My daughter has understood, for some time now, a few basic applications of the distributive law. However, it became clear yesterday that she did not really comprehend why the law is true and what it means. I have never seen a book properly tackle the concept -- for very young kids. The law is usually stated as such. Applications are introduced. An algebraic justification may be given. However, I have never seen good motivational examples. This was precisely the problem with my daughter. She saw no connection between the real world and the law.
Food is a big deal in our family. So are parties and friends. So, it occurred to me to use food and children to explain the distributive law.
Distributive Law for Division - Get a bunch of M&Ms. Get 20 in total. Put them in a pile. Tell you child that you want to divide the 20 M&Ms evenly between 2 children. Clearly, the answer is 10. Now, ask you child to split the 20 M&Ms into two piles: one with 6 and one with 14. Ask your child if you get the same number of candies per child if you divide each of the two smaller piles first. Does each child get the same number of M&Ms? Try this with varying number of children and M&Ms.
The key here is to make you kid understand that you could divide the whole pile of candies or just divide any two smaller piles and then put everything together. You may have to explain this multiple times. Do it. Repetition works wonders. Have your child do this over and over with M&Ms until it becomes natural. Have the kid choose the sizes of the two piles.
It is time to work on paper. Set up another M&M pile. Break it into two smaller piles. Divide each pile evenly among the desired number of children. Now, write the equivalent arithmetic equation. Ask the child to match the various piles of M&Ms and the correct parts of the equation. Do this several times.
It is now time to use variables to see if your kid really understood the distributive law and its interpretation in terms of candy and children. Ask him or her to solve the following problem:
(2m + 4n) / 2 = ?
If your kid gets stuck, tell him or her that you have a total of 2m + 4n M&Ms. Suggest that he or she think of the big pile with 2m + 4n M&Ms as two smaller piles put together: one with 2m and one with 4n. If your child cannot do this on his own, it is time to start explaining things again. Give lots of written examples. Use simple examples. Always revert back to the interpretation using candy/children. Do not get discouraged. Keep trying because the distributive law is incredibly important.
Distributive Law for Multiplication - Use the same method as for division. For instance,
2 * 10 = 2 * (4 + 6) = 2*4 + 2*6 = 20
can be interpreted as two children, each getting 10 candies. This is the same as doubling each of the two piles of 4 and 6 M&Ms. It should not matter if you double the big pile of 10 or proceed to double the smaller piles first.
You may be tempted to explain the distributive law of multiplication by using the definition of multiplication as repeated addition. However, I advise against it because children like to relate things to concrete, familiar concepts. Candy and children are definitely concrete, so they ought to work well.