aiMOOC

Decimal place value to thousandths and rounding - Mathematics Year 5

QR-Code

Introduction

A flat, a ten-part rod and a small cube. If the flat represents one whole, the rod represents one tenth and the small cube one hundredth. State the reference whole before assigning values.

Which is greater: 0.8 or 0.75? Looking only at the whole-number strings eight and seventy-five gives the wrong comparison. Eight tenths is eighty hundredths, so 0.8 is greater. The position of each digit tells you its value.

Read and represent decimals to thousandths, compare and order numbers with different numbers of decimal places, and round a two-place decimal to a whole number or tenth. Explain rounding by neighbouring values and a midpoint, then use the digit shortcut with understanding. Your final work contains ten decimal cards in order, eight rounding decisions and a short measurement report.

The Year 5 focus follows England's mathematics programme. Rounding to hundredths from longer decimals is a later extension, not a requirement of this course. Recall whole-number place value, tenths and hundredths; use Place value to one million and Roman numerals - Mathematics Year 5 and Equivalent fractions and comparing fractions - Mathematics Year 5 for retrieval where needed.

Learning goals

Read, write and partition decimals to three places, including zero placeholders.
Compare and order decimals by place value rather than the number of digits.
Round a two-place decimal to the nearest whole or tenth with the stated halfway rule.
Explain a method, check with another representation and distinguish an exact supplied value from a rounded report.

All numerical measurement records below are invented for learning. You do not need a real instrument or any personal data.

Read and partition decimals to thousandths

The decimal point separates the whole-number part from the fractional places. Immediately to its right are tenths, then hundredths, then thousandths. Moving one place right makes the place unit one tenth as large: ten thousandths make one hundredth, and ten hundredths make one tenth. There is no extra 'oneths' column after the point.

In the opening block picture, we chose the large hundred-square flat to represent one whole. Each ten-square rod then represents one tenth and each small cube one hundredth. The blocks have no permanent numerical value independent of that choice. Choosing the small cube as one would instead make the rod ten and the flat one hundred. Always declare the reference whole.

A place-value chart for 6.407

Ones	Tenths	Hundredths	Thousandths
6	4	0	7

Read the number as 'six point four zero seven' to preserve the digits. You can also describe it as six wholes and four hundred and seven thousandths. The four contributes four tenths, the zero holds the hundredths place and the seven contributes seven thousandths. The digit seven is not worth seven hundredths in this number.

$6.407 = 6 + \frac{4}{10} + \frac{0}{100} + \frac{7}{1000} = 6 + \frac{407}{1000}$

Removing the zero from the middle would produce 6.47. That has seven hundredths, or seventy thousandths, and is a different value. By contrast, adding a zero at the far right gives 6.4070, an equivalent name with an extra decimal place. Our core comparison work stops at thousandths; the extra zero simply illustrates value preservation.

Connect decimals and fractions

One tenth is 0.1, one hundredth is 0.01 and one thousandth is 0.001. Seven thousandths is 0.007. Seventy thousandths is 0.070, equivalent to 0.07. Seven hundred thousandths would be 0.700, equivalent to 0.7. All three use the digit seven, but their positions give different amounts.

$0.407 = \frac{407}{1000} = \frac{400}{1000} + \frac{7}{1000}$

For a thousandth model, imagine a whole cube divided into ten layers, each with one hundred equal small cubes. That makes one thousand small cubes in the whole. A single small cube then represents one thousandth. This is a different reference whole from the flat-as-one opening model; changing the whole must be explained, not done silently.

The chart places tens and tenths on opposite sides of the ones column to help remember names. Ten and one tenth are not equal or negative versions of each other. Also, whole numbers can be written in decimal notation; the chart's 'Decimal Numbers' heading should be read here as the fractional columns, not a claim that whole numbers are outside base-ten notation.

Try it: Write six wholes and seven thousandths as 6.007. Partition 2.305 as two wholes, three tenths, zero hundredths and five thousandths. Explain why 2.350 has a different value. Check: its fractional part is thirty-five hundredths, not three hundred and five thousandths.

Compare and order without a digit-length shortcut

Compare the whole-number parts first. If they match, compare tenths, then hundredths, then thousandths. Stop at the first place that differs. You may append zeros on the right to make the alignment easier, but do not insert zeros between existing digits or move the decimal point.

Worked comparison: 0.8 and 0.75

Write 0.8 as 0.80. Both have zero whole units. In the tenths place, eight tenths exceeds seven tenths, so 0.8 is greater. The five hundredths in 0.75 do not outweigh the missing tenth. Alternatively, compare eighty hundredths with seventy-five hundredths.

