Checkerboard Multiplication

Materials

• The checkerboard mat and colored bead bars 1–9 from the introduction (pony beads on pipe cleaners work fine)
• Two sets of small digit cards 0–9: one regular for the multiplicand, one gray for the multiplier
• Pencil and squared paper so the child can record the problem and the answer
• A basket or dish to hold the bead bars not in use

No materials at home? Use the virtual Checkerboard.

Before this lesson

• Introduction to the Checkerboard

Aims

Presentation

1. 1 Begin with a single-digit multiplier so the whole problem lives on the bottom row: 4,357 × 3. Lay the multiplicand cards 4, 3, 5, 7 along the bottom edge, one under each column, and the single gray multiplier card 3 beside the bottom row on the right edge. (On the virtual board, type 4357 and 3 into the multiplicand and multiplier boxes and press Set.)We are going to multiply four thousand three hundred fifty-seven by three.
2. 2 Point to the square where the 7 column meets the 3 row — the units square — and place three 7-bars on it. On the virtual board, the child taps that square and the bars appear.Seven taken three times. Three bars of seven.
3. 3 Continue across the bottom row: three 5-bars on the tens square, three 3-bars on the hundreds, three 4-bars on the thousands. Say what each crossing means as you go.Now fifty times three — three bars of five on the tens square.
4. 4 Combine the bottom row starting at the right. Count the beads on the units square: 21. Twenty-one is one unit and two tens, so a 1-bar stays and 2 carries to the tens square. On the virtual board, tap the square and watch the exchange.Twenty-one: the one stays here, the twenty moves one square to the left.
5. 5 Work left square by square, combining and carrying, until every square holds a single bar or is empty. Read the bottom row right to left — 1, 3, 0, 7, 1 — and write the product on paper with commas: 13,071.Four thousand three hundred fifty-seven times three is thirteen thousand, seventy-one.
6. 6 Now the same multiplicand with a two-digit multiplier: 4,357 × 23 (the starting problem on the virtual board). Keep the multiplicand cards in place; set the gray 3 beside the bottom row and the gray 2 beside the tens row.This time we multiply by twenty-three — three units and two tens.
7. 7 Fill the units row exactly as before: three 7-bars, three 5-bars, three 3-bars, three 4-bars. Then move to the tens row for the multiplier digit 2: at each crossing place two bars of the multiplicand digit — two 7-bars, two 5-bars, two 3-bars, two 4-bars.This whole row is "times twenty," so everything lands one place higher.
8. 8 When every crossing is filled, slide the bars in the tens row down-left along their diagonals to the bottom row. Remind your child of the introduction: diagonal squares are worth the same.Slide, don’t lift — along the diagonal nothing changes its value.
9. 9 Combine the bottom row from the right just as before, carrying one square to the left whenever a square goes past nine. Then read the bottom row right to left — 1, 1, 2, 0, 0, 1 — and write the product on paper with commas: 100,211. Let the child read it aloud.Four thousand three hundred fifty-seven times twenty-three is one hundred thousand, two hundred eleven.
10. 10 Give the child a new problem — another 4-digit × 1-digit if they want the easier round, then 4-digit × 2-digit — and let them run the whole cycle themselves: place, slide, combine, read, and record the result on paper.Your turn: place the bars, slide the diagonals, make the exchanges, and read me the answer.

Points of interest

• Watching a big product assemble itself out of small, known facts like 7 × 3
• The satisfying diagonal slide — a whole row of beads glides down and the total does not budge
• Squares piling up past 9 and the carry moving left, just like the golden bead exchanges years earlier
• Reading a six-figure answer off the board and checking it against pencil-and-paper work

Control of error

• Every square must end with nine or fewer beads — a crowded square shows there is still exchanging to do
• Sliding along diagonals cannot change the total, so an answer that disagrees with a re-count means a bar was lifted off its diagonal
• The child can verify the product by re-running the same problem or by multiplying on paper; the guided virtual board only places the correct bars for each crossing, so a skipped square is visible as an empty crossing

Vocabulary

• multiplicand
• multiplier
• partial product
• product
• diagonal
• carry
• exchange

Variations

• Include a zero in the multiplicand (for example 4,057 × 23) and notice the empty column
• Let the child slide one row at a time and re-count the board total after each slide to prove nothing changed
• Once two-digit multipliers feel easy, try a 3-digit multiplier — the hundreds row adds a second diagonal slide

Extensions

• After the board answer is read, write the same problem vertically on paper and match each written partial product to a row of the board
• Try a 4-digit × 4-digit problem and read a product in the tens of millions
• Estimate first: round 4,357 × 23 to 4,000 × 20 on paper, then compare the estimate with the board’s exact answer

Follow-up work (pencil & paper)

• Print a long-multiplication worksheet and let your child solve a few problems on the board, then the same problems with pencil only, comparing answers.Printable: Long Multiplication
• Have the child write one checkerboard problem vertically on squared paper, recording each row of the board as a written partial product before adding them up.
• Ask the child to make up a "story problem" for a large multiplication (boxes of beads, seats in stadiums), solve it on the board, and write the full sentence answer on paper.

What comes next

When the checkerboard feels easy and your child starts writing partial products on paper without the beads, the passage to abstraction is nearly complete for multiplication. The next material in this strand is Racks & Tubes, which does for long division what the checkerboard does for long multiplication.

Next in this strand: Long Division with Racks & Tubes