÷ 1/4 There are 2 halves in 1, so there must be 20 times that many in 20. 2 x 20 = 40. 10 ÷ 1/4 = 10 x 4/1 = 40 There are 4 quarters in 1, so there must be 6 times that many in 6. 4 x 6 = 24. 6 ÷ ¼ = 6 x 4/1 = 24 3 ÷ 1/8 7 ÷ 1/6 There are 8 eighths in 1, so there must be 3 times that many in 3. 8 x 3 = 24. 3 ÷ 1/8 = 3 x 8/1 = 24 There are 6 sixths in 1, so there must be 7 times that many in 7. 7 x 6 = 42. 7 ÷ 1/6 = 7 x 6/1 = 42 30 ÷ 1/5 There are 5 fifths in 1, so there must be 30 times that many in 30. 5 x 30 = 150.30 ÷ 1/5,30 x 5/1= 150 9 ÷ 1/10 There are 10 tenths in 1, so there must be 9 times that many in 9. 10 x 9 = 90. 9 ÷ 1/10 = 9 x 10/1 = 90 HSNP © Hamilton 2013 Shining Term 2 Page 5 Subtraction 1 If you have internet access, try this Play Adding and Subtracting Fractions Challenge at bit.ly/13zR7P1. Click to roll the dice. Draw a card. Add or subtract the fractions. Carry on until you reach the finish. My score was __________ Play Fruit Shoot Fraction Subtractions at bit.ly/14rZPwq. Choose ‘Level 3’ and ‘Relaxed mode’. Subtract the pair of given fractions and click on the fruit with the answer. Record the subtractions and answers below. What was your score? This is a computer game. Level 3a 2/3 Level 3b + 2/6 = 1 __ 10 just right __ 10 How did you find these problems? too hard too easy Page 6 Multiplication 1 1. Find the squares of the numbers below. Then find the digital root of each answer. For example, for 342 = 1156. Adding 1 + 1 + 5 + 6 gives 13. To get to a single digit, add 1 and 3 to get the digital root of 4. What do you notice about the digital roots of these numbers? 12 12 x 12 = 144 1+4+4 =9 9 23 23 x 23 = 529 5+2+9 = 16 1+6 =7 7 34 34 x 34 = 1156 1+1+5 +6 = 13 1+3= 4 4 45 45 x 45 = 2025 2+0+2 +5=9 56 56 x 56 = 3136 3+1+3 + 6 = 13 1+3 =4 4 67 67 x 67 = 4489 4+4+8 + 9 = 25 2+5 =7 7 78 78 x 78 = 6084 6+8+4 = 18 1+8 =9 9 9 I notice that______________All digital roots are 1, 4, 7 or 9___________. Does the pattern work with 89?______It is 1___ Does it matter whether the larger digit is first or second?______No____ HSNP © Hamilton 2013 Shining Term 2 Page 7 Multiplication 1 If you have internet access, try this Play the video about Alex’s Number Plumber at bit.ly/12F8PnE. Click on the picture under the video and enter the same number as on the video. Keep pressing ‘drop’ so that the last output becomes the next input. Click on ‘results table’ on the far right. What do you notice about the final digits of each number? This is a computer game. I noticed that… Choose your own number to enter and see what happens. Keep re-entering the output as the next input. Look at the results table. Can you predict the pattern for a new number? I predict the pattern to be… just right How did you find these problems? too hard too easy Page 8 Division 1 1. Work to find the biggest four-digit number you can that is divisible by each of its digits. Each digit must be different. E.g. 1236 is divisible by 1, 2, 3 and 6, but you can do better than that! 1236 1236 ÷ 1 = 1236 1236 ÷ 2 = 618 1236 ÷ 3 = 412 1236 ÷ 6 = 206 1296 1296 ÷ 1 = 1296 1296 ÷ 2 = 648 1296 ÷ 9 = 144 1296 ÷ 6 = 216 1395 1395 ÷ 1 = 1395 1395 ÷ 3 = 465 1395 ÷ 9 = 155 1395 ÷ 5 = 279 3816 3816 ÷ 3 = 1272 3816 ÷ 8 = 477 3816 ÷ 1 = 3816 3816 ÷ 6 = 636 3195 3195 ÷ 3 = 1065 3195 ÷ 1 = 3195 3195 ÷ 9 = 355 3195 ÷ 5 = 639 HSNP © Hamilton 2013 9864 9864 ÷ 9 = 1096 9864 ÷ 8 = 1233 9864 ÷ 6 = 1644 9864 ÷ 4 = 2466 9864 is the biggest possible, there are lots of possibilities. 