Variables, expressions and everyday formulas

The letter n represents the number of people in a group. Which statement is correct?

Use operation order before replacing numbers with letters

Work inside brackets first, then evaluate powers such as a square

Next carry out multiplication and division from left to right. They have equal priority, so multiplication does not always come before division

Finally carry out addition and subtraction from left to right. They also have equal priority

Brackets can deliberately change the quantity on which an operation acts

These examples use the same numbers but describe different calculations. Keep the original operations and grouping when writing a formula

First

\(\displaystyle 6+2\times3^2\)

Second

\(\displaystyle (6+2)\times3^2\)

Which pair gives the values in the displayed order?

Calculation

\(\displaystyle 12\div3\times2\)

Evaluate from left to right at the multiplication-and-division level

Calculation

\(\displaystyle 15-6+2\)

Evaluate from left to right at the addition-and-subtraction level

Define what each letter measures or counts

A variable is a symbol representing a number whose value is not fixed in the relationship being considered

State what the letter means and include units when it represents a measurement

If each notebook costs three dollars, the cost changes when the notebook count changes

The number three is a constant in this model. The letters n and C can vary

Use the same value for every occurrence of the same letter within one calculation or situation

In a formula, l is defined as a length measured in centimetres. What does l represent?

An expression names a quantity; a formula relates quantities

An expression combines numbers, variables and operations without making an equality statement

A formula states a relationship between quantities using an equals sign

This formula says the value of C equals three times n plus four

A formula is an equation used to describe a relationship. It does not automatically ask you to solve for a particular numerical value

In the construction tasks, enter the complete formula, including the result letter and equals sign

First form

\(\displaystyle 5n+2\)

Second form

\(\displaystyle T=5n+2\)

Which statement correctly distinguishes the two forms shown?

Read multiplication without a multiplication sign

A number written directly before a letter means multiplication

Three is the numerical coefficient of n in this term

The coefficient of n written on its own is one

Letters next to each other also indicate multiplication, not addition

A power has a different meaning from a coefficient

Use a fraction bar for division when it makes the grouping clearer

Which statement describes the expression shown?

Translate an increase without fixing the variable

Seven more than x means start with x and add seven

If y names the result, write the complete relationship

The formula works for any allowed value of x. Do not replace x with a guessed number

Construct a formula for a greater quantity

Let y be 7 more than x. Write a complete formula for y in terms of x

Start with x

More than means add 7

Use y for the result and join the two equal quantities with an equals sign

The letters remain variables. No particular numerical value has been assigned to x

Keep the changing input as a letter

The word order can differ from subtraction order

Four less than n starts with n and subtracts four

The quantity being made smaller comes first in the subtraction

By contrast, n subtracted from four starts with four

These expressions usually have different values. Do not reverse the order merely to follow the order of the words

Translate less than

Let R be 4 less than n. Write a complete formula for R in terms of n

Start with n, the quantity being reduced

Subtract 4, keeping this order

Use R for the result and join the two equal quantities with an equals sign

The letters remain variables. No particular numerical value has been assigned to n

Subtract the stated amount from the variable

Identify the starting amount after from

Translate equal copies using a coefficient

Five copies of n have total five times n

If C names that total, the formula is

Five plus n would add five once instead of multiplying n by five

Construct an equal-group formula

Let C be 5 times n. Write a complete formula for C in terms of n

There are 5 copies of n

This can also be written as

with multiplication understood

Use C for the result and join the two equal quantities with an equals sign

The letters remain variables. No particular numerical value has been assigned to n

Enter multiplication with or without its sign

Keep the dividend above the divisor

If a total n is divided into four equal shares, the amount per share is n divided by four

Four divided by n is a different relationship

Use brackets with a typed slash whenever the complete numerator has more than one term

Construct an equal-share formula

Let R be n divided by 4. Write a complete formula for R in terms of n

The complete amount is n

Divide that amount into 4 equal parts

Use R for the result and join the two equal quantities with an equals sign

The letters remain variables. No particular numerical value has been assigned to n

Divide the changing amount by the constant

Brackets keep a whole quantity together

Three times the sum of x and four means first form the sum, then multiply the entire sum by three

Three times x, then add four, places the four outside the multiplication

The location of the brackets changes which quantity is multiplied

Construct a formula with a grouped sum

Let y be 3 times the whole sum of x and 4. Write a complete formula for y in terms of x

First form the entire sum

Keep that sum together in brackets and multiply all of it by 3

Use y for the result and join the two equal quantities with an equals sign

The letters remain variables. No particular numerical value has been assigned to x

Multiply the entire sum

Multiply first and add once afterwards

The table compares three times n plus four with three times the whole sum of n and four. Change n and notice which parts of the rules stay fixed

At n equal to two, which comparison is correct?

Define two input quantities and add their contributions

Suppose a counts two-dollar items and b counts five-dollar items. Let T be the total cost in dollars

The first contribution is two times a, and the second is five times b

Add the separate costs to form the total

The counts a and b do not have to be equal. Different letters can represent independent quantities

Keep each coefficient attached to its own count

Add a fixed charge once, not once per item

A fictional activity costs four dollars per participant plus one fixed seven-dollar equipment charge. Let n count participants and C be the total cost in dollars

The seven-dollar fee is added once after the per-person cost

Putting n plus seven inside brackets after the multiplier four would multiply the fixed fee too, which is not the stated situation

Construct the formula from the original charges

Use a formula for a measured share

Which expression means the sum of a number n and another copy of that same number?

Formula

\(\displaystyle C=3n+5\)

Here n counts participants and C is the cost in dollars. Which description matches the formula?

Preserve the quantities and the structure

Define each letter, including units or what is counted. A variable can change, while a stated constant stays fixed in the model

A coefficient multiplies a variable. An expression names a quantity; a formula uses equality to relate quantities

Read operation words carefully, especially less than and subtracted from

Use brackets when an operation applies to a whole sum, and keep a fixed added amount outside a product when the context requires it

Construct the complete formula with the requested result letter and equals sign. Do not guess a numerical value for an unassigned variable