Extend and Use Properties for RIT under 215: comparing decimals with place value columns, hundred grids and number lines, and rounding decimals to the nearest whole, tenth or hundredth.
Subject: NWEA MAP Growth Math · 60 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
MAP Growth · The Real and Complex Number Systems · RIT under 215
Deciding which decimal is larger, with columns, grids and number lines — and rounding
Objectives
Four MAP skills sit here, and three of them are the same comparison shown three ways.
The one idea: compare column by column from the left, and stop at the first column where the digits differ.
Section
Part 1
Concept
Start at the biggest column both numbers have and work rightwards. The first column where they differ decides everything.
Figure (svg): Two decimals lined up in a place value chart with the tenths column highlighted
Once a column differs, nothing to its right can change the result.
Concept
This is the single most damaging habit carried over from whole numbers.
Figure (svg): Two hundred squares shaded to compare zero point four with zero point three five
0.35 has more digits than 0.4, but 0.4 is larger — because 4 tenths beats 3 tenths.
Warm-up
Answer quickly, before any method is taught.
Discussion prompt
Which is bigger, 0.4 or 0.35? Write your first instinct, then say why you thought it.
Hint: Notice what your brain compared — the digits, or the columns?
Answer:
0.4 is bigger. If you said 0.35, the reason is almost certainly that 35 is bigger than 4.
That instinct comes from whole numbers, where more digits does mean bigger. Past the decimal point it stops being true, and this deck is largely about unlearning it.
Pattern
Three steps, and the third is the one that saves time.
Adding a trailing zero never changes a number's value, which is what makes step one safe.
Intuition
Writing 0.5 as 0.50 adds zero hundredths, which changes nothing at all.
Figure (svg): Zero point five and zero point five zero shown as identically shaded grids
So you may always pad the shorter decimal to match the longer one.
Trap
A student says 0.35 is greater than 0.4, because 35 is greater than 4.
Pad both to two places: 0.40 against 0.35. Now 40 hundredths clearly beats 35 hundredths, so 0.4 is greater.
The number of digits after the point says nothing about size. Only the columns do.
Socratic
Understanding this makes the method obvious rather than memorised.
Discussion prompt
Why can a difference in the tenths column never be overturned by the hundredths column?
Hint: Compare the largest hundredths digit with one tenth.
Answer:
Because one tenth is worth ten hundredths. Even the largest possible hundredths digit, 9, is worth less than a single tenth.
So a lead in a bigger column can never be caught up by anything to its right — which is exactly why you stop as soon as a column differs.
Definition probe
Find the deciding column without working out which number is bigger.
Sort into buckets
Sort each pair by the first column where the digits differ.
Explain it to yourself
Explaining it is what displaces the old whole-number instinct.
Discussion prompt
In your own words, why is 0.4 bigger than 0.35 even though 35 is bigger than 4?
Answer:
Because the 4 is four tenths and the 3 is only three tenths, and tenths are the biggest column in play.
The 5 in the hundredths column is worth too little to make up the gap — it would need to be worth a whole tenth, and it is not.
Analogy
Nobody gets this wrong when the decimals are prices.
Match the pairs
Why: Writing a decimal as money forces you to read the columns, because pence and pounds are different things. That is exactly the reading the padding rule produces.
Discrimination
Some pairs punish the digit-count instinct and some do not.
Sort into buckets
Sort each pair by whether the longer number is also the larger one.
Section
Part 2
Worked example
Which is greater, 0.4 or 0.35?
Figure (svg): Two decimals lined up in a place value chart with the tenths column highlighted
Pad to the same length
Why: 0.4 becomes 0.40, so both have two decimal places.
Compare the ones
Why: Both are 0, so move right.
Compare the tenths
Why: 4 against 3. They differ, so this column decides.
\[ 0.40 > 0.35 \]
Verify: as hundredths
Why: 0.40 is 40 hundredths and 0.35 is 35 hundredths, and 40 beats 35 — the same answer reached a second way.
Worked example
Which is greater, 0.62 or 0.68?
