Flexible Thinking Young Mathematicians

Thinking flexibly about numbers is one of our goals for students in third grade. Throughout our MI Unit 3, Addition, Subtraction, and the Number System, we’ve been highlighting multiple strategies in Closing Session. We do this because subtraction is the distance between two numbers, and based on the problem, the most efficient strategy isn't always the same strategy. 

In addition, we want students to be able to check their work using a different strategy than the one they used to originally solve the problem. Many times, if a student checks their work with the same strategy, it’s common for them to make the same computational error they did the first time. However, if they make an error in the first solution and then check their work with a second strategy, they are more likely to catch their error.

We are working toward students' ability to recognize, based on the situation, the most efficient strategy with the least likelihood of error, and with the idea that mental math can be one of the most effective ways to solve. For example, we don’t want students to use the traditional algorithm to solve 1000-989, because it would be easy for them to make a computational error when regrouping multiple times. Rather, we want students to recognize that the distance between these two numbers can easily be done by counting up, 989+1=990 and 990+10=1000, therefore the difference is 11.  Of course, in other situations, it’s simply easier to solve using the traditional algorithm like 876-563.  Flexible thinking based on the situation is key.

In order to develop number flexibility, we’ve been working on several strategies in class.

Sample Problem:  245 - 178 = m

Adding up

Turn the equation into a missing addend 178 + m = 67. Put the number 178 on a number line and count up to the next landmark number. (Landmark numbers have a O or 5 in the ones place.) 178 count up 2 to 180, count up 20 to 200, and then jump 45 from 200 to 245. Adding the jumps gives you the answer, m = 67.

Decomposing

Decompose the number by place value. This is also known as expanded notation. Then, subtract each place value. In this problem, 200-100 = 100, 40-70 = -30, 5-8 = -3, therefore 100-30-3= 67.  Sometimes, this strategy has you in negative numbers, but students know that 0 is the middle of the number system and can flexibly use negative numbers. Some students use this strategy and regroup from the larger place value. If they did that in this problem, they would take a group of 100 from 200 and put 140 in the tens place.

            245  :     200  +   40  +   5

          -178  :    -100  +   70  +   8

                            100   - 30  - 3 = 67

Counting backward

245 - 178 = m                                                                    

Put 245 on the open number line and count backward by 178. You can make the jump of 178 any way you want. Most kids jump backward to landmark numbers. 245 jump back 45 is 100, and then jump back 30 is 70, then jump back 3 is 67.  Left to Right

Students think 200 – 100 = 100 and 40 – 70 = -30 and 5 – 8 = -3.

            245  :    

          -178  :   

            100 – 30 – 3

                70 – 3 = 67

Furthermore, in some situations, we also encourage students to compensate. For example,

  56   + 3        59

-47     -3       -44

                        15

Remember, the purpose of exposing students to multiple strategies is two-fold. First, students need to be able to solve using two different strategies to check their work, and secondly students will be able to identify the strategy that is most efficient based on the problem. Students who successfully accomplish this have number sense and are able to work with numbers mentally and flexibly. Our students are busy every day becoming young mathematicians.