$0.8 = 0.80 = \frac{80}{100} > \frac{75}{100} = 0.75$

The number of written digits is not a size rule. 0.075 has more decimal places than 0.8 but is much smaller. A longer whole-number part can matter through place value, but 'more characters means a larger decimal' is unreliable.

Worked comparison: 6.407 and 6.47

Align them as 6.407 and 6.470. Ones and tenths match. At hundredths, zero is less than seven, so 6.407 is smaller. Equivalently, the fractional amounts are 407 thousandths and 470 thousandths. You do not need to compare the final seven and zero once a larger place has already decided.

Equal values with different written forms

0.7, 0.70 and 0.700 represent the same number. Seven tenths, seventy hundredths and seven hundred thousandths describe equal values. Writing 0.700 can communicate a requested number of decimal places, but extra written zeros alone do not prove that a physical measurement was accurate to that precision.

For pure-number ordering, place equivalent values together and mark equality. If separate cards read 0.75 and 0.750, keep both cards; they have equal value, so either order within that equal group is acceptable.

Use a number line to check a close pair

Mark 0.70 and 0.71 on a line, then split the interval into ten equal steps. Each step is 0.001, one thousandth. The midpoint is 0.705. The number 0.709 is one thousandth before 0.710, so it lies to the right of 0.705. State the interval and step unit; arbitrary visual spacing is not a measured comparison.

Independent ten-card ordering task

Put all ten cards in increasing order, retaining the two equal-valued cards: 0.705, 0.75, 0.075, 0.7, 0.507, 0.57, 0.750, 0.709, 0.079, 0.8. First try a place-value chart; then verify with equivalent thousandths.

Check after attempting: 0.075 < 0.079 < 0.507 < 0.57 < 0.7 < 0.705 < 0.709 < 0.75 = 0.750 < 0.8. The thousandth counts are 75, 79, 507, 570, 700, 705, 709, 750, 750 and 800. Both 750 cards are retained. The comparison assumes the same underlying unit if the numbers represent measurements.

Error clinic: A learner says 0.507 is greater than 0.57 because 507 is greater than 57. The comparison mixed thousandths and hundredths. Rename 0.57 as 0.570; now 507 thousandths is less than 570 thousandths. The correction keeps the original numbers' values instead of deleting the internal zero.

Round to a requested place using neighbours and midpoints

Rounding replaces a number with a nearby value at a stated reporting level. 'Round 8.46' is incomplete unless you know whether to use the nearest whole, tenth or another unit. Here we use non-negative numbers and the school convention that an exact halfway case goes to the higher of the two neighbouring values.

Nearest tenth using a number line

The neighbouring tenths around 8.46 are 8.4 and 8.5. Their midpoint is 8.45. The original value is 0.06 above 8.4 and 0.04 below 8.5, so it is nearer 8.5. We write that 8.46 is approximately 8.5 to one decimal place, not exactly equal to it.

For 8.44, the distance to 8.4 is 0.04 and the distance to 8.5 is 0.06. It rounds to 8.4. Comparing the two distances supplies the reason behind the digit shortcut. The exact halfway number 8.45 rounds to 8.5 under the stated convention.

The place-value shortcut

Identify the place you are rounding to: tenths for one decimal place.
Inspect the digit immediately to its right: the hundredths digit for this task.
If that digit is zero through four, retain the tenths digit; if it is five through nine, increase the retained amount by one tenth, carrying if needed.
Write the result at the requested reporting level. Do not keep the discarded digits as though the answer were exact.

For 8.46, the hundredths digit six selects the upper tenth, 8.5. For 8.44, the hundredths digit four selects 8.4. The digit you inspect is not the digit you keep, and moving the decimal point is not rounding.

Carry across a whole-number boundary

For 9.96 to one decimal place, the neighbouring tenths are 9.9 and 10.0, with midpoint 9.95. The original is closer to 10.0. Increasing the retained nine tenths by one tenth carries to the next whole. Write 10.0 to show the one-decimal-place request; do not write 9.10 or 10.96.

Similarly, 6.95 is exactly halfway between 6.9 and 7.0 and rounds to 7.0 under this convention. A trailing zero in the rounded report shows the requested decimal place; it does not change the numerical value from seven.

Watch the original Corbettmaths rounding explanation. Focus first on the one-decimal-place examples and identify the retained place, the deciding digit and any carry. The film also covers two decimal places; that part is optional extension here. Explain a number-line midpoint as well as the shortcut so the procedure has a mathematical reason.

