Base 10 Block Teaching Ideas

Base 10 blocks consist of unit cubes (1/2" x 1/2" x 1/2"), ten rods (1/2" x 1/2" x 5"), and hundred flats (5" x 5" x 1/2"). I also made thousand cubes (5" x 5" x 5") for demonstration. I made ten of them so children can stack them to show the value of ten thousand.

Here are ideas for using them:

Introducing the Concept of Place Value

Show a unit cube and define it as a "unit" or "one." Put ten of them together.
Put ten unit cubes together and then put a ten rod next to it to show they are the same length. Define the rod as a "ten."
Put ten tens side by side and 100 units. Show that the hundred flat is the same size. Define the flat as a "hundred."
Stack ten hundred flats together. Show that a thousand cube is the same size. Define the large block as a "thousand." Notice that the shape of the thousand cube is the same as the shape of the unit cube, only the thousand is a lot bigger.
Stack ten of the thousand blocks to show ten thousand. Notice that the shape of ten thousand is the same as the shape of the ten rod, only ten thousand is a lot bigger.
Mark off or try to imagine what a hundred thousand and a million would look like.

For older children: teach them how to fill out the place value chart and have them do it daily until they have mastered it. Write ones, tens, hundreds, one thousands, ten thousands, hundred thousands, one millions, ten millions, hundred millions, etc. Notice the pattern: one, ten, hundred.
Learn place values through the decillions. See Facts Plus p. 201.

Reading and Writing Numbers

Make a chart showing ones, tens and hundreds on a 12" x 18" piece of paper.
Teacher assembles pieces for a variety of numbers on the chart: 9; 35; 273; 762; etc. The child writes the number in the place value chart.
The teacher writes a number and the child assembles the appropriate pieces on the large piece of paper.

Zero as Place Holder

Assemble pieces for a variety of numbers using zero as a place holder: 106; 30; 520; 200. The child writes the number on a place value chart, using the zero as needed.
You write a number. The child assembles the appropriate pieces to show the value, placing no pieces in the columns with a zero.

Addition Facts

Use the base 10 blocks (or beans, pretzels, any manipulatives) to generate the table of facts.
Show that 3 + 5 and 5 + 3 are exactly the same.

Addition Facts Masters

Practice with flash cards the ones; then twos; then zeros, ones and twos; then threes; then zeros, ones, twos and threes; then fours; and so on.
Do daily facts practice, aiming for a mastery level of 100 facts in three minutes or less. See sample worksheets in Addition Facts in Five Minutes a Day.

Addition with No Regrouping

For example: 35 + 23. The child lays out 35, then 23. Combine the ones and write the total in the ones column. Combine the tens and write the total in the tens column.
Continue with problems not requiring regrouping into the 100s and even 1000s place (if you have 1000s).

Addition with Regrouping

75 + 17. Child lays out 75 and 17. Combine the units. Count them and when you get to ten, exchange for a ten bar and put it in with the tens. Start counting the units again from one.
Write the ones in the ones column. Then count the tens and write the total in the tens column.
Continue until this process is automatic.
Continue with problems requiring regrouping into the 100s place.
Transfer to paper by doing each step with manipulatives and showing how it is represented on paper. Once a child understands, have him use paper exclusively. The manipulatives can be used for review if needed.

Column addition. Adding by endings.

This doesn't require manipulatives, but this is a good time to teach it. It enables children to do huge addition problems correctly. See Marvelous Math handout for directions.

Subtraction Facts

Use the unit cubes to generate a table of subtraction facts.
Show that 3 + 5 = 8, 5 + 3 = 8, 8 - 3 = 5 and 8 - 5 = 3.

Subtraction Facts Masters

Practice with flash cards the ones; then twos; then zeros, ones and twos; then threes; then zeros, ones, twos and threes; then fours; and so on.
Do daily facts practice, aiming for a mastery level of 100 facts in three minutes or less. See sample worksheets in Subtraction Facts in Five Minutes a Day.

Subtraction without Regrouping

55 - 32. Child lays out 55. Take away 2 units and write the number of the remaining units (3) in the ones column.
Take away 3 of the tens and writing the number of remaining tens (2) in the tens column.
Continue. Later add problems not requiring regrouping into the 100s place.

Subtraction with Regrouping

34 - 18. Child lays out 34. Try to take away 8 ones. There aren't enough. Where can you get more? Take one of the tens and exchange it for 10 units (you could cut it into ten units, but exchanging does the same thing). Now take away 8 ones and write the remainder, then take away one ten from the 2 tens that are left. The answer is 16.
Do lots of these without pencil and paper when introducing. Make sure the child really understands the concept and can explain it. Continue to the 100s place.
Transfer to paper and pencil by showing each step first with the manipulatives and then with the representation on paper. Use manipulatives to check problems done with pencil and paper and return to review if problems surface.

Subtraction across Zeros

103 - 57. Child lays out 203. There aren't enough ones to take 7 away. Where can you get more? No tens, either. Where can you get some tens?
Take a hundred and exchange it for ten tens, then take a ten and exchange it for ten ones.
Now take 7 away and write the remainder in the ones column. Take 5 tens away and write the remainder in the tens column.
Do lots of these with manipulatives before going to just paper and pencil. Transfer gradually once the child really understands the concept.

