I had shim both the large insert the Dremel is mounted to and the circular inset because I cut too deep for both.
The groundwork for multiplication was laid earlier for students when they used objects, number lines, and arrays to multiply. The Commutative Property of Multiplication states that changing the order of the factors does not change the product.
Following is one of the more dramatic multiplication patterns that you can help students discover by using a multiplication table. Notice also that the products on one side of the diagonal mirror the products on the other side.
Students can explore square numbers as well as other patterns by using multiplication tables.

The products in the row and column for some factors are double the products in the row and column for other factors. The products in the rows and columns for 1, 3, 5, 7, and 9 alternate between even and odd numbers. Once students understand how to multiply two factors, they are ready to proceed to multiplying three factors. Explore with students how the relationships between the factors in the rows and columns affect the relationship between the products of those factors. While this may sound challenging to students at first, they will discover that by grouping factors together, it is really quite easy.
Students are now ready to use their understanding of multiplication to explore number patterns.

For example, since 8 is the double of 4, the products in the row for 8 are double the products in the row for 4. Students should grasp the Associative Property of Multiplication easily because it works in much the same way as the Associative Property of Addition, which they studied earlier.

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