This worksheet asks a student to do two things with the same set of dots, stars, and crescent moons: write an addition problem for each group, and then go back and turn the repeated ones into multiplication. 

Not easy for a child who has challenges even identifying what “3” is, right?

We asked Randy Ewart how he'd introduce this worksheet (above) to a student for the first time, starting with the dot groups in the first two rows (upper left).

Randy's first move is to set aside the vocabulary entirely. "The addition symbol and the term 'addition' are not important at this level," he said. "What is important is the concept of pulling quantities together for a greater amount."

Before any worksheet, he starts with something a student can hold. "You could present 8 small cookies he can eat and separate them into groups of 3 and 5. Talk about 3 cookies, then 5 cookies. And show them this:

Once that concept is in place, the worksheet becomes a script. Row by row, walk through the dots, the stars, and the moons.

Then say: "I have 5 cookies (now dots) and 5 dots and 5 dots, that is 5 + 5 + 5," he said. This is “repeated addition.” 

From there, he hands the pattern to the student. "Now you write out how many stars are in each box," he said, pointing to the first row. 

Now do the next row. 

Now do the moons.

The row of moons above is where the worksheet's lesson shows up. "The 4 + 4 + 4 + 4 +4 is 'repeated addition' — it has 5 FOURS, so write 5 × 4," Randy said. 

"We do this for repeated. That doesn't work for 3 + 5 — it is not repeated."

That distinction — repeated versus not repeated — is the whole worksheet in one sentence. 4 + 4 + 4 + 4 + 4 turns into 4 × 5 because the same number shows up five times. 

This 4 + 3 stays addition because nothing repeats.

A student who can sort a page of dots and stars into those two buckets has the actual concept multiplication is built on, before a single times-table fact enters the picture.