Simplify the expression. Is the following work correct? If not, please correct it. Can you imagine why a person made this mistake?
2 – 8m + 2m + 6 + 8m = 8 – 18m
One possible reason for this mistake:
- The person does not know the sign in front of a term goes with the term.
Understanding Negative numbers and terms:
The use of negative numbers: we are in debt. we need to subtract a larger number from a smaller number.
Ask: why negative sign is the same as minus sign? Why positive sign is the same as plus sign? Why does “+” or “-” have to go with the number in front of them? What are we doing when we apply operations on those numbers?
Before we explain why 5 – (-2) = 5 + 2, let’s reach some agreements and facts as below.
- negative is the opposite of positive. vice versa.
- subtraction is the opposite of addition. vice versa
- to give away is opposite of to get
- exactly two opposite things get canceled out. The 2nd opposite reverse the 1st opposite = positive. It is like you say double no or why not, which means yes.
- -999 +999=0. You owed bank $999 last month. You earned and paid bank $999 back. So your balance is $0.
- 1+(-1) = 0. You have one apple, then you give this one apple to your sister. You have 0.
- 2.5 + (-2.5) = 0
Explanation One: Tile methods
Correction: The above question should be 5 – (-2) instead of -5 – (-2)
Multiplications with negative numbers
- A positive number Times a positive number: 5 x 2
2. A positive number Times a negative number: 5 x (-2)
3. A negative number Times a negative number: (- 5) x ( – 2 )
Step 1) (- 5) x ( – 2 ) = (-1) X 5 X (-2) = 5 x (-2) x (-1) — Commutative property
So we have 5 x (-2) = -10;
Step 2 ) Then we multiply -10 by -1 which means we change -10 to its opposite.
So we get (-10) x (-1) = 10
Therefore: (-5) x (-2) = 5 x 2 = 10. The product of two negative numbers is positive. negative negative = positive.
negatives as “opposite” and subtraction as “opposite of addition”, therefore “opposite of opposite of addition” = “addition”
Explanation Two: Walk Step Method
Walking steps: addition / subtraction = (face forward / face backward); positive / negative = regular steps / backwards steps
- imagine going on a walk, you are facing forward first, and take 5 steps forward. This is 0 + 8 = 8; (0 is your starting point. “+” means facing forward; “8” means 8 steps in the direct you are facing. )
- then keep facing forward and take 6 more step. That will be 8 + 6 = 14; this gives us 14 steps from our starting point.
- Now imagine in step 2 we had faced backwards and took 6 more step. 8 – 6 = 2 or 8 + (-6) = 4. We are close to our starting point.
- How about we had faced backwards but walked backwards 1 more step. 8 – (-6) = 8 + 6 = 14
Do you also see we actually always start from 0, and if you put 0 in the beginning of your equation, and put “+” sign to connect this “0” with the rest of the equation, you will see why a sign has to go with the number after it. It tells what we do from the starting point and by what extent we change. (tentative explanation)
J explains that negative and subtraction both mean “take away.” Particularly, -3 means you take away 3 from origin 0. Because we started with counting numbers, that is why we assume a number without “+/-” is positive. Therefore, 1) negative sign is meaningful and cannot be omitted. 2) as a negative number -3 has the negative sign in front of the number, we must associate the number with the sign in front of it. This will be important when we regroup terms in algebra.
Explanation Three: Number lines
- -3 + 5 = 2: Start at -3, move to your right by 5; you reach 2.
- -3 – 5 = -8; Start at -3, move to your left by 5; you reach -8
- -3 + (-5) = -8; Start at -3, move to your right by the opposite of 5 (= move to the left by 5) ; you reach -8.
- -3 – (-5) = 2; Start at -3, move to your left by the opposite of 5 (= move to the right by 5); you reach 2.
A mother’s words: If your child made a similar mistake before 6th grade, help him. At this stage, don’t say that your child might not be good at maths anyway. Instead, help him to develop some number sense little by little by using visual toys first, then drawing on the paper, then writing numbers on the paper. Further, if you can dig in to see why with your child, then maths will become more interesting.
Maths is very abstract for young children. No matter we like it or not, no matter what we do, we need to put various factors together as we grow up. Maths helps us train our brains to think complicated issues altogether, carry out complicated tasks step by step, and “divide and conquer” to succeed.