Description and Lesson Plan for the Base 10 Blocks Program

 

 

I want to start by thanking Margo Mankus for suggesting that I develop this applet as well as for her very useful suggestions and feedback. Visit her base 10 blocks activities page.

This page is specifically for teachers and describes a complete lesson plan on how to teach base 10 place value as well as basic arithmetic operations using this online version of the base 10 blocks manipulatives. The topics covered in this lesson include:

Description of the Base 10 Blocks Program

This program consists of a panel, as shown on the right, where one can click on any of three different block sizes that represent 1 unit, 10 units, and 100 units and drag them into the working panel. Once inside the panel, students can click on the blocks so they can move, rotate, break, and glue the blocks to do all types of arithmetic (and even extended to algebraic) operations.

Check the detailed base 10 blocks program instructions to learn about rotate, break, glue, and place value mats!

The blocks available in the program are the standard base 10 blocks used in many classrooms today and they are:

Block ShapeBlock Name Size
1-block  1 unit
10-block, rod, or long 10 units
100-block or flat  100 units

This program is very simple to use. Click on the arrow and then click on the blocks on the left panel to make new blocks which can then be dragged into the main working panel. Manipulation of these blocks will allow you to do basic arithmetic operations, all of which are described in the sections below. It is a great tool to teach, learn, and practice, through hands-on manipulation and visualization, basic base 10 arithmetic.

Two advantages that this program has over real physical blocks are, first, the ability to break the 10- and 100-blocks into the next smaller size and glue them back together. The second advantage is that once the student grasps the idea of how the system works, she can take shortcuts. Lets say that she needs to make the number 19, she could pull one 10-block and nine 1-blocks.  But she could also pull two 10-blocks, break one into ten 1-blocks, and recycle one of the 1-blocks. They start doing more math trying to be efficient! You should encourage this method.

Uses of the Base 10 Blocks Program

One of the best uses for this program is to present the concepts to the class. If you are able to display the computer image on a large screen TV for the class to watch, it can be used instead of overheads, making it a very nice, quick, and simple way to teach the material.

A second use is obviously for the students to practice problems in class or in labs with many computers. They can always work in groups or individually.

Finally, the students can also practice at home. You could give them assignments for them to use this same program from home. No need to carry real blocks home and parents can also get involved!

Introduction to Base 10 Numerical Systems (Place Value)

I think that a great way to introduce the concept of base 10 place value is through a hands-on role play activity which I call Counting the Rice. Read the description, try it out in class, and please let me know how it worked. Please send feedback of how this worked in your class. As I receive this feedback I'll collect it here for reference.

The activity can be complemented by using the program to go over the same ideas. Use the [100, 10, 1] place value backdrop and move 1-blocks to the 1 column. When reaching 10, glue them together to form a 10-block and move that over to the 10 column.  Do the same for the 10-blocks, i.e., collect 10, glue them together to form a 100-block and move it to the 100 column.

You can also ask the students to give you random numbers from 1 to 999 and you translate them into blocks in the right columns. You can then give numbers back to them to exercise the concept. You should write on the board all of the numbers that you come up with in this and all other activities.

Addition

The basic method to do addition consists of pulling the correct type and number of blocks that represent the addends, combining them together, and adding the total to obtain the sum. The best backdrop to use is the [100, 10, 1] place value one and each block type should be kept in its corresponding column.

Subtraction

To do subtraction you start by pulling out the correct type and number of blocks that represent the minuend. You then proceed to take away and recycle the blocks that represent the subtrahend. To do this you may have to break blocks into the next smaller unit. Once the blocks have been recycled you count the remaining blocks to obtain the result or difference. The best backdrop to use for subtraction is the [100, 10, 1] place value one.

Multiplication

With multiplication it starts to get more interesting as the students can visualize the area represented by the multiplication of the two factors. To do multiplication, you essentially do repeated addition. For example, to multiply 7 x 12, you group together 7 sets of 12 units.  Multiplications can be done using the backdrop shown in the picture below.  Notice that it looks like the first quadrant of the XY plane. I call this the XY backdrop.

You put one of the factors (12) along the Y axis, the other (7) along the X axis, and proceed to fill in the area delimited by both.

7 x 12 = 12 + 12 + 12 + 12 + 12 + 12 + 12

7 x 12 = (7 x 10) + (7 x 2) = 70 + 14 = 84

When you are done completing the multiplication, you only have to count all the blocks which make up the area of the 7 by 12 rectangle to get the product of the two factors. Remember that you can also show the commutative property for multiplication, i.e, 7 x 12 = 12 x 7. 7 groups of 12 is the same quantity as 12 groups of 7! Notice that the blocks that represent the 12 as well as the ones in the rectangle are not touching. This is done to make it easier to count the blocks.

