Ratios and proportional relationships for RIT under 215: comparing and converting metric length, mass and volume, converting with decimals, and reading metric conversion tables.
Subject: NWEA MAP Growth Math · 61 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
Comparing and converting metric length, mass and volume, including decimals
Objectives
Six MAP skills sit here, and the metric system makes all six easier than their customary cousins.
The reason metric is easier: every conversion factor is a power of ten, so converting is really just moving a decimal point.
Section
Part 1
Concept
Customary units use 12, 3, 16, 2,000 and 5,280. Metric units use 10, 100 and 1,000 and nothing else.
Figure (svg): A ladder of metric prefixes from kilo down to milli, each step being ten times the one below
Prefix — The word stuck on the front of a unit that says how big it is: kilo means a thousand, centi means a hundredth, milli means a thousandth.
Learn the prefixes once and they work identically for length, mass and volume.
Warm-up
These prefixes are already in your vocabulary, just not in a maths lesson.
Discussion prompt
Write down every word you know that starts with kilo, centi or milli, from any subject at all.
Hint: Think about computers and about time, not just measurement.
Answer:
Kilo: kilogram, kilometre, kilobyte — all mean a thousand of something.
Centi: centimetre, century, per cent — all involve a hundred.
Milli: millimetre, millilitre, millennium, millisecond — all mean a thousandth, or a thousand.
The prefix carries the number, whatever it is attached to.
Concept
Everything you learned about customary conversions still applies. Only the factors have got friendlier.
Figure (svg): The three metric length facts: a kilometre is a thousand metres, a metre is a hundred centimetres, a centimetre is ten millimetres
Pattern
Same three questions as the customary deck, with one extra convenience at the end.
Because the factor is always a power of ten, you can move the decimal point instead of doing long multiplication.
Definition probe
Get the prefixes secure and every family comes for free.
Sort into buckets
Sort each prefix by the number it stands for.
Socratic
There is a practical reason, not just a historical one.
Discussion prompt
Why would a scientist prefer to work in metric units rather than customary ones?
Hint: Compare converting 3.5 km to metres with converting 3.5 miles to feet.
Answer:
Because converting never requires memorising an awkward factor. Everything is a power of ten, so a conversion is a decimal shift rather than a multiplication.
That means far fewer arithmetic errors, which matters enormously when a measurement is used in a further calculation.
Discrimination
MAP mixes the families, so naming the family is always the first move.
Sort into buckets
Put each unit under what it measures.
Section
Part 2
Concept
Metric length has four units and three facts, and every factor is a power of ten.
Figure (svg): The three metric length facts: a kilometre is a thousand metres, a metre is a hundred centimetres, a centimetre is ten millimetres
| fact | a feel for it |
|---|---|
| 1 km = 1,000 m | a kilometre is a ten minute walk |
| 1 m = 100 cm | a metre is about a long stride |
| 1 cm = 10 mm | a centimetre is about a fingernail |
Worked example
A table is 4 metres long. How many centimetres is that?
Which unit is bigger
Why: A metre is bigger than a centimetre.
Which direction
Why: Metres to centimetres is big to small, so multiply.
Which power of ten
Why: There are 100 centimetres in a metre.
\[ 4 \times 100 = 400 \text{ cm} \]
Verify: the direction
Why: 400 is bigger than 4, exactly as a move to a smaller unit requires. A four metre table really would measure 400 on a centimetre tape.
Intuition
You do not have to do long multiplication. Multiplying by 100 slides every digit two places to the left.
Figure (svg): The digits of a number shifting two places as metres are converted to centimetres
This is the single biggest time saver in the whole metric topic.
Faded example
Supply the factor that does the work.
Fill in the blanks
6 \text1000 = 6 \times ___ = 6000 \text___
Why: A kilometre is a thousand metres, and going from the bigger unit to the smaller one multiplies, so 6 kilometres becomes 6,000 metres.
Prediction
Same two units, opposite direction.
Predict first
How many metres are in 250 centimetres?
Correct: 2.5 m
Dividing by 100 slides the digits two places to the right, which is where the decimal point comes from.
Why: A metre is bigger than a centimetre, so this is small to big and you divide by 100. Dividing 250 by 100 gives 2.5 metres.
Trap
Converting 250 cm to metres, a student multiplies by 100 and writes 25,000 m.
A metre is bigger than a centimetre, so the answer must be a smaller number than 250. Dividing gives 2.5 m.
25,000 metres is twenty five kilometres — the size of the answer alone rules it out.
