Mirko Dejić and Branka Dejić
Mirko Dejić and Branka Dejić
Activities for developing creativity and giftedness
Observations................................................................................................... 8 Quips............................................................................................................. 13 Numbers and calculation............................................................................ 18 Geometry...................................................................................................... 31 Combinatorics.............................................................................................. 40 Brain-twisters.............................................................................................. 45 Measurement............................................................................................... 53 Answer Key................................................................................................... 59
NOTE TO CHILDREN Why do we learn maths? Many students ask themselves this question without realizing the many benefits of problem solving. Whatever we choose to do in life, we won’t be able to do it without maths. Without maths, there would be no airplanes, bridges, toys, trade and many other things. Maths is applied even where we don’t expect it – in painting, music and literature. Maths teaches us how to think logically, and we become smarter when we learn it. This book contains a variety of interesting tasks, most of which you won’t see during your maths classes in school. Not only will solving these problems become a pleasure, but you will also be nurturing your mathematical giftedness. It is very important to be patient when solving problems. Those that might seem difficult at first can usually be solved in a simple way. If you’re having trouble with one problem, move onto the next one. Success will encourage you. Your reward will be feeling joy and accomplishment because of a job well done. Try not to ask adults for help; keep going until you solve the problem on your own. At the end of the book, you will find a key that contains either full answers, step-by-step explanations or solutions for most of the problems. Only look at the answer key after you’ve finished solving the problem. Compare it to your answer and, if needed, try to establish where the error occurred. Try to understand the reasoning behind the answer.
NOTE TO TEACHERS AND PARENTS The book presented before you is intended for first-graders, but if children of younger age are able to solve these problems, this might mean they could become great mathematicians in the future. The tasks contained in this book are engaging, unorthodox and dedicated to problem solving. Children are presented with various problematic situation for which they need to find solutions. By independently seeking ideas for solutions and anticipating results, the children are developing both creativity and intuition needed for solving mathematical problems. Brief confusion that occurs at the beginning of the activity will motivate them to find where the problem lies. Then, a solution will pop up, causing the children to have an a-ha! moment. This will bring them joy and desire to keep going. The children will then begin to resemble real mathematicians and researchers. Ensure your child has favourable conditions for problem solving: yy Accept every attempt at problem solving, even when incorrect. These efforts of seeking answers are also expressions of children’s creativity; yy Convince your child they can solve the problem all the way to the end; yy Express genuine joy when your child is successful and praise them; yy Help only by offering them advice when necessary; in most cases, a short “you’re on the right path” will do. Avoid: yy Causing fear in children: ”You are too stupid for this, you will never figure it out”; yy Frustration: when the child is making an effort and we don’t pay attention to their work; yy Forcing children to solve problems – this will cause an adverse effect; yy Words: replace “let’s do some maths” with “let’s play, so we can see how the wolf, the goat and 6
the cabbage managed to cross the riverâ€Śâ€? The problems are useful for discovering and developing mathematical giftedness. It is especially important to pay attention to the following indicators of mathematical giftedness in children: yy Did the child solve the problem in multiple ways? yy Do they fill in the cognitive blanks independently while solving maths problems? yy Do they ask for help while solving problems? yy Are they persistent when solving problems? yy Are they offering unorthodox answers? yy Are the answers concise? yy Are they quick in problem solving? yy Are they using a wide range of ideas acquired through earlier problem solving? yy Do they express exceptional inventiveness in problem solving? yy Do they find pleasure in solving more demanding problems? yy Are they able to utilise drawings and models? yy Do they stick to their original plan of solving the problem all the way to the end? yy Are they quick to notice new relations? yy Are they able to differentiate between important and unimportant elements in a problem? yy Are they quick to understand the problem at hand and lay out a plan for solving it? The problems in this book have varied aims: some are useful for developing logical and abstract thinking, some are related to spatial orientation, others deal with ways of behaving in certain situations, while many are, simply, fun and interesting tasks â€“ ones that will make us fall in love with maths and motivate us to work constantly. All of them can greatly develop mathematical abilities and intelligence. The most intense period of intellectual development in children is until the age of 13. This is when tasks aimed at advancing cognitive skills are at their most effective. The activities in this book are notably varied, so as to avoid problem solving through repeated patterns. Every problem will present the child with a new situation, so seeking answers will be equal to finding your way in unique circumstances. This requires intelligence, which will simultaneously be utilised and developed.
1. W hich path should the mouse take in order to safely reach the cheese? Circle the correct answer.
2. Draw the other half of the picture as shown.
3. W hich of the figures marked with numbers should be placed instead of the question mark? Circle the correct number. а)
4. Нацртај фигуре које недостају. а)
5. Which two figures are the same? Circle the correct numbers.
6. Figure C is equal to figure 10. Which numbers correspond to figures A, B, D and E? Write in the correct answers on the lines. 1
2 7 10
В 10 Д
7. Use the attached pieces to assemble a chessboard.
7. Work out in which order the pearls have been organised and draw two additional necklaces.
1. Five candles were lit. Two burned out. How many candles were left? 2. Farmer had 18 sheep. All sheep, apart from seven, have died. How many sheep does the farmer have now?
3. O n the chandelier, three light bulbs are lit and two are switched off. How many light bulbs are on the chandelier?
4. If two out of nine light bulbs are switched off, how many light bulbs will remain? 5. A granny was walking home from town when she ran into two grannies who were walking towards the town. How many grannies went home from town?
6. O ne thermometer is showing 20 degrees Celsius. How many degrees will two thermometers show?
7. Four birds were perched on the branch. A hunter came along and shot one of them. How many birds were left on the branch?
