The bridge between the fraction decks and the decimal decks: reading a decimal off the place-value chart as a fraction, simplifying and checking it, scaling a denominator to a power of ten, long division when that fails, which fractions repeat and why, and comparing a mixed set by converting everything into one notation.
Subject: NWEA MAP Growth Math · 63 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
Session 2
Reading one as the other, both directions, without guessing
Objectives
You already know how to multiply and divide fractions. This session connects them to the other way of writing the same numbers, which is where sixth grade spends most of its time.
NWEA MAP Growth learning continuum — The Real and Complex Number Systems: convert between fractions and decimals — the MAP strand this session targets
Section
Section 1
Warm-up
One minute, no working out.
Discussion prompt
You know that 3 divided by 4 is 0.75. What does the fraction bar in three-quarters actually tell you to do?
Hint: What operation is hiding in the fraction bar?
Answer:
It tells you to divide. A fraction is a division problem that has not been carried out yet.
That single sentence is the whole of this session. Every conversion in either direction is that idea used carefully.
Concept
Three-quarters and 0.75 are not two numbers that happen to be close. They are one number written two ways, the way a name can be written in print or in cursive.
Shade three of four strips, or shade seventy-five of a hundred squares, and you have shaded the same amount of paper.
Illustrative Mathematics — grade 5 and 6 tasks on decimal and fraction equivalence — grade 5 tasks are built around exactly this equivalence
Picture it
Grid, bar, circle. Different shapes, identical amount shaded.
Figure (svg): A hundredths grid with seventy-five squares shaded, a bar with three of four strips shaded, and a circle with three quarters shaded
When a question feels hard in one notation, it is often easy in the other. Switching is allowed and it is usually the smart move.
Prediction
No calculating.
Predict first
Is 0.5 bigger than, smaller than, or equal to one half?
Correct: Exactly equal.
Why: 0.5 means five tenths, and five tenths simplifies to one half. Comparing the digit 5 against the digit 1 ignores what the digits mean, which is the single most common decimal mistake at this level.
Section
Section 2
Concept
Every decimal place has a name, and every name is a fraction: tenths, hundredths, thousandths.
Whichever column the decimal ends in gives you the denominator. The digits themselves give you the numerator.
denominator — The bottom number of a fraction. It says how many equal pieces the whole was cut into.
Common Core State Standards for Mathematics, 5.NBT.A.3 (read and write decimals as fractions) and 7.NS.A.2d (fractions convert to terminating or repeating decimals) 5.NBT.A.3
Picture it
0.375 ends in the thousandths column.
Figure (svg): A place value chart with columns for ones, tenths, hundredths and thousandths holding the digits of 0.375, with the fraction value of each column written underneath
Worked example
Say the number out loud first. Three hundred seventy-five thousandths.
Write what you just said as a fraction
Why: Saying it aloud does the whole first step for you, which is why this is worth the two seconds.
\[ 0.375 = \frac{375}{1000} \]
Find the greatest common factor of top and bottom
Why: Both end in 5 and both are divisible by 125.
\[ \frac{375 \div 125}{1000 \div 125} = \frac{3}{8} \]
Verify: by dividing 3 by 8
Why: Long division gives 0.375 exactly, which is the number you started from. If it does not match, the simplifying went wrong.
Picture it
Divide top and bottom by the same number, then divide the answer out to make sure.
Figure (svg): The fraction 375 over 1000 with an arrow labelled divide both by 125 leading to 3 over 8, and a second arrow to the check that 3 divided by 8 is 0.375
Trap
Converting 0.375.
\[ 0.375 = \frac{375}{100} \]
Use hundredths because hundredths sounds familiar
Why: The number has three decimal places, so the last one is thousandths, not hundredths.
Get a fraction bigger than 1
Why: 375 over 100 is more than three whole units. The original number was less than one, so this cannot be right.
Converting 0.375.
\[ 0.375 = \frac{375}{1000} \]
Count the decimal places, then use that many zeros
Why: Three places means three zeros: one thousand.
Sanity-check against 1
Why: The numerator is smaller than the denominator, so the fraction is less than one, which matches the decimal.
Matching
Before simplifying anything, just name the bottom number.
