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# Unit 2 learning overview

You can use this overview to inform your planning and help secure children's learning of the mathematics covered in the unit. It includes understanding the value of each digit in decimal numbers, multiplying and dividing decimals by 10, 100 or 1000, developing written methods to include multiplication and division, and deciding when to use mental, written or calculator methods to solve problems.

Children secure understanding of the value of each digit in decimal numbers with up to two places. For example, they use coins (£1, 10 p and 1 p) or base-10 apparatus (with a 'flat' representing one whole) to model the number 2.45, recognising that this number is made up of two wholes, four tenths and five hundredths.

They understand the relationship between hundredths, tenths and wholes and use this to answer questions such as:

• 'Which of these decimals is equal to $\frac{19}{100}$?'
• 1.9
• 10.19
• 0.19
• 19.1
• 'How many hundredths are the same as three tenths?'

Children use images such as bead strings or number lines to help them count in tenths and hundredths from various start numbers. They position decimals on number lines, explaining for example that 2.85 lies halfway between 2.8 and 2.9. They suggest numbers that lie between, say, 13.5 and 13.6.

Children create and continue sequences of decimals, for example counting up from zero in steps of 0.2 or backwards from 3 in steps of 0.3. They identify the rule for a given sequence and use this to find the next or missing terms, e.g. finding the missing numbers in the sequence: 1.4, ☐, 1.8, 2, 2.2, ☐. They use counting to answer questions such as $0.2×6 \text{or} 1.8÷0.3$, explaining how they worked out the answer.

Children partition decimals using both decimal and fraction notation, for example, recording 6.38 as $6+\frac{3}{10}+\frac{8}{100}$ and as $6+0.3+0.08$. They write a decimal given its parts: e.g. they record the number that is made from 4 wholes, 2 tenths and 7 hundredths as 4.27. They apply their understanding in activities such as:

• 'Find the missing number in $17.82-\square =17.22$'
• 'Play "Zap the digit": in pairs, choose a decimal to enter into a calculator, e.g. 47.25. Take turns to "zap" (remove) a particular digit using subtraction. For example, to "zap" the 2 in 47.25, subtract 0.2 to leave 47.05.'

Children extend their understanding of multiplying and dividing by 10, 100 or 1000 to decimals. They use digit cards and a place value grid to practise multiplying and dividing numbers by 10, 100 and 1000, e.g. moving each digit two columns to the right to work out that $132÷100=1.32$. They recognise that as each digit moves one column to the right, its value becomes 10 times smaller (and the reverse for multiplication). They apply this understanding in a range of activities such as:

• 'Find the missing number in $0.42×\square =42$'
• 'Play 'Stepping stones': work out what operation to enter into a calculator to turn the number in one stepping stone into the number in the next stepping stone.'
Children extend written methods for addition to include numbers with one and two decimal places. They use their understanding that ten tenths make one whole and ten hundredths make one tenth to explain each stage of their calculation, for example, to add 72.8 km and 54.6 km.

8 tenths add 6 tenths makes 14 tenths, or 1 whole and 4 tenths. The one whole is 'carried' into the units column and the four tenths is written in the tenths column.

With subtraction of three-digit numbers and decimals, some children may be ready to use more compact methods. The number of steps in the vertical recording of the 'counting up' method is reduced.

For $326-178$, they extend their understanding of 'difference' by counting up from 178 to 326, initially using an empty number line and then moving on to vertical recording.

$\begin{array}{ccc}& 326& \\ -& \underset{——}{178}& \\ & \phantom{22}2& \phantom{\rule{0.1em}{0ex}}\to 180\\ \phantom{-}& \phantom{2}20& \phantom{\rule{0.1em}{0ex}}\to 200\\ & 100& \phantom{\rule{0.1em}{0ex}}\to 300\\ & \underset{——}{\phantom{2}26}& \phantom{\rule{0.1em}{0ex}}\to 326\\ & 148& \end{array}$
$\begin{array}{ccc}& 326& \\ -& \underset{——}{178}& \\ & \phantom{2}22& \phantom{\rule{0.1em}{0ex}}\to 200\\ \phantom{-}& \underset{——}{126}& \phantom{\rule{0.1em}{0ex}}\to 326\\ & 148& \end{array}$

The examples below work towards the decomposition method. For example: $563-248$, adjustment from the tens to the ones, or 'borrowing ten'

$\begin{array}{cc}& 500+60+3\\ -& \underset{———————}{200+40+8}\end{array}$
$\begin{array}{ccc}\hfill \phantom{-}5& \hfill \stackrel{5}{\overline{)6}}& \hfill \stackrel{13}{\overline{)3}}\\ \hfill -\underline{2}& \hfill \underline{\phantom{\rule{0.5em}{0ex}}\stackrel{\phantom{5}}{4}}& \hfill \underline{\phantom{\rule{0.5em}{0ex}}\stackrel{\phantom{13}}{8}}\\ \hfill \phantom{-}3& \hfill \stackrel{\phantom{5}}{1}& \hfill \stackrel{\phantom{13}}{5}\end{array}$

Discuss how $60+3$ can be partitioned into $50+13$. The subtraction of the ones becomes 'thirteen minus eight', a known fact.