8. There were eight chickens in the yard. Three of them left the yard. Later, three lambs entered the yard. The number of legs in the yard increased by how many? 13
9. A bug has two left and two right legs, two front legs and two hind legs. How many legs does the bug have?
10. If a rooster is standing on one leg, he weighs four kilograms. How much does the rooster weigh when he stands on two legs?
11. Our hands have ten fingers. How many fingers do ten hands have? 12. How many pairs of fingers do four friends have? 13. If a brick weighs one kilogram and half a brick, how much do two bricks weigh? 14. A brick and a half weighs three kilograms. How much do two bricks weigh? 15. There are six glasses on the table â€“ three of them full and placed next to each,
and three of them empty, also placed next to each other. How can you have the glasses stand in alternate order (one empty, one full, one empty, etc.) if you can only take one glass in your hand?
16. It has been 30 hours since midnight. What time is it now? 17. What is the current time if the big hand has made 12 full circles since noon? 18. W hat day and time will it be if the little hand makes two full circles from Wednesday 8 a.m.?
Numbers and calculation
1. If you connect the dots from 1 to 38, you will see what Fred the fisherman caught.
2. Find the right paths through this labyrinth and help Andy solve the problem.
3. In the table, find the numbers from 1 to 25 in the right order. Time yourself to see how much it takes you to complete the task. If you can do it in under 30 seconds, you are a true genius!
4. The fortress is secured with four walls. A princess is captured inside. You can save her by opening a gate on each wall. You can reach the princess only if you go through the gates whose numbersâ€™ sum is 100. Find three different ways to the princess. Write the correct numbers in the blank squares.
1. D raw the following figures without lifting your pencil from the paper or drawing a same line twice
2. Connect the nine dots in one move without lifting your pencil from the paper. You can only go through each dot once.
3. Н е подижући оловку с папира, подели фигуру на слици тако да добијеш шест једнаких троуглова.
3. Continue drawing the path bunny needs to take in order to get to the carrot. The bunny
can step on each tile only once. If he steps on a white tile, the next one has to be black. If he steps on a black tile, the next one has to be white.
4. Are these boys the same height? Circle the correct answer. Yes No
Are line segments a and b the same length?
1. P lace three kittens, three puppies
and three bunnies in nine houses. Every horizontal and vertical line should have each of the three animals. Write the names of the animals on their matching houses.
2. A rrange red and blue circle and red and blue square in all the possible ways without having
two figures of the same colour next to each other. Continue colouring in the picture as shown.
3. Draw the cups on trays so that the order of colours is different on each one.
4. A nna, Molly and Ruth are supposed to sit on three chairs. Find all
the ways in which they can sit. Continue as shown: write the letter A for Anna, the letter M for Molly and the letter R for Ruth.
5. B eth is getting ready for her friendâ€™s birthday party. She is supposed to pick out one shirt and one skirt. There are two shirts (red and white) and two skirts (black and blue) in the closet. What kind of outfits can Beth make? Colour them in.
6. P erry had three hats, a blue one, a yellow one and a green one. He also had two pairs of sweatpants, black and white. In how many different ways could have Perry combined his hats and sweatpants?
7. Jen, Sophie, Chris and Matt can sing really well. Their teacher decided the children should sing in pairs, one boy and one girl. How many singing pairs could she assemble? Write the childrenâ€™s names in the blank fields.
1. M ike’s brother is 6 months old. Danny’s brother is three months old. How old will Mike’s brother be when Danny’s brother turns one?
2. Last year, Mike was the same age Danny is now. Who is younger, Mike or Danny? 3. P aul is 12, Claire is 13 and Jake is 15.
Find out what is the sum of their ages. What was the sum of their ages last year?
4. Twins Andrew and Andrea were born five years ago. What will be the sum of their ages in five years?
5. Lisa, Sarah and Susan are best friends. The youngest girl is eight years old
and the oldest one is 13. Sarah is not the youngest. Susan is two years older than Sarah. How old is each of the girls? Lisa:
6. The sum of Rachel’s and Emily’s ages is 12. Rachel is two years younger than Emily. How old is Rachel and how old is Emily? Rachel:
7. W hen the mother was twenty years old, her son was two. His mother is now 40 years old. How old is the son?
8. M y sister is now seven years old. When she was born, I was two years and seven months old. How old am I now, in years and months?
9. There is an equal number of apples in two baskets. When ten apples are added to one of the baskets, there will be 60 in total. How many apples were in each basket originally?
10. O ne plate has six apples more than the other. How many apples should be moved from one plate to another in order for the two plates to have an equal number of apples?
11. If two apples each are put on plates, one plate will remain empty. If one apple is put on
each plate, there will be a shortage of one plate. How many apples and plates are there?
12. O ne bucket is filled with 22 litres of water, the other with 12 litres.
How much water should be transferred from one bucket into another in order for the two buckets to have the same amount of water?
13. W ill has an equal number of sweets in three different bags. When he added 10 sweets in each of the bags, he had 45 sweets in total. How many sweets were in each bag to begin with?
14. D avid, Ben and Tom were winners of the race organized by their school. Tom was faster than David and Ben was slower than David. They are all standing on the podium. Who won the first, who won the second and who won the third place? Write in the first letters of their names in the blank squares.
15. Three friends, Josh, Tim and Mark, were born the same year. One of the boys was born in January, one in April and one in August. Josh is not older than Mark and Tim is not older than Josh. In which month was each of the boys born? Write the months below the boys’ names. Влада Никола Саша
This book is intended for the gifted children.
The tasks in these books are not found in regular textbooks â€“ some tasks are for developing logical and abstract thinking, some are related to special orientation. There are also tasks related to dealing with different situations and some are just beautiful exercises which make mathematics interesting. All exercises encourage and develop mathematical skills and giftedness.
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