Base Six Math

Have you ever thought about using numbers with bases other than 10? 

The concept of place value involves the notion of grouping numbers or amounts of objects.

This is true for all bases.

We played a game in which we grouped by sixes instead of tens, and then used this to establish a more broad concept of place value.

The rule for this game is that no one can use the word six (or whatever number you use as your base.) You use a nonsense word instead. Have them make up the word.

My boys came up with the word googa.

Then we placed counters down and counted to five and then the next counter was called googa.

Each time this counter was reached, we cleared the table and the counting started over.

Once they got used to counting in this way, we started counting on a dry-erase board instead, with a line down the middle. We counted directly on the board with the ones on one side and the googas on the other. We then wrote numerically on the side.

Next we practiced counting forwards and backwards in base 6 using cups and beans.

After we had done this for awhile, we recorded these numbers on paper so that they may be examined for  patterns.

Can you see where he made a mistake, and then tried to erase it?
It is easy to see when you make a mistake when you look at the patterns.

This can also be done with colors, each place value equalling a different color.

It can be done with colored counters, colored squares of paper or on a sheet of large-block graph paper.

source:

• Mathematics; a Way of Thinking, Bob Baratta-Lorton

Nim

is such an easy math game to set up and play because you can use almost anything and it only takes minutes to play. You use an agreed upon set of objects. If you are outside, you can play with sticks or small stones. If you are home, you can use any objects you have collected around the house. Today we used toothpicks. We decided to use 10 to start with. (Incidently, it took ten toothpicks to spell the word NIM at the top of this post.) For each turn, each person is allowed to pick up either 1 or 2 objects from the pile. The person who picks up the last object, even if it is part of two, wins. You can modify the game to have more objects or allow a different amounts for pick-up. You could also have the last person to pick-up loses instead of wins. The origins of this game are unknown, but a game similar to this was played in ancient China, and so many attribute China to it's origins.

Also, if you turn the word NIM, like as written in toothpicks at the top of this post, 180 degrees, it spells WIN.

More Math Games with Jellybeans

"Students may be skilled at addition, yet not understand in what situations that skill might be applied. This failure to extrapolate is most apparent when students are asked to solve word problems. They may have the technical ability to solve problems when numbers are provided, but be lost when asked to extract the same numbers from words. " We have been making up math word problems with pictures for about a week. These pictures come from a book called Instant Math Storymats, but you could sketch your own.

You can take turns making up stories that involve counting, adding, subtracting...
and skills like multiplication and division are simple for even a five-year old...



if they are part of a story, like dividing fruit represented by colorful jellybeans between two story friends.
Quentin particularly likes stories in which he plays one of the characters.

"To discover for themselves the workings of (multiplication) students have to think; this thinking is the point of all our questions and, in the end, is the point of all mathematics."

Both quotes are from Mathematics is a Way of Thinking, by Robert Baratta-Lorton.

Easter Egg Math

For a warm-up you can have your child place a jellybean in each egg carton cup.

He can count them while he is putting them in or after he has put them all in, or both.

For the next activity have him fill plastic Easter egg with 1-10 jellybeans.
While he is putting them in, have him count how many he is putting in and have him write that first number down on a blank sheet of paper. Then have him open the egg up, being careful not to drop the jellybeans out, and count how many jellybeans have fallen in one side of the cup. Have him write that number down as the second numeral of a subtraction problem. Then have him count how many have fallen on the other side of the egg and use that number as the answer numeral of the equation.

Then have him close the egg up again and shake them. When he opens the egg up this time have him count how many jellybeans have fallen in one side of the cup. Have him write that number down as the first numeral of an addition equation. Then have him count how many have fallen on the other side of the egg and use that number as the second numeral of the equation. Then have him count the total, and write down the answer the the equation.

Then have him close up the egg again, and see if a new equation comes up for this same number of jellybeans total. How many different equations can he come up with in the same egg?

This process is a fun way for children to learn "fact families," or what problems can be made from three numbers.He can then go on to another egg, with a new fact family to explore. This is also practice in conservation in number, or the concept that 3 + 4 is the same as 4 + 3.

Quentin also discovered that he could write the equation backwards, such as 7=4+3 or 1=0+1.
Confidence in the ability to manipulate numbers is an important foundation for future mathematical concepts.

Assessing K-2 Math Skills, Part 2:Operations (Simple Addition and Subtraction)

These simple games (for they can be fun) can help to to assess accurately your child's skill at number operations with simple addition and subtraction.

Concept Level

For this assessment game, ask him to place five objects in your hands. Hide some of the objects in one hand. Show the remaining objects in the other hand and ask him to tell you how many objects are hiding. Repeat this process several times. Try six objects if he is able to answer the questions correctly for the combinations of five objects confidently and with little hesitation. Increase the number of objects until the his responses begin to slow down. Try a smaller number of objects if he was hesitant or responded incorrectly with five objects. Decrease the number of objects until the child can respond correctly.

