Mirko Dejić and Branka Dejić
9 788652 906864
ADVENTURES IN MATHS 4
T
Кhis book was intended for fourth-graders who are able to learn faster than their peers. It contains many interesting activities, rarely encountered in your regular maths classes. Different problems serve different aims: some to promote abstract thinking, others to develop spatial awareness, and some to teach you how to find your way out of puzzling situations. Many are, simply put, interesting problems, the kind that will make us love maths and drive us to keep practicing it. They are all suitable for discovering and developing mathematical giftedness and promoting creativity.
Mirko Dejić and Branka Dejić
ADVENTURES
IN MATHS 4 4
Mirko Dejić and Branka Dejić
ADVENTURES
IN MATHS 4 4
MA1
Activities for developing creativity and giftedness
Fourth grade
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Contents
Observations..................................................................................................... 8 Quips............................................................................................................... 15 Numbers and calculation.............................................................................. 21 Geometry........................................................................................................ 36 Combinatorics................................................................................................ 50 Brain-twisters................................................................................................ 55 Measurement................................................................................................. 66 Answer key..................................................................................................... 70
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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.
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NOTE TO TEACHERS AND PARENTS The book presented before you is intended for fourth-graders, but can also be used by older children. 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;
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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 the cabbage managed to cross the river…” 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 – 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 takes place before 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.
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Observations
1. E very morning, the forester goes to the woods and visits each tree. There is a
path leading to each tree. Help the forester find his way to every tree, without ever stopping twice at the same tree, then return to the same house from which he set off.
2. W ithout lifting your pen, trace a path that goes through all the sides of all the rectangles. You are not allowed to trace the same path twice.
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3. T he spider made a square grid web for catching flies. The flies could be trapped in white
squares. Every day, the spider went through each place where a fly could have been caught. He moved vertically and horizontally, passing through squares only once. He could not move diagonally. He was also not able to move across black squares. Continue tracing spider’s path so he comes back to his starting point.
4. B efore PE class, the students from the school on island A have to go to the gym hall on island
B. However, the teacher who leads them there does not take the shortest way, but warms up the students first by taking them across all eight bridges, marked with numbers 1-8, before they go to the gym hall. The students have to walk across all the bridges, but cannot cross the same bridge twice. Draw their path.
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5. O nly one of the figures marked with numbers can be placed instead of the question mark. Which figure is that? Circle the correct answer.
a)
1
2
3
4
?
b)
??
10
1
2
3
4
5
6
6. W hich number is missing? Write it in the blank figure. 3
4
0
5
4
6
4
3
7. D raw the second half of the picture.
8. D raw the same giraffe as shown in the picture, but facing right.
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9. D raw the car as shown in the picture. The two cars should be facing each
10. I n this activity, you will test your concentration and memory. Try to count the
clocks, airplanes and pencils. You have to count them the following way: first clock, first airplane, second clock, first pencil, second airplane, third clock, etc. You can cover the lines you’ve already counted with a piece of paper, so they don’t confuse you. There aren’t that many clocks, airplanes and pencils, but it’s still not easy to count them.
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11. O nly one shadow matches this clown. Find it and circle the correct answer.
Đ?
B
C
D
E
12. F igure out how the order of figures in squares changes. Fill in the blanks in the last square.
13. T here are nine dots in this picture. Connect them with eight lines so there are exactly three dots on each line.
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14. E very circle should be filled in with one of the colours of the sticks. The arrows point to circles that should be filled in with the same colours as the shorter sticks.
15. W hich figure should be the next in line? Circle the correct answer. ?
Đ°)
b)
c)
d)
e)
f)
16. T ake a look at this row of numbers. In which order have they been assorted? 2, 1, 0, 8, 5, 7, 3, 4, 6
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Quips
1. H ow can you get to number 50 by removing 10 out of 40? 2. T wo men are standing in the street. One always tells the truth and the other one always tells lies. Ask them a question to which they will both give the same answer.
3. T he glass is full of water. How will you get exactly half the amount of water in the glass by only pouring it out? Draw it.
4. D an and Pete got into an argument about whether this glass is less or more than half full. How can you help them out and give the correct answer to this question without measuring or pouring over the water? Draw it.
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5. T wo equal, 10-centimeter-tall glasses are shown in the picture. They contain equal amounts of water. How tall is the amount of water in the first glass?
cm
6. T here is a cube on this table. How will you measure the length of the AB diagonal with a ruler? Explain your method.
7. T he carpenter had four planks. Every plank was several meters long. After cutting all the
planks into one-metre pieces, the carpenter had 25 pieces of wood. How many cuts did he have to make with his saw? The planks were cut one at a time.
8. T wo fathers, two sons, a grandfather and a grandson – how many people are there in total? 9. T wo sons and two fathers split three apples among each other. Each of them got an apple. How is that possible? Explain.
10. T hree mothers and three daughters went to the market. There were four of them in total. How is that possible?
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Mirko Dejić and Branka Dejić
9 788652 906864
ADVENTURES IN MATHS 4
T
Кhis book was intended for fourth-graders who are able to learn faster than their peers. It contains many interesting activities, rarely encountered in your regular maths classes. Different problems serve different aims: some to promote abstract thinking, others to develop spatial awareness, and some to teach you how to find your way out of puzzling situations. Many are, simply put, interesting problems, the kind that will make us love maths and drive us to keep practicing it. They are all suitable for discovering and developing mathematical giftedness and promoting creativity.
Mirko Dejić and Branka Dejić
ADVENTURES
IN MATHS 4 4
