Grade 2 · 2.MD.C.8 · 4.MD.A.2

Money

A dime is smaller than a nickel but worth more. How do you know what each coin and bill is worth, make the same amount in different ways, and hand back the right change?

You need

Mental strategiesGrades 2 to 5 Subtraction as an unknown addendGrade 1

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Compensation when adding and subtractingGrade 3 Counting mixed coinsGrade 2 Counting up to find a differenceGrade 3 Money written as decimalsGrade 5 +1 more

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When you reach for it

A child who can count to a hundred can count a handful of coins and say “seven”. That answers how many coins there are. It does not answer how much money there is, because coins are not all worth the same. Seven pennies buy almost nothing; seven quarters buy a lot. Money is the first place a child meets things that are counted by their worth, not by how many there are.

So three jobs come together in this lesson, one for each day.

Day 1: what each coin and bill is worth, how to write an amount with the cent sign or the dollar sign, and why 100 cents make one dollar.
Day 2: the same amount in different ways. Two piles can look nothing alike and still be worth the same. Some ways use fewer coins than others.
Day 3: making change. When someone pays more than the price, the change is the part that goes back. It is found by counting up from the price to the amount paid.

All three rest on two things the child already has. One is Skip counting by fives, tens and twenty-fives. The other is the missing part: the price and the change together make the amount paid, which is subtraction as a missing part. Amounts here stay within 100 cents, or are whole dollars. Writing dollars and cents together, as in \text{\textdollar}3.40, comes later in Money written as decimals.

Real coins help more than anything printed. A small dish of pennies, nickels, dimes and quarters, and a few play bills, is enough for every example below.

The procedure

Part 1: Coins, bills, and what each is worth

The four coins. The coins in everyday use in the United States are the penny, the nickel, the dime and the quarter. The United States stopped making new pennies in 2025, but the pennies already made are still money and still turn up in change.

Coin	Worth	Written
penny	1 cent	1\text{¢}
nickel	5 cents	5\text{¢}
dime	10 cents	10\text{¢}
quarter	25 cents	25\text{¢}

Figure 1The four coins. Size does not tell you the worth; the dime is the smallest and is worth more than the penny and the nickel.

The worth of a coin is not something a child can see. It is a fact about the coin, learned the way a name is learned. Size does not decide it: the dime is the smallest coin of the four and is worth more than the penny and the nickel, and the nickel is bigger than the penny and the dime. What a child can see helps tell the coins apart. The penny is copper-colored and the other three are silver-colored. The penny and the nickel have smooth edges; the dime and the quarter have ridges around the edge. Each has a different face on the front: Abraham Lincoln on the penny, Thomas Jefferson on the nickel, Franklin Roosevelt on the dime, George Washington on the quarter. Let the child sort a pile into four little stacks, one kind of coin to a stack, before any adding.

The cent sign and the dollar sign. A cent is the smallest amount of money. The cent sign \text{¢} goes after the number: 25 cents is written 25\text{¢}. A dollar is a larger unit, and the dollar sign goes before the number: five dollars is written \text{\textdollar}5. Read both aloud the same way, number first: “twenty-five cents”, “five dollars”. At this stage an amount is written in cents or in whole dollars, never both at once.

The four bills. Paper money comes in bills. The ones a child handles most are worth \text{\textdollar}1, \text{\textdollar}5, \text{\textdollar}10 and \text{\textdollar}20.

Figure 2Bills are all the same size. The number printed on them is the only way to tell them apart.

Bills are all the same size and shape, so again the worth has to be read: it is printed as a number in the corners and written in words. A \text{\textdollar}5 bill is worth five \text{\textdollar}1 bills; a \text{\textdollar}20 bill is worth twenty of them.

A hundred cents make a dollar. One dollar is worth exactly 100 cents. That fact ties the coins to the bills: 100 pennies, 20 nickels, 10 dimes or 4 quarters are each worth one dollar, the same as one \text{\textdollar}1 bill.