Connecting Level

Make up some cards with simple addition and subtraction equations. Also have some sort of counters on the table.

Show the child an equation card and ask him to use the counters to show what the card means. If the child is comfortable doing that, continue by giving the child several more addition and subtraction equations. Be sure the child has an opportunity to solve vertical and horizontal equations while subtracting and adding.

Symbolic Level

For this level, you only need the counters. Verbally tell the child an addition equations. Ask him to write the equation on a piece of paper and solve it with manipulatives. Repeat the process with a subtraction equations.

Word Problems

Ask him a word problem that he can visualize. Ask him to close his or her eyes and visualize the story in his or her head. The story might be something like:

- “Imagine that you had five gumdrops in your hand. If you gave me three, how many would you have left?”

- “Imagine that you found four shells while walking on the beach. I found two and decided to give them to you. How many do you

have altogether?”

The Abacus and the Soroban

This is a Chinese abacus (Latin meaning "sand tray") or "Saun pan", which means calculating plate, in Chinese. It has two sections of bead separated by a "reckoning bar." The top section is called heaven, and the beads are worth 5. The bottom section is called earth, and the beads are worth one. There are two beads on each rod in the heaven section and five beads each in the earth section. The beads are counted by moving them up or down towards the bar. If you move them toward the beam, you count their value. If you move away, you subtract their value. Addition and subtraction however are not the only things the Chinese do on the abacus. Techniques have been developed to do multiplication, division, addition, subtraction, square root and cube root operations, although I do not know how to do these techniques and they are not necessary for a simple introduction the abacus.

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This one is from Japan and is called a "Soroban" or "counting tray." Use of the Chinese abacus spread to Korea, and then to Japan during the latter part of the 15th century. Originally the soroban looked very much like its Chinese cousin having two beads above the reckoning bar and five beads below. Around 1850, it was modified to have only one bead above the reckoning bar while maintaining the five beads below. It was further changed by removing one lower bead in 1930. This one bead above and four beads below, an arrangement which remains to this day. The soroban is still being used in Japanese schools. The beads have retained their same value, but using it is a little different. Also, another unique feature is that it can calculate percentages, or numbers less than 1.

And if you'd like to make one...

Breathing Life into a Workbook Page

Let me say first that Quentin loves workbooks. He loves the tidiness of them. He loves the bright colors. He loves to feel that he can do things that other kids his age are "supposed" to do. I avoid workbooks, but since he loves them, he gets them and he does them as much as he wants to. With that said, I would like to tell you of a game we made out of one of his workbook pages. He wanted to be more creative with a particular page than the page allowed for, so we cut out the pictures of items and pictures of price tags from his workbook page. He put them together however he wanted. I got out some coins and then we played store. We each had a store and we took turns buying things, which therefore added to our own store stock. He had to read the prices and translate that into coins. He had to make change. He got to be creative. We played for quite some time, and laughed and had a lovely time. A time where learning and play are one and the same.

"How Did Numbers Begin?"

"1. Matching could have been the first important step in the story of numbers.
2. The second important step in the story of numbers might have come about from matching. It is these three ideas: "as many as," "less than," and "more than."
3. The third step in the story of numbbers also probably came from matching. It is the naming of numbers.4. Numbers must be put into their proper order.
5. Last of all in our story of numbers: counting."

-How Did Numbers Begin? by Mindel and Harry Sitomer

"Imagine that you are living at a time when your people have just learned to tame wild animals. They drive their herds out to pasture each day. Since they need these animals for food and clothing, it is important not to lose any. You are the herder. You know nothing about numbers, and you cannot count. What can you do to make sure all your animals are back each night? You might use a scheme like this: You put down a pebble for each animal as you drive it out to pasture. Later, you pick up a pebble for each animal that returns." -How Did Numbers Begin? by Mindel and Harry Sitomer, p. 6.

"If you have one or more pebbles left over when your animals are back, you know you have to go out to look for the strays. If you have one or more animals and no pebbles left to match them, you know you have a bigger herd than you started with."-How Did Numbers Begin? by Mindel and Harry Sitomer, p. 9.

Math Games: War Subtraction

You will need dice and counters for this simple counting game. Have each player start with an equal amount of counters and rolled to see how many they could take away from their chosen opponent. Winner is the one who is left last.

Math Games: My Turn, Your Turn

This game was designed to be played outside with large squares chalked out in a row large enough for the players to step in but it was too wet and cold outside, so I modified it to the kitchen tabletop and used game men to represent people. In this game each player has two men which are put in the middle space of the playing area. A die is rolled and the player can choose whether to move his man forward toward his side of the board, or to send one of the other player's men back. It takes a lot of die rolling for this game, but they did not seem to mind. The object of the game is to get both of your men off the board first.

