Today Aiden and I spent the lesson diving into the whys behind the properties we've recently learned: distributive, commutative, and associative. We started with yesterday's example of 2(3x)=6x. We replaced x with 5 this time and considered 2(3*5). We discussed how the 3*5 is already 15, so when we multiply the 2 to the 3, it's really not just 3, it's 15. The goal was to clarify why we don't multiply the 2 to the 3 and 5. Aiden then connected it to the commutative property. He realized that if we multiply 2 to the 3 to get 6 and then apply the x, that would be the same as if we multiplied the 2 to the x to get 2x, and then apply the 3 to get 6x. Putting numbers into the variables so Aiden can SEE it has been helpful for him. We also explored the idea of combining like terms like 5x+2x and WHY it equals 7x. In the past I explained to Aiden to think about it like he is combining items: "5 x-rays plus 2 x-rays equals 7 x-rays." Today we considered the cost of the x-rays, what if it was $100/per x-ray. We thought, "5 x-rays at $100 each plus 2 x-rays at $100 each would give us 7 x-rays at $100 each, for a total cost of $700." Then I asked, "What is 5 fourths plus 2 fourths?" Aiden answered, "7 fourths". Then I said, "What is 5 thousand plus 2 thousand?" He answered, "7 thousand." He could start to see that whether it was x-rays, cost of x-rays, fourths, or thousands, since we are combining "like" objects/amounts, we are able to look at them in this way. We considered it a step further. What if we used the idea of factoring: We've previously learned that that we could factor x out of each term so that 5x+2x=x(5+2). Then using the order of operations x(5+2)=x(7)=7x. These explanations helped Aiden to conclude on his own WHY 2(3x)=6x and 5x+2x=7x.
**I told Aiden today that his questions are making me a better teacher. I am enjoying his approach to learning. :)