Investigating systematic changes in formulas

For C = pn + f, let p = 2, n = 3 and f = 4. What is C?

Separate inputs, output and the model assumption

A fictional kit order uses this supplied pricing rule

Here n is the whole-number quantity of kits, p is the price in dollars per kit, f is one fixed fee in dollars, and C is the total cost in dollars

The letters p, n and f are inputs that can be chosen for an investigation. C is recalculated from those inputs

The model assumes one fixed fee per order, with no discounts, taxes or other charges. It is not an actual supplier offer

Unlike a recorded-data graph, these outputs are consequences of a stated formula. Conclusions about the formula do not establish a real supplier policy

Change one input and hold the others fixed

To investigate the effect of quantity n, choose a unit price p and fee f and leave them unchanged

Use an organised list of n-values, calculate each cost, then compare the outputs

For a separate investigation of p, keep n and f fixed instead. For an investigation of f, keep n and p fixed

Record which values are fixed. If two inputs change at once, their separate effects cannot usually be read from that comparison alone

You want to investigate what happens when the unit price p changes in C = pn + f. Which inputs must stay fixed?

Increase quantity by one at a fixed price

A fictional kit-order rule gives total cost C in dollars for quantity n, unit price p dollars per kit and one fixed fee f dollars

All listed quantities are whole numbers of kits; the fee is added once, not once per kit

Only n varies. Keep p = 3 and f = 2 fixed throughout this table

Complete every cost in the table before comparing the settings

Use C = pn + f for every row, multiplying quantity by unit price before adding the one fixed fee

For n = 1, p = 3, f = 2

For n = 2, p = 3, f = 2

For n = 3, p = 3, f = 2

For n = 4, p = 3, f = 2

Find the change in total cost between the first two rows. This is the change for each equal n-step in this table

The first selected setting has total cost 5 dollars

The second selected setting has total cost 8 dollars

Signed change equals second cost minus first cost

The fixed unit price p multiplies the change in n

Match the output change to the actual input step

At a fixed unit price of three dollars, one extra kit adds three dollars to the total. The fee does not get charged again

The symbol Δ means change in the variable that follows it

Two extra kits add twice the unit price

Read the step between the table inputs. Do not assume it is always one

Use a two-kit input step

Increase the unit price for a fixed quantity

A fictional kit-order rule gives total cost C in dollars for quantity n, unit price p dollars per kit and one fixed fee f dollars

All listed quantities are whole numbers of kits; the fee is added once, not once per kit

Only p varies. Keep n = 4 and f = 3 fixed throughout this table

Complete every cost in the table before comparing the settings

Use C = pn + f for every row, multiplying quantity by unit price before adding the one fixed fee

For n = 4, p = 1, f = 3

For n = 4, p = 2, f = 3

For n = 4, p = 3, f = 3

For n = 4, p = 4, f = 3

Find the change in total cost between the first two rows. This is the change for each equal p-step in this table

The first selected setting has total cost 7 dollars

The second selected setting has total cost 11 dollars

Signed change equals second cost minus first cost

The fixed quantity n multiplies the change in p

A price change applies to every kit

If the quantity is fixed at four kits, a one-dollar increase in unit price adds one dollar for each of the four kits

The increase in total cost is not just one dollar because the price is per kit

The fixed fee stays unchanged in this comparison

Vary the unit price in two-dollar steps

Vary only the one fixed fee

A fictional kit-order rule gives total cost C in dollars for quantity n, unit price p dollars per kit and one fixed fee f dollars

All listed quantities are whole numbers of kits; the fee is added once, not once per kit

Only f varies. Keep n = 5 and p = 2 fixed throughout this table

Complete every cost in the table before comparing the settings

Use C = pn + f for every row, multiplying quantity by unit price before adding the one fixed fee

For n = 5, p = 2, f = 0

For n = 5, p = 2, f = 1

For n = 5, p = 2, f = 2

For n = 5, p = 2, f = 3

Find the change in total cost between the first two rows. This is the change for each equal f-step in this table

The first selected setting has total cost 10 dollars

The second selected setting has total cost 11 dollars

Signed change equals second cost minus first cost

The added fee changes the total by the same amount because pn stays fixed

An added fee is not a price per item

With p and n fixed, their product pn stays unchanged

Adding two dollars to f adds two dollars to the total, regardless of how many kits there are

Multiplying the fee increase by n would change the stated pricing rule

Compare equal fee increases

Change one control at a time while the selected fee stays at three dollars. The selected row shows your quantity, price and fee, and each comparison row increases just one of them by one. Predict the change first, then compare the recalculated totals

In the f + 1 row, how does the total compare with the selected row for every allowed setting?