Example: $563-271$, adjustment from the hundreds to the tens, or 'borrowing one hundred'

$\begin{array}{cc}& 500+60+3\\ -& \underline{200+70+1}\end{array}$
$\begin{array}{c}\hfill \phantom{-}\stackrel{400}{\overline{)500}}+\stackrel{160}{\overline{)60}}+3\\ \hfill -\underline{\phantom{\rule{0.5em}{0ex}}\stackrel{\phantom{400}}{200}+\stackrel{\phantom{160}}{70}+1}\\ \hfill \phantom{-}\stackrel{\phantom{500}}{200}+\stackrel{\phantom{113}}{90}+2\end{array}$
$\begin{array}{ccc}\hfill \phantom{-}\stackrel{4}{\overline{)5}}& \hfill \stackrel{16}{\overline{)6}}& \hfill 3\\ \hfill -\underline{2}& \hfill \underline{\phantom{\rule{0.5em}{0ex}}\stackrel{\phantom{16}}{7}}& \hfill \underline{\phantom{\rule{0.5em}{0ex}}1}\\ \hfill \phantom{-}2& \hfill \stackrel{\phantom{16}}{9}& \hfill 2\end{array}$

Discuss how $500+60$ can be partitioned into $400+160$. The subtraction of the tens becomes '160 minus 70', an application of subtraction of multiples of ten.

Children continue to rehearse their recall of multiplication and division facts and use these facts and their knowledge of place value to multiply and divide multiples of 10 and 100. They use jottings to record, support or explain mental multiplication and division of TU by U, forging links to the written methods that they are developing and refining. For example: $38×7$, $38×7=\left(30×7\right)+\left(8×7\right)=210+56=266$.

 × 7 30 210 8 56 266

The number with the most digits is placed in the left-hand column of the grid so that it is easier to add the partial products.

 30 + 8 × 7 210 56 266

The next step is to move the number being multiplied (38) to an extra row at the top of the grid. Presenting the grid like this helps children to set out and add the partial products 210 and 56.

$\begin{array}{ccc}& 30+8& \\ ×& \underline{\phantom{30}\phantom{+}7}& \\ & \phantom{30}210& 30×7=210\\ & \underline{\phantom{\rule{0.5em}{0ex}}\phantom{30}56}& \phantom{3}8×7=\phantom{2}56\\ & \underline{\phantom{30}266}& \end{array}$

The next step is to represent the method of recording to a column format, but showing the working. Point out the links with the grid method.

$\begin{array}{cc}& \phantom{2}38\\ ×& \underline{\phantom{32}7}\\ & 210\\ & \underline{\phantom{3}56}\\ & \underline{266}\end{array}$

Children should describe what they do by referring to the actual values of the digits in the columns (e.g. the first step in $38×7$ is 'thirty multiplied by seven', not 'three times seven', although the relationship to $3×7$ should be stressed).

Children use the multiplication and division facts that they know to find factors of numbers, for example, determining that 35 has a factor pair of 7 and 5, so 350 has a factor pair of 70 and 5 or 7 and 50. They use their knowledge of factors for special cases of multiplication and division calculations. For example, to multiply 15 by 6, they work out $15×3×2=45×2=90$, and to divide 72 by 6 they halve it to get 36, then divide by 3. They find common multiples, investigating questions such as:

• 'What is the smallest whole number that is divisible by five and by three?'
• 'Tell me a number that is both a multiple of four and a multiple of six. Are there any other possibilities?'

Children solve a range of one-step and two-step word problems, choosing whether to use mental, written or calculator methods. They record their method in a clear and logical way, using jottings and diagrams where appropriate. They compare their methods with others, recognising where another method is more efficient than the one that they chose. They solve inverse operation problems such as $3.42+\square =10$, and word problems such as:

• 'Emma saves £3.50 each week. How much has she saved after 16 weeks?'
• 'I buy presents costing £9.63, £5.27 and £3.72. How much change do I have from £20?'
• '1 bag of sugar weighs 2.2 pounds. How much will ten bags of sugar weigh?'
• 'Zak saves half of his pocket money each month. In one year he saves £51. How much pocket money does he get each month?'