Counting mixed coins

When you reach for it

A child who has met Money knows what each coin is worth and can count a stack of one kind: quarters by twenty-fives, dimes by tens. They can also count two kinds, bigger first. Real money does not come that tidy. A pocket, a purse or a piggy bank gives up a jumble of three or four kinds at once, and the question “how much is this?” has to be answered from the jumble.

The move this lesson teaches is to sort first, then count once, biggest worth first, switching the skip count each time the kind of coin changes. It is Skip counting put to work, with one new demand: the count does not always start from zero. After two quarters the dimes are counted on from 50; after one quarter they are counted on from 25, and “35, 45, 55” is a skip count by tens that never lands on a ten. The breaking apart and counting on of Mental strategies carry the child through those spots.

The lesson runs over four days.

• Day 1: sort and count. Three or four kinds of coin, sorted into stacks and counted biggest first, switching from twenty-fives to tens to fives to ones.
• Day 2: counting on from an awkward spot. Coins that arrive one at a time and cannot be sorted, and a quarter or a dime added to a count that is not on a ten.
• Day 3: past a dollar. A count that goes beyond 100 cents, said as dollars and cents, and bills and coins counted together.
• Day 4: using the count. Comparing two handfuls, and deciding whether there is enough to buy something.

Real coins on the table do more than any picture. Every example below can be laid out with a dish of pennies, nickels, dimes and quarters and a few play bills.

The procedure

Part 1: Sort, then count biggest first

Sort before counting. Tip the coins out and push them into four little stacks, one kind to a stack: quarters, dimes, nickels, pennies. Then line the stacks up from the biggest worth to the smallest. Sorting is not counting, and it is worth doing as its own step, because the counting that follows only ever has to remember one thing: the number just said.

Count once, switching the step. Point at the first quarter and say its worth. Touch each coin in turn, and add that coin’s worth to the number just said. While the coins are quarters the count goes up by twenty-fives; when the dimes start it goes up by tens; then fives for the nickels and ones for the pennies. The last number said is what the whole handful is worth.

Why biggest first. The order of the coins does not change the total: adding in any order gives the same sum, which is the commutative and associative properties of addition. So the order is chosen to make the counting easy. Counting by twenty-fives is only easy from 0: 25, 50, 75, 100 is a chant a child can learn. Counting by tens is easy from anywhere, since only the tens digit moves. Counting by ones is easy from anywhere. So the hardest step goes first, from 0, and the easy steps come after, from wherever the count has reached. Pennies counted first would leave the quarters to be counted on from something like 3, which is hard for anyone.

Say each switch out loud. At the moment the kind of coin changes, the child should notice it. A helpful habit is to pause and say the new step: “quarters: 25, 50. Now tens: 60, 70. Now fives: 75, 80, 85. Now ones: 86, 87.” The pause is where the place is most easily lost.

Part 2: Counting on from an awkward spot

Counting by tens from a number ending in 5. After one quarter or three quarters, the count ends in 5, and the dimes have to be counted on from there: 25, 35, 45, 55. Only the tens digit moves; the 5 in the ones place stays. A hundred chart shows this well: the count walks straight down the column of numbers ending in 5.

Adding a quarter in the middle. Sometimes the coins cannot be sorted first, because they arrive one at a time: dropped into a jar, handed across a counter. Then a quarter can turn up when the count is at 20 or 35. Adding 25 to a number like that is not one easy hop, so break the quarter apart in the head. A quarter is worth the same as two dimes and a nickel, so adding a quarter is two tens and a five: from 35, “45, 55, 60”. Or a quarter is two tens and five ones: from 20, “30, 40, 45”. This is the break-apart strategy of Mental strategies, and it is trading coins in the head, which Money already justified: a trade never changes the worth.

Quarters from a dime. If the count is at 10 and quarters follow, the totals run 35, 60, 85. From 0 they run 25, 50, 75. In both lists the ones digit goes 5, 0, 5, 0, because two quarters make 50, a whole number of tens.

Part 3: Past a dollar

Crossing 100. A count can run past 100 cents with no change in method: 95, 105, 115 is still counting by tens. What changes is how the amount is said. One hundred cents is one dollar, so 120 cents is one dollar and 20 more cents. The child says and writes it in words, “1 dollar and 20 cents”. It can also be written 120\text{¢}, which is correct but not how amounts are usually said.

Writing such an amount as one number with a decimal point waits for Money written as decimals. Here the dollars and the cents are kept as two separate counts joined by “and”.

Bills and coins together. Count the bills first and keep that number of dollars. Then count the coins, as in Part 1, in cents. Put the two together: “6 dollars and 40 cents”. If the coins come to 100 cents or more, a dollar is hiding in them: take it out and add it to the dollars, so 6 dollars and 105 cents is said “7 dollars and 5 cents”. The bills and the coins are counted separately because they are counted in different units, dollars and cents, the way tens and ones are kept apart in a two-digit number.