Comparison
Each row uses one of the three facts.
Comparison matrix
| from | to | operation | answer |
|---|---|---|---|
| 3 m | cm | multiply by 100 | 300 |
| 7 cm | mm | multiply by 10 | 70 |
| 5,000 m | km | divide by 1,000 | 5 |
| 80 mm | cm | divide by 10 | 8 |
Every operation in that column came from which unit was bigger, never from memory.
Check
Solve it on paper before you click.
Check your understanding
A rope is 2.5 metres long. How many centimetres is that?
Answer: B
Why: A metre is bigger than a centimetre, so multiply by the 100 centimetres in a metre. Multiplying 2.5 by 100 slides the digits two places, giving 250 centimetres.
Ranking
Convert everything to the smallest unit first, then order.
Put in order
Why: In millimetres these are 45, 60, 100 and 150. Converting everything to the smallest unit turns a mixed comparison into a plain whole-number one.
Explain it to yourself
Explaining it is what makes it yours.
Discussion prompt
In your own words, how do you know how many places to slide the digits, and in which direction?
Hint: Separate the two decisions: how far, and which way.
Answer:
Count the zeros in the factor. One zero means one place, two zeros two places, three zeros three places.
The direction comes from the units: going to a smaller unit slides left and makes the number bigger; going to a bigger unit slides right and makes it smaller.
Analogy
The method is identical; only the difficulty of the factor changes.
Match the pairs
Why: Every pair is big unit to small unit, so every one multiplies. The only difference is that the metric factors are powers of ten and can be done by sliding digits instead of multiplying.
Section
Part 3
Concept
Metric mass has three units and two facts, and both factors are a thousand.
Figure (svg): Metric mass units: a kilogram is a thousand grams and a gram is a thousand milligrams
| fact | a feel for it |
|---|---|
| 1 kg = 1,000 g | a kilogram is a bag of sugar |
| 1 g = 1,000 mg | a gram is a paperclip |
Worked example
A parcel has a mass of 3 kilograms. How many grams is that?
Which unit is bigger
Why: A kilogram is bigger than a gram.
Which direction
Why: Kilograms to grams is big to small, so multiply.
Which power of ten
Why: There are 1,000 grams in a kilogram.
\[ 3 \times 1000 = 3000 \text{ g} \]
Verify: the direction
Why: 3,000 is bigger than 3, which is right for a move to a smaller unit. Three thousand paperclips is a believable mass for a small parcel.
Intuition
Nothing new is happening here. The prefixes mean exactly what they meant for length.
Figure (svg): A ladder of metric prefixes from kilo down to milli, each step being ten times the one below
Once the prefixes are secure, a new family costs you nothing to learn.
Estimation
A rough check is faster than a careful calculation, and catches direction errors.
Predict first
Roughly how many grams is 2.4 kilograms?
Correct: about 2,400
An estimate that lands in the right thousand is enough to rule out three of the four options.
Why: Each kilogram is a thousand grams, so a bit over two kilograms is a bit over two thousand grams. The exact value is 2,400 grams.
Fill the middle
Going up the ladder now, so the operation flips.
Fill in the blanks
4500 \text\div = 4500 ___ 1000 = 4.5 \text___
Why: A kilogram is bigger than a gram, so going from grams up to kilograms divides. 4,500 divided by 1,000 is 4.5 kilograms.
Anomaly
Two conversions of the same quantity, and only one could describe a real object.
\[ 2 \text{ kg} = 2000 \text{ g} \quad \text{versus} \quad 2 \text{ kg} = 0.002 \text{ g} \]
Predict first
Which conversion is correct, and how can you tell without checking the rule?
Correct: 2,000 grams, because a kilogram is far bigger than a gram
Attaching a physical picture to the unit is the fastest error check in the whole topic.
Why: Two kilograms is about two bags of sugar. Two thousandths of a gram is less than a grain of sand, which no bag of sugar could be. The size of the answer settles it before any rule is applied.
Check
Solve it on paper before you click.
Check your understanding
A book has a mass of 850 grams. What is its mass in kilograms?
Answer: B
Why: A kilogram is bigger than a gram, so this is small to big and you divide by 1,000. Dividing 850 by 1,000 gives 0.85 kilograms, which is a sensible mass for a book.
Section
Part 4
Concept
Metric volume is the shortest list in the topic: litres and millilitres, joined by a thousand.
