Split a productAbout 5 min · Untimed

Discover

Split a rectangle

Make a difficult product out of two easy products. For 23 × 7, calculate 20 × 7 and 3 × 7, then add. Every part must be multiplied.

Why it works

Splitting each group into two parts preserves the total number of objects.

(a + b) × c = a × c + b × c

Spot it: One factor splits into easy tens and ones.

Technique reference and sources

Imagine seven rows of 23 squares. Split each row into 20 squares and 3 squares. The two rectangles contain 140 and 21 squares: 161 altogether.

When this method fits

  • One factor splits into easy tens and ones.
  • A factor is near a round number, so subtracting a small product is easier.
  • You know the small products but not the whole answer.
  1. Choose a split. For 23, choose 20 + 3. For 49, 50 - 1 may be easier.
  2. Multiply the large part. For 23 × 7, hold 20 × 7 = 140.
  3. Multiply the small part. 3 × 7 = 21. Do not leave the 3 unmultiplied.
  4. Combine with the right sign. Add for 20 + 3; subtract for 50 - 1.
Tens plus ones: Multiply both parts.
  • 23 × 7 = 140 + 21 = 161
Round and remove: Remove a whole group for each unit rounded up.
  • 49 × 8 = 400 - 8 = 392

Common traps

23 × 7 becomes 140 + 3. Multiply the 3 by 7 too: 140 + 21.

Always splitting into tens and ones. When a factor ends in 9, compare the nearby-round-number route.

Another way

Round and compensate

Compare a tens-and-ones split with a nearby round number. Choose the route whose small products and final addition or subtraction you can do reliably.

  1. 1.49 × 8 = 50 × 8 - 8 = 392.
  2. 2.23 × 7 = 20 × 7 + 3 × 7 = 161.

Wins when: A factor is very close to a convenient round number and the correction is small.

Loses when: Rounding creates an awkward subtraction, or the original small products are already fluent.

Source notes

These explanations and questions are written for QuantArc. Links show the underlying mathematics and learning design.

OpenStax: Distributive Property
Free textbook reference for distributing and regrouping. QuantArc's explanations and exercises are original.

IES: Organizing Instruction and Study to Improve Student Learning
Supports worked examples, retrieval, explanations and spaced study. Does not establish QuantArc's exact lesson length, mastery threshold or review intervals.