Problem
Count numbers formed from digit cards
There are three number cards: 5, 9 and 6. Each two-digit number uses two different cards. Count how many different two-digit numbers are possible.
Reasoning · Grade 2-2 Applying the Multiplication Facts
Your answer
Watch out Choosing two cards looks like just 3 ways, but 59 and 95 are different numbers, so order counts.
How to solve
Strategy Make a Systematic List — There are only a handful of two-digit numbers, so I Make a Systematic List by fixing the tens digit first and listing each possible ones digit. Drawing the two slots (tens box, ones box) as a Diagram makes it clear that once the tens card is chosen there are only 2 cards left for the ones place, which lets me count by multiplying 3 x 2.
1STEP 1
Set up the two place-value slots
A two-digit number has a tens slot and a ones slot.
= (tens)(ones)
Knowing a two-digit number is just a tens digit and a ones digit tells me exactly two cards are needed.
1.NBT.B.2Draw A Diagram2STEP 2
Count choices for the tens place
Any of the 3 cards can take the tens place.
tens choices = 3
All three cards are real choices for the first slot, so the first slot has 3 options.
4.OA.A.3Make A Systematic List3STEP 3
Count choices for the ones place
That leaves 2 cards for the ones place.
ones choices = 2
Using a card up leaves one fewer card, so the second slot always has 2 options.
4.OA.A.3Make A Systematic List4STEP 4
Multiply the choices (and check by listing)
Multiplying gives 3 × 2 = 6, and the list 59, 56, 95, 96, 65, 69 confirms it.
3 × 2 = 6
Each of the 3 tens choices branches into 2 ones choices, so 3 groups of 2 makes 6 in total.
4.OA.A.3Make A Systematic ListEach of the 3 choices for the tens place opens 2 choices for the ones place, so 3 groups of 2 make 6 numbers.
Why?
Whichever card takes the tens place, exactly two cards remain for the ones place, so the second choice never depends on the first.
🧱Independent choices multiplyIf a first choice can go a ways and each of those leaves b ways for a second, the pair can be made a times b ways.
Why?
Three branches each holding two numbers is three equal groups of two, which is what multiplying counts.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Answer
6 numbers
3 × 2 = 6
The full list 59, 56, 95, 96, 65, 69 has 6 entries with no repeats and no card used twice in one number, matching 3 x 2 = 6. A count of 6 is sensible: it is more than the 3 cards but small, exactly what we expect for ordered pairs from 3 items.
Takeaway
Pick the tens card (3 ways), then a ones card from what is left (2 ways), and multiply: 3 x 2 = 6 numbers!
- Set up the two place-value slots
- Count choices for the tens place
- Count choices for the ones place
- Multiply the choices (and check by listing)