Even though she is not taking precalculus until next semester, Jay came to me for help this week. She has set a goal of earning a 90 or above in all her classes in her last semester of high school. We both know she has gaps in understanding, relies too much on the calculator, and mixes up formulas. Rather than revisiting difficult topics while under the pressure of mastering new material, she is going to build a solid base now when she can relax and learn.
I asked Jay what concept she would like to study first. "Negative numbers! I know what to do when you multiply and divide. I get confused about what happens when you add a negative and a positive. I never know what the sign should be."
I drew a number line, and we did some simple problems. I said, "Explain to me what happens when you add two positive numbers like +5 and +3." I illustrated the problem on the number line.
She told me, "When you add two positives, you head in a positive direction. They stay positive."
We did the same thing for adding two negative numbers like -5 and -3. Jay gave a similar explanation of heading in a negative direction and staying positive.
Then, we worked on the piece of the puzzle that had long mystified Jay. I plotted adding +5 and -3. Jay looked at it for a moment and said, "Oh, I see. The negative number is not large enough to cross zero. So, the answer is positive."
We then studied what would yield a negative answer: adding +3 and -5. Jay smiled, "I get it! Since 5 is larger than 3, it is going to cross zero and the answer is negative! Wow!!"
We continued pursuing this line thought by adding -5 and +3 and then -3 and +5. I gave her a couple of problems to make sure she could apply what she understood.
I like to give students other ways to understand a problem. Jacob's Algebra offered an alternative view of adding positive and negative numbers. The book depicts a number as a set of circles. A positive number has circles with no filling (white), and a negative number has filled-in circles. The picture below shows two numbers: +6 is the first row of circles and -6 is the second row. I drew a picture like this for Jay, and she saw immediately how adding them together results in an answer of 0.
I drew another picture with -2 in the first row of circles and +2 in the second row. Again, she saw immediately how adding them together results in an answer of 0.
The picture below illustrates -5 plus +16. Jay had no trouble explaining that the answer had to be positive since the number of positive circles is greater than the number of negative circles. Since there are 11 circles left, the answer must be 11.
The picture below illustrates +3 plus -8. Jay easily explained that the answer had to be negative since the number of negative circles is greater than the number of positive circles. Since there are 5 circles left, the answer must be -5.
I assigned several addition problems for her to solve. Jay got them all correct. Even better, she smiled the whole time because she didn't have to guess the sign of the resulting sum. Then I picked a more difficult problem from the book:
1 + -3 + 5 + -7 + 9 + -11
She drew a picture like the one below and said the answer had to be -6.
Since Jay needs practice with mental math, I processed the problem aloud in a different way for her, "Do you see that 1 plus 5 plus 9 is 6 plus 9, and that is 15?" Then, I wrote down the number 15. I said, "And, -3 plus -7 plus -11 is -10 plus -11 and that is 21." I wrote "+ -21 = -6" to finish the solution. To practice both ways of adding positive and negative numbers, I assigned the next problem in the book.
-1 + 3 + -5 + 7 + -9 + 11
She drew the following picture and told me the answer was 6. Then, she did some mental math and wrote down the equation, "21 + -15 = 6."
Then, she paused and said, "That's funny. You get the same answer only the signs are reversed. Oh, wait! All the signs are opposite. Had I thought about it, I wouldn't have had to do all that work. That's cool." She warmed my heart when she said, "I want to keep this paper! It finally makes sense!"
Before closing, I will summarize some of the important ideas that help people who may not care for math to experience joy.
- Figure out what the student knows. I usually do this by asking them to explain things to me. When they get, I ask a question about a more foundational concept until I find the student on solid ground.
- Take small steps of logic by demonstrating the process with objects or in pictures. Too many students think in memorized procedures and formulas but cannot explain the rationale behind them. Once they understand why, procedures and formulas become effortless.
- Ask the student to explain the rationale behind what they are doing. If they cannot explain it, then they lack a solid foundation.
- Show how to solve a problem in different ways. It helps students see how math can be a creative process.
- If a student struggles with mental math, assign problems with simple numbers and avoid using the calculator. I find that students who rely on calculators for simple problems usually lack number sense.