Figure 3One dollar, three ways. The bars are the same length because each row is worth 100 cents.

Each row in the picture can be checked by skip counting. Quarters: 25, 50, 75, 100, four counts. Dimes: 10, 20, 30 and on to 100, ten counts. Nickels count by fives and take twenty counts to reach 100. This is the same kind of fact as ten ones making a ten in Tens and ones: a big unit that is worth a set number of small ones. Here the big unit is the dollar and the small one is the cent.

Counting coins of one kind. Since every coin in a stack is worth the same, count the stack by that coin’s worth. Pennies count by ones, nickels by fives, dimes by tens and quarters by twenty-fives. Point at each coin as you say the number; the last number said is the worth of the stack. Bills of one kind count the same way: three \text{\textdollar}5 bills are “five, ten, fifteen”, \text{\textdollar}15.

Counting two kinds. With two kinds, count the coins worth more first, then carry on from there by the smaller worth. Two dimes and three pennies: “ten, twenty”, then “twenty-one, twenty-two, twenty-three”, 23\text{¢}. Starting with the bigger coins means the long skip counts happen first and the small steps come last. Counting a handful of three or four kinds is its own lesson, Counting mixed coins.

Part 2: The same amount in different ways

Equal worth. Two piles of money are worth the same when they count to the same amount. They need not have the same number of coins. One dime and two nickels are both worth 10\text{¢}, so a dime can be traded for two nickels and nobody loses anything. These trades are worth knowing by heart:

Trade	because
1 nickel for 5 pennies	5 = 1 + 1 + 1 + 1 + 1
1 dime for 2 nickels	10 = 5 + 5
1 quarter for 5 nickels	25 = 5 + 5 + 5 + 5 + 5
1 dollar for 4 quarters	100 = 25 + 25 + 25 + 25
\text{\textdollar}10 for two \text{\textdollar}5 bills	10 = 5 + 5
\text{\textdollar}20 for two \text{\textdollar}10 bills	20 = 10 + 10

A trade changes how many coins or bills there are. It never changes the worth.

Every way, with two kinds. To list every way to make an amount with two kinds of coin, start with as many of the bigger coin as fit, then trade them away one at a time. With dimes and nickels, 20\text{¢} is 2 dimes. Trade one dime for two nickels: 1 dime and 2 nickels. Trade the other: 4 nickels. No dimes are left to trade, so there are 3 ways. Each trade moves down one step, so no way is skipped and none is listed twice. This is the same staircase as listing every split of a number in number bonds.

The fewest coins. Among all the ways, the one with the fewest coins uses as many of the bigger coin as it can. Each bigger coin does the work of several smaller ones, so swapping it in can only shrink the pile. With quarters and dimes, for example, every quarter used in place of dimes means fewer coins in all.

One catch: the part left after the big coins must be something the small coins can make. Make 30\text{¢} with quarters and dimes. One quarter leaves 5\text{¢}, and no number of dimes makes 5\text{¢}. So no quarter can be used, and the only way is 3 dimes. The rule is: use as many big coins as you can while the rest can still be made exactly.

Part 3: Making change by counting up

What change is. When the amount paid is more than the price, the seller gives some money back. That money is the change. The price and the change together make the amount paid:

\text{price} + \text{change} = \text{amount paid}

So the change is a missing part. It could be found by subtracting the price from the amount paid, but there is an easier way, and it is the way a cashier does it.

Counting up. Start at the price and count up to the amount paid, handing over one coin or bill for each hop. The hops that are easiest to make are hops to a landmark: the next 5 or 10 cents, then the next quarter (25, 50 or 75 cents), then the whole dollar. For whole dollars the landmarks are \text{\textdollar}5, \text{\textdollar}10 and \text{\textdollar}20. Each hop is one coin or bill, so the coins in the child’s hand at the end are the change, and adding up the hops tells how much it is.

