Add & Subtract
Rational Numbers
Numbers can describe both WHERE something is and HOW it changes.
Yesterday, numbers belonged to families. Today, those rational numbers are going to MOVE.
What are we learning?
We are learning to add and subtract rational numbers and use rational numbers to describe position, change, and distance.
Success Criteria
By the exit ticket, you should be able to explain the sign before you calculate.
Get your brain moving
Which is greater?
−4 or 3
Order least to greatest
−5, 0, 2, −1
A diver is 6 meters below sea level.
Write the signed number.
Which of these are rational?
3/4 −8 0.5 π
Every number we will operate with today is rational. The sign tells us position or direction.
Numbers can tell us WHERE
Read the sign as location
+3.5 mi → 3.5 miles above sea level
Zero is the reference point.
−2.75 mi → 2.75 miles below sea level
Negative does not mean “bad.” It tells location or direction relative to zero.
Read the position
What signed number describes this position?
1. Move the marker
Students respond before the value is revealed.
If students cannot reliably read position here, do not begin signed addition yet.
Numbers can also tell us HOW WE MOVE
Example
Start at −2. The +5 is not another location. It tells us the change.
The second number in an addition problem can describe a change.
What do the signs tell us?
−3 + (−4)
Both movements point negative.
SAME DIRECTION: the result moves farther below 0.
−7 + 4
The movements oppose each other.
OPPOSITE DIRECTIONS: the result moves back toward 0.
Now that we understand what the arrows are doing, we can write a shortcut rule.
Like Signs
Definition
The numbers have the same sign.
What to do
1. Add the absolute values.
2. Keep the common sign.
Like signs: add and keep the sign.
Example
Same sign = same direction
Definition
The numbers have the same sign.
Add the absolute values.
Keep the common sign.
Both changes move in the negative direction.
Opposite Signs
What to do
1. Find the absolute values.
2. Subtract the smaller absolute value from the larger.
3. Keep the sign of the number with the greater absolute value.
Opposite signs: subtract. Keep the sign of the number farther from 0.
Choose the sign from the greater absolute value, not from the digit that merely looks larger.
9 − 4 = 5
Answer: −5
Why negative? −9 is farther from zero than +4.
Opposite directions compete
−9 + 4
Compare the distances from zero: 9 and 4.
9 − 4 = 5
Because −9 is farther from zero, the result keeps the negative sign.
−5
How do I decide what to do?
Keep the sign.
Keep the sign of the number farther from 0.
Quick checks
Tap a problem. Follow its path before solving.
Students should begin saying the rule automatically when they see an addition problem.
Where is Point B?
Point A is at −40½ ft.
The trail rises 120½ ft.
Step 1: Opposite signs.
Step 2: 120½ − 40½ = 80.
Step 3: +120½ has greater absolute value.
Point B = +80 ft
Ask before arithmetic: Is the hiker starting above or below sea level? Does RISE mean positive or negative change? Should the answer move toward or away from sea level?
Can you predict the new position?
Start −6 m; rise 9 m.
Reconnect before we subtract
Add.
−6 + (−3)
Add.
−8 + 11
A diver starts at +5 m and dives 12 m.
What is the new position?
Look carefully. Do not solve yet.
6 − (−4)
Which symbol is the operation? Which sign belongs to the second number?
Yesterday's addition rules still matter. Today we learn how to rewrite subtraction so those same rules can do the work.
What are we learning today?
We are learning to subtract rational numbers and use rational-number operations to describe position, change, and distance.
Success Criteria
Keep the sign.
Keep the sign farther from 0.
Today, the key move is not memorizing a second set of sign rules. It is learning to rewrite subtraction correctly, then use the addition reasoning you already know.
What happens when forces act in opposite directions?
Expression
Opposite signs: compare 1.50 and 0.49.
1.50 − 0.49 = 1.01
The positive force has the greater magnitude, so the result is positive.
Addition is now familiar. The new problem today is what to do when the operation symbol itself is subtraction.
Subtraction creates a problem we must rewrite
Operation: addition
Operation: subtraction
Are these the same problem?
We need a way to turn Problem B into an addition problem before using our sign rules.
Two nearby minus signs do not mean “automatically positive.” First identify what each sign is doing.
Add the Opposite
Subtracting a number is the same as adding its opposite.
Second example
Language first → transformation → sign rule → solution.
Do not let the entire expression jump from start to finish. Change one mathematical idea at a time.
KEEP → CHANGE → CHANGE
KEEP
Keep the first number.
4
CHANGE
Change subtraction to addition.
− → +
CHANGE
Change the second number to its opposite.
−7 → +7
Rewrite first. Then solve.
7 − (−4)
−3.5 − 2.1
6¼ − 9½
Problem 1: Model.
Problem 2: We Do.
Problem 3: Students lead.
If students are making errors, identify whether the error occurs during REWRITE or during ADDITION. They are different misconceptions.
From below sea level to above sea level
Start: −2¾ miles
Change: +5¼ miles
Opposite signs.
5¼ − 2¾ = 2½
The positive value has the greater absolute value.
+2½ miles
The scenery earns its place here: students should see why crossing zero changes the final sign.
“Where did I end?” is not the same as “How far apart?”
A fish moves from −0.4 to +0.5.
Where did it end?
+0.5
How far apart are −0.4 and +0.5?
0.9
Distance cannot be negative.
To find how far apart two positions are, we need the absolute value of their difference.
Distance = absolute value of the difference
NET POSITION
FISH POSITION
distance = |first position − second position|
If the subtraction produces a negative difference, the absolute value makes the distance nonnegative.
What kind of problem is this?
One diver is at −12.5 m. Another is at −4 m.
How far apart are they?
A shark starts at −18 m and rises 6.75 m.
What is its new depth?
Now use the class assignment
Tap any tool to enlarge it.
Identify whether you are adding, subtracting, or finding distance. Rewrite subtraction. Then choose the sign rule. Explain the sign when the problem asks for position.
A subtraction sign is not automatically a negative-number sign. Opposite signs do not automatically make a negative answer. Fractions and decimals use the same sign rules as integers.
Show what you know
Work independently. Explain your reasoning.
−7.2 + 3.9
Explain why the answer is positive or negative.
4½ − 8¼
Show the Add the Opposite rewrite.
Point A is at −1.6.
Point B is at +0.7.
How far apart are they?
Choose the correct rule • rewrite subtraction correctly • determine the correct sign • find nonnegative distance.