Adding Integers

Adding Integers exercise

• Determine the correct value of each arrow on the number line.

  Hints

  There are two important things to consider:

  • First, focus on the length of the arrow.
  • Then, take a look at the direction in which the arrow is pointing.

  There are two arrows of the same length, but whose directions differ.

  Solution

  As we know, the number line consists of a positive side and a negative side. The positive and negative numbers are separated by zero on the number line. We can display arrows on the number line that differ in both length (number of units) and orientation (if they are positive or negative).

  When we examine the different arrows, we can see that two arrows can have the same length. The only difference is their orientation. One points to the left while the other points to the right, depending on whether the number is positive or negative. Additionally, it doesn't matter where on the number line we find the arrow.

  Let's take a closer look at the arrows in the picture from top to bottom. We have three different arrows:

  • The first arrow differs in its orientation from the other two. It's pointing to the right, in the positive direction, with a length of $3$ units. The correct value for this arrow is $+3$.
  • The next arrow is a little shorter. Its length is $2$, but since it's pointing to the left, in the negative direction, the correct value is $-2$.
  • The last arrow has a length of $3$ units. It's also pointing to the left, also in the negative direction, so the correct value is $-3$.

• Explain why Negatron's power up decides the game.

  Hints

  Moving left on the number line indicates a negative number.

  Moving right on the number line indicates a positive number.

  Solution

  What happens when we add a positive and a negative number with the same absolute value? (Remember, absolute value is the value of a number without the sign.) To answer this question, we can imagine moving along the number line. Let's try this with $+3$ and $-3$:

  • $+3$ means moving three steps to the right on the number line.
  • $-3$ means moving three steps to the left on the number line.

  Both numbers of steps can be illustrated using arrows. Adding equal and opposite arrows equals $0$. We can remember that adding opposite numbers always equals zero.

  A power up of $-2$ means moving two steps further to the left on the number line. Since we have already moved three steps left, the power up can be represented by the expression $-3 + (-2)$. In this example, power up means we have to add two numbers. Evaluating this expression gives us $-3-2=-5$. Now we can confidently say that adding two negative numbers equals a negative sum. When adding a positive and a negative number together, the sign of the number with the greater absolute value wins.

• Assign the value of the arrows to expressions using addition of intergers.

  Hints

  The lower arrow represents the left number while the upper arrow represents the right number.

  Take a closer look at the algebraic sign of each number.

  Adding two negative numbers equals a negative sum.

  Solution

  The arrows on the number line represent both a calculation and the result. Let's take a look at this example. The upper arrow goes from $-3$ to $-4$. Its length is $1$ unit and it is negative because the arrow is pointing to the left. We can see that the lower arrow goes from $0$ to $-3$. So it has a length of $3$ units. Because the arrow points to the left, it also represents a negative number. This represents the mathematical expression $-3+(-1)$ which can be simplified to $-3-1$. Evaluating this expression gives us $-4$.

  These are the other expressions represented in the pictures from left to right:

  • The second picture represents $-1+(-2)=-1-2=-3$.
  • The third picture represents $+2+(-3)=+2-3=-1$.
  • The fourth picture represents $-2+(+1)=-2+1=-1$.

• Evaluate the expressions by adding the integers.

  Hints

  Adding opposite numbers equals zero:

  • $-1+(+1)=0$
  • $+5+(-5)=0$

  When adding a positive and a negative number, the sign of the number with the greater absolute value 'wins.' Absolute value is the value of a number without the sign.

  Solution

  We can simplify all mathematical expressions where negative and positive numbers are added into a single number. Let's take a closer look at $+7+(-3)$. We can imagine moving seven steps to the right from $0$ and then moving three steps to the left. Doing so leaves us with $+4$, or simply $4$.

  Here are the results of the other expressions from left to right:

  • $-3+(-4)=-3-4=-7$
  • $-2+(+2)=-2+2=0$
  • $+5+(-2)=+5-2=3$

• Find the missing numbers on the number line.

  Hints

  You can move to the left or to the right of a known number.

  Moving left means adding a negative number, while moving right means adding a positive number.

  Solution

  There are some numbers missing on our number line. But that's no problem since we can determine them by moving to the left or to the right.

  There are two known numbers: $-3$ and $6$. From one of these numbers, we can move left or right to find out which number belongs in each blank. It makes sense to start with a number we already know.

  From $-3$ we can go three steps to the right to reach one blank. Moving three steps to the right means adding $+3$. So that gives us $-3+(+3)=-3+3=0$. In the same way, we can move one step to the left from $6$ to reach another blank. This is represented by the expression $6+(-1)=6-1=5$. So the unknown number was $5$.

  The other missing numbers were $-6$, $-2$ and $3$.

• Determine the value of these expressions by adding the intergers.

  Hints

  When adding a positive and a negative number, the sign of the number with the greater absolute value 'wins.' Absolute value is the value of a number without the sign.

  Calculate from left to right.

  Imagine moving along the number line. Adding a negative number means moving left, while adding a positive number means moving right.

  Solution

  We had to calculate each expression before we could arrange them in order. For each, imagine walking on the number line: adding a negative number means walking left and adding a positive number means walking right. We could do that for the expression $+2+(-5)+(-1)$: starting at zero, we walk two steps to the right, reaching $2$. Then, we walk $5$ steps left getting to $-3$. Finally, we come to $-4$ after going $1$ final step to the left.

  These are the other results arranged in the right order:

  • $-3+(+2)+(-1)=-2$
  • $+6+(-3)+(-3)=0$.
  • $-7 + (+8)+(+2)=3$

  In general, we can remember that adding two negative numbers equals a negative sum. When adding a positive and a negative number, the sign of the number with the greater absolute value defines the sign of the result.