Math Stories and Games: Logic, Patterns and Mathematical Thinking Anna Shevyakova, Alexey Shevyakov............... Lesson 1. Attributes of Objects Dad, play with me, I am bored! Nicky called his father. OK. What shall we play? Let s play with Lego! Sounds good. I can show you something interesting. Dad sat in front of Nicky, and they poured all the Lego on the floor between them. Then Dad took four toy trucks from the shelf, and said: We have Lego blocks of four colours. Let each truck carry blocks of one colour. The first truck will take red ones, the second blue ones, and the two other trucks yellow and green blocks. We will take them to the other side of the room and make four depots red, blue, yellow and green. OK, great! shouted Nicky. The work began. They loaded the trucks, hauled the blocks, and built the depots. While Dad and Nicky were busy with the construction, Nicky s older sister Kate came in. She watched and said: The houses are cute! I d like to build such for my dolls! These are depots not houses, said Nicky. We can build something together if you d like, said Dad. I d like to make a house, Kate responded, But my two Lego sets are mixed together. Then we will sort them. Bring the box! Sure! Just a moment! Kate brought her Lego box. It contained Lego blocks of various colours, sizes, and shapes. They began to sort them. Large blocks were put in one pile, and small ones in the other. They put the small-sized set away, and separated the large-sized set into a pile of squares and a pile of rectangles. Eah pile was further sorted according to the color. After that, it was easy to build a house! 1
Problem 1. Games with Lego blocks. Sort a Lego set by separating blocks into piles of different colours. Now mix them together again and sort according to the shape: make a pile of blocks of each shape (squares, rectangles, etc.). Problem 2. Color balloons of the same shape using the same colour. 2
Problem 3. Use one colour to color shapes that have the same shape and the same size. Problem 4. Color mushroom caps so that the blue circle has mushrooms with red caps, and the green circle contains mushrooms with brown caps. Color stipes of large mushrooms yellow, and stipes of small mushrooms orange. 3
1 Problem 5. Draw a path between the car on the left, and a car of the same colour on the right. 2 Draw a path between the car on the left, and a car of the same shape on the right. Problem 6. Pin large green fruits on the big hedgehog. Pin small yellow and orange fruits on the small hedgehog. Put the remaining fruits into the basket. 4
Lesson 3. Relative Positions of Objects Nicky was sitting on the floor and playing his toy cars. He moved them across the room and then parked in a parking lot. It looked nice. Nicky decided to draw a picture showing the parking lot and all the cars exactly as they were arranged. But it was difficult. Nicky often could not figure out what the color of the next car should be. He tried to look at his parking lot and memorize it but it didn t really work. Dad came by. What are you so busy with? he asked. Nicky explained and asked for help. Dad agreed: OK, you do the drawing, and I will be telling you what to draw. Nicky was excited, and they began. Dad said: First, draw a green car. Then a blue car next to it on the right, and a then a yellow one further to the right. What does it mean to the right? asked Nicky. Show me your right hand, which is holding the pencil. Then stretch it all the way to the side. That way means to the right. Oh, I see, answered Nicky. Following Dad s guidance, he completed the upper row of the cars, then the middle one, and all the way until he finished the picture of the parking lot. It turned out just like it was on the floor! 5
Problem 1. 1 Color all cars to the left of the cube: 2 Color all dolls to the right of the ball: 3 Color all ducks between the ball and the cube: 6
Problem 2. 1 Color all stars above 2 Color all planets 3 Color the clouds between the airplane: below the rocket: the airplane and the rocket: 1 7
1 Problem 3. Color all fishes to the left of the brown one red. 2 3 4 5 6 Color starfishes to the right of the purple one yellow. Color all sea creatures above the red fishes blue. Color all uncoloured sea creatures that are below fishes green. Color the fish to the right of the brown one orange. Color all sea creatures above the brown and the orange fishes green. 8
Problem 4. Games with parents (optional). 1 Put toys (for example, cars) in a row. Ask the child to show you toys that are specifically located with respect to a given one, for example, as follows. Show the car to the left of the blue car. (This task aimed at relative positions of objects.) Show the car to the right of the big red car. (This task aimed at relative positions and properties of objects.) Show the car between the green car and the white car. (An advanced problem on relative positions.) 2 Place the toys in a column and play similarly to practice above-below relations. 9
Lesson 3. Algorithms Nicky and Kate s family decided to do some home renovations. They picked a new color for their kitchen. The kids helped to paint the walls. When all was done, and the paint dried up, Nicky and Dad pulled off painter s tape. Dad started from the top and Nicky started from the bottom. Nicky crumpled the tape in a ball and tried to kick it, but it did not go. It was sticky and it stuck to Nicky s foot. Too bad you can t play with it, said Nicky. Dad looked at him and said: Oh yes you can. Look. Dad took the tape and began sticking it to the floor. He made five parallel strips. What are we going to do with them? asked Nicky in surprise. Patience! Can you guess what I am doing? asked Dad and added some strips across, making a square grid. Is it a sort of a hopscotch game? Not at all, said Dad. We are going to play the Maze game. I want to play the Maze game, too! shouted Kate as she walked into the kitchen. I have never tried it before! Excellent, Dad replied. Then would both of you go and bring some big toys? While the kids were gone, Dad added a couple of arrows at the sides of the Maze. Coming back, Nicky brought a dump truck and an airplane; and Kate got a Teddy bear, a bunny and a kitten. Dad took the truck and the bear and placed them in two squares, like this: Nicky, imagine that you are a robot. Stand here, on the Entrance arrow. You can only move when you receive a command, one step at a time. That s fun! But how will I receive commands? I have no antennas! You will hear voice commands. Kate will be your Operator. Kids were excited to begin. Dad put Nicky the Robot in front the Maze, on the Entrance arrow, and Kate sat beside. Dad explained that the Robot could only understand very simple commands. Forward, Back, Left, and Right. Following each command, the Robot makes one step a certain direction, and stops, waiting for a new command. The Robot does not turn. And he cannot step on the squares with obstacles. 10
