When should kids know their combinations?

I have had many parents ask me when their child should know their addition or multiplication facts. I would first like to say that Investigations prefers to call them combinations. Calling them facts implies that you can't learn them through reasoning. However, if I know 8 x 3 = 24, then I should be able to reason that 8 x 6 is double 8 x 3 or 48.

What does it mean to know your combinations? Fluency means that you can immediately recall an answer or quickly perform a calculation to get the answer. If it takes longer than two seconds to figure a combination, a student is not fluent. This includes single digit addition and multiplication pairs and their counterparts for subtraction and division. We encourage students to work on combinations that they don't immediately know. This can be done during downtime in class or at home. I also encourage 3rd, 4th, and 5th graders to use arrays when working on their combinations.

The following are guidelines for learning combinations through the grades:

Addition - fluent by the end of Grade 2, with review and practice in Grade 3.

Subtraction - fluent by the end of Grade 3, with review and practice in Grade 4.

Multiplication - fluent with multiplication combinations with products to 50 by the end of Grade 3; up to 12 x 12 by the middle of Grade 4, with continued review and practice.

Division - fluent by the end of Grade 5.

Adding by Place Value

How would you solve the problem 254 + 763?

Many of us would line the numbers up like this:

However, there are other ways to solve this problem. One way is to break the numbers down by place value.

• We can start with the hundreds place and solve 200 + 700 to get 900.
  
• Moving to the tens place, we can solve 50 + 60 to get 110. If you have trouble adding these two numbers, think of the problem as 50 + 50 + 10.

• Finally, move to the ones place and solve 4 + 3 to get 7.

• Now we have the numbers 900 + 110 + 7. Another way to think of this problem is 900 + 100 + 10 + 7 which is 1,017.
  

Why does this strategy make more sense than lining up the numbers like above? Let's talk our way through solving that problem. The first step is pretty easy. 4 + 3 = 7. Now have a student explain the second step. You will probably hear something like "5 + 6 is 11, so put down a 1 and carry the other 1." What does this mean? The actual step is adding 50 + 60 to get 110. So, you are putting 10 down and carrying the 100. This explanation get a little confusing for kids.

While lining up your numbers and following the steps will work, it doesn't give students any meaning about what they are really doing. Adding, or subtracting, by place value makes sense of the problem. As our students move through Investigations, they will become more efficient at solving problems like this and may even begin doing them in their head.