Figure (svg): A litre bottle beside a thousand millilitre marks, showing one litre is a thousand millilitres
A millilitre is exactly the same size as a cubic centimetre, which is why medicine syringes are marked in both.
Worked example
A bottle holds 2.5 litres. How many millilitres is that?
Which unit is bigger
Why: A litre is bigger than a millilitre.
Which direction
Why: Litres to millilitres is big to small, so multiply.
Which power of ten
Why: There are 1,000 millilitres in a litre, so slide the digits three places left.
\[ 2.5 \times 1000 = 2500 \text{ mL} \]
Verify: the digit shift
Why: The 2 moved from the ones place to the thousands place, three columns to the left, which is exactly what multiplying by a thousand should do.
Intuition
A decimal in the starting number changes nothing. The digits slide the same number of places either way.
Figure (svg): The digits of a number shifting two places as metres are converted to centimetres
Count the zeros in the factor, and slide that many places. Three zeros means three places.
Pattern
Each frame slides the digits one more place.
Step through it
How many places would the point move going from millimetres back to kilometres?
The number of places equals the number of zeros in the factor. That is the whole decimal rule.
Notation
Notation trips students up more than the arithmetic does.
Annotate
On: \( 0.75 \text{ L} = \square \text{ mL} \)
Three quarters of a litre is 750 mL, which is exactly what a wine bottle is labelled.
Trap
Converting 0.75 L, a student slides only two places and writes 75 mL.
The factor is 1,000, which has three zeros, so the digits must slide three places, giving 750 mL.
Count the zeros before sliding, and add a placeholder zero if you run out of digits.
Elimination
Convert 1.2 litres to millilitres by elimination.
Eliminate the wrong options
How many millilitres are in 1.2 litres?
Survives elimination: v2
Why: A litre is bigger than a millilitre, so multiply by 1,000. Sliding 1.2 three places left gives 1,200 millilitres, a little over one bottle.
Check
Solve it on paper before you click.
Check your understanding
A jug holds 0.4 litres. How many millilitres is that?
Answer: C
Why: Multiply by the 1,000 millilitres in a litre, which slides the digits three places left. 0.4 becomes 400 millilitres.
Invariant
The digits shuffle around, but something underneath does not budge.
Step through it
What is identical on all three lines?
The amount is invariant. Only the size of the counting piece, and therefore the number, changes.
Section
Part 5
Concept
To compare two metric quantities, put them both in the same unit first — usually the smaller one, to keep the numbers whole.
Figure (svg): Two quantities converted to the same unit before being compared
Comparing 2.5 m with 230 cm directly is impossible; comparing 250 cm with 230 cm is trivial.
Step zero
A mixed item rewards sorting before calculating.
Discussion prompt
You are asked which is greater: 3,500 g or 4 kg. What should you settle before doing any arithmetic?
Hint: Pick the smaller unit so neither number needs a decimal.
Answer:
First, that both are masses, so the comparison is meaningful at all.
Second, which unit to convert into. Grams is smaller, so convert the 4 kg into 4,000 g. Then 4,000 beats 3,500.
Worked example
Which is greater: 3,500 grams, or 4 kilograms?
Name the family
Why: Both are masses, so they can be compared.
Choose one unit
Why: Grams is the smaller unit, so convert the kilograms into grams.
Convert
Why: 4 kilograms is 4 times 1,000, which is 4,000 grams.
\[ 3500 \text{ g} \quad \text{versus} \quad 4000 \text{ g} \]
Verify: the comparison
Why: 4,000 is greater than 3,500, so 4 kilograms is the greater mass. Converting downwards kept both numbers whole, which is why no decimal errors were possible.
Intuition
A table hands you the rule; you only have to spot it.
Figure (svg): The digits of a number shifting two places as metres are converted to centimetres
Take a complete row, work out what was done to the left number, and apply the same thing to the missing row.
Reverse engineer
The table shows kilometres on the left and metres on the right.
Fill in the blanks
3 \text7500 \rightarrow 3000 \text___, \quad 7.5 \text___ \rightarrow ___ \text___
Why: From the visible row, 3,000 divided by 3 is 1,000, so the rule is multiply by 1,000. Applying it to 7.5 kilometres gives 7,500 metres.
Comparison
All three families in one table, as MAP mixes them.
Comparison matrix
| quantity | family | converted |
|---|---|---|
| 2 km | length | 2000 m |
| 1.5 kg | mass | 1500 g |
| 3 L | volume | 3000 mL |
| 40 mm | length | 4 cm |
Same method every row, and every factor a power of ten.