Why it works. Every hop is added on top of the price, and the hops stop exactly at the amount paid. So the price and all the hops together make the amount paid, which is just what the change must do. Counting up is finding the missing part of \text{price} + \square = \text{amount paid}, the same move as in Counting up to find a difference.

Checking. Add the change back to the price. If the sum is the amount paid, the change is right.

Worked examples

Examples 1 and 2 go with Day 1, Examples 3 to 5 with Day 2, and Examples 6 and 7 with Day 3. Lay real or play coins on the table and let the child move them.

Example 1

Dimes in the fare jar

Ihaka keeps a jar of dimes for his bus fare. He tips out 7 dimes. How much money is that? How many more dimes would make one dollar?

Solution

All the coins are dimes, so count by tens, touching each dime once: 10, 20, 30, 40, 50, 60, 70. The jar holds

70\text{¢}

A dollar is 100 cents. Keep counting by tens from 70: “80, 90, 100”. That is three more counts, so 3 more dimes make a dollar. As a missing part, 70 + 30 = 100, and 30\text{¢} is 3 dimes. A child who knows that 10 dimes make a dollar can also see it as 7 + 3 = 10 dimes.

Example 2

Bills in the cash box

At the school bake table, Boyana opens the cash box. It holds three \text{\textdollar}5 bills and two \text{\textdollar}1 bills. How much money is in the box?

Solution

Count the bills worth more first. The \text{\textdollar}5 bills count by fives: “five, ten, fifteen”. Then carry on by ones for the \text{\textdollar}1 bills: “sixteen, seventeen”. The box holds

\text{\textdollar}17

Check by adding the two stacks: three fives are 5 + 5 + 5 = 15, and 15 + 2 = 17. Counting the ones first, “one, two”, and then the fives, “seven, twelve, seventeen”, gives the same total, but it means skip counting by fives from an awkward start. Biggest first keeps the counting easy.

Example 3

Every way to make twenty-five cents

At a seed swap, a packet of bean seeds costs 25\text{¢}. Akmaral has only dimes and nickels. Find every way she can pay exactly 25\text{¢}.

Solution

Start with as many dimes as fit. Two dimes are 20\text{¢}; three would be 30\text{¢}, too much. Two dimes leave 5\text{¢}, which is one nickel. So the first way is 2 dimes and 1 nickel: “10, 20, 25”.

Now trade one dime for two nickels. The worth stays the same: 1 dime and 3 nickels, “10, 15, 20, 25”.

Trade the last dime: 0 dimes and 5 nickels, “5, 10, 15, 20, 25”.

Dimes	Nickels	Count
2	1	10 + 10 + 5 = 25
1	3	10 + 5 + 5 + 5 = 25
0	5	5 + 5 + 5 + 5 + 5 = 25

There are 3 ways. No dimes are left to trade, so the list is complete. Notice that every way has an odd number of nickels. That is because 25 ends in 5 and dimes only ever make amounts ending in 0, so the nickels have to supply the 5 on the end. Of course, a single quarter also pays 25\text{¢}, but quarters were not in her hand.

Example 4

The fewest coins

At the seed swap, a bag of sunflower seeds costs 60\text{¢}. Ihaka has quarters and dimes. He wants to pay with as few coins as he can. Which coins should he use?

Solution

Use as many quarters as can fit while the rest can still be made with dimes.

3 quarters are 75\text{¢}, more than 60\text{¢}. Too many.
2 quarters are 50\text{¢}. That leaves 60 - 50 = 10 cents, which is 1 dime. This works: 2 quarters and 1 dime, 3 coins.

25 + 25 + 10 = 60

To see that nothing beats it, look at the other choices. 1 quarter leaves 35\text{¢}, and dimes make 30\text{¢} or 40\text{¢} but never 35\text{¢}, so that does not work at all. 0 quarters means 6 dimes, which is 6 coins. So the fewest is 2 quarters and 1 dime, 3 coins.

