Place Value with Decimals

1. Review place value in general. Read How Much is a Million? by David Schwartz. Practice reading and writing large numbers in numeral and written form. David Schwartz tells us that it takes one gallon of water to keep a one-inch goldfish alive. How many gallons of water will it take to keep 10 goldfish alive? 100 goldfish? 10, 000 goldfish? One million goldfish? There are 7 1/2 gallons of water in a cubic foot of water. How many cubic feet of water are necessary to keep 10 goldfish alive? 100 goldfish? 10, 000 goldfish? One million goldfish?
2. Begin demonstrating decimals on a trading board. If you used one of these to introduce addition and subtraction, now you can just shift your child's thinking by telling him the one cup on the trading board is now one, instead of ten, then one bean is 1/10. I told him to think of it as if the cup represented a class at co-op which had 10 students in it. The class would be one, and each student would be 1/10 of the class. If we decided that the bowl represented one (instead of 100 as it had in the past), then we could look at it as one co-op school, in which each class was 1/10 of the co-op (assuming that the co-op had 10 classes in it) and each student was 1/100 of the co-op students. After this introduction, we practiced ways of writing fractions whose denominators are powers of ten. We wrote them as fractions and we wrote them as decimals.
3. Next, practice the same concept using different materials. Assign fractional values using plastic colored chips on the counting/trading board. Together create decimal problems for addition and subtraction using the counting/trading board by rolling a die for each column. Since he already knows how to create addition and subtraction problems in this way, the only difficulty he might encounter is where to place the decimal. 
4. Next get out a piece of 10 x 10 graph paper to make a matrix. You can use Math-U- See blocks, Unifix cube sticks, bean sticks or whatever you have as a base-10 manipulative, since we are working with decimals, which are all working with divisions of ten. Use whatever you had used to represent 10 and call it one. Using a series of questions, you can guide your student to see the relationships.
5. Have the student read decimal fractions through ten thousandths and decimal fractions with whole numbers. Have your student write the following rules in his math journal: 1.) The decimal point is read as "and." 2.) The first decimal place is tenths, the second decimal place is hundredths, the third decimal place is thousandths and the fourth is ten thousandths. 3,) Read the decimal number as a whole number followed by the name of the ending place.
6. Play Decimal Place Value with Playing Cards. Instructions and a printable score keeping sheet can be found at Games for Gains. Basically, you have a deck of playing cards without the face cards, and in this game the 10 card represents 0. Each player takes a card from the pile and decides which place value to assign the number on the "Round 1" portion of their score sheet. The player can only write that number under one place value column (hundreds, tens, ones, tenths, hundredths, or thousandths). Once each player has written down the number in a place value column, it cannot be changed at any point during the game. This is repeated until all 6 columns are filled. The winner of that round is whoever has the largest number.
7. Connect tenths and hundredths to dimes and pennies. Questions like, which is larger, 1.8 or 1.09? may suddenly seem easier if translated into money terms, $1.80 or $1.09.
8. Connect decimal numbers meterstick, since it is already divided into hundredths.

Resources: 

• How Much is a Million? by David Schwartz
• Mathematics; a Way of Thinking, Bob Baratta-Lorton
• The Math Maniac
• Games for Gains

Decimals (3-6)

If you get students to search problems and answers for patterns leading to rules for the correct placement of the decimal point in decimal problems and they are allowed to discover the rules for themselves, their learning will be a product of understanding, not memorization.

In the past we have assigned values to our counting board and coordinated amounts and colors.
At first we used beans and grouped them in cup, bowls and large #20 tin cans to represents ones, tens, hundreds and thousands. We next made bean sticks and rafts so we didn't have to always count out the beans. We then matched them to Math-U-See blocks, matching the colors of the counting board to the block amounts.
Now we are taking this one step further and assigning fractional values to the beans, cups and bowls on the bean trading board in  preparation for using it to represent decimal fractions.

Beans, Cups, Bowls and Cans

James has begun to learn about decimals. We began by assigning values to beans, cups and bowls on a bean trading board in preparation for using it to represent decimal fractions. When we were working with fractions, we made up different Unifix cube sticks and called them one. We now did the same thing with beans on the trading board. If one cup on the trading board is now one, instead of ten, then one bean is 1/10. We thought of it as if the cup represented a class at co-op which had 10 students in it. The class would be one, and each student would be 1/10 of the class. If we decided that the bowl represented one (instead of 100 as it had in the past), then we could look at it as one co-op school, in which each class was 1/10 of the co-op (assuming that the co-op had 10 classes in it) and each student was 1/100 of the co-op students. After this introduction, we practiced ways of writing fractions whose denominators are powers of ten. We wrote them as fractions and we wrote them as decimals.

Chips

We then practiced the same concept using different materials. We assigned fractional values using plastic colored chips on the counting/trading board. We then created decimal problems for addition and subtraction using the counting/trading board by rolling a die for each column. Since he already knows how to create addition and subtraction problems in this way, the only difficulty he might encounter is where to place the decimal. Some students may find placing the decimal point correctly is obvious, but others might need to be asked questions to come to this understanding.