3612 3612 ÷ 3 = 1204 3612 ÷ 6 = 602 3612 ÷ 1 = 3612 3612 ÷ 2 = 1806 2196 2196 ÷ 2 =1098 2196 ÷ 1 = 2196 2196 ÷ 9 = 244 2196 ÷ 6 =366 4872 4872 ÷ 4 = 1218 4872 ÷ 8 = 609 4872 ÷ 7 = 696 4872 ÷ 2 = 2436 Shining Term 2 9648 9648 ÷ 9 = 1072 9648 ÷ 6 = 1608 9648 ÷ 4 = 2412 9648 ÷ 8 = 1206 9135 9135 ÷ 9 = 1015 9135 ÷ 1 = 9135 9135 ÷ 3 = 3045 9135 ÷ 5 = 1827 4236 4236 ÷ 4 = 1059 4236 ÷ 2 = 2118 4236 ÷ 3 = 1412 4236 ÷ 6 = 706 9162 9162 ÷ 9 = 1018 9162 ÷ 1 = 9162 9162 ÷ 6 = 1527 9162 ÷ 2 = 4581 1248 1248 ÷ 1 = 1248 1248 ÷ 2 = 624 1248 ÷ 4 = 312 1248 ÷ 8 = 156 Page 9 Division 1 2. Roll two dice (or choose numbers using 1-6 cards) to make a fraction less than 1. Divide it by 1/4. For example, if you roll a 5 and a 2, divide 2/5 by 1/4. How many can you do in two minutes? 2 ÷ 1 = 8 5 4 5 3 ÷ 1 = 12 4 4 4 3 ÷ 1 = 12 5 4 5 1 ÷ 1 = 4 2 4 2 2 ÷ 1 = 8 3 4 3 4 ÷ 1 = 16 6 4 6 2 ÷ 1 = 8 6 4 6 3 ÷ 1 = 12 6 4 6 4 ÷ 1 = 16 5 4 5 1 ÷ 1 = 4 5 4 5 1 ÷ 1 = 4 4 4 4 2 ÷ 1 = 8 4 4 4 3. Two people are thinking of the same number less than 100. One divides it by 3 and gets a remainder of 1; the other divides it by 20 and gets a remainder of 3. What is the number? 43 ÷ 20 = 2 r3, 43 ÷ 3 = 14 r1 63 ÷ 20 = 3 r3 63 ÷ 3 = 21 just right X How did you find these problems? too hard too easy Page 10 Decimals and Fractions 2 1. Use a calculator to find the decimal equivalents for 1/13, 2/13, 3/13 and so on up to 12/13. 1/13 = 1 ÷ 13 0.076923 7/13 = 7 ÷ 13 0.53846153 2/13 = 2 ÷ 13 0.15384615 8/13 = 8 ÷ 13 0.61538461 3/13 = 3 ÷ 13 0.23076923 9/13 = 9 ÷ 13 0.69230769 4/13 = 4 ÷ 13 0.30769230 10/13 = 10 ÷ 13 0.76923076 5/13 = 5 ÷ 13 0.38461538 11/13 = 11 ÷ 13 0.84615384 6/13 = 6 ÷ 13 0.46153846 12/13 = 12 ÷ 13 0.92307692 Is there a pattern of recurring digits? Yes 076923 and 615384 Which fractions have the same pattern? 1,3,4,9,10,12 2,5,6,7,8,11 What do you notice about the sum of the digits? Both 27 Can you find any other interesting digit sums? The sum of the fractions that share a pattern equal 39. i.e. 1 + 3 + 4 + 9 + 10 + 12 = 2 + 5 + 6 + 7 + 8 + 11 = 39. The fractions that share patterns are pair bonds to 13. i.e. 1 + 12, 3 + 10, 4 + 9, 2 + 11, 5 + 8, 6 + 7. HSNP © Hamilton 2013 Shining Term 2 Page 11 Decimals and Fractions 2 If you have internet access, try this Play Fruit Shoot at bit.ly/12S4dHe. Start with ‘Level 4’. Click on fruits with decimal equivalents to the given fractions. Record your score. Now have a go at ‘Level 5’! This is a computer game. Level 4 score Level 5 score __________ __________ just right How did you find these problems? too hard too easy Page 12 Addition 2 1. Carry on Pascal's triangle so that you have at least 12 rows. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1 Look at the second number in each row. Does this go into each number in the row (apart from 1)? Look to see if there is pattern for this rule. I noticed that… The rule is that if the second number in the row is a prime number, it will go into each number in the row (apart from 1). HSNP © Hamilton 2013 Shining Term 2 Page 13 Addition 2 2. Use the internet to research Fibonacci sequences and spirals in nature. Write about one fact which you found interesting. I found out… E.g. Leaves on a stem, fruitlets of a pineapple, flowering of artichoke…. 