The middle case is the one to notice. “Fewer quarters” did not just give more coins; it gave no way at all, because the part left over did not end in 0.

Example 5

Same worth, different piles

Ihaka has 3 quarters. Boyana has 7 dimes and 1 nickel. Boyana has more coins. Does she have more money?

Solution

Count each pile by its worth. Ihaka’s quarters: 25, 50, 75. He has 75\text{¢}.

Boyana’s pile has two kinds, so count the dimes first: 10, 20, 30, 40, 50, 60, 70. Then the nickel: 75. She has 75\text{¢}.

25 + 25 + 25 = 75 \qquad 70 + 5 = 75

The piles are worth the same. Boyana has 8 coins and Ihaka has 3, but the number of coins says nothing about the worth. Each of Ihaka’s quarters could be traded for 2 dimes and 1 nickel, so his 3 quarters could become 6 dimes and 3 nickels; trading two of those nickels for a dime gives exactly Boyana’s pile.

Example 6

Change from a dollar

At the bake table a muffin costs 35\text{¢}. Akmaral pays with a \text{\textdollar}1 bill. How much change does Boyana give her, and which coins could it be?

Solution

The amount paid is \text{\textdollar}1, which is 100 cents. Count up from the price to 100, hopping to landmarks.

Figure 4Counting up from the price to the dollar that was paid. Each hop is one coin of change.

From 35 to 40 is 5 cents: a nickel. Now the count is on a ten.
From 40 to 50 is 10 cents: a dime. Now it is on a quarter mark, 50.
From 50 to 75 is 25 cents: a quarter.
From 75 to 100 is 25 cents: another quarter.

The change is a nickel, a dime and two quarters. Count it, biggest first: 25, 50, 60, 65.

5 + 10 + 25 + 25 = 65

Akmaral gets 65\text{¢}. Check: 35 + 65 = 100, the dollar she paid. Boyana, handing over the coins, says the running total out loud: “35; 40, 50, 75, and one dollar.” She never has to find 65 before she starts; the coins in Akmaral’s hand are the answer.

Example 7

Change in whole dollars

Ihaka buys a bus pass for \text{\textdollar}14 and pays with a \text{\textdollar}20 bill. How much change does he get?

Solution

Count up from \text{\textdollar}14 to \text{\textdollar}20. The nearest landmark is \text{\textdollar}15.

From \text{\textdollar}14 to \text{\textdollar}15 is \text{\textdollar}1: a \text{\textdollar}1 bill.
From \text{\textdollar}15 to \text{\textdollar}20 is \text{\textdollar}5: a \text{\textdollar}5 bill.

1 + 5 = 6

The change is \text{\textdollar}6. Check: 14 + 6 = 20.

Counting up chose the bills too: one \text{\textdollar}1 and one \text{\textdollar}5, two bills. Six \text{\textdollar}1 bills would be the same \text{\textdollar}6, only in more pieces.

Boundaries

Other coins and bills exist. There are half-dollar and one-dollar coins, and \text{\textdollar}2, \text{\textdollar}50 and \text{\textdollar}100 bills. They are seldom handled, so this lesson leaves them out, but every idea here works for them the same way.
Worth is not size, and not the number of coins. A pile of 8 coins can be worth the same as a pile of 3 (Example 5). Only counting by worth compares two piles.
Some amounts cannot be made with the coins at hand. Dimes and nickels only make amounts that end in 0 or 5, since every count by fives or tens lands on one of those. So 23\text{¢} needs pennies. Quarters and dimes cannot make 5\text{¢} or 15\text{¢} at all.
“Fewest coins” depends on which coins are allowed. With quarters and dimes, the fewest coins for 30\text{¢} is 3 dimes. If nickels are allowed too, a quarter and a nickel do it in 2. Say which coins are allowed before asking for the fewest.
Change needs the amount paid to be at least the price. If it is less, there is no change; the buyer still owes the missing part. If it is equal, the change is 0.
Amounts stay within 100 cents, or are whole dollars. Six quarters are 150 cents, which is one dollar and 50 cents. Writing that as one amount needs the decimal point of Money written as decimals, so it waits until then.