Part 4: Using the count

Comparing two handfuls. Count each handful, then compare the two totals as numbers, using tens and ones. The number of coins says nothing about which handful is worth more. To find how much more, count up from the smaller total to the bigger one.

Is there enough? Count the handful and compare it with the price. If the handful is worth the price or more, there is enough. If it is worth less, the difference is how much more is needed, found by counting up from the total to the price. A shortcut: count biggest first and stop as soon as the count reaches the price. Whatever coins are left over are not needed.

Worked examples

Examples 1 and 2 go with Day 1, Examples 3 and 4 with Day 2, Examples 5 and 6 with Day 3, and Examples 7 and 8 with Day 4.

Boundaries

• Every coin is counted once. The count is only right if each coin is touched exactly once. Sliding each coin aside as it is counted keeps that true; a coin counted twice or skipped changes the total.
• Any order gives the same total. Biggest first is a choice made for ease, not a rule of arithmetic. Counted in any order, the same coins make the same sum (Example 3). What changes is how hard the steps are.
• Other coins. Half-dollar coins (50 cents) and dollar coins (100 cents) exist but are seldom handled. They fit the method unchanged: they go at the front of the line, biggest first.
• Past a dollar, this lesson uses words. An amount over 100 cents is said and written as dollars and cents in words, “1 dollar and 20 cents”. The single-number form with a decimal point, and adding or subtracting such amounts, belong to Money written as decimals and later lessons.
• Bills and coins are counted in different units. Bills count dollars and coins count cents. Adding a number of dollars to a number of cents as if they were the same unit gives nonsense: 5 dollars and 25 cents is not 30 of anything.
• Enough means at least the price. A handful worth exactly the price is enough. A handful worth even one cent less is not, however many coins are in it.

Where this goes next

Counting a handful is the first step of nearly every money question. Story problems with dollars and cents use it in stories about buying things and putting money together, with the dollar sign and the cent sign. Finding how much more is needed, or how much is left, by counting up grows into Counting up to find a difference, which counts across hundreds. Writing 1 dollar and 20 cents as one number, with the cents as hundredths of a dollar, is Money written as decimals. And the habit of counting the big units first and the small ones after is the same habit as counting hundreds, then tens, then ones with base ten blocks.

Exercises

Sort and count (Day 1)

1. How much money is in handful A?
2. How much money is in handful B?
3. How much money is in handful C?
4. Count 3 dimes, 3 nickels and 3 pennies. How much money is that?
5. Count 2 quarters, 4 nickels and 1 penny. How much money is that?
6. Write the numbers you say, coin by coin, when you count 1 quarter, 2 dimes and 2 nickels biggest first. What is the total?

Counting on from an awkward spot (Day 2)

7. Coins are dropped into a jar in this order: a nickel, a dime, a quarter, a dime. Say the total after each coin. How much is in the jar?
8. The count so far is 45\text{¢}. A quarter is added. What is the count now?
9. The count so far is 30\text{¢}. Two quarters are added, one at a time. Say the count after each.
10. The count so far is 15\text{¢}. A quarter and then a dime are added. Say the count after each.
11. Two dimes are on the table. Three quarters are added, one at a time. Say the total after each quarter.

Past a dollar (Day 3)

12. How much are 5 quarters worth? Say it in dollars and cents.
13. Count 3 quarters, 3 dimes and 2 pennies. Say the amount in dollars and cents.
14. Count 4 quarters, 2 dimes and 3 nickels. Say the amount in dollars and cents.
15. Count one \text{\textdollar}5 bill, three \text{\textdollar}1 bills, 1 quarter and 1 dime.
16. Count one \text{\textdollar}10 bill, one \text{\textdollar}5 bill, 2 quarters, 3 dimes and 2 nickels.
17. Count one \text{\textdollar}1 bill, 4 quarters and 1 dime.

Using the count (Day 4)

18. Radoslava has 2 quarters and 2 pennies. Nurzhan has 4 dimes and 3 nickels. Who has more money, and how much more?
19. Leimomi has 1 quarter and 5 dimes. Nurzhan has 2 quarters, 4 nickels and 3 pennies. Who has more money, and how much more?
20. At the toy shop a yo-yo costs 60\text{¢}. Radoslava has 1 quarter, 2 dimes, 2 nickels and 3 pennies. Does she have enough? If not, how much more does she need?
21. At the class store a notebook costs 95\text{¢}. Nurzhan has 3 quarters, 1 dime and 2 nickels. Does he have enough?
22. At the toy shop a toy truck costs \text{\textdollar}4. Leimomi has three \text{\textdollar}1 bills, 2 quarters, 4 dimes and 3 nickels. Does she have enough? How much will be left over?

Explain

23. Why is it easier to count the quarters first and the pennies last? Would the total be different the other way round?
24. When you count dimes on from 25, why does every number you say end in 5?
25. Can a handful of 4 coins be worth more than a handful of 10 coins? Give an example, and say why.