Figure (svg): Two decimals lined up in a place value chart with the tenths column highlighted
Compare the ones
Why: Both are 0, so move right.
Compare the tenths
Why: Both are 6, so still tied. Move right again.
Compare the hundredths
Why: 8 beats 2, so 0.68 is greater.
\[ 0.68 > 0.62 \]
Verify: as hundredths
Why: 68 hundredths against 62 hundredths confirms it directly.
Prediction
Use the column method, not instinct.
Predict first
Which is greater, 2.07 or 2.7?
Correct: 2.7
Padding gives 2.70 against 2.07 — seventy hundredths against seven.
Why: The ones match at 2. In the tenths column, 2.7 has 7 and 2.07 has 0, so 2.7 wins there and nothing further needs checking.
Faded example
Pad and compare.
Fill in the blanks
0.9 \;>\; 0.89
Why: Padding gives 0.90 against 0.89. The tenths are 9 and 8, so 0.9 is greater, despite having fewer digits.
Check
Solve it on paper before you click.
Check your understanding
Which symbol makes this true: 0.5 ___ 0.45?
Answer: A
Why: Padding gives 0.50 against 0.45. The tenths are 5 and 4, so 0.5 is the greater number even though it has fewer digits.
Ranking
Pad them all first, then order.
Put in order
Why: As hundredths these are 9, 20, 25 and 30. Padding every number to two places turns a confusing mixed list into a plain whole-number ordering.
Error analysis
A student ordered three decimals from smallest to largest.
Annotate
On: \( 0.7 \;<\; 0.15 \;<\; 0.283 \)
Padding to a common length converts every one of these into an ordinary whole-number comparison.
Notation
The symbols are easy to write the wrong way round under pressure.
Annotate
On: \( 0.4 > 0.35 \qquad 0.35 < 0.4 \)
Write the two numbers down first, decide which is bigger, and only then choose the symbol.
Step zero
One move makes every comparison in this deck routine.
Discussion prompt
You are given 0.6 and 0.58 to compare. What is the very first thing to do?
Hint: The first move involves writing something, not deciding anything.
Answer:
Pad the shorter one so both have the same number of decimal places: 0.60 and 0.58.
Now the comparison is 60 against 58, which needs no decimal reasoning at all.
Section
Part 3
Concept
A hundred square turns a two-place decimal into something you can literally count.
Figure (svg): Two hundred squares shaded to compare zero point four with zero point three five
Shade both and the larger is obvious without any column reasoning.
Worked example
Use hundred squares to compare 0.4 and 0.35.
Figure (svg): Two hundred squares shaded to compare zero point four with zero point three five
Shade the first
Why: 0.4 is 4 full columns, which is 40 squares.
Shade the second
Why: 0.35 is 3 full columns plus 5 more, which is 35 squares.
Compare the counts
Why: 40 shaded squares beats 35.
\[ 0.4 > 0.35 \]
Verify: against the column method
Why: The tenths column gave the same answer, so the picture and the rule agree.
Intuition
On a number line, the number further to the right is always the larger one.
Figure (svg): A number line from zero to one marked in tenths with two decimals plotted
This works for any decimals, however many places they have.
Worked example
Use a number line to compare 0.35 and 0.4.
Figure (svg): A number line from zero to one marked in tenths with two decimals plotted
Draw the line in tenths
Why: Mark 0 to 1 with a tick every tenth.
Plot the easy one
Why: 0.4 is exactly on the fourth tick.
Plot the other
Why: 0.35 is halfway between the third and fourth ticks, so it sits left of 0.4.
\[ 0.35 < 0.4 \]
Verify: across all three methods
Why: Columns, grids and the number line all give the same answer, which is what you would expect of three views of one fact.
Matching
All three work; they suit different situations.
Match the pairs
Why: Padding is not a fourth method — it is the preparation that makes the other three straightforward, which is why it appears in every worked example.
Check
Solve it on paper before you click.
Check your understanding
On a hundred square, 0.6 is shaded on one grid and 0.55 on another. Which statement is true?