Where this goes next

The next step with coins is Counting mixed coins: a handful of three or four kinds, sorted and counted biggest first, which is Example 2’s habit stretched to more stacks. Story problems that use the cent sign and the dollar sign, adding prices and finding what is left, are Story problems with dollars and cents. Counting up for change grows into Counting up to find a difference, which counts up across hundreds and makes change in dollars and cents together. Later, Money written as decimals writes a dollar and its cents as one number, with the cents as hundredths of a dollar, and Compensation when adding and subtracting adds a price such as 99\text{¢} by adding a dollar and taking a cent back. Comparing what one item costs across sizes is Unit price and comparing rates.

Exercises

Coins, bills and what they are worth (Day 1)

Which coin is worth 10 cents? Which is worth 5 cents? Which of the four coins is the biggest, and what is it worth?
How much are 6 nickels worth?
How much are 3 quarters and 1 dime worth?
Boyana has 8 dimes. How much money is that? How many more dimes would make a dollar?
How much are four \text{\textdollar}5 bills worth?
Akmaral has two \text{\textdollar}10 bills and three \text{\textdollar}1 bills. How much money does she have?
How many pennies make a dollar? How many nickels?
Write forty-five cents with the cent sign, and nine dollars with the dollar sign.

The same amount in different ways (Day 2)

Find every way to make 30\text{¢} with dimes and nickels.
What is the fewest number of coins that makes 50\text{¢} with quarters and dimes? Which coins?
What is the fewest number of coins that makes 35\text{¢} with dimes and nickels? Which coins?
Ihaka has 2 quarters. Akmaral has 5 dimes. Who has more money?
Boyana has 9 nickels. Ihaka has 4 dimes. Who has more money, and how much more?
Find every way to make \text{\textdollar}15 with \text{\textdollar}5 bills and \text{\textdollar}1 bills.
What is the fewest number of coins that makes 80\text{¢} with quarters and nickels? Which coins?
Using only quarters and dimes, can you make 45\text{¢}? Can you make 15\text{¢}?

Making change (Day 3)

A pencil costs 85\text{¢}. You pay with a \text{\textdollar}1 bill. How much change do you get?
A sticker costs 45\text{¢}. You pay with 2 quarters. How much change do you get?
A packet of seeds costs 38\text{¢}. You pay with 5 dimes. How much change do you get, and which coins could it be?
A bag of pretzels costs 55\text{¢}. You pay with a \text{\textdollar}1 bill. How much change do you get? Count up to choose the coins.
A seed tray costs \text{\textdollar}12. You pay with a \text{\textdollar}20 bill. How much change do you get?
A book costs \text{\textdollar}26. You pay with a \text{\textdollar}20 bill and a \text{\textdollar}10 bill. How much change do you get?

Explain

A dime is smaller than a nickel. How can it be worth more?
Why does counting up from the price to the amount paid give the change?
Which amounts of cents can be made with only nickels and dimes? How do you know?