Match the pairs
Why: Count the digits after the point and use that many zeros. The last one is the sneaky case: 0.06 has two decimal places, so it is 6 over 100, and the zero in the tenths column is doing real work rather than being decoration.
Fill the middle
The decimal and the final answer are given. Fill the middle.
Fill in the blanks
0.24 = \frac24}100} = \frac______
Why: Two decimal places means hundredths, so 0.24 is 24 over 100. The greatest common factor of 24 and 100 is 4, and dividing both by 4 gives 6 over 25. Checking: 6 divided by 25 is 0.24.
Error analysis
A classmate converted 0.6 and then simplified it.
Annotate
On: \( 0.6 = \frac{6}{10} = \frac{3}{10} \)
The check catches this instantly: 3 divided by 10 is 0.3, not 0.6.
Pattern
Four steps, every single time. The last one is not optional.
| decimal | straight from the chart | simplified |
|---|---|---|
| 0.5 | 5/10 | 1/2 |
| 0.6 | 6/10 | 3/5 |
| 0.24 | 24/100 | 6/25 |
| 0.35 | 35/100 | 7/20 |
| 0.375 | 375/1000 | 3/8 |
Check
Solve it on paper before you click.
Check your understanding
Write 0.35 as a fraction in lowest terms.
Answer: A
Why: Two decimal places means 35 over 100. The greatest common factor of 35 and 100 is 5, giving 7 over 20. Checking, 7 divided by 20 is 0.35.
Section
Section 3
Concept
If the denominator divides evenly into 10, 100 or 1000, you can turn the fraction into a decimal with one multiplication and no division at all.
Multiply the top and the bottom by whatever number turns the denominator into that power of ten. Multiplying both by the same number never changes the value.
Common Core State Standards for Mathematics, 5.NBT.A.3 (read and write decimals as fractions) and 7.NS.A.2d (fractions convert to terminating or repeating decimals) 7.NS.A.2d
Picture it
Four goes into a hundred twenty-five times, so multiply both parts by twenty-five.
Figure (svg): The fraction three over four with arrows labelled times twenty-five on the top and the bottom, becoming seventy-five over one hundred, then 0.75
Worked example
The denominator is 20. Does it divide a power of ten?
Find the scale factor
Why: Twenty times five is one hundred, so the factor is five.
\[ \frac{7}{20} = \frac{7 \times 5}{20 \times 5} = \frac{35}{100} \]
Read the decimal off the hundredths denominator
Why: Thirty-five hundredths is written with two decimal places.
\[ \frac{35}{100} = 0.35 \]
Verify: by converting back
Why: 0.35 read off the chart is 35 over 100, which simplifies by 5 to 7 over 20. That is the fraction you started with.
Picture it
The grid makes the hundredths denominator visible.
Figure (svg): A ten by ten grid with thirty-five squares shaded, labelled 0.35
Discrimination
The first question is always whether the denominator divides a power of ten.
Sort into buckets
Which method gets there faster?
Concept
The fraction bar means divide. When scaling will not work, carry out the division the fraction was always asking for.
Put the numerator inside the division bracket, add a decimal point and as many zeros as you need, and divide.
Worked example
Eight does divide a thousand, so scaling would work here too. Do it by division anyway, because this is the method that always works.
Set up 3 divided by 8 with a decimal point and zeros
Why: Three point zero zero zero. You can always add more zeros; they do not change the value.
Eight into thirty goes three times, remainder six
Why: Write the 3 in the tenths place above the line.
Bring down a zero: eight into sixty goes seven times, remainder four
Why: Write the 7 in the hundredths place.
Bring down a zero: eight into forty goes five times, remainder zero
Why: Write the 5 in the thousandths place. The remainder is zero, so the division is finished.
\[ \frac{3}{8} = 0.375 \]
Verify: by scaling instead
Why: Eight times 125 is 1000, and 3 times 125 is 375, giving 375 over 1000, which is 0.375. Two different methods, same answer.
Picture it
Follow the remainders down the left side: six, then four, then zero.
Figure (svg): Long division of three by eight showing the quotient 0.375 with remainders of six, four and finally zero
If the remainder never reaches zero, the decimal never stops. That is the next idea.
Prediction
Try dividing 2 by 3 in your head for a few steps.
Predict first
What does the decimal for two-thirds do?
Correct: It repeats the digit 6 forever.