Calculate a before-and-after change

When the input moves by more than one step, calculate the two complete outputs using the same formula

Then explain the change using the input that varied and the inputs that stayed fixed

A lower second cost gives a negative signed change. The costs themselves can both remain positive

Calculate both costs when quantity increases

A fictional kit-order rule gives total cost C in dollars for quantity n, unit price p dollars per kit and one fixed fee f dollars

All listed quantities are whole numbers of kits; the fee is added once, not once per kit

Only n varies. Keep p = 4 and f = 2 fixed throughout this table

Complete every cost in the table before comparing the settings

Use C = pn + f for every row, multiplying quantity by unit price before adding the one fixed fee

For n = 3, p = 4, f = 2

For n = 6, p = 4, f = 2

Calculate the signed change in total cost when n changes from 3 to 6. Subtract the first cost from the second cost

The first selected setting has total cost 14 dollars

The second selected setting has total cost 26 dollars

Signed change equals second cost minus first cost

The fixed unit price p multiplies the change in n

Keep the sign when quantity decreases

Double the unit price without doubling the fee

A fictional kit-order rule gives total cost C in dollars for quantity n, unit price p dollars per kit and one fixed fee f dollars

All listed quantities are whole numbers of kits; the fee is added once, not once per kit

Only p varies. Keep n = 4 and f = 2 fixed throughout this table

Complete every cost in the table before comparing the settings

Use C = pn + f for every row, multiplying quantity by unit price before adding the one fixed fee

For n = 4, p = 3, f = 2

For n = 4, p = 6, f = 2

Calculate the signed change in total cost when p changes from 3 to 6. Subtract the first cost from the second cost

The first selected setting has total cost 14 dollars

The second selected setting has total cost 26 dollars

Signed change equals second cost minus first cost

The fixed quantity n multiplies the change in p

Calculate the actual change when the price doubles

Doubling one input need not double the entire output

For four kits at three dollars each with a two-dollar fee, the total is fourteen dollars

Doubling only the unit price to six dollars gives

Twice the old total would be twenty-eight dollars, which would also double the fee. That is not the controlled change we made

If the fee is zero, doubling p while keeping n fixed does double C. A claim must state its conditions

For C = pn + f with a positive fixed fee, what happens if only p doubles?

Check the special case with no fixed fee

Reduce the fixed fee while preserving the product

A fictional kit-order rule gives total cost C in dollars for quantity n, unit price p dollars per kit and one fixed fee f dollars

All listed quantities are whole numbers of kits; the fee is added once, not once per kit

Only f varies. Keep n = 6 and p = 2 fixed throughout this table

Complete every cost in the table before comparing the settings

Use C = pn + f for every row, multiplying quantity by unit price before adding the one fixed fee

For n = 6, p = 2, f = 5

For n = 6, p = 2, f = 1

Calculate the signed change in total cost when f changes from 5 to 1. Subtract the first cost from the second cost

The first selected setting has total cost 17 dollars

The second selected setting has total cost 13 dollars

Signed change equals second cost minus first cost

The added fee changes the total by the same amount because pn stays fixed

Change the fee without charging it per kit

Organise combinations rather than changing inputs randomly

To investigate n and p together, choose a short list of quantities and a short list of prices, keeping f fixed

Use every chosen quantity with the first price, then repeat the same quantities with the second price

This organised grid of combinations lets you compare rows where only one input differs

It also lets you find different settings with equal outputs. Equal total costs do not imply that the quantities and prices are the same

Build all combinations and compare an equal-product pair

A fictional kit-order rule gives total cost C in dollars for quantity n, unit price p dollars per kit and one fixed fee f dollars

All listed quantities are whole numbers of kits; the fee is added once, not once per kit

The fee f stays at 4 dollars while the listed quantities and prices are combined systematically

Complete every cost in the table before comparing the settings

Use C = pn + f for every row, multiplying quantity by unit price before adding the one fixed fee

For n = 2, p = 3, f = 4

For n = 4, p = 3, f = 4

For n = 6, p = 3, f = 4

For n = 2, p = 6, f = 4

For n = 4, p = 6, f = 4

For n = 6, p = 6, f = 4

Compare n = 4, p = 3 with n = 2, p = 6, keeping f = 4. Calculate the second cost minus the first cost

The first selected setting has total cost 16 dollars

The second selected setting has total cost 16 dollars

Signed change equals second cost minus first cost

Doubling the unit price while halving the quantity preserves the product pn here; the unchanged fee then preserves the total

Calculate every combination before claiming a pattern

Compare the same product when the fee is zero

Implement the same rule in a spreadsheet

A spreadsheet can perform the same controlled calculation as the digital table

Put n in column A, p in column B, f in column C and the calculated total in column D

For row two, put this formula in D2

=A2*B2+C2

The asterisk means multiplication. The formula multiplies the quantity and price before adding the fee

Use a simple known row to check the formula. With A2 = 3, B2 = 2 and C2 = 4, D2 should give 10

For an n-investigation, vary the values in column A while repeating the same p and f on each row. Fill the formula down so that each output refers to its own row

Do not type the old output as a fixed number: it would not recalculate when an input changes

A2 contains n, B2 contains p and C2 contains f. Which formula correctly calculates C = pn + f in D2?

One trial uses n = 2, p = 3, f = 1. The next uses n = 4, p = 3, f = 5. Can the cost difference be attributed only to changing n?

Plan, vary, recalculate and explain

State the formula, meanings, units and assumptions before calculating

Choose which input varies and hold the other inputs fixed for a controlled comparison

Use an organised sequence or grid, calculate every output and compare the relevant settings

Check a digital formula against a simple hand calculation before trusting a table of results

Explain output changes using the operation affected by the input change, including the actual step size

Do not extend a doubling claim to an unchanged added term, and do not assign a two-input change to one input alone