3. 0 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 5768 10,609 19,513 35,890 66,012 121,415 223,317 410,744 755,476 1,389,537 This sequence is a twist on Fibonacci sequence. The fourth number is the sum of the first three numbers, the fifth number is the sum of the previous three numbers and so on. Continue the sequence so that you write at least 17 numbers. What patterns can you find? Is there a pattern of odd and even numbers? I noticed… that the sequence alternates with two odd numbers, then two even numbers and so on. just right How did you find these problems? too hard too easy Page 14 Subtraction 2 1. Josh says if you subtract a positive number from a positive number, you will always get a positive answer. What do you think? Explain your thinking with some examples. 15 – 8 = 7 If the number subtracted is smaller than the other number, it will be positive, if the number subtracted is larger than the other number, it will be negative. J 2. Roll a 0 to 9 dice (roll again if you roll a 0) and flip a coin to determine whether the number is positive (heads) or negative (tails). Repeat, then find the difference between the two numbers. Draw a number line jotting if it helps. Record five subtractions. 4 4 – (-7) = 11 7 -7 0 4 E.g. -3 - 4 = -7 E.g. -7 - 2 = -9 E.g. -1 - (-2) = 1 E.g. 6 - 8 = -2 E.g. 5 - (-3) = 8 HSNP © Hamilton 2013 Shining Term 2 Page 15 Subtraction 2 If you have internet access, try this Play Walk the Plank at bit.ly/16d1IQY. Choose hair and skin colours for the person you want to walk the plank. This is a computer game. You will be asked a question. Roll the mouse over the pirates to see their answers. Click the one you think is right. If correct, you’ll be asked to click on the dice to move the person forward on the plank. Click for the next question. Carry on until the game is complete. Did you make the person walk the plank? ________ Score __________ just right How did you find these problems? too hard too easy Page 16 Multiplication 2 Work out 1! 2! 3! up to 10! and record the answers below. Remember to use your previous answer to help work out the next one. 1! 1 2! 2 × 1 = 2 Are all the answers odd or even? All even (after 1!) 3! 3 × 2 × 1 = 6 Why? They all include x2 4! 4 × 3 × 2 × 1 = 24 5! 5 x 4 x 3 x 2 x 1 = 120 6! 6 x 5 x 4 x 3 x 2 x 1 = 720 Will all further factorials be the 7! 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040 same? Yes 8! 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320 9! 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880 10! 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800 2. Now look at the digital roots. For example, 9! 3 + 6 + 2 + 8 + 8 + 0 = 27. Add 2 + 7 to get 9, so the digital root is 9. What do the digital roots have in common? From 6! onwards the digital root is 9. What do you think will happen to further factorials? It will always be 9. Why? The digital root of a number divisible by 9 is 9 and all further factorials will include x9. What do you notice about the digital roots of multiples of 9? It is always 9. HSNP © Hamilton 2013 Shining Term 2 Page 17 Multiplication 2 3. Use a written method, to work out the following multiplications. Work out the digital root for each. Can you spot any