Why: Every step of the division leaves a remainder of 2, so every step produces another 6. The remainder never reaches zero, so the digits never stop. Writing 0.67 is a rounded answer, useful but not exact.
Concept
A fraction in lowest terms gives a decimal that stops exactly when its denominator is built only from 2s and 5s. Any other prime factor makes it repeat forever.
The reason is simple: 10 is 2 times 5. A denominator made only of those two primes can always be scaled up to a power of ten, and a power of ten is what a terminating decimal is.
Common Core State Standards for Mathematics, 5.NBT.A.3 (read and write decimals as fractions) and 7.NS.A.2d (fractions convert to terminating or repeating decimals) 7.NS.A.2d
Picture it
Look at the denominators, not the numerators.
Figure (svg): Two panels listing fractions whose decimals stop on the left and fractions whose decimals repeat on the right, sorted by the prime factors of the denominator
Sorting
Each fraction is already in lowest terms. Decide from the denominator alone.
Sort into buckets
Does the decimal stop, or does it repeat forever?
This works only when the fraction is already in lowest terms. Six over twelve looks like it has a 3 in it, but it is really one half.
Trap
Converting two-thirds.
\[ \frac{2}{3} = 0.67 \]
Round after two places and write an equals sign
Why: The rounding is fine. The equals sign is not: 0.67 is a different number from two-thirds.
Carry the rounded value into the next calculation
Why: Small errors compound, and on a multi-step problem the final answer drifts off the mark.
Converting two-thirds.
\[ \frac{2}{3} = 0.666\ldots \]
Show that the digits continue
Why: Dots, or a bar over the repeating digit, both say the same thing: this goes on forever.
Keep the fraction when you need exactness
Why: In a multi-step problem, work in fractions and convert at the very end. Two-thirds is exact; 0.67 is an approximation.
Picture it
These come up constantly. Knowing them cold saves a division every time.
Figure (svg): A number line from zero to one with the fractions one eighth, one quarter, one half and three quarters marked above and their decimal values below
Add one fifth equals 0.2 and one third is about 0.333, and you can handle most sixth-grade problems without any working.
Ranking
They are deliberately mixed between the two notations.
Put in order
Why: Convert everything to decimals first: 0.2, then 0.25, then 0.375, then 0.5, then 0.75. Putting them all in one notation is the whole technique, and it is much safer than trying to compare a decimal against a fraction directly.
Section
Section 4
Concept
Two and three fifths is a whole-number part and a fraction part sitting side by side. Only the fraction part needs any work.
Convert the fraction, then put the whole number back in front of it. The whole number never touches the decimal point.
Picture it
Two complete bars, then three of five strips of a third bar.
Figure (svg): Two fully shaded bars of five parts each, followed by a third bar with three of its five parts shaded, labelled two and three fifths equals 2.6
Worked example
Split the number into its two parts before doing anything.
Set the whole number aside
Why: The 2 will be the digits in front of the point. It needs no conversion at all.
Convert the fraction part by scaling
Why: Five times two is ten, so multiply top and bottom by two.
\[ \frac{3}{5} = \frac{6}{10} = 0.6 \]
Put the whole number back in front
Why: Two, point, six.
\[ 2\tfrac{3}{5} = 2.6 \]
Verify: using the improper fraction instead
Why: Two and three fifths is thirteen fifths. Thirteen divided by five is 2.6. Two routes, same answer.
Picture it
Between 2 and 3 there are five fifths, and each one is two tenths wide.
Figure (svg): A number line from two to three with fifths marked above and decimal values marked below, and two point six highlighted
Worked example
Now the other direction, starting from a decimal bigger than one.
Split at the decimal point
Why: The 1 in front is the whole number. Only the digits after the point become the fraction.
Read the fraction part off the chart
Why: One decimal place means tenths.
\[ 0.4 = \frac{4}{10} \]
Simplify the fraction part
Why: The greatest common factor of 4 and 10 is 2.
\[ 1.4 = 1\tfrac{2}{5} \]
Verify: by converting back
Why: Two fifths scales to four tenths, which is 0.4, and putting the 1 back in front gives 1.4.
Picture it
Ten tenths make the first bar. Four more tenths is two of the five strips on the second.
