July 2, 2013 at 9:48 pm #2090
We are trying to make sense of integer subtraction on the number line. In the case of –6 – -4 is –2 the only explanations that work start by stating that the direction between the second number (-4) to the first number (-6) is in a negative direction resulting in a negative answer whereas, 7-5, distance from second (5) to first (7) is in a positive direction, resulting in a positive answer. This seems to be challenging in the sense that students are not used to starting with the second number when subtracting. Is this the only way to explain these cases using a number line? Also, what about using the number line for multiplication when a negative factor comes first. How do you show -3(7) or negative 3 groups of positive 7 without switching the problem to be 7(-3) or 7 groups of negative 3. Lastly, negative integers on the number line with division: how do we show 8 divided by negative four? “How many -4’s are in 8?” doesn’t make sense on the number line either. Do we have to skip these cases when teaching these topics? Thank you!July 3, 2013 at 10:04 pm #2099
Great questions, I wonder if these would help?
For subtraction, I always tell my students it’s a loss of a debt. In terms of the numberline, I ask students to start with the first number, -6 and subtract -4. If they move toward the left I say, that looks like you’re subtracting a positive four. Negative 4 is the opposite of positive 4 so you need to move to the right.
For multiplication, I think in terms of football yards. If a team loses 3 yards 7 times, they’ve lost a total of 21 yards.
Division…I just tell them division doesn’t exist. It’s multiplying by the reciprocal. so 8 / -4 is 8 * -1/4. same rules apply as multiplication…July 5, 2013 at 6:31 am #2110
There are two big steps in students’ understanding of number in Grade 6. The first is the unification of whole numbers, fractions, decimals, and negative numbers into a single number system as represented by the number line. But the number line doesn’t do everything you need. The other big step is a systematic use of properties of operations to understand how operations can be extended to include negative numbers.
For example, the relation between addition and subtraction helps in understanding subtraction with negative numbers. In earlier grades, students understand $6-4$ as the number you add to $4$ in order to get $6$, that is, the missing addend in the equation $4 + ? = 6$. In Grade 6 they understand $(-6) – (-4)$ as the number you need to add to $-4$ in order to get $-6$, the missing addend in $(-4) + ? = -6$. Since $-6$ is two units to the left of $-4$ on the number line, the missing addend is $-2$. So $(-6) – (-4) = -2$.
By the same token, the relation between multiplication and division helps with division of negative numbers. So $8\div (-4)$ is the missing factor in the equation $? \times (-4) = 8$.
In general in Grade 6 there is a consolidation of operational understanding of rational numbers, and a move away from concrete models, although concrete models like the ones suggested by molleyk are still useful.