Use arithmetic laws to calculate and form expressions

Which calculation is always equal to seventeen plus eight?

Commutative means the order can change

Addition is commutative: reversing the order of two addends preserves their sum

Multiplication is also commutative: reversing the order of two factors preserves their product

These rules apply to the numbers represented by the letters, including the fractions and decimals already studied. They do not mean every operation is commutative

Product

\(\displaystyle 12\times0.5\)

Which calculation is guaranteed to have the same value as the product shown?

Associative means the grouping can change

For three addends, changing which pair is added first preserves the total

Associativity changes brackets. Commutativity changes order. They can be used together to make a convenient pair

The calculation uses every original addend once. The intermediate grouped sum is one hundred, and the final total is one hundred and eighteen

Make one hundred before adding the remaining amount

Regroup 37 with 63, then add 18. Enter the regrouped sum and the final sum

Addition permits changing order and grouping

The same three addends are used exactly once

Enter the useful grouped sum and final total

Regroup factors into a convenient product

Multiplication is associative as well as commutative

A convenient pair such as four and twenty-five can be multiplied first

Keep the same three factors. Changing the grouping does not mean changing multiplication into addition

Use a factor pair that makes one hundred

Regroup 4 with 25, then multiply by 7. Enter the regrouped product and the final product

Multiplication permits changing order and grouping

No factor is added, removed or changed

Construct the regrouped product and final product

Subtraction and division do not generally commute

Reversing subtraction usually changes its value

Reversing division usually changes its value

A particular coincidental equality does not make an operation commutative in general. Check the operation, not just whether the numbers look familiar

First

\(\displaystyle 10\div2\)

Second

\(\displaystyle 2\div10\)

Which statement correctly explains the two divisions shown?

Subtraction and division do not generally associate

Changing subtraction brackets can change which amount is subtracted

Changing division brackets can also change the quotient

Do not move brackets across a subtraction or division as though it were addition or multiplication

Calculation

\(\displaystyle 15-(8-3)\)

What is the value of the bracketed calculation shown?

The same laws can make rational calculations easier

Choose fractions that complete a whole before doing a longer common-denominator calculation

A fraction and a suitable factor can also make a whole-number product

The values and operations are preserved; only order or grouping changes

Distribute a multiplier to every term in a sum

Multiplying a whole sum is the same as multiplying each addend and adding the products

This is the distributive law. It can turn a product into easier place-value products

Both forty and seven are multiplied by six. Multiplying only the first term would change the value

Split one factor into tens and ones

Use 47 = 40 + 7. Enter both positive products, then add them

Replace the right factor by the stated equal quantity

Multiply both terms by the outside factor, keeping the addition

The outside factor applies to both terms. Keep the addition between the two products

Enter both partial products before the total

The inside quantities stay seven and three. Change the outside multiplier k and compare the two calculations

At k equal to four, which working applies the multiplier to the complete sum?

Distribution also preserves a difference

An outside positive factor multiplies both quantities in a difference, with subtraction kept between the products

A nearby multiple of ten can make this efficient

Subtract the whole second product, not just the small adjustment before multiplication

Multiply a nearby ten and subtract the adjustment

Use 49 = 50 - 1. Enter both positive products, then subtract them

Replace the right factor by the stated equal quantity

Multiply both terms by the outside factor, keeping the subtraction

The outside factor applies to both terms. Keep the subtraction between the two products

Keep the subtraction between the products

Use a common factor to reverse distribution

When both products have the same factor, combine their other multipliers before multiplying

This uses the distributive law in reverse. There are six groups and four groups of the same thirty-seven, making ten such groups

The shared factor must really be the same in both products

Combine equal-sized groups

Use 37 as the common factor. Enter the sum of its multipliers and the final total

Both products have the same factor

This is the distributive law used in reverse

Enter the combined multiplier and final total

A variable can remain inside the same law

The distributive law does not require a numerical value for every letter

Three multiplies x and also multiplies four

The expanded expression has coefficient three for x and constant term twelve

The coefficient-table tasks ask you to construct those values. They do not ask for a guessed value of x

Multiply both a variable and a constant

Multiply out the bracket. Enter the coefficient of x and the signed constant term in the expanded expression

Multiply each term by the same positive outside factor

The coefficient of x is 3 and the signed constant term is 12

Enter both coefficients after expansion

Multiply the existing coefficient as well

If a term already has a coefficient, multiply that coefficient by the outside number

The variable n remains a single factor. This is not a square, and no new power is introduced

Preserve the variable while multiplying its coefficient

Keep a subtracted constant negative in the expanded expression

Apply the same positive factor to both quantities, then keep the subtraction

The coefficient of x is eight. The signed constant term is negative twelve because it is subtracted

Entering positive twelve would describe eight x plus twelve, which is a different expression

Construct the signed constant term

Multiply out the bracket. Enter the coefficient of x and the signed constant term in the expanded expression

Multiply each term by the same positive outside factor

The coefficient of x is 8 and the signed constant term is -12

Multiply every term and retain the subtraction sign

Use a convenient grouping in a count

Apply a multiplier to a complete measured expression

A learner changes five times the sum of x and three into five x plus three. Which correction is needed?

Preserve the operation and every original quantity

Commutativity changes order and associativity changes grouping. Addition and multiplication have both properties

Subtraction and division cannot generally be reordered or regrouped in those ways

Distribution multiplies every term of a sum or difference by the outside factor. Reversing the law can combine products with a common factor

Choose convenient pairs, place-value splits or common factors and show the intermediate values before the result

For a linear algebraic expression, multiply the numerical coefficients while keeping the variable and the signs. Do not invent a value for the variable or a new power