They began to play. First, Kate guided her Robot-brother through the Maze. Then they moved the toy obstacles and changed roles. When Nicky was the Operator, he sometimes confused left and right, but quickly figured it out. Then Dad suggested another challenge. He brought some Lego pieces of four colors. He made four paper arrows and put them as shown. The arrows were pointing like this: Red = Forward, Blue = Back, Yellow = Left, Green = Right. Now the Operator could not just shout the commands. He had to prepare a full Lego program for the Robot to use. A Lego program is a line of Lego blocks with colors chosen so that the Robot reads them as commands, one by one, and makes steps accordingly. They also added more toy obstacles : FORWARD LEFT RIGHT BACK 11
One of their programs that successfully led Kate the Robot through the Maze looked like this: Can you draw Kate s path in the Maze? Dad and kids played a number of times and liked it a lot. They decided to call the game Lego programming. They created another maze landscape, with some more real obstacles: pictures of forests, swamps, mountains, and fire. FORWARD LEFT RIGHT BACK If the Robot follows the program below, will it pass the Maze successfully? Then supper time came, and Mom called everyone to the table. While they are eating, try to solve the following problems. 12
Problem 1............ Problem 2. Create your own Maze. Place obstacles. Show Entrance and Exit with arrows. FORWARD LEFT RIGHT BACK Write a program for a Robot to go through your following Maze. 13
............... Lesson 5. Attributes of Objects The next day was a working day. Kids were at school. After school, they had some rest, read a book with Mom, and then Dad came home. As soon as supper was over, Kate and Nicky asked Dad: Daddy, let s play as yesterday! You want to play the Maze game again? Yes! Dad thought a little, and then said: I have a new game for you! Let s use your toy railroad. What s the game called? asked Nicky. It s called «the Magic Tunnels», Dad replied. The kids were excited to begin. They put together the wooden tracks as follows. Dad made three tunnels from large Lego pieces. Then Dad explained the rules: The train will carry cards with geometrical shapes. After a train passes one of the tunnels, the shape will change one of its attributes. Tunnel 1 changes the shape to any other shape. Tunnel 2 changes color of the shape to any other color. And Tunnel 3 changes the size of the shape it may grow or shrink, but cannot stay the same. Tunnel 1: shape Tunnel 2: color Tunnel 3: size They played for a long time, trying different shapes and looking at possible results. The Problems below include some possible games for «the Magic Tunnels». 14
Problem 1. The train goes through Tunnel 1. Draw the possible shape after passing the tunnel, for each initial shape. Problem 2. Now the train goes through Tunnel 2. Think of the possible new shapes. Problem 3. Think of examples of what could happen to the following shapes if the train takes them first through Tunnel 2, then Tunnel 3, and finally, Tunnel 1? Tunnel 1 Tunnel 2 Tunnel 3 15
............... Lesson 16. Logic and Arrangements The kids room was a mess. The toy shelves were all empty, and the toys were scattered all over the floor. In the midst of this sat Kate and Nicky. They were sad and bored. Cleaning up the room and putting toys on their places did not seem like a fun thing to do, and they were working slowly. Suddenly, Dad came in. At once he understood the situation, and said: Do you know that clean-up can be fun? You only have to come up with rules. What kind of rules? The kids were surprised. For example, Dad continued, try to guess which rule I am going to use to fill the shelves. Out of the heap of toys, Dad picked the smallest ones, and put them on the top shelf. He then put medium toys on the middle shelf and large toys on the bottom shelf. I know! said Kate. You increased the size of the toys. First you put the smallest ones on the top, and then bigger and bigger ones on the shelves below. Good guess, said Dad, but this was a very simple rule. Now try to guess the next one. 16
Dad removed all the toys, and once again began to fill the shelves. This time, in the left corner the top shelf, he put the biggest toy. Next to it, he placed a smaller toy, and then, at the right end of the shelf, he put was a little toy bunny. On the second shelf, Dad did the same thing: on the left, he put the biggest toy (it was a teddy bear), then a medium toy, and on the right end of the shelf went a little car. This is simple too! I see it. smiled Nicky. Great. Now the third rule. It is harder, so look carefully! And Dad arranged the toys like this. 17
Kids guessed this one quickly, too. Kate explained: First, you put dolls, then cars or trucks in the middle, and animals on the right. So every shelf has a doll, a car, and an animal. Good work! Now I will show you the most complicated arrangement. He placed toys like this: I know! shouted Kate. Me too, said Nicky, and it is not very simple! You put them so that on each shelf, there is one car, one doll, and one stuffed animal. And in each vertical row, there is again one car, one doll, and one stuffed animal! You are very good at this! Dad was happy. The kids liked the Dad s game and began to think of their won rules and ways to arrange toys. While playing, they did not notice how the whole room was cleaned, and there was not a singe toy left one the floor! 18
Problem 1. Find objects that fit in empty cells. Draw arrows to show where they belong. 19
Problem 2. In each picture, draw and color the missing shape. 20
Problem 3. Look at the first three pictures. Finish the fourth one, so that it follows continues the pattern. 21