Expressions using Letter-Numbers · Lesson 7 of 11
Pick Patterns and Reveal Relationships — Formula Detective
“Discover number-machine rules and build expressions by tracking what each quantity contributes.”
• Infer a simple number-machine rule and check it against all supplied inputs. • Explain input order and describe a rule in words before symbols. • Track an expression through successive operations. • Model disjoint customer groups and repeated day-night changes. • Count repeated travel and cycling contributions before writing a formula.
Discovering a Number-Machine Rule
A number machine takes inputs, follows the same instructions each time, and produces an output. Your task is to discover an instruction that explains the examples. Start by comparing the inputs and output in ordinary language, then use letters to express your proposed rule.
| First input a | Second input b | Output |
|---|---|---|
| 5 | 2 | 8 |
| 8 | 1 | 15 |
| 9 | 11 | 7 |
| 10 | 10 | 10 |
| 6 | 4 | 8 |
The first row suggests doubling 5 and subtracting 2: 10 − 2 = 8. That suggests “twice the first input minus the second”. Let a be the first input and b the second; the rule is 2a − b. A proposed rule must work with all the supplied rows, so test it rather than stopping after the first success.
Problem
Check the rule 2a − b against the input table.
- 1.For (5, 2), the rule gives 2 × 5 − 2 = 8. For (8, 1), it gives 16 − 1 = 15.
- 2.For (9, 11), it gives 18 − 11 = 7. For (10, 10), it gives 20 − 10 = 10.
- 3.For (6, 4), it gives 12 − 4 = 8. Every supplied row agrees with the proposed rule.
Input order matters. If the inputs 5 and 2 are reversed, 2a − b gives 2 × 2 − 5 = −1 rather than 8. Give letters distinct meanings even when two inputs happen to have the same value.
Problem
Machine A maps (5,2), (8,1), (9,11), (10,10) to 5, 7, 18, 18. Machine B maps (4,1), (6,0), (3,2) to 5, 1, 7. Suggest simple rules and use B for (10,3).
- 1.For A, adding the two inputs gives 7, 9, 20, 20; each is 2 more than the output. A fitting rule is a + b − 2.
- 2.For B, multiplying the inputs gives 4, 0, 6; each is 1 less than the output. A fitting rule is ab + 1, where ab means a × b.
- 3.For B with a = 10, b = 3: ab + 1 = 10 × 3 + 1 = 31.
A simple rule fitting all given examples is a useful discovery. A finite table can sometimes be fitted by other rules too. If the machine’s actual instructions are supplied, use them; when inferring from examples, state the rule you propose and verify every listed case.
Follow an Expression Through Operations
A machine can also start with an expression instead of a numerical input. Each new operation acts on the current whole expression. Writing the intermediate result prevents a later multiplier from being applied only to the letter term.
Problem
Starting from w + 2, find the results of these paths: +3 then ×4; −5 then ×3; −8 then −4; −4 then ×3.
- 1.Path 1: (w + 2) + 3 = w + 5; then 4(w + 5) = 4w + 20.
- 2.Path 2: (w + 2) − 5 = w − 3; then 3(w − 3) = 3w − 9.
- 3.Path 3: (w + 2) − 8 = w − 6; then (w − 6) − 4 = w − 10.
- 4.Path 4: (w + 2) − 4 = w − 2; then 3(w − 2) = 3w − 6. Every path uses the updated expression as its next input.
To create a machine of your own, choose a rule first, compute several examples, then invite someone to infer the rule. Include different kinds of inputs, such as zero or equal inputs, when they help distinguish competing suggestions. Creating and testing a problem develops the same pattern reasoning as solving one.
Count People and Changes Carefully
A flower seller has p customers who buy only champak, q who buy only marigold, and r who buy both. These are three separate groups of people. If every customer receives one flag, the total is p + q + r. Buying two types of flowers does not turn one person into two customers.