Figure (svg): One fully shaded bar of ten parts followed by a second bar of five parts with two shaded, labelled 1.4 equals one and two fifths
Trap
Converting two and three fifths.
\[ 2\tfrac{3}{5} = 2.35 \]
Write the 3 and the 5 after the point
Why: The digits of the fraction were copied across without being divided.
Get a number that is nowhere near right
Why: Two and three fifths is more than two and a half. 2.35 is less than two and a half, so the answer fails the roughest possible check.
Converting two and three fifths.
\[ \frac{3}{5} = 0.6, \quad \text{so} \quad 2\tfrac{3}{5} = 2.6 \]
Divide the fraction part, then attach the whole number
Why: Three fifths is a division, not two digits.
Check against one half
Why: Three fifths is more than a half, so the decimal part must be above 0.5. It is.
Estimation
Do not calculate. Compare against a half.
Predict first
Is five eighths above or below 0.5, and roughly by how much?
Correct: Above, by about 0.125.
Why: Four eighths is exactly a half, so five eighths is one eighth more, and one eighth is 0.125. That makes five eighths 0.625. Comparing against a benchmark like this takes two seconds and catches most wrong answers before you do any work.
Notation
Textbooks and calculators write repeating decimals in a special way. The notation is worth ten seconds.
Annotate
On: \( \frac{2}{3} = 0.6666\ldots = 0.\overline{6} \)
Neither notation is a rounded value. Both are exact.
Pattern
Dividing 2 by 3, one step at a time. Keep your eye on the remainder column.
Step through it
What would have to happen for the division to stop, and can it happen here?
A decimal stops only when a remainder of zero appears. Here the remainder is 2 at every step, so zero never comes.
Socratic
Take a real minute on this one.
Discussion prompt
Why do denominators of 2 and 5 give decimals that stop, while 3 and 7 do not?
Hint: What are the prime factors of 10, of 100, and of 1000?
Answer:
Because a decimal is a fraction whose denominator is a power of ten, and ten is 2 times 5.
A hundred is 2 times 2 times 5 times 5. A thousand is three 2s and three 5s. So any denominator built only from 2s and 5s can be scaled up to fit.
A 3 or a 7 in the denominator can never be cancelled by a power of ten, because no power of ten contains those factors. The division has nowhere to land.
Common Core State Standards for Mathematics, 5.NBT.A.3 (read and write decimals as fractions) and 7.NS.A.2d (fractions convert to terminating or repeating decimals) 7.NS.A.2d
Definition probe
The rule says: in lowest terms, a denominator built only from 2s and 5s gives a decimal that stops.
Sort into buckets
Does each fraction satisfy the rule, or break it?
Elimination
Converting three and one quarter into a decimal.
Eliminate the wrong options
Which answer would you write down?
Survives elimination: A
Why: One quarter scales to twenty-five hundredths, which is 0.25, and the whole number 3 goes in front unchanged. Both wrong-digit answers come from copying a digit across instead of dividing, which is the single commonest error with mixed numbers.
Trade off
Neither one is always better. Knowing which to reach for is part of the skill.
Comparison matrix
| situation | reach for |
|---|---|
| adding thirds and sixths | fractions, because thirds have no exact decimal |
| comparing a long list of numbers | decimals, because you compare digit by digit |
| money and measurement | decimals, because that is how they are written |
| a multi-step problem needing an exact answer | fractions, converting only at the end |
The habit worth building: work in fractions when exactness matters, convert to decimals when comparing or reporting.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Which of these is exactly equal to 0.125?
Correct: One eighth.
Why: Three decimal places means 125 over 1000. The greatest common factor of 125 and 1000 is 125, giving 1 over 8. Checking, 1 divided by 8 is 0.125.
Section
Section 5
Concept
A question that mixes the two is really asking whether you can convert. Pick one notation, move everything into it, then compare.
Decimals are usually the easier target, because comparing decimals is just comparing digits from the left.
Illustrative Mathematics — grade 5 and 6 tasks on decimal and fraction equivalence — grade 6 comparison tasks are built this way
Worked example
They are in different notations, so nothing can be compared yet.
Convert the fraction to a decimal
Why: Three divided by eight is 0.375, which you worked out earlier in this deck.
\[ \frac{3}{8} = 0.375 \]
Line up the decimal places
Why: 0.400 against 0.375. Adding the trailing zeros makes the comparison a digit-by-digit one.
