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Hamilton Secondary Numeracy Project Shining Term 2 Homework 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= 0.1111111 1/9 1/99 1/999 1/9999 1/99999 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 HSNP © Hamilton 2013 Shining Term 2 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. 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 HSNP © Hamilton 2013 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? 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 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 3 ÷ 1/8 7 ÷ 1/6 30 ÷ 1/5 9 ÷ 1/10 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? 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 23 34 34 x 34 = 1156 1+1+5 +6 = 13 1+3= 4 4 45 56 67 78 I notice that________________________________________________. Does the pattern work with 89?_________ Does it matter whether the larger digit is first or second?____________ 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? 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 HSNP © Hamilton 2013 Shining Term 2 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. 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? 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 2/13 = 2 ÷ 13 8/13 = 8 ÷ 13 3/13 = 3 ÷ 13 9/13 = 9 ÷ 13 4/13 = 4 ÷ 13 10/13 = 10 ÷ 13 5/13 = 5 ÷ 13 11/13 = 11 ÷ 13 6/13 = 6 ÷ 13 12/13 = 12 ÷ 13 Is there a pattern of recurring digits? Which fractions have the same pattern? What do you notice about the sum of the digits? Can you find any other interesting digit sums? 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’! 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 ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ 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… 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… 3. 0 1 1 2 4 7 13 24 44 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… 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 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 HSNP © Hamilton 2013 4 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. 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. Are all the answers odd or even? 1! 1 2! 2 × 1 = 2 3! 3 × 2 × 1 = 4! 4 × 3 × 2 × 1 = 5! 5 x 4 x 3 x 2 x 1 = 6! 6 x 5 x 4 x 3 x 2 x 1 = 7! 7 x 6 x 5 x 4 x 3 x 2 x 1 = 8! 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 9! 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 10! 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = Why? Will all further factorials be the same? 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? What do you think will happen to further factorials? Why? What do you notice about the digital roots of multiples of 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? ________________________________________________ 12 x 21 Digital root x 10 2 20 200 40 1 10 2 34 x 43 252 = 2 + 5 +2=9 Digital root 45 x 54 Digital root 67 x 76 Digital root 89 x 98 Digital root x Digital root x 23 x 32 x 56 x 65 Digital root x x 78 x 87 x Digital root x just right 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. 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! just right 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