"How can you tell where the decimal point goes? Could you make up a rule to predict where it goes for addition and subtraction problems?"

Matrices

Today we got out a piece of 10 x 10 graph paper to make a matrix.

Just like when we played with fractions and worked with Unifix cube sticks, and called them one, we used Math-U-See blocks today, and called them one. You could use Unifix cube sticks, bean sticks or whatever you have as a base-10 manipulative, since we are working with decimals, which are all working with divisions of ten. We then chose the blue ten rod and called it one.
"What is the fraction of one division of this rod?"
"One-tenth."

"Since the fraction has ten as a denominator, you can write it as a decimal. Let's write it along the side beside the first block on this graph paper."
"What fraction would two spaces or divisions be?"
"Two tenths."
"Let's write that next to the second block on our matrix."
We continued on, filling in the whole side. He especially enjoyed writing in the last square (1.0) because we called the whole rod one, and he found delight in ending up with both a numeral which represented one and .9 +.1 =1.0, or 10 tenths.
 We then completed the same notations on top of the blocks along the row of spaces at the top of the graph.
Now we looked at the whole graph paper as a multiplication matrix, and began the multiplication of decimals.
"How many squares are there altogether on the matrix?"
"100."
"If I say the whole matrix is one big square, what fraction of the big square be to the little square?"
"1/100."
"How would you write that as a decimal?"
".01"
"Put that in the first square."
"Can you fill out other squares in the matrix?"
He picked out a few and added their numbers into the matrix. We shall work on this again, until it is all filled in, but I don't anticipate that being too hard for him at this point.
Then I will give him a new matrix of 100 x100, but it won't be necessary for him to completely fill this one too. I will give him a few problems to work out on it instead such as
.01 x .01= or .12 x .12=
He will be able to learn how to add, subtract and multiply decimals, not because he was told how, but because he was directed where to look and saw for himself what needed to be done.

source: Mathematics; a Way of Thinking, Bob Baratta-Lorton

Potion Making: A Math Game



at Time to Craft



When I saw this math game at Time to Craft, I was very intrigued. I left a comment there, telling of my interest to know more and Cheryl treated us to full explanations and even print-outs of the game that her daughter, Arabella, made. She even game me permission to share the idea with you. I love the blogisphere because you can meet the nicest people. I showed what she sent me to my boys, and they were very interested and even had some ideas of their own. The point of the game is to be able to match the different ways a fraction can be written whether it be a decimal, percent or non-reduced fraction.

Their game board is similar to snakes and ladders, with about 26 squares, 20 of which have shapes which are divided into fractions (1/2, 1/3, 1/4, 1/5 and 1/6.) One square is a "Miss a Turn" square. You can make it any style you would like, to personalize it as it suits your needs. Cheryl's daughter is interested in Harry Potter, so she chose to make it like a moving staircase, and kept the snake but added a trap door for one square, that leads to the tail of the same snake. We made ours a Magician's castle and had the path going every direction, and ended up having many more squares. It is interesting what happens when a whole group of people work on a project together!

The potion list that Cheryl sent me was very cute and clever but it was based on the Harry Potter books, which we don't read, so we modified it to suit us. Sam made me a list of some interesting potions we could make such as "Sleep", ""Fly" and "Fireball." We started out with a sheet with several circles divided into 8ths, 6ths and 5ths. These would be the parts or components list for the potions.  Players need to pick up components of the potions according to what fractions they land on, picking up various parts of the ingredient until he made a whole. This added an element of adding/subtracting fractions.
Next you need to make a deck of cards which have each fraction written out as a decimal, percentage and another fraction (eg. 1/2, 0.5, 50%, 2/4). Arabella used an old pack of playing cards. We used cut out pieces of cardstock.

Cheryl and her daughter adapted a regular 6-sided die so that there were two sides with one, two sides with two and two side with three as the game is too quick with a regular 6-sided die. We decided to use a ten-sided die since we had many more squares to travel.

Now that all the materials are ready, you are ready to play.

The Rules of the Game:

I have adapted this just a bit, but most of it was created by Cheryl's daughter.

1. Each player takes a spellbook sheet with circle divided into different fractions and some colored pencils (although you could use just a regular pencil.)

2. Give each player 7 cards.
3. Youngest goes first. Roll the dice and move your counter that number of squares.
4. If the player has a card that matches the square's diagram, then fill in  the amount on the  potion graph for a potion of choice. Discard the card and pick up a new one. Pass the die to the next player.

5. If a player offers a card that does not match, and the other players catch it, they must go back two squares.
6. If there is no match, pass the die to the next player.
7. If a player has two cards that add up to the square, he can use both cards at once. (For example, if you land on a square with a shape that is half-filled in, and you have cards with 0.25 and 1/4 on them, you can use both cards to equal the 1/2).
8. First one to get all the ingredients for two of the spells, has the ability to begin heading back home to the beginning square. Once the first person returns to the home square, everyone tells how many spells they have all the ingredients for, and the one with the most wins!

More here.