patterns? ____Digital roots are 1, 4, 7 or 9_______________ 12 x 21 Digital root x 10 2 20 200 40 1 10 2 34 x 43 x 30 4 1200 160 3 90 12 56 x 65 1462 = 1 + 4 + 6 + 2 = 13 = 4 20 3 30 600 90 2 40 6 6 60 3000 360 5 250 30 78 x 87 3640 = 3 + 6 + 4 + 0 = 13 = 4 70 8 80 5600 640 7 490 56 just right 6786 = 6 + 7 + 8 + 6 = 27 = 9 Digital root x 40 5 50 2000 250 4 160 20 2430 = 2 + 4 +3=9 67 x 76 Digital root x 60 7 70 4200 490 6 360 42 Digital root x 736 = 7 + 3 + 6 = 16 = 7 45 x 54 Digital root 50 Digital root x Digital root 40 x 252 = 2 + 5 +2=9 23 x 32 5092 = 5 + 0 + 9 + 2 = 16 = 7 89 x 98 Digital root x 80 9 90 7200 810 8 640 72 8722 = 8 + 7 + 2 + 2 = 19 = 10 = 1 How did you find these problems? too hard too easy Page 18 Division 2 The factors of 48 are: 1. If we add all the factors of 1 and 48 48 less than 48, we get 76. 2 and 24 1+2+3+4+6+8+12+16+24=76 3 and 16 48 is called an abundant 4 and 12 number because it is less than 6 and 8 the sum of its factors (without itself). 32 has factors 1, 2, 4, 8 and 16 (apart from 32) and the sum of these factors is 31, so 32 is not an abundant number See if you can find some more abundant numbers! The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24 X 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36 36 > 24, so 24 is an abundant number. The abundant numbers under 100 are: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56 60, 66, 70, 72, 78, 80, 84, 88, 90 and 96 HSNP © Hamilton 2013 Shining Term 2 Page 19 Division 2 2. Use factor trees to find the prime factors of five two-digit numbers. Try and make the biggest tree that you can! 36 12 6 2 48 3 2 6 . 3 2 . 3 2 . . 4 2 54 2 8 2 80 27 3 . 9 3 just right 4 2 3 . 96 20 2 2 4 10 2 2 5 . 24 2 4 6 2 2 2 3 How did you find these problems? too hard too easy Page 20 Shining websites Fractone bit.ly/13xZBVl or www.coolmath-games.com/0-fractone/index.html Fruit Shoot Fraction Additions bit.ly/11fmvhA or www.sheppardsoftware.com/mathgames/fractions/FruitShootFractionsAddition.htm Adding and Subtracting Fractions Challenge bit.ly/13zR7P1 or www.math-play.com/adding-and-subtracting-fractions-game.html Fruit Shoot Fraction Subtractions bit.ly/14rZPwq or www.sheppardsoftware.com/mathgames/fractions/FruitShootFractionsSubtraction. htm Alex’s Number Plumber bit.ly/12F8PnE or nrich.maths.org/8387 Fruit Shoot bit.ly/12S4dHe or sheppardsoftware.com/mathgames/fractions/FractionsToDecimals.htm Walk the Plank bit.ly/16d1IQY or www.math-play.com/integers-game.html The links to the websites and the contents of the web pages associated with such links specified on this list (hereafter collectively referred to as the ‘Links’) have been checked by Hamilton Trust and to the best of Hamilton Trust’s knowledge, are correct and accurate at the time of publication. Notwithstanding the foregoing or any other terms and conditions on the Hamilton Trust website, you acknowledge that Hamilton Trust has no control over such Links and indeed, the owners of such Links may have removed such Links, changed such Links and/or contents associated with such Links. Therefore, it is your sole responsibility to verify any of the Links which you wish you use. Hamilton Trust excludes all responsibility and liability for any loss or damage arising from the use of any Links. Well done! You’ve finished Shining Term 2