Sorting
Direction only. No arithmetic yet.
Sort into buckets
Sort each conversion by whether the number will grow or shrink.
Settling the direction first makes the power of ten almost impossible to misapply.
Matching
Each pair describes the same amount in two units.
Match the pairs
Why: Three of these four use a thousand and only the metre to centimetre fact uses a hundred, which is exactly the fact students most often misremember.
Error analysis
A student converted 6 centimetres to millimetres.
Annotate
On: \( 6 \text{ cm} = 6 \div 10 = 0.6 \text{ mm} \)
The direction check would have caught this before any factor was even chosen.
Two truths and a lie
Two of these are sound; one is not.
Eliminate the wrong options
Which statement about the metric system is false?
Survives elimination: t2
Why: A kilogram is 1,000 grams, not 100. The prefix kilo always means a thousand, whichever base unit it is attached to, so the same 1,000 appears in kilometres and kilograms alike.
Explain it
A classmate can convert whole numbers but freezes when a decimal appears.
Discussion prompt
How would you show them that a decimal changes nothing about the method?
Answer:
Have them convert 3 metres to centimetres, then 3.5 metres. Both slide two places; the only difference is that the second one had a digit after the point to slide.
Counting the zeros in the factor tells you how many places to slide, whether or not a decimal point is present.
Commit first
Decide your answer and your confidence before revealing.
Predict first
Which is greater: 0.5 km or 400 m?
Correct: 0.5 km
Converting to the smaller unit is nearly always the easier direction for a comparison.
Why: Half a kilometre is 500 metres, and 500 is greater than 400. Converting the kilometres down to metres made both numbers whole and the comparison immediate.
Real world
This is the arithmetic behind medicine doses and food labels.
Discussion prompt
A medicine label says a dose is 0.005 grams. Why would a pharmacist rewrite that as milligrams, and what number would they write?
Answer:
0.005 grams is 5 milligrams. Multiplying by 1,000 slides the digits three places.
They rewrite it because a number with three leading zeros is easy to misread, and a misplaced decimal in a dose is dangerous. Choosing the unit that makes the number whole is a safety measure.
Edge cases
The prefixes do not stop at kilo and milli.
Discussion prompt
A kilobyte is a thousand bytes and a megabyte is a million. If the same pattern applied to metres, how many metres would a megametre be, and would you ever use it?
Hint: Apply the same power-of-ten logic you have used all deck.
Answer:
A megametre would be a million metres, or a thousand kilometres.
It is a legal unit but nobody uses it, because kilometres are already convenient at that scale. The ladder extends forever; we only name the rungs we need.
Exit ticket
One item that tells you whether the deck landed.
Predict first
A wire is 3.2 metres long. How many centimetres is that?
Correct: 320 cm
Why: A metre is bigger than a centimetre, so multiply by 100. The factor has two zeros, so the digits slide two places left, turning 3.2 into 320 centimetres.
Check
Solve it on paper before you click.
Check your understanding
Which of these is the greatest mass?
Answer: B
Why: Converting everything to grams gives 2,500, 3,000, 2,900 and 2,800. The greatest of those is 3,000 grams, so 3 kilograms is the largest mass.
Missing information
Not every comparison can be completed.
Discussion prompt
A question asks which is more: 2 kilograms or 2 litres. What is wrong with it?
Answer:
Kilograms measure mass and litres measure volume, so they are different families and cannot be converted into one another.
You would need to know what substance is involved. For water they happen to match, since a litre of water has a mass of one kilogram — but that is a fact about water, not about the units.
Counterexample
A rule is worth more once you have tried to break it.
Discussion prompt
Does kilo always mean a thousand? Try to find a metric unit where it does not, or explain why none exists.
Answer:
It always does. A kilometre is a thousand metres, a kilogram a thousand grams, a kilolitre a thousand litres.
That consistency is the whole design of the metric system: the prefix carries the number, and the base unit carries the family. Learning a prefix once buys you every family at no extra cost.
Connect it up
One page in your handwriting beats re-reading this deck.
Draw it
Draw the prefix ladder from kilo down to milli, with the factor on every rung. Beside it, write the three length facts, the two mass facts and the one volume fact. Mark the down arrow multiply and the up arrow divide.
If you can draw this from memory, every metric item becomes a decimal shift.
Recap
Six skills, one ladder, three families.
The prefix carries the number: kilo is always a thousand, centi is always a hundredth, milli is always a thousandth.
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