Nearest whole number

For 2.49, the neighbouring wholes are two and three, and the midpoint is 2.50. Since 2.49 is below the midpoint, the rounded result is two. Exactly 2.50 rounds to three under the stated convention. To round directly to a whole number, the tenths digit decides; do not first round the hundredths to create a different intermediate number.

Why double rounding can fail

Take the original 2.46. Directly to the nearest whole, it rounds to two because it is below 2.50. If you first round it to one decimal place, it becomes 2.5; rounding that new value to a whole gives three. The second route has lost information at the first step and answers a different question. Always use the original supplied value for each separately requested rounding.

Try it: Round 0.04 to one decimal place. It lies between 0.0 and 0.1 and below their midpoint 0.05, so the result is 0.0. This does not mean the original amount was zero; a small nonzero value can have a rounded report of zero at a coarse reporting level.

Keep meaning when decimals report measurements

Reporting a rounded value and making a decision from an original value are different jobs. Keep the supplied unrounded record when a threshold, comparison or later calculation needs it. Rounding cannot add information that the record never contained.

An invented measurement report

A fictional length record lists 8.44 m and 8.46 m. To the nearest tenth of a metre, report 8.4 m and 8.5 m. The original difference is 0.02 m. Subtracting only the rounded reports gives 0.1 m, which is not the original difference. Rounding each input first can alter a later calculation.

If a different pair is 8.41 m and 8.44 m, both report as 8.4 m to the nearest tenth. The identical reports do not prove that the original lengths were equal. In a pure mathematical ordering task, trailing-zero equivalents are equal; in measurement reporting, a common rounded label can stand for several different original values.

A threshold question

Suppose a fictional design brief accepts a supplied value only when it is at most 8.45 m. Compare the original 8.44 m and 8.46 m with that threshold: the first meets it, the second does not. Their coarse rounded labels are not a substitute for the original data. This is an arithmetic example, not a real engineering inspection or guarantee of instrument accuracy.

Independent eight-rounding check

For every row, use the original value directly. Write the target neighbours and midpoint for rows 3 and 4. Use the digit shortcut for the remaining rows and justify one with a number line.

Question	Original value	Requested reporting level
1	8.46	Nearest tenth
2	8.44	Nearest tenth
3	3.45	Nearest tenth
4	9.96	Nearest tenth
5	2.49	Nearest whole
6	2.50	Nearest whole
7	0.04	Nearest tenth
8	6.95	Nearest tenth

Worked results after attempting

8.5: the hundredths digit six selects the higher tenth.
8.4: the hundredths digit four selects the lower tenth.
3.5: 3.45 is the midpoint of 3.4 and 3.5, so the stated convention selects the higher value.
10.0: the neighbours are 9.9 and 10.0; 9.96 is above their midpoint 9.95.
2: 2.49 is below the whole-number midpoint 2.50.
3: 2.50 is exactly halfway between two and three, so the higher whole is used.
0.0: 0.04 is below the midpoint 0.05.
7.0: 6.95 is halfway between 6.9 and 7.0; carry to seven while retaining the requested one decimal place.

Choose the method and explain its limits

A place-value chart is useful when zeros or unequal decimal lengths are confusing. Equivalent thousandth counts can efficiently check the ten-card ordering task. A number line with a midpoint explains why a rounded value is nearest; the deciding-digit method is quicker for many routine calculations. None of these methods permits replacing supplied evidence with invented more precise readings.

Your final portfolio

Keep a partition of 6.407, the complete ten-card ordering including equality, all eight rounding decisions and one fictional measurement explanation with units. Include one corrected error about digit length and one about double rounding. Explain which representation helped you and why. A partner should be able to follow the first place that decides a comparison or the midpoint that decides a rounding result.

Transfer question: Can a value reported as 8.4 m to the nearest tenth be assumed to equal exactly 8.400 m? No: the report alone does not recover the original unrounded value. State what extra information would be needed rather than manufacturing digits from a shorter display.

