Problem
Infer and use the cross multiplication method
A layout called cross multiplication is shown only through 48 × 67. Nobody explains the rule. Work out what each of the three lines under the bar is. Use the same layout on 74 × 98.
Reasoning · Grade 4-1 Multiplication
Your answer
Watch out Adding the four products flat is tempting, but the crossing lines must be shifted one place.
How to solve
Strategy Look for a Pattern — With only one example to go on, the way in is to list every small product the four digits can make and match those numbers against the three lines. Once each line is identified I check the place value of each line — what each digit is really worth — so I can be sure the shifts are not a coincidence but the method's whole point. Then applying the rule to 74 x 98 is just four small multiplications and one addition.
1STEP 1
List every digit-times-digit product in the example
List all four digit products in the example.
4 × 6 = 24, 4 × 7 = 28, 8 × 6 = 48, 8 × 7 = 56
Four digits can only be paired up four ways, so a complete list is short — and a short complete list turns 'guess the rule' into 'match the numbers'.
3.OA.B.5Make A Systematic List2STEP 2
Match the list against the three printed lines
The first line pairs outer with outer, inner with inner.
2456 = 24₄ × 6 56₈ × 7
Once you see 28 and 48 as the crossing products, the leftover four-digit line has to be built from the two products you have not used yet, and 24 followed by 56 is exactly 2456.
3.OA.B.5Look For A Pattern3STEP 3
Explain the shifts with place value
The shifts come from place value.
2400 + 56 + 280 + 480 = 3216
Shifting a number one column to the left is the same as multiplying it by 10, so the layout is quietly doing the tens bookkeeping for you and you never have to write the zeros.
4.NBT.A.2Analyze The Units4STEP 4
Build the first line for 74 x 98
For 74 × 98 the first line is 6332.
7 × 9 = 63, 4 × 8 = 32 → 6332
The example's first line came from the two straight pairs, so copying that step needs only two facts from the multiplication table you already know.
4.NBT.B.5Look For A Pattern5STEP 5
Build the two crossing lines
The crossing lines are 560 and 360.
7 × 8 = 56 → 560, 4 × 9 = 36 → 360
Breaking one hard two-digit multiplication into four easy one-digit ones is what makes the whole method something a fourth grader can do in their head.
4.NBT.B.5Identify Subproblems6STEP 6
Add the three lines
Adding the three lines gives 7252.
6332 + 560 + 360 = 6332 + 920 = 7252
Adding the two shifted lines together first keeps the numbers small, and column addition of whole numbers is a skill already in hand.
4.NBT.B.4Identify Subproblems7STEP 7
Confirm with the full place-value picture
The full place-value expansion also gives 7252.
6300 + 560 + 360 + 32 = 7252
Splitting each factor into tens and ones and multiplying every piece by every piece is the distributive property, the same idea behind the area-model rectangles used in class.
3.OA.B.5Analyze The UnitsSplitting each factor into tens and ones and multiplying every piece by every piece is what the cross layout is really doing.
Why?
Multiplying a split-up number is the same as multiplying each piece and adding the results, so the four small products rebuild the whole one.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
Why?
Shifting a line one column to the left multiplies it by ten, so the layout keeps the tens bookkeeping without writing any zeros.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Answer
7252
6332 + 560 + 360 = 7252
98 is 2 less than 100, so 74 x 98 should be a little under 74 x 100 = 7400. Exactly, it should be 7400 - 74 x 2 = 7400 - 148 = 7252, which is the answer found. The size is right too: a number in the seventies times a number in the nineties has to land in the seven thousands. The result is a plain product of two counts, so it carries no units.
Takeaway
A two-digit times two-digit multiplication is really just four times-table facts — the columns you write them in take care of all the tens for you.
- List every digit-times-digit product in the example
- Match the list against the three printed lines
- Explain the shifts with place value
- Build the first line for 74 x 98
- Build the two crossing lines
- Add the three lines
- Confirm with the full place-value picture