Answers

The dime is worth 10 cents and the nickel 5 cents. The quarter is the biggest, and it is worth 25 cents.
30\text{¢}. Count by fives, six times: 5, 10, 15, 20, 25, 30.
85\text{¢}. Quarters first, by twenty-fives: 25, 50, 75. Then the dime: 85.
80\text{¢}, and 2 more dimes. Count by tens to 80; then “90, 100” is two more dimes, since 80 + 20 = 100.
\text{\textdollar}20. Count by fives: “five, ten, fifteen, twenty”.
\text{\textdollar}23. Biggest first: “ten, twenty”, then “twenty-one, twenty-two, twenty-three”. So 10 + 10 + 3 = 23.
100 pennies and 20 nickels. A penny is 1 cent, so a dollar is 100 of them. Counting by fives reaches 100 after twenty counts.
45\text{¢} and \text{\textdollar}9. The cent sign goes after the number, the dollar sign before it.
4 ways: 3 dimes; 2 dimes and 2 nickels; 1 dime and 4 nickels; 6 nickels. Start with 3 dimes and trade one dime for two nickels at a time.
2 coins: 2 quarters, since 25 + 25 = 50. Any way with dimes uses more coins: 1 quarter leaves 25\text{¢}, which dimes cannot make, and 5 dimes is 5 coins.
4 coins: 3 dimes and 1 nickel. Three dimes are 30\text{¢}, four would be too many, and the 5\text{¢} left is one nickel: 10 + 10 + 10 + 5 = 35.
Neither; they have the same. 2 quarters are 25 + 25 = 50 cents, and 5 dimes are 50 cents too.
Boyana, by 5\text{¢}. Her 9 nickels count by fives to 45\text{¢}; Ihaka’s 4 dimes are 40\text{¢}. Counting up from 40 to 45 is 5.
4 ways: three \text{\textdollar}5 bills; two \text{\textdollar}5 bills and five \text{\textdollar}1 bills; one \text{\textdollar}5 bill and ten \text{\textdollar}1 bills; fifteen \text{\textdollar}1 bills. Each step trades one \text{\textdollar}5 bill for five \text{\textdollar}1 bills.
4 coins: 3 quarters and 1 nickel. Three quarters are 75\text{¢}, four would be 100\text{¢}, too much, and 80 - 75 = 5 is one nickel.
45\text{¢}, yes: 1 quarter and 2 dimes, since 25 + 10 + 10 = 45. This is the only way, because with no quarter the dimes would have to make 45\text{¢}, which does not end in 0. 15\text{¢}, no: a quarter is already too much, and dimes make 10\text{¢} or 20\text{¢}, never 15\text{¢}.
15\text{¢}. Count up: 85 to 90 is a nickel, 90 to 100 is a dime, and 5 + 10 = 15. Check: 85 + 15 = 100.
5\text{¢}. Two quarters are 50\text{¢}, and 45 to 50 is one nickel. Check: 45 + 5 = 50.
12\text{¢}: 2 pennies and 1 dime. Five dimes are 50\text{¢}. Count up: 38 to 40 is 2 pennies, 40 to 50 is a dime, and 2 + 10 = 12. Check: 38 + 12 = 50.
45\text{¢}. Count up: 55 to 75 is 20 cents, two dimes (65, 75); 75 to 100 is a quarter. So 10 + 10 + 25 = 45. Check: 55 + 45 = 100.
\text{\textdollar}8. Count up: \text{\textdollar}12 to \text{\textdollar}15 is three \text{\textdollar}1 bills, and \text{\textdollar}15 to \text{\textdollar}20 is a \text{\textdollar}5 bill. So 3 + 5 = 8. Check: 12 + 8 = 20.
\text{\textdollar}4. The two bills are 20 + 10 = 30 dollars. Count up from \text{\textdollar}26 to \text{\textdollar}30: four \text{\textdollar}1 bills. Check: 26 + 4 = 30.
A coin’s worth is not set by its size. It is a fact about that coin, the way a word has a meaning, and the dime was made to be worth 10 cents. Ten pennies or two nickels are worth the same as one dime, however big they are.
Each hop is added to the price, and the hops stop exactly at the amount paid. So the price and the hops together make the amount paid. That is exactly what the change has to do, so the hops added up are the change.
Only amounts that end in 0 or 5: 5, 10, 15, 20 and so on. Nickels count by fives and dimes count by tens, and every count by fives or by tens lands on a number ending in 0 or 5. Every such amount can be made, since nickels alone reach each one.