Division -- Measurement Method

One of the methods to do division using base 10 blocks consists of breaking the dividend into groups the size of the divisor and counting the number of groups. The basic steps, using (84/12) as the example, are:

  1. Bring into the working area the blocks that represent the dividend.  In this case it would be 8 10-blocks and 4 1-blocks,
  2. Group the blocks into sizes that represent the divisor and start pulling them off such that you can count the number of groups. In this case the size of each group should be 12 and you should end up with 7 separate groups giving you a quotient or result of 7.
  3. You will typically have to break the larger blocks into smaller sizes to be able to pull apart groups of the correct sizes. The lasso will also come in handy to be able to pull the groups apart efficiently.
  4. The default backdrop, i.e. the one with only the simple grid, might be the best to use for this method of division.
You probably want to start to teach division by choosing numbers that divide exactly with no remainder. However, once the students understand the concept of division, this method (as well as the partition one) are great to introduce the concept of the remainder, i.e., examples when the divisor does not divide the dividend exactly and you end up with a group of blocks that does not fill up the group size requirement.

But before going into the concept of the remainder, I think that this is a great opportunity to introduce to the students the Prime Numbers. I recommend that you assign the following homework exercise using the base 10 blocks program:

Make all the numbers between 10 and 20. Find all the factors for each number. Write down your observations about each number 10 through 20.

Do not tell them about remainders or prime numbers. Have them discover it by themselves. Also, if you can, make sure that they work on this assignment on their own. The purpose is for them to discover the problem. Next day I would start the class with a 5 or 10 minute discussion, where everyone is able to participate, on the issues and problems associated with the homework. You can now go into describing prime numbers and remainders illustrating the concepts with the base 10 blocks program.

Division -- Partition Method

The partition method is a variation of the measurement method. Instead of grouping the blocks into groups of 12 and moving them off, you distribute the blocks, one by one, into 12 different places in the working area.  When done moving off all the blocks, you simply count how many blocks ended up in each pile.

You can distribute more than one block at the time as long as you divide them fairly, that is, every pile receives the same number of blocks. You can also use this method to illustrate very clearly the concept of remainder with the leftovers.

Division -- Scaffold Method

This other method to do division is a bit more complicated and it is basically the inverse of the multiplication method described above. The scaffold method consists of the following steps, using again the (84/12) as an example:

  1. Choose the XY axis type backdrop.
  2. Put blocks that represent the divisor along the X axis. In this case 12.
  3. Bring into the working area enough blocks that represent the dividend and arrange them in vertical columns such that they align with the blocks along the X axis and are all of the exact same size. In this case you bring out 84 blocks (or one 100-block and recycle back 16) and arrange them into 12 identical height columns.
  4. You read the result by counting the blocks that form the columns. You could also align an equivalent size column along the Y axis for clarity.

The picture below shows the working area after doing this example:

In this example we are dividing a total of 84 into12. The quotient or height of the columns is 7.

 

Beyond Whole Numbers -- Working with Decimals

This program will also allow you to work with decimal numbers by using the [1, 0.1, 0.01] place value backdrop provided. In this case, the red block represents a 1, the blue one a tenth of that or 0.1, and the green one represents a hundredth or 0.01. You can use this system to introduce decimal numbers and the place value with decimal numbers. You can do additions and subtractions using the same method that was used for whole numbers.

You can do multiplication as repeated addition. For example, to multiply 3 x 0.4 you pull out 3 sets of 4 blue bars, combine them all, and move them into the right columns to represent 1.2.

Division can be done using the measurement method which consists of bringing the dividend into the working area and pulling off groups the size of the divisor and counting the number of groups.

Bar Charts

Another thing that one can do with base 10 blocks is bar graphs. You can choose easy examples of numbers that the students can relate to and represent those in a bar chart.

Basic Algebra

There are manipulatives that are better tailored to do algebra than the base 10 blocks, however, these can be used to do basic algebra with a single variable. To do this the blocks get different meanings. The green 1-block represents a unit, so 5 of these blocks represent the number 5. The blue 10-block, is no longer a 10 unit but an unknown variable such as x, so 3 of these blocks represent 3x. Finally, the red 100-block does not represent 100 units but the square of the unknown variable x, so 7 of these red blocks would be used to represent 7x2.

The multiplication of simple algebraic expressions is done exactly as it was done above with whole numbers; the only difference being that one uses the new algebraic meaning for the blocks. This is shown in the figure below:

We are doing (x + 1) (2x + 3) by putting the first factor along the Y axis and the second along the X axis. By adding the blocks in the area represented by the multiplication we get the correct answer of: 2x2 + 5x + 3!

Concluding Comments and Suggestions

Using manipulatives helps students gain a solid understanding of the basic arithmetic operations, however, the real objective is for them to learn the written or symbolic method of doing these operations. I strongly suggest that you, as well as the students always write down all of the numerical representations of the exercises.

When starting to work on the different operations, it makes sense to start with simple one digit operations before moving into two or three digit ones.

The ability to do many of these exercises in a visual or graphical context is very helpful, especially for young minds, to get a good sense of numbers and a solid understanding of the basic arithmetic operations. The use of hands-on manipulative tools like this to solve a variety of problems will make the person mathematically stronger in his or her:

  1. Conceptual understanding,
  2. Basic skills, and
  3. Problem solving abilities.
Finally, I would really like to hear back from you on any comments that you may have about this lesson or the actual program. If there are additional topics or material that you'd like to see included here for you or other teachers to use, feel free to contact me. You can either send me direct e-mail or use the mail form provided.

You can now:

Check Feedback Received Go to Program InstructionsStart the Base 10 Blocks Program

Last Updated: Sun, 17 Oct 1999 21:22:44 GMT



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