Lattice Multiplication Worksheets - Free Printable

Let’s solve both problems using lattice multiplication, step by step.

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Problem a: Find 34 × 17

Step 1: Set up the lattice grid.
We have two digits in each number, so we draw a 2x2 grid.

Write 3 and 4 on top (for 34).
Write 1 and 7 on the right side (for 17).

Now fill in each box by multiplying the digit on top with the digit on the side:

- Top-left box: 3 × 1 = 03 → write 0 above diagonal, 3 below
- Top-right box: 4 × 1 = 04 → write 0 above, 4 below
- Bottom-left box: 3 × 7 = 21 → write 2 above, 1 below
- Bottom-right box: 4 × 7 = 28 → write 2 above, 8 below

Your grid should look like this:

```
3 4
+-----+-----+
1 | 0/3 | 0/4 |
+-----+-----+
7 | 2/1 | 2/8 |
+-----+-----+
```

Step 2: Add along the diagonals (from bottom right to top left).

There are 4 diagonals:

- Diagonal 1 (bottom right): only 8 → write down 8
- Diagonal 2: 1 + 2 + 4 = 7 → write down 7
- Diagonal 3: 0 + 2 + 0 = 2 → write down 2
- Diagonal 4 (top left): only 0 → but since it’s leading zero, ignore it

Wait — let’s double-check the diagonals carefully.

Actually, label the diagonals from right to left:

Diagonal 1 (far right): just the bottom-right corner → 8
Diagonal 2: goes through bottom-left (1) and top-right (4) and also the “above” part of bottom-right? Wait — no.

In lattice multiplication, you add along the diagonals that go from bottom-right to top-left. Each diagonal includes numbers that lie on the same slanted line.

Let me redraw mentally:

Grid positions:

Top row: [3×1=0|3] , [4×1=0|4]
Bottom row: [3×7=2|1], [4×7=2|8]

So the four boxes contain:

Box A (top-left): 0 / 3
Box B (top-right): 0 / 4
Box C (bottom-left): 2 / 1
Box D (bottom-right): 2 / 8

Now, diagonals (starting from bottom right):

- Diagonal 1: only the lower triangle of Box D → 8
- Diagonal 2: upper triangle of Box D (2) + lower triangle of Box B (4) + lower triangle of Box C (1) → 2 + 4 + 1 = 7
- Diagonal 3: upper triangle of Box B (0) + lower triangle of Box A (3) + upper triangle of Box C (2) → 0 + 3 + 2 = 5
- Diagonal 4: upper triangle of Box A (0) → 0 (ignore)

Wait — I think I messed up the diagonal grouping.

Standard way: imagine drawing lines from bottom-right corner going up-left.

The diagonals are:

1. Only the very bottom-right cell’s lower half → 8
2. The bottom-left cell’s lower half (1) + top-right cell’s lower half (4) + bottom-right cell’s upper half (2) → 1 + 4 + 2 = 7
3. The top-left cell’s lower half (3) + bottom-left cell’s upper half (2) + top-right cell’s upper half (0) → 3 + 2 + 0 = 5
4. The top-left cell’s upper half (0) → 0

So reading from left to right (diagonals 4 to 1), we get: 0, 5, 7, 8 → drop leading zero → 578

But wait — 34 × 17 is actually 578? Let me check with regular multiplication:

34 × 10 = 340
34 × 7 = 238
340 + 238 = 578 → YES!

So answer for a is 578.

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Problem b: Find 86 × 39

Set up 2x2 grid.

Top: 8 and 6
Side: 3 and 9

Fill in boxes:

- Top-left: 8 × 3 = 24 → 2 / 4
- Top-right: 6 × 3 = 18 → 1 / 8
- Bottom-left: 8 × 9 = 72 → 7 / 2
- Bottom-right: 6 × 9 = 54 → 5 / 4

Grid:

```
8 6
+-----+-----+
3 | 2/4 | 1/8 |
+-----+-----+
9 | 7/2 | 5/4 |
+-----+-----+
```

Now add diagonals (from bottom right to top left):

Diagonal 1 (bottom right): lower half of bottom-right → 4
Diagonal 2: upper half of bottom-right (5) + lower half of top-right (8) + lower half of bottom-left (2) → 5 + 8 + 2 = 15 → write down 5, carry 1
Diagonal 3: upper half of top-right (1) + lower half of top-left (4) + upper half of bottom-left (7) + carried 1 → 1 + 4 + 7 + 1 = 13 → write down 3, carry 1
Diagonal 4: upper half of top-left (2) + carried 1 → 2 + 1 = 3

So reading from left to right: 3, 3, 5, 4 → 3354

Check with normal math:

86 × 39 = ?

Break it down:

86 × 40 = 3440
Minus 86 → 3440 - 86 = 3354 → Correct!

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Final Answer:
a. 578
b. 3354