Answer: A
Why: 0.6 is 60 hundredths and 0.55 is 55 hundredths, so 0.6 has 60 squares against 55, which is 5 more.
Elimination
Only one of these lists is in increasing order.
Eliminate the wrong options
Which list is ordered from smallest to largest?
Survives elimination: e1
Why: As hundredths the values are 8, 40 and 62, so the increasing order is 0.08, then 0.4, then 0.62.
Discrimination
Trailing zeros are the trap here.
Sort into buckets
Sort each pair by whether the two numbers are equal.
Pattern
Each frame pads one more number to a common length.
Step through it
Why is it safe to compare them as whole numbers once they are padded?
Padding converts a decimal ordering into a whole-number ordering, which is why it is worth doing every time.
Explain it to yourself
The number line is the method that never needs a special case.
Discussion prompt
In your own words, why does position on a number line settle any comparison?
Answer:
Because a number line is built so that moving right always means getting bigger, with no exceptions.
So once both numbers are plotted, the comparison is decided by which is further right — no columns, no padding, no rules to misremember.
Section
Part 4
Concept
To round, find the two candidates the number sits between and decide which it is closer to.
Figure (svg): A number line showing three point six seven sitting between three point six and three point seven
Rounding — Replacing a number with the nearest value at a chosen level of precision.
Concept
Look at the digit immediately to the right of the place you are rounding to.
Figure (svg): A number line showing three point six seven sitting between three point six and three point seven
Worked example
Round 3.67 to the nearest tenth.
Figure (svg): A number line showing three point six seven sitting between three point six and three point seven
Find the two candidates
Why: The tenths either side are 3.6 and 3.7.
Look at the deciding digit
Why: The hundredths digit is 7.
Apply the rule
Why: 7 is 5 or more, so round up to 3.7.
\[ 3.67 \approx 3.7 \]
Verify: on the number line
Why: 3.67 sits past the halfway mark of 3.65, so it really is nearer to 3.7 than to 3.6.
Worked example
Round 12.4 to the nearest whole number.
Figure (svg): A number line showing three point six seven sitting between three point six and three point seven
Find the two candidates
Why: The whole numbers either side are 12 and 13.
Look at the deciding digit
Why: The tenths digit is 4.
Apply the rule
Why: 4 is 4 or less, so round down to 12.
\[ 12.4 \approx 12 \]
Verify: on the number line
Why: 12.4 is short of the halfway point of 12.5, so 12 really is the nearer whole number.
Prediction
Find the deciding digit before anything else.
Predict first
Round 5.48 to the nearest tenth.
Correct: 5.5
Note that the tenths digit itself is 4 — but it is not the digit that decides.
Why: The tenths digit is 4 and the deciding digit is the hundredths, which is 8. Since 8 is 5 or more, the tenths round up from 4 to 5, giving 5.5.
Trap
Rounding 5.48 to the nearest tenth, a student sees the 4 and rounds down to 5.4.
The 4 is the digit being rounded, not the one that decides. The decider is the next digit right, which is 8, so it rounds up to 5.5.
Name the place you are rounding to, then look one column further right. Always one column, never the same one.
Faded example
Round to the nearest whole number.
Fill in the blanks
7.62 \approx 8
Why: The candidates are 7 and 8, and the deciding digit is the tenths, which is 6. Since 6 is 5 or more, it rounds up to 8.
Comparison
The same number rounded to three different levels.
Comparison matrix
| number | nearest whole | nearest tenth |
|---|---|---|
| 4.68 | 5 | 4.7 |
| 9.31 | 9 | 9.3 |
| 2.55 | 3 | 2.6 |
The same number can round to different values depending on the precision asked for, so read that part of the question carefully.
Check
Solve it on paper before you click.
Check your understanding
Round 8.35 to the nearest tenth.
Answer: A
Why: The candidates are 8.3 and 8.4, and the deciding digit is the hundredths, which is 5. Since 5 rounds up, the answer is 8.4.
Error analysis
A student rounded 4.96 to the nearest tenth.