Problem
There are 12 champak-only customers, 9 marigold-only customers, and 4 customers buying both. How many flags are needed?
- 1.The three groups are separate, so no customer appears in two group counts.
- 2.Use p + q + r = 12 + 9 + 4.
- 3.The total is 25 flags. The expression p + q + 2r would count the four customers buying both twice.
For a snail climbing u centimetres in the day and slipping d centimetres in the night, one complete day-night cycle changes its position by u − d centimetres. Choose upward as the positive direction. Repeating exactly ten complete cycles gives ten copies of that change.
Problem
Find the change after ten complete cycles for u = 12, d = 5, and then for u = 4, d = 6.
- 1.For u = 12, d = 5, one cycle gains 12 − 5 = 7 cm, so ten cycles gain 70 cm.
- 2.For u = 4, d = 6, one cycle changes position by 4 − 6 = −2 cm, so ten cycles give −20 cm.
- 3.The negative result means 20 cm below the start in this repeated-change model. It is a signed change; the magnitude of the separation from the start is 20 cm.
If d > u, the model gives a downward net change per complete cycle. This alone does not prove the snail can never reach the top: it might reach it during a daytime climb before slipping. The ten-cycle expression assumes the described cycles actually continue, with no earlier exit or boundary stopping the motion.
Repeated Contributions over Weeks and Journeys
A cyclist rides 5 km daily in week 1 and increases the daily distance by z km each week. Week 2 therefore has 5 + z km each day and week 3 has 5 + 2z km each day. Include seven days in every week before combining the totals.
Problem
Write Radha’s total distance over three weeks and evaluate it for z = 2.
- 1.Weekly distances are 7 × 5, 7(5 + z), and 7(5 + 2z).
- 2.Add and expand: 35 + 35 + 7z + 35 + 14z = 105 + 21z km.
- 3.For z = 2, the total is 105 + 42 = 147 km. Directly, the weekly totals are 35, 49, and 63 km.
A train that stops at three intermediate stations has four travel legs: start to stop 1, stop 1 to stop 2, stop 2 to stop 3, and stop 3 to the destination. If every leg takes t minutes and each intermediate wait takes 2 minutes, total time is 4t + 6 minutes. For t = 4, this gives 16 + 6 = 22 minutes. Counting only three travel legs would miss the final journey to the destination.
Check Your Understanding
Use the ideas from this lesson to choose an answer. Explain your choice to yourself before opening the explanations below.
Quiz
A machine doubles the first input a and subtracts the second input b. What is its rule?
A machine follows ab + 1. What is the output for a = 6, b = 0?
Customers are p champak-only, q marigold-only, and r buying both. One flag is given per customer. How many flags?
A train has three intermediate stops, each with a 2-minute wait, and every travel leg takes t minutes. What is its total time?
Which expression gives the total cycling distance over the three weeks described?
Double only a, then subtract b once.
Practice Problems
- Use the number-machine rule 2a − b for inputs (6,4) and (4,6).
- Find outputs of a + b − 2 for (5,2), (8,1), (9,11), (10,10).
- Find outputs of ab + 1 for (4,1), (6,0), (3,2), (10,3).
- Create a two-input rule, write three input-output examples, and explain how you checked them.
- Fill intermediate and final expressions for w + 2 followed by −8 then −4.
- What is the snail’s signed change after ten full cycles if u = d? What if d > u?
- Find the cycling total when z = 3 km.
- Find the train time for t = 7 minutes and explain the number 6 in the formula.
- Describe a situation for 15x − 2x and simplify it.
12 − 4 = 8 and 8 − 6 = 2. Reversing inputs changes this machine’s output.
Key Takeaways
• Describe a rule in words before writing letters. • Check a proposed machine rule against every supplied example. • Keep the meanings and order of the inputs clear. • Each operation acts on the current whole expression. • Count people, intervals, and repetitions before assigning coefficients. • A model’s conclusions depend on the conditions under which it continues to apply.