The students in one of my SAT classes recently worked on the following Level 3 math problem:
When the positive integer H is divided by 5, the remainder is 3. When the positive integer K is divided by 5, the remainder is 4. What is the remainder when H + K is divided by 5?
Several students read the problem and then blurted out, “What is a remainder?” Still others used their calculator and confidently announced that the correct answer was .4. My students’ difficulty with this problem did not surprise me. Over the years I have learned that today’s students do not understand the term “remainder.” What was a very easy problem for the pre-calculator generation is a challenging problem for today’s calculator generation.
Let’s begin by explaining how to calculate the values of H and K. One simple approach is to add the divisor 5 and the remainder 3. This will give you 8, the lowest possible value of H. If you repeat this process, you will see that 9 is the lowest possible value of K.
The next step is deceptively simple. The students who used their calculators divided 17 by 5 and got 3.4. They therefore concluded that the remainder was .4. WRONG! As we have learned, the calculator is a powerful tool but it is not invincible. Good old-fashioned long division tells us that 5 will go into 17 three times with a remainder (ie what’s left over) of 2. So the correct answer is 2.
As I have previously noted, there are usually several ways to solve an SAT math problem. Nick stunned the class by correctly answering this remainder problem in less than 5 seconds. He simply added the two given remainders (3 + 4 = 7) and then divided by 5. This immediately gave Nick the correct answer of 2!