Choose the
right operation.
The hard part is no longer just doing the arithmetic. It is deciding WHAT the problem is asking and choosing the operation that matches.
This week: we learned the four operations with rational numbers. Today: we decide when and how to use them together.
What are we learning?
We are learning to make sense of rational-number problems, choose efficient operations, and check whether our answers are reasonable.
Success Criteria
Operation words can help, but the relationship matters more than a keyword.
Four operations. Four decisions.
Start + change
−7 + 12.5
Distance between positions
from −4 to +3
Repeated change
−1.5 each hour for 6 hours
Change per unit
−8.4 over 4 days
The operation is determined by what is known and what is missing.
Choose the relationship — not a keyword
“Per” does not always mean divide. If the rate is already given per hour and you need several hours, you multiply.
Should Stefan rent the store?
| Type | Amount | Frequency |
|---|---|---|
| Sales | +$950 | each week |
| Services | +$2,875 | each month |
| Rent | −$4,500 | each month |
| Travel | −$7.50 | each day |
| Merchandise | −$1,650 | each month |
First problem: the frequencies do not match
Weekly, monthly, and daily amounts cannot be added yet. We need a common time period.
Predict before calculating
Do you expect the monthly result to be a profit or a loss?
Do not calculate every line immediately. First decide what must be converted and what signs the amounts should have.
Estimate first. Then make the decision.
Build the estimate
With this simple estimate, Stefan has a small +$300 monthly cushion. Mathematically, the estimate is positive — but it is small enough that a real business decision would need more information.
If your result said Stefan earns thousands each month, that would conflict with the rough estimate. Estimation helps catch bad operation or sign choices.
What is the boat’s change in position each minute?
What do we know?
Total change: −3 ft
Total time: 4 1/2 minutes
What are we finding?
The boat’s change in position in 1 minute.
Total change ÷ total minutes = change per 1 minute.
Represent the 4 1/2 minutes before calculating
Solve
How far did the weather balloon rise?
Ask the right question
We are finding distance risen, not final position.
Simply doing 19.5 + (−18) finds a signed combination, not the distance between the two positions.
Which relationship belongs to each situation?
A. Savings account
Start with $153, make withdrawals, then add a deposit.
B. Coffee cooling
Total change is −54°F over 22 1/2 minutes. Find change each minute.
C. Airplane temperature
Temperature drops 1.1°F per 1,000 ft. Find total change over 11,000 ft.
D. Plant growth
Growth is 1 1/6 in/week. How many weeks to grow 6 1/6 in?
Kevin’s trivia score: two correct methods
What is the structure?
There are 15 of each type. That repeated 15 makes the Distributive Property useful.
How are the two methods alike? How are they different?
Same mathematics. Different organization.
Multiply first, then add
Factor out the common 15
Why are both correct?
The Distributive Property says:
Same quantities. Same result. The second method is more efficient because the number of correct and incorrect answers is the same.
Rashida’s score
Rashida had:
The counts are not equal, so factoring out one common count is not as convenient as it was for Kevin.
Build the expression
Fraction or decimal — which representation helps?
Fractions
Useful when values are exact parts, simplify cleanly, or reciprocals are involved.
Decimals
Useful for money, measurements written as decimals, or calculator-ready rates.
Choose strategically
Changing representations is allowed when it makes the arithmetic easier. The goal is not to stay loyal to one format; the goal is to preserve the value and solve efficiently.
Temperature from 4:00 p.m. to 8:00 p.m.
Step 1 — Find total change
Rate is given per hour and we need 4 hours.
Step 2 — Apply the change
Start at 2.5°F and add the signed change.
A 3-degree drop from 2.5°F should end slightly below zero. −0.5°F makes sense.
What is the temperature at 3:00 p.m.?
Given
What must happen first?
Can you choose the mathematics before calculating?
A. Choose the primary operation
B. Predict the sign of the result
C. Solve
If the operation is wrong, do not reteach arithmetic yet. Revisit the relationship between the knowns and the unknown.
Mean win/loss margin across six games
Game results
What was the mean amount by which the team won or lost?
Across all six games, the signed margins total +12. Dividing by 6 gives an average margin of +2.
Show that you can choose, solve, and justify
Choose + solve
A hole is −8.25 ft after 3 hours of digging. What is the change in location each hour?
Multi-step
A bonus starts at $155. Twelve complaints each reduce it by $2.50. Find the final bonus.
Explain
How can estimation help you decide whether a rational-number answer is reasonable?
Independent practice — Lesson 1.6
#13: write two equivalent expressions. #16: STAAR mean win/loss margin.