Multiplication Facts

Use unit cubes to generate the times tables in a grid. Do no more than one table a day.
Practice counting by 2's, 3's, 4's, etc.

Multiplication Facts Masters

Practice with flash cards the ones; then twos; then zeros, ones and twos; then threes; then zeros, ones, twos and threes; then fours; and so on.
Do daily facts practice, aiming for a mastery level of 100 facts in three minutes or less. See sample worksheets in Multiplication Facts in Five Minutes a Day.

Multiplication with Tens

Do 10 x 3, 40 x 5, etc. with the manipulatives. Note the pattern.
Do 100 x 3, 40 x 30, etc., with zeros on both numbers. Note and learn the pattern.

2 Place x 1 Place Multiplication

15 x 4. Lay out a grid 15 tall and 4 wide.
Exchange any group of 10 units for a ten bar.
Exchange any group of 10 tens for a hundred flat if needed.
Write the total of each place value correctly on the place value chart.
Later move on to 3 place x 1 place multiplication, etc.

2 Place x 2 Place Multiplication

21 x 42. To introduce, make a grid 21 high and 42 wide and fill it in. Move quickly to paper and pencil as students won't have enough manipulatives to do this in their individual kits.
Practice this until it is mastered.
Later move on to 3 place x 2 place multiplication, etc.

Gelosia Multiplication

No manipulatives are needed for this. It enables children to do huge multiplication problems if they know their facts. Directions are in the Marvelous Math Tricks handout.

Division Facts

Use unit cubes to generate the division tables by laying out the dividend and "passing out" the pieces into equal piles.
Practice 1's with flash cards, then 2's, then 1's and 2's, then 3's, etc.

Division Facts Masters

Practice with flash cards the ones; then twos; then zeros, ones and twos; then threes; then zeros, ones, twos and threes; then fours; and so on.
Do daily facts practice, aiming for a mastery level of 100 facts in three minutes or less. See sample worksheets in Division Facts in Five Minutes a Day.

Division with Remainders

23 ÷ 5, for example. Lay out 23, then "pass them out" into 5 piles. How many piles? That is the quotient. How many are left over? That is the remainder.

Do lots and lots of these with manipulatives before going to paper, then lots with both manipulatives and paper. Careful work now will make future work easy.

Division with Remainders and Two-Digit Answers

73 ÷ 4, for example. Lay out 73 (7 tens and 3 ones). "Pass out" as many tens as possible into 4 equal piles. Then take the ten left over and exchange it for ten units.
Next, "pass out" the ones. Any left over are the remainder, as before.
Continue with numbers into the hundreds when this has been mastered.

Fractions

Children should understand fractions before they move on to decimals because otherwise they may memorize the "rules" for working with decimals but they probably won't understand the concept. Click here for fractions lessons with manipulatives.

Decimals

Place value is the most important and sometimes the most difficult concept in decimals.
Have children imagine that the unit cube "grows and grows and grows" until it becomes the size of the thousand cube (5" x 5" x 5"). Redefine the large block as a "unit." Emphasize that it is the same shape, only larger. The pieces would be too small if we tried to cut up the small unit cubes into tenths, hundredths, and thousandths.
Ask how many tenths are in a whole (from fractions). Ten. Whick of the blocks would represent a tenth, in that case? (The flat. Demonstrate that 10/10 equals one unit.)
Ask how many hundredths are in a whole. One hundred. Which of the blocks would represent a hundredth? (The bar. Demonstrate that if you have 100 bars, it equals one unit cube.
It would be a good idea to work extensively with tenths and hundredths before introducing thousandths.

Decimal Place Value Chart

Lay out a number and have the child write it in the decimal place value chart. Emphasize saying "and" at the decimal point and saying tenth, hundredth, etc.
Write a number with decimals and have the child lay it out.
Say a number and have the child lay it out.
43/100. Four tenths and three hundredths. This could also be forty-three hundredth if "broken down."

Expanded Numeration

Lay out a number such as 5.38. Show that there are 5 units, 3 tenths, and 8 hundredths.
Have the child write 5 + 3/10 + 8/100.
Do several with manipulatives until the concept is understood, then transfer to paper and pencil.

Zero as Place Holder

This concept is often confusing. How would you write 3/10? Write it in the decimal place value chart: .3
How would you write 3/100? Write a 3 in the hundredths place. What goes in the tenths place? A zero because there are no tenths. It's .03
There is no need for zeros after the final numeral in a decimal number. They don't mean anything. .3 = .30 = .300 and so on. (Later, this concept will be useful when subtracting. Zeros may be added without changing the value when necessary for some purpose.)

Adding Decimals

Use base 10 blocks and decimal place value charts to add decimals. Notice how everything centers on the decimal point. That is the reference point for place value.
When adding decimals on paper, make sure the decimal points are aligned.

Subtracting Decimals

Use base 10 blocks and a decimal place value chart. Lay out the pieces for the minuend. Take away the appropriate number of each and write the difference in that column. Regroup just as in regular subtraction.
Note that 3.5 - 2.86. 3.5 = 3.50. The child will need to add a zero and then borrow to be able to take away 6. Exchange a tenth for hundredths.

Multiplying Decimals

Because decimals are a way of writing fractional values, children should already understand why the answer is smaller than either of the numbers being multiplied.