I have strayed away from the traditional scope and sequence for math for my boys. I have found that multiplication is just as easy as addition for younger students if they are provided experience in constructing multiplication tables through the use of a variety of concrete objects. Through their use, they begin to understand the derivation of the numbers. If given the opportunity, they need little encouragement for them to examine multiplication tables in search of countless patterns. I begin by teaching them a system for finding answers to individual multiplication problems, first for problems whose products are less than 100.

I begin multiplication lessons by building a rectangle with tiles.

**Multiplication with Tiles**I begin multiplication lessons by building a rectangle with tiles.

We take a blank sheet of paper and write at the top

Rows, Columns and Area

and we list these as we make them.

.

Rows, Columns and Area

and we list these as we make them.

.

Sometimes we make random rectangles and sometimes we make two rows and then change the number of tiles in the columns, showing a pattern in the area.

Sometimes the column remains constant and the row amount changes, also showing a pattern in the area.

After we have done this for awhile, we realize that if the column stays constant we can just put that as a heading and record just the rows and areas.

After some time we can begin to record these patterns on a sheet of graph paper, essentially making a multiplication table.

We are always looking for patterns in the rows, in the columns.

Once we have completed our multiplication table, and learned a lot through the process, we can switch materials.

.Multiplication with Cups and BeansNow we can use cups and beans. We use a new sheet of paper and list at the top: Beans in Cup, Number of Cups and Total Number of Beans. We begin with just two cups and a pile of beans, putting first one bean in each cup. We record this on our sheet. Then we add one more bean in each cup; two beans in two cups for a total of four beans. |

The next time we do math, we can abbreviate it just like we did the first time, keeping the cups a constant at the top, and varying the beans and therefore the totals.

We look again for patterns. What would happen if we added two beans instead of one bean at a time?

What if we used 5 cups?

Explorations are encouraged.

Will the cups and beans produce the same chart as the tiles? Let's try it.

Now let's look at the chart again, seeing the numbers in all sorts of ways.

Can you answer questions with the table such as this:

If you had 24 beans altogether, and you have the beans in 4 different cups, how many beans would you have in each cup?

If you have 30 beans altogether and I ask you to put 6 in each cup, how many cups would you need?

It is more important to think about a solution than to actually find it. The learning is in the process, not in the answer.

For this example, this illustrates the problem 3 x 2 = 6. |

**On the Geoboard**

We look at multiplication problems on the geoboard, noting the number of spaces between the posts on the length and width sides. You can record the lengths, widths and areas on a sheet of paper. Can any patterns be seen?

We record these problems, one column recording the lines going across, one column recording the line going down and the third column recording the dots.

At this point, he is free to make up problems himself, making the cross-line pictures on index cards.

Once we have a few of these index cards made, we can then take them and cover up where they cross with a piece of paper, only leaving the lines sticking out at the top and left hand side. He has to visualize the dot matrix that is now under the paper, internalizing multiplication facts.

Now he is ready for advanced multiplication.

Now he is ready for advanced multiplication.

**"The child may learn the multiplication-table and do a subtraction sum without any insight into the rationale of either. He may even become a good arithmetician, applying rules aptly, without seeing the reason of them." -CM, Home Education, p. 255.**
I love the exploration with a constant and variables! Great way to keep it interesting instead of this is 2 x 4. I used to use boxes but the lines and dots are so much easier to use. You can also use tiles for tiling projects and use models to tile a scaled down kitchen floor. Might be a good time to discuss scaling! You gave me such good ideas. Now I can't wait to introduce multiplication!

ReplyDeleteThis will be perfect for me to use with my 9 yo as we are working on on this. I need to make math more fun instead of just worksheets and repetition. Thanks for the ideas.

ReplyDeleteVery nice post. In the multiplication/division class for parents we've just been discussing some of these techniques. We use the rectangular model with tile (we call them arrays) a lot in our curriculum. We also try to look at different strategies for solving various multiplication problems.

ReplyDeleteGreat work!!!! And thanks for posting at Math Monday.

I remember learning about doing this in college. Great way of expanding on a few different ideas.

ReplyDeleteWonderful ideas for hands on learning.

ReplyDeleteWhat a great way to make multiplication concepts into concrete activities!

ReplyDeleteYou are a great teacher. I agree - multiplication is pretty natural with concrete objects. When Anna played DreamBox, there was a game there where she had to estimate a number of boxes to fill the area. After a few tries she was multiplying naturally very easily when the problem was presented in boxes :)

ReplyDelete