Multiplying multi-digit numbers (e.g., 36 x 24) means finding the total when you have 24 groups of 36 (or equivalently, the area of a 36-by-24 rectangle). The standard algorithm breaks this into partial products using place value: 36 x 4 and 36 x 20, then adds the results. Each partial product is itself a multi-digit-by-single-digit multiplication with possible regrouping. Understanding that the algorithm is an organized application of the distributive property -- (30 + 6) x (20 + 4) -- gives students insight into why it works rather than just how.
Begin with area models (drawing a rectangle partitioned into place-value sections) to make partial products visible. Transition to the partial products written method, then to the compact standard algorithm. The area model and partial products should be used long enough that students see the standard algorithm as a shortcut for what they already understand, not a mysterious procedure.
You have already learned that multiplication means equal groups or, equivalently, the area of a rectangle. You also know your single-digit multiplication facts and how multiples of ten work. Multi-digit multiplication combines these ideas: multiplying 36 × 24 is exactly the same kind of thing as multiplying 6 × 4, just with larger numbers. The question is how to organize the work so you do not lose track of anything.
The area model makes the structure visible. Draw a rectangle that is 36 wide and 24 tall. Now split the width into 30 and 6, and the height into 20 and 4. You have divided the big rectangle into four smaller ones. Their areas are: 30 × 20 = 600, 30 × 4 = 120, 6 × 20 = 120, and 6 × 4 = 24. Add all four: 600 + 120 + 120 + 24 = 864. This is the *partial products* method — you have computed the same thing the standard algorithm computes, just with every step written out explicitly.
The connection to the distributive property is worth pausing on: 36 × 24 = (30 + 6) × (20 + 4). The distributive property says you must multiply every piece of one factor by every piece of the other — four multiplications, not two. This is the precise reason you must treat the 2 in 24 as 20, not as 2. When you write the second partial product in the standard algorithm on its own indented line, the indentation is a shorthand for that multiplication by 10. Forgetting the indentation — or thinking 36 × 2 instead of 36 × 20 — is the most common error, and it comes from ignoring place value.
The standard algorithm compresses the area model into a more compact procedure. Instead of labeling four sub-rectangles, you write two rows of partial products and add. The first row is the bottom strip (× ones digit), the second row is the left strip (× tens digit), shifted one place left to honor place value. Once this compression makes sense to you — because you have seen the area model enough times — the algorithm is fast and reliable. If you can ever not remember why you are shifting left, draw the rectangle.
Estimation is your best check. Before computing 36 × 24, round to 40 × 25 = 1000. Your answer should be near 1000 — not 216 or 8640. If your answer is off by a factor of 10, you almost certainly made the place-value error on a partial product. If it is off by a smaller amount, check your regrouping. Building the habit of estimating first means that large errors announce themselves immediately.