Compare from the left
Why: Both have 0 ones. In the tenths column, 4 beats 3, so the comparison is decided right there.
\[ 0.4 > \frac{3}{8} \]
Verify: the other direction
Why: Converting 0.4 to a fraction gives 4 over 10, which is 2 over 5, and scaling to fortieths gives 16 over 40 against 15 over 40. Same conclusion.
Picture it
Once they are in the same notation, the picture answers the question.
Figure (svg): A number line from 0.3 to 0.5 with three eighths marked at 0.375 and 0.4 marked slightly to its right
Trap
Which is bigger, 0.4 or three-eighths?
Compare 4 against 8 and pick the eighths
Why: The 8 is a denominator, not a size. A bigger denominator means smaller pieces, not a bigger number.
Answer three-eighths
Why: Wrong, and wrong for a reason that will repeat on every question of this type until the habit changes.
Which is bigger, 0.4 or three-eighths?
Convert first, compare second
Why: Three-eighths is 0.375. Now both numbers are the same kind of thing.
Compare place by place from the left
Why: Tenths column: 4 against 3. So 0.4 is larger.
Real world
A recipe calls for 0.75 of a cup. Your measuring cups are marked in halves, thirds and quarters.
Discussion prompt
Which cup do you reach for, and how did you decide?
Hint: Convert the decimal into a fraction and see which marking it matches.
Answer:
Three-quarters. 0.75 is 75 over 100, which simplifies to 3 over 4, and that is exactly the marking on the cup.
This is the everyday version of the whole session: the world writes the same number in whichever notation suits the tool, and you have to move between them without thinking about it.
Prices, measurements, sports statistics and test scores all mix the two, usually in the same sentence.
Comparison
The rows you should eventually know by heart.
Comparison matrix
| fraction | decimal | percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
Percent is just a third notation for the same thing: hundredths with a special symbol. Sixth grade adds this column, and it comes free once the first two are solid.
Pattern
Two directions, and one check that works for both.
| going this way | do this |
|---|---|
| decimal into fraction | say it aloud, write digits over the place value, simplify |
| fraction into decimal, denominator of 2s and 5s | scale up to 10, 100 or 1000, then read it off |
| fraction into decimal, any other denominator | long division, and expect it to repeat |
| comparing a mix | convert everything to decimals, then compare from the left |
| checking any answer | convert it back the other way and see if you land where you started |
NWEA MAP Growth learning continuum — The Real and Complex Number Systems: convert between fractions and decimals — everything above sits in the RIT 201-220 band
Check
Solve it on paper before you click.
Check your understanding
Write nine twenty-fifths as a decimal.
Answer: A
Why: Twenty-five goes into a hundred four times, so multiply top and bottom by 4: 36 over 100, which is 0.36. Checking, 0.36 is 36 over 100, which divides by 4 to give 9 over 25.
Check
Solve it on paper before you click.
Check your understanding
Which of these fractions, all in lowest terms, has a decimal that repeats forever?
Answer: A
Why: Six is 2 times 3, and the 3 is the problem: no power of ten is divisible by 3. Five sixths is 0.8333 continuing forever.
Exit ticket
One honest answer, and it decides what we open with next time.
Predict first
Which of these still feels shakiest?
Correct: Whichever you picked is where we start next session.
Why: Naming the weak step is worth more than another pass over the strong ones. Every item on that list is a ten-minute fix once it is named.
Connect it up
Index card, both sides, your handwriting. This is the thing you will actually use during homework.
Draw it
On one side write the six benchmark pairs from the number line. On the other side write the four-step recipe for each direction. Add one example of your own to each.
Keep it in your maths folder. If you use it three times this week you will stop needing it.
Recap
Both directions, plus the check that proves either one.
| fraction | decimal |
|---|---|
| 1/2 | 0.5 |
| 1/4 | 0.25 |
| 3/4 | 0.75 |
| 1/5 | 0.2 |
| 1/8 | 0.125 |
| 3/8 | 0.375 |
| 7/20 | 0.35 |
| 2/3 | 0.666 repeating |
OpenStax, Prealgebra 2e, Ch. 5 — Decimals Ch. 5 — more practice than one session can hold
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