(See comments below for the rules to the outside/many players game.)

sources:

• Math Their Way

Frog Pond Math Game

In this game you want to be the last frog to hop out of the pond. We got our board out of this book but you can make your own game board with 10 spaces just on a piece of paper. Just circles would be fine, or you could use frog stickers or draw them yourself. Fill the board with counters. We used gummy frogs just for fun. Taking turns, each player is allowed to take either one or two frogs out of the pond, whichever they choose. Can you be the last frog to hop out of the pond? This game builds thinking skills as well as mathematics awareness.

sources:

• Frog  Pond Math, GEMS guide

"You have the peas ready. It must be math time."

This is two math activities in one. First you show them unshelled peas and ask them how many peas on the average are in each pea pod. Record their guesses.

Now open the pea pods and have them count how many peas are on each side, counting one side at a time and recording the numbers as an addition problem. (Number of peas on left side plus number of peas on the right side equals how many peas altogether.)

After you have done a few of these, have them figure out the average number of peas in pods by adding together all the answers to the equations and dividing by the number of equations you did.

Line Them Up; A Math Game for Beginning Multiplication and Concentration

Each child starts with an equal amount of counters. We used 15 to start. Ask one child to pick a number within a range, say 3-5. Let's say he picked "3." You give each child 3 small cups. We used the 4-ounce cups you use in the bathroom. Have them line up a counter in front of each cup. If it comes out evenly, they can then put them in the cups. When all the counters have been used then you ask them how many groups of how many they have. In this case it is 3 groups of 5. If the number the child picked had been 4, for example, however, they would line up 3 times but the fourth time, there would be only 3 and not enough to put it. Help them to express that they have 4 groups of 3 with 3 left over. You can at some point show them how these are put into a pattern or formula for a division problem. The 4 being the denominator, and 15 being the numerator, 3 being the quotient, with 3 in remainder. I went over this verbally with them, but did not show them on the board yet. They will indicate when they want to know this, and I will show them then.

Concentration
You have played this game before, I am sure. This time we used half-gallon milk cartons cut in half. Only use the bottom half and turn them upside down. Start by putting counters in pairs of amounts. We used Unifix cubes in pairs of 2, 3, 4 and 5. Each child takes a turn lifting two milk cartons trying to locate identical stacks. The game is played until all stacks have been discovered. You can later change the counters for the cards with numbers on them, or even problems with identical answers.

sources:

• Math Their Way, Robert Baretta

Math Games: Connecting the Concrete to the Abstract

Numerical and mathematical symbols are introduced at the connecting level. Mathematical symbols(e.g., +, –, =) are even more abstract than the mathematical language they represent. Both vertical and horizontal equations should be experienced. Children visualize symbols as they solve number problems using manipulatives. They show their understanding at this level by building concrete models with various types of manipulatives to match written equations and relating equations to manipulatives or to a word problem they have created. The student or teacher records the mathematical symbols.

Recording Number Stations

Remember when we learned how to do the Number Stations? The next step is to record the results of this game. Take cups of any fixed number or beans, for example 8 white beans and 8 red/brown beans. You could also use white beans that have been spray painted a color on one side. (In that case you would only use 8 beans, instead of 16 total.) To play the game, the student pulls out 8 beans without looking at them and using a worksheet with shapes of beans colors the red/brown beans, leaving the white beans alone. Write the ratios on a white board and then add the symbols "+" and "=."

Recording explorations with Unifix cubes can also be recorded. Take whatever number you wish and have them make patterns using that amount of unifix cubes in two colors They then recorded them by coloring them on a worksheet.

Unifix Trains
Again pick out a number of Unifix Cubes connected together into one stack. I then asked them to divide the stack evenly into two. "Did it break evenly into two's?" "Are there any left over?" "Does that make 9 an even or an odd number?" "How about if we break it into three's?"...

Lift the Bowl-Connecting Level
Remember when we learned the "Lift the Bowl" game? When we first played it, we only used counters. Now we begin connecting the counters to numbers but they are not yet asked to write the numerals. Using cards with numbers on them, and the bowl template, they work out simple addition and subtraction problems.

Recording Toothpick Number Station

Remember when we made patterns with pasta? Well, this time you can "record" toothpick patterns by gluing toothpicks to pieces of cardstock.

The Cave Game
You can play the cave game initially with just counters of your choice and your child's hand. Have him up a number of counters out and then have him put some of the counters in a "cave" made by your child cupping his hand into a backwards "C" shape. Once he can both say what the problem he made can be (such as 4 minus 1 equals 3 or 3 plus 1 equals 4) then he can move onto the connecting level. You can then lay out the counters and then have him put the numeral cards under the counters. Once this becomes comfortable, then he can look at problems on a worksheet that laid out just like the cave game. Lastly, he can use flashcard problems and can work them out himself using his cave game skills.

sources:
Math Their Way, Robert Baretta