Interactive Tasks

Quiz: Test Your Knowledge

What is the value contributed by the digit 4 in 6.407? (Four tenths) (!Four hundredths) (!Four thousandths) (!Four whole units)

Which decimal is seven thousandths? (0.007) (!0.07) (!0.7) (!7.000)

Which value is greatest? (0.8) (!0.75) (!0.705) (!0.079)

Which decimal has exactly the same numerical value as 0.75? (0.750) (!0.075) (!0.705) (!0.507)

What is 8.46 rounded to the nearest tenth? (8.5) (!8.4) (!8.46) (!8.0)

Using the stated higher-value halfway convention, what is 3.45 to the nearest tenth? (3.5) (!3.4) (!3.45) (!3.0)

What is 9.96 rounded to one decimal place? (10.0) (!9.9) (!9.10) (!10.96)

What is 2.49 rounded directly to the nearest whole number? (2) (!3) (!2.5) (!2.4)

Why should 2.46 be rounded directly from its original value when a nearest-whole report is requested? (Rounding to tenths first can change the later whole-number result) (!The original value must always be discarded immediately) (!A longer decimal is automatically larger than every shorter decimal) (!Rounding means moving the decimal point left)

Two supplied lengths both round to 8.4 m to the nearest tenth. What follows? (Their rounded reports match, but their original lengths need not be equal) (!Their original lengths must both be exactly 8.400 m) (!Their instruments must have equal accuracy) (!Their original difference must be exactly zero)

Memory Game

0.1	One tenth
0.01	One hundredth
0.001	One thousandth
0.8 and 0.80	Equal values with different written decimal lengths
Neighbour midpoint	Boundary used to choose the nearest reported value
Internal zero	Placeholder that cannot generally be removed without changing value

Drag and Drop

Digit 4 in 6.407	Four tenths
Digit 7 in 6.407	Seven thousandths
Digit 3 in 2.305	Three tenths
Digit 5 in 2.305	Five thousandths
Digit 5 in 2.350	Five hundredths

Crossword Puzzle

TENTHS	First fractional place immediately right of the decimal point
HUNDREDTHS	Place unit ten times smaller than a tenth
THOUSANDTHS	Place unit ten times smaller than a hundredth
MIDPOINT	Point halfway between neighbouring values
ROUNDING	Reporting a number at a stated nearby level
PLACEHOLDER	Role of a zero that preserves another digit's position

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Label a decimal place-value chart: Place the digits of 6.407 in ones, tenths, hundredths and thousandths. State each digit's value and read the number without losing the internal zero.
Connect decimal and fraction names: Represent 0.1, 0.01 and 0.001 with unit fractions. Explain what ten of each smaller part makes.
Distinguish internal and trailing zeros: Compare 6.407, 6.47 and 6.4070. Explain which zero change preserves numerical value and which does not.
Compare unequal decimal lengths: Explain why 0.8 exceeds 0.75 using equivalent hundredths and a labelled place-value chart.

Standard

Order all ten decimal cards: Complete the ten-card ordering in the teaching chapter. Keep both equal-valued cards, mark equality and verify with thousandth counts.
Explain a nearest-tenth decision: Round 8.46 and 8.44 to tenths with their two neighbouring tenths, midpoint and distances. Compare with the deciding-digit shortcut.
Complete eight independent rounding decisions: Attempt all eight rows in the final teaching chapter before checking. Use original values directly, and explain the two midpoint cases and two carry examples.
Correct a decimal digit-length error: A learner compares 0.507 with 0.57 by comparing 507 with 57. Identify the mixed units, rename without changing values and correct the order.

Advanced

Demonstrate a double-rounding failure: Use 2.46 to compare rounding directly to a whole with first rounding to a tenth. Explain why the routes can disagree.
Preserve an original measurement difference: Using the fictional 8.44 m and 8.46 m pair, compare the original difference with the difference of the nearest-tenth reports. Explain which calculation answers the original question.
Make a threshold decision from unrounded data: Apply the supplied at-most-8.45-m threshold to 8.44 m and 8.46 m. Explain why a coarse report cannot replace the original record.
Evaluate the limits of a decimal representation: Explain why trailing-zero equality in pure arithmetic does not prove instrument precision, and why a report of 8.4 m does not recover an original reading of exactly 8.400 m.

Learning Check

Read and partition decimals to thousandths: Partition 6.407 and write six wholes and seven thousandths. Compare 2.305 with 2.350.
Order ten decimals with an equal pair: Order the exact ten cards from the teaching chapter, retaining both 0.75 and 0.750, and explain the first differing place in one close comparison.
Complete eight rounding decisions: Give the results for the eight-row independent rounding table and justify one midpoint and one carry.
Compare rounding methods and information loss: Explain the number-line and deciding-digit methods and demonstrate why 2.46 should be rounded directly to a whole when that is the request.
Interpret fictional measurement reports with units: Find the difference between 8.46 m and 8.44 m before and after nearest-tenth reporting; explain why equal coarse reports need not mean equal original lengths.
Check a supplied threshold: A fictional limit accepts values at most 8.45 m. Classify 8.44 m and 8.46 m and explain what extra information a rounded-only report lacks.
Evaluate the broader chart and block model: Explain the reference whole for the opening blocks and one limit of the broader naming chart.

