Calculate with positive fractions and decimals

Fraction

\(\displaystyle \frac{3}{5}\)

What does the denominator tell us in the fraction shown?

Add counts of equal-sized parts

When the denominators are equal, the parts are the same size. Add the numerators and keep that part size

Two sevenths and three sevenths make five sevenths, not five fourteenths

Simplify the result if its numerator and denominator share a factor

Keep the common denominator

Calculate exactly. Give a fraction in simplest form

The parts must have the same size before their counts can be combined

Add the numerators and keep the common denominator

Keep the common denominator, then simplify the result

Add, then simplify

Make unlike parts the same size

Different denominators describe different-sized parts. Replace each fraction with an equivalent fraction using a common denominator

For quarters and sixths, twelfths give a common part size

Multiply each numerator by the same factor used for its denominator. Changing only the denominator changes the value

The lowest common multiple gives the smallest common denominator, although a larger common multiple also works

Use twelfths to combine quarters and thirds

Calculate exactly. Give a fraction in simplest form

The parts must have the same size before their counts can be combined

Use the lowest common denominator, 12, and scale both numerators with their denominators

Add the numerators and keep the common denominator

Keep the common denominator, then simplify the result

Find a common denominator

The second amount is one third, which is two sixths. Change the number of sixths in the first amount. Every entry in the table counts sixths

When the first amount is three sixths, what is the sum?

Subtract counts in the original order

For subtraction, use a common denominator and subtract the second numerator from the first

Keep the denominator because the parts still have the same size

Check by adding the amount subtracted back to the result

The subtraction questions here have positive operands and non-negative results

Use a common part size before subtracting

Calculate exactly. Give a fraction in simplest form

The parts must have the same size before their counts can be combined

Use the lowest common denominator, 12, and scale both numerators with their denominators

Subtract the second numerator from the first and keep the common denominator

Keep the common denominator, then simplify the result

Check by adding the subtracted quantity back to the result

Subtract and check by addition

Convert mixed numbers to avoid losing a whole

Converting mixed numbers to improper fractions gives a consistent method for addition and subtraction

If you subtract whole and fractional parts separately, you may need to regroup one whole into fractional parts

Do not subtract the smaller fractional numerator from the larger one independently of the whole-number subtraction

Subtract when the first fractional part is smaller

Calculate exactly. Give a fraction in simplest form

The parts must have the same size before their counts can be combined

Write the first mixed number as an improper fraction, keeping its sign

Write the second mixed number as an improper fraction, keeping its sign

Use the lowest common denominator, 12, and scale both numerators with their denominators

Subtract the second numerator from the first and keep the common denominator

Keep the common denominator, then simplify the result

Check by adding the subtracted quantity back to the result

Keep the whole and fraction parts together

Add two mixed numbers

Equal amounts have zero difference

Subtracting an amount from itself leaves zero, even when the amount is not a whole number

Equivalent decimal writings describe the same amount

A zero difference is valid. It is not a reason to change the order or invent a non-zero fraction

Recognise complete subtraction

Align decimal points, not the last digits

Ones must combine with ones, tenths with tenths, and hundredths with hundredths

Append zeros to show the same final place without changing a value

When a column reaches ten of its units, regroup them into one of the next larger unit

Add unlike numbers of decimal places

Calculate exactly. Give a decimal or whole-number answer without rounding

Align the decimal points so each column has the same place value

Append zeros where needed without changing either value

Express both quantities in units of one over 100

The sum is greater than each positive addend

Regroup tenths into a whole

Regroup place values when subtracting

Align decimal points and append zeros before subtracting

In hundredths, this is four hundred and twenty hundredths minus eighty-five hundredths

Adding the subtracted amount back checks the result

Subtract through a zero place

Calculate exactly. Give a decimal or whole-number answer without rounding

Align the decimal points so each column has the same place value

Append zeros where needed without changing either value

Express both quantities in units of one over 1000

Check by adding back the original amount subtracted

Keep the thousandths place aligned

Equivalent decimals subtract to zero

Group compatible quantities

Addition can be reordered and regrouped. Look for pairs that make whole numbers

Multiplication can also be reordered and regrouped

Subtraction and division cannot generally be reordered or regrouped this way

For successive subtractions, subtracting each amount is equivalent to subtracting their sum

Sum

\(\displaystyle 0.35+2.4+0.65\)

Which rewrite keeps the value and makes a whole-number pair first?

Choose an equivalent representation that simplifies the work

Fractions and decimals can describe the same exact value. Choose the form that makes the operation clearer

A quarter of sixteen is four, so three quarters is twelve

For adding tenths to hundredths, decimal place-value alignment may be more direct

Do not replace a recurring fraction by a rounded decimal when an exact answer is required

Calculation

\(\displaystyle 0.125\times24\)

Which conversion gives an exact fraction useful for the multiplication?

Choose the operation from the quantities

Combining two measured lengths calls for addition. Finding what remains after a length is used calls for subtraction

Taking a fraction of a quantity calls for multiplication. Counting equal-sized portions calls for division

Name the requested quantity and its unit before calculating. A length has a length unit, while a count of portions is a whole-number count

For a multi-step problem, find each intermediate quantity in order and keep its meaning visible

Combine measured lengths

Find the length remaining

Use an exact fraction for a decimal part

Count fractional portions

Find an intermediate quantity before multiplying

One decoration uses two ribbon pieces, of lengths zero point four metres and zero point eight metres. Three identical decorations are made

First add the two lengths for one decoration

Then multiply by the number of decorations

Multiplying just one of the two lengths would leave some of the required ribbon uncounted

Account for every piece in each decoration

Keep equal parts, place values and quantities clear

For fraction addition and subtraction, use a common denominator and combine the numerators in the stated order

Convert mixed numbers when that makes the work clearer, and simplify exact fraction results

For decimal addition and subtraction, align decimal points and regroup place values as needed

Use commutative and associative properties for addition and multiplication, not as unrestricted rules for subtraction and division

Choose exact equivalent representations, then select the operations that match the situation and check the result in context