LEARNER JOURNEY Taxi fare
Name the goalLet m be miles. The fare is a $4 start fee plus $3 per mile, totaling $25.
Choose the next moveTranslate as 4 + 3m = 25.
Carry out the mathematics3m = 21, so m = 7.
Check and interpretSeven miles gives 4 + 3(7) = 25 dollars.
Start here
Build the concept from the beginning
Starting point
No prior equation-solving skill is assumed. Arithmetic operations, variables, equality, and every new algebra decision are explained on this page.
Learning target
Translate a situation into a linear equation, solve it, and interpret the solution with units.
Evidence of mastery
Construct the reasoning path, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.
Essential terms
- equation
- A statement that two mathematical expressions have the same value.
- solution
- A value that makes an equation true when substituted for the variable.
- coefficient
- A numerical factor multiplying a variable.
- constant
- A number without a variable.
Look first, then name the mathematics
Build the visual meaning
- Starting amount
- Rate × x
- Total
- Solve
- Interpret units
Read the complete visual relationship as text
- Define x
- Initial amount
- Rate × x
- Total
- Interpret units
Definition in plain language
A linear model represents a starting amount plus or minus a constant rate times an unknown quantity.
Why this matters
The model connects algebraic operations to real quantities, so units and context determine whether a numerical solution makes sense.
Reasoning path
See the method as a sequence
Move from left to right. Each decision prepares the next one; on a phone, follow the numbered cards from top to bottom.
Define the unknown with units.
Translate each quantity and relationship.
Solve the resulting linear equation.
Check the value in the situation and state the units.
The same method in words
- Define the unknown with units.
- Translate each quantity and relationship.
- Solve the resulting linear equation.
- Check the value in the situation and state the units.
Interactive decision model
Build the reasoning, one decision at a time
Model a gym cost of $25 to join plus $12 per month for a $97 total.
Choose this step, then check the construction.
Choose this step, then check the construction.
Choose this step, then check the construction.
The reasoning path changes as each decision is selected.
Choose every step, then check the construction.
Model before practice
Three fully explained examples
Every example identifies the goal, explains the next move, carries out the algebra, and checks or interprets the result.
Consecutive integers
Name the goalLet n be the smaller integer; the next is n + 1. Their sum is 31.
Choose the next moveWrite n + (n + 1) = 31.
Carry out the mathematics2n + 1 = 31, so 2n = 30 and n = 15.
Check and interpretThe integers 15 and 16 sum to 31.
Rectangle perimeter
Name the goalLet width be w and length be w + 4. The perimeter is 40.
Choose the next moveWrite 2w + 2(w + 4) = 40.
Carry out the mathematics4w + 8 = 40, so w = 8.
Check and interpretThe length is 12; 2(8) + 2(12) = 40.
Practice with decreasing support
Guided practice
Practice immediately after the model
Predict first. Then use the visual relationship and reasoned feedback to check two decisions.
1. A taxi charges $4 plus $3 per mile. The fare is $25. How many miles?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
2. A gym charges $20 to join and $15 per month. The total is $95. How many months?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Partially completed problem
Finish the missing step
A prompt supplies part of the reasoning. Predict the missing mathematical move before choices appear.
1. A number increased by 9 is 23. What is the number?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Independent practice
Six problems for repetition
Solve all six. Use the specific feedback to revise a method or sign error.
1. Three times a number minus 4 is 20. What is the number?
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Write a prediction before viewing the choices.
2. A 42-inch ribbon is cut into two pieces. One is 8 inches longer. What is the shorter piece?
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3. A phone plan costs $30 plus $0.10 per text. The bill is $42. How many texts?
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Write a prediction before viewing the choices.
4. The perimeter of a rectangle is 50 cm. Its length is 3 cm more than its width. What is the width?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
5. After a $12 discount, a jacket costs $48. What was the original price?
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6. A tank contains 18 L and fills at 4 L per minute. When will it contain 50 L?
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Not completed yet.
Mastery and transfer
Mastery check
Demonstrate understanding
Attempt all five. At least four correct answers are required for mastery.
1. The sum of two consecutive integers is 41. What is the smaller integer?
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Write a prediction before viewing the choices.
2. A salesperson earns $300 plus $25 per sale. Earnings were $575. How many sales?
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Write a prediction before viewing the choices.
3. A temperature is 18°C and drops 2°C per hour. When will it be 6°C?
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Write a prediction before viewing the choices.
4. One angle is twice another. Their sum is 90°. What is the smaller angle?
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Write a prediction before viewing the choices.
5. A savings account starts with $150 and receives $40 weekly. When will it reach $430?
Your prediction is locked so the record preserves what you thought before recognition choices appeared.
Write a prediction before viewing the choices.
Not completed yet.
Unfamiliar transfer
Explain, compare, and revise
Create and solve a linear equation for a situation with a fixed starting amount and a constant rate.
Guided self-check
Judge the locked first attempt against every criterion. Each judgment gives revision guidance.
Choose Met or Needs revision.
Choose Met or Needs revision.
Choose Met or Needs revision.
Model reasoning
Example: A savings account starts with $80 and receives $25 each week. To reach $230, let w be weeks: 80 + 25w = 230. Then 25w = 150, so w = 6. Check: 80 + 25(6) = 230 dollars.
Write a first attempt before opening the self-check.
The record reveals exact answers and model reasoning only after the corresponding attempt or self-check has occurred.