The Multiplication Symbol

Time to give "groups of" a symbol. Once your child can read 3 × 4 as either "3 groups of 4" or "4 + 4 + 4," they've crossed into multiplication proper.

What we're learning

The symbol × and its name (it's a "times")
Two correct ways to read 3 × 4 (both mean 12)
The vocabulary "factor" and "product" (worth knowing, not memorizing)

Reading the symbol

Show your child: 3 × 4 = 12.

Read it together TWO ways:

"Three times four equals twelve." This is how it's read in most classrooms. "Times" is just the name we say when we see ×.
"Three groups of four is twelve." This is what it means.

Both are correct. Use both, often. The second gives the meaning; the first gives the shorthand.

The vocabulary

3  ×  4  =  12
↑     ↑     ↑
factor factor product

The numbers being multiplied are called factors.
The answer is the product.

You don't need to drill these terms. They show up later, in division and factoring. For now, just use them in passing — "4 is one of the factors here" — so the words become familiar.

Try it together

Round 1: read aloud. Write 6 multiplication sentences and ask your child to read each one in BOTH ways:

2 × 5 → "two times five" / "two groups of five"
4 × 3 → "four times three" / "four groups of three"
5 × 2 → "five times two" / "five groups of two"
1 × 7 → "one times seven" / "one group of seven"
3 × 3 → "three times three" / "three groups of three"
0 × 8 → "zero times eight" / "zero groups of eight"

The last two are interesting:

3 × 3 — both factors the same. We call that a "square" later.
0 × 8 — zero groups of anything is zero. (You have no groups, so the total is no objects.) This trips kids up; keep it concrete: "If I have zero piles, how many things do I have? None.")

Round 2: write the symbol. For each story, your child writes the matching multiplication sentence:

"There are 4 cars. Each car has 2 wheels visible." → 4 × 2 = 8
"Lucy has 3 jars. Each jar has 5 marbles." → 3 × 5 = 15
"Each shelf has 4 books. There are 6 shelves." → 6 × 4 = 24

Round 3: commutativity, gently. Show that 3 × 4 and 4 × 3 are both 12, with the picture:

3 × 4 (3 rows of 4)         4 × 3 (4 rows of 3)
• • • •                      • • •
• • • •                      • • •
• • • •                      • • •
                             • • •

Twelve dots either way. The arrangement looks different. The total is the same.

This will save them a lot of times-table memorization in Grade 3 — they only need to learn each pair once.

Watch for

Reading × as a plus or an x. Some children, especially if they're reading-curious, will read × as the letter "x." Gently correct: "That's the times symbol. Different from the letter x." Later in algebra they'll learn that 3x (no symbol between) means multiplication too — but that's years away.
Skipping the meaning. A child who can recite "three times four equals twelve" but can't draw the picture or explain why has memorized a sound. Always ask "show me with objects" or "draw the picture." If they can't, return to the previous lesson.
Confusing factor and product. Don't drill these terms. Use them in passing. The kids who get the meaning will pick up the vocabulary later effortlessly; the kids who don't get the meaning won't be helped by knowing the words.

Where this is going

You've reached the start of multiplicative thinking. From here:

Times tables — Grade 3 will memorize the most-common products (1×1 through 12×12). The work you've done makes this easier and more meaningful.
Division — the inverse: "12 = 3 × 4 means 12 ÷ 3 = 4." Same three numbers, different operation.
Multi-digit multiplication — eventually, 23 × 47. The same place-value reasoning from K-2 carries forward.

Each layer reuses what came before. None of it is magic.