Worked checks and assessment guidance

6.407 is six wholes plus four tenths, zero hundredths and seven thousandths, or six plus 407/1000. Six wholes and seven thousandths is 6.007. In 2.305 the five is thousandths; in 2.350 it is hundredths, so 2.305 is smaller. Removing an internal zero shifts a later digit's place value.
The order is 0.075 < 0.079 < 0.507 < 0.57 < 0.7 < 0.705 < 0.709 < 0.75 = 0.750 < 0.8. In thousandths the counts are 75, 79, 507, 570, 700, 705, 709, 750, 750, 800. Both 750 cards remain. For 0.705 versus 0.709, tenths and hundredths match and the thousandths five is smaller than nine.
The results in order are 8.5, 8.4, 3.5, 10.0, 2, 3, 0.0 and 7.0. At 3.45 the neighbours are 3.4 and 3.5 and the stated halfway convention chooses 3.5. At 9.96, the midpoint between 9.9 and 10.0 is 9.95, so 10.0 is nearer. 6.95 is halfway between 6.9 and 7.0, selecting 7.0 with a carry. Trailing zeros show the requested one decimal place.
A number line identifies neighbouring target values and their midpoint; the digit immediately right of the target place provides a quicker equivalent decision for these non-negative values under the stated convention. Directly, 2.46 is below 2.50 and rounds to two. Rounding first to 2.5 and then to a whole gives three, because the intermediate report discarded information. Use the original value for each separate request.
The original difference is 0.02 m. The rounded reports 8.5 m and 8.4 m differ by 0.1 m, so their difference does not preserve the original calculation. Values 8.41 m and 8.44 m both round to 8.4 m despite differing originally. These are fictional supplied records and do not demonstrate a real instrument's accuracy.
The original 8.44 m meets the supplied limit and 8.46 m exceeds it. A rounded value alone generally does not reveal the exact original or its relation to a finer threshold. Keep original records where available; do not infer extra precision or engineering safety from a short decimal report.
With the hundred-square flat as one, the rod is one tenth and a small cube one hundredth. Choosing a different whole changes the assigned values, so that choice must be stated. The chart's mirror aid concerns place names, not equality of tens and tenths. Whole numbers can also be represented in base-ten decimal notation; the right-hand columns are fractional places, not a separate number system.

Learning evidence

Knowledge: Explain tenths, hundredths, thousandths, internal zero, equivalent trailing-zero forms, target place, neighbouring values, midpoint and rounded report.

Skills: Read and partition decimals to three places, order different written lengths, round two-place non-negative decimals directly to a whole or tenth, handle a carry and justify a halfway decision under the stated convention.

Products: Keep the partition of 6.407, the ten-card order retaining its equal pair, eight independent rounding decisions, a midpoint diagram and an invented measurement report with original values and units preserved.

Assessment: Require place-value reasoning, complete notation and an independent fraction, number-line or exact-value check. A correct digit shortcut alone does not demonstrate why the nearest value is chosen. Accept tactile or spoken representations with the same specified whole and scale. No quiz score establishes actual measurement accuracy.

Transfer: Explain why matching coarse reports need not mean equal originals, why double rounding can change an answer and why a threshold may need the original unrounded record. Extra written digits alone cannot prove physical precision.

OER and Wikipedia

Original explanations, fictional records and independent tasks were prepared against Department for Education: mathematics programmes of study, Year 5 decimal work. The core combines comparison to three places with rounding two-place values to a whole or tenth. Later extensions in the external film and broader chart are not presented as extra compulsory Year-5 objectives.

Commons images

Base Ten Block.png, BarbourLA, CC BY-SA 4.0. Original illustration unchanged; reference whole assigned explicitly in the course. This is an image of a representation, not an embedded interactive application.
Decimal Place Value Chart emphasizing the ones column, GregWelch8, CC0. Original unchanged. Its naming analogy is explained with limits, and farther decimal places are optional reference.

Original video

Corbettmaths: Rounding to 1 or 2 Decimal Places links the original YouTube video embedded in the rounding chapter. Focus on one decimal place for this course. No video or transcript copied.

All length values and thresholds are invented mathematical data. Physical accuracy, real instrument readings and professional safety certification are not inferred. The encyclopedia below is optional reference; the course supplies its own worked answers.