Annotate
On: \( 4.96 \approx 4.10 \)
When the digit being rounded is a 9, expect a carry — exactly as in ordinary addition.
Ranking
Put the method in the order that avoids the wrong-digit error.
Put in order
Why: Naming the target place first is what stops you looking at the wrong digit, which is the single commonest rounding error.
Edge cases
The rule meets its awkward case here.
Discussion prompt
What happens when you round 2.97 to the nearest tenth, and why is it not 2.10?
Hint: Think about what happens when a column overflows.
Answer:
The deciding digit is 7, so the tenths digit 9 rounds up. But 9 plus 1 is 10, which will not fit in one column.
The ten tenths carry into the ones column, giving 3.0. Rounding a 9 behaves exactly like carrying in ordinary addition.
Section
Part 5
Concept
Both ask you to look at one particular column and ignore the rest.
Figure (svg): A number line showing three point six seven sitting between three point six and three point seven
Comparing looks for the first column that differs; rounding looks at the column just right of the target.
Sorting
MAP mixes the two, and they want different answers.
Sort into buckets
Sort each question by what it is asking for.
A comparison answer is a symbol or an ordering; a rounding answer is a number.
Estimation
Rounding is most useful as a fast estimate.
Predict first
Roughly what is 4.82 plus 3.11, rounding each to the nearest whole first?
Correct: about 8
Rounding before adding is how you check a decimal calculation in a couple of seconds.
Why: 4.82 rounds to 5 and 3.11 rounds to 3, and 5 plus 3 is 8. The exact answer is 7.93, so the estimate is close.
Two truths and a lie
Two of these are sound; one is not.
Eliminate the wrong options
Which statement about decimals is false?
Survives elimination: u2
Why: 0.35 has more digits than 0.4 but is smaller. Only the leftmost differing column decides size, and digit count is irrelevant.
Explain it
A classmate keeps saying 0.35 is bigger than 0.4.
Discussion prompt
How would you fix this in one move, without a long explanation?
Answer:
Get them to write both with two decimal places: 0.40 and 0.35.
Now they are comparing 40 against 35, which they will get right instantly. The padding does the teaching for you.
Missing information
Not every rounding question is complete.
Discussion prompt
A problem says: round 6.482. What is missing, and why does it matter?
Answer:
The precision — nearest whole, nearest tenth, or nearest hundredth.
It matters because the three answers are 6, 6.5 and 6.48, which are all different. Rounding without a stated precision is meaningless.
Real world
Money is rounded to hundredths by law, and measurement by convention.
Discussion prompt
A till calculates 4.678 pounds. Why must it round, and to which place?
Answer:
To the nearest hundredth, because the smallest coin is one penny, which is a hundredth of a pound.
The deciding digit is the thousandths, which is 8, so it rounds up to 4.68 pounds. You cannot pay a fraction of a penny.
Commit first
Decide your answer and your confidence before revealing.
Predict first
Which is the largest: 0.7, 0.68, or 0.695?
Correct: 0.7
This is the digit-count trap in its hardest form — the shortest number is the biggest.
Why: Padding to three places gives 0.700, 0.680 and 0.695. Comparing as thousandths, 700 beats 695 and 680, so 0.7 is the largest despite having the fewest digits.
Exit ticket
One item that tells you whether the deck landed.
Predict first
Round 7.451 to the nearest tenth.
Correct: 7.5
Why: The candidates are 7.4 and 7.5, and the deciding digit is the hundredths, which is 5. Since 5 or more rounds up, the answer is 7.5.
Connect it up
Your handwriting, one page, from memory.
Draw it
Draw two hundred squares comparing 0.4 with 0.35, and a number line showing 3.67 sitting between 3.6 and 3.7 with the halfway mark labelled. Beside them write the padding rule, the compare-from-the-left rule and the rounding decider rule.
If you can draw this from memory, all four skills in this deck are covered.
Recap
Four skills, one habit.
Almost every error on this topic comes from the whole-number instinct that a longer number is a bigger number.
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