Find the place-value error in a product

diagnostic. Calculate 46 × 20. Enter the product and explain what the digit 2 means in the multiplier 20.

Answer: 920

Teacher response: The 2 means two tens, or twenty groups: 46 × 2 = 92, then 92 × 10 = 920. Forty-six times two would count only two groups.

model. There are 32 packs with 246 labels in every pack. A draft shows 246 × 2 = 492, 246 × 3 = 738, then adds 492 + 738 = 1,230 as the answer to 246 × 32. What is the correct total, and where did the draft lose place value?

1. Predict a useful range before repairing the work. Thirty packs contain 246 × 30 = 7,380 labels, while forty packs contain 9,840. A total for 32 packs must lie between those values. The draft's 1,230 labels cannot account for even thirty packs.
2. Expand the multiplier: 32 = 30 + 2. Draw two columns labelled multiplier part and number of labels. Retain the correct ones row, 246 × 2 = 492, and say that it counts two full packs.
3. Calculate the tens row using the standard algorithm: 246 × 3 = 738, but three tens require 738 tens, or 7,380 ones. In place-value columns, align this row one position to the left of the ones row; a zero in the ones column records that place value.
4. Add the aligned partial products to obtain 7,872. Reconcile the algorithm with 246 × (30 + 2), then confirm that 7,872 lies between 7,380 and 9,840. Identify the original error as treating three tens as three ones; the final addition of the faulty rows was itself correct.

Worked example: The correct total is 7,872 labels. The first row counts two packs. The digit 3 in 32 represents thirty packs, so the second partial product must be 246 × 30 = 7,380. Add 492 + 7,380 = 7,872.

p1. Support: a draft for 124 × 12 adds 248 and 124 to get 372. Enter the corrected product, and label which row must count ten groups.

Answer: 1488

Teacher response: The ones row is 248 and the tens row is 1,240, so 1,240 + 248 = 1,488. The written 1 in 12 represents ten groups.

p2. Core: audit 307 × 24. A draft adds 1,228 and 614 to get 1,842. Enter the correct product and show the two correct partial products, including the zero inside 307.

Answer: 7368

Teacher response: 307 × 4 = 1,228 and 307 × 20 = 6,140. The total is 7,368. The tens multiplier is twenty even though 307 has zero tens within it.

p3. Challenge: a draft for 409 × 36 gives 409 × 6 = 2,454 and 409 × 30 = 12,027, then adds them to get 14,481. By how much must that total increase? Identify the faulty row and calculate its correction.

Answer: 243

Teacher response: The ones row is correct. The tens row should be 12,270 because 409 × 3 = 1,227 and ten times that is 12,270. The correct total is 14,724, which is 243 above 14,481.

e1. Independent check: a draft for 218 × 43 adds 654 and 872 to get 1,526. Enter the correct product, write both corrected partial products in columns, and give a bound that rejects the draft total.

Answer: 9374

Look for:

• Names 3 in 32 as three tens and connects the shifted row to thirty groups.
• Separates a partial-product error from an addition error and verifies the corrected total.
• Independent rubric: one criterion each for rows 654 and 8,720, total 9,374, and a reason such as 43 groups must exceed 40 groups; all three demonstrate the audit.

Teacher response: The rows are 218 × 3 = 654 and 218 × 40 = 8,720, giving 9,374. A total for 43 groups must exceed 8,720, so 1,526 is too small.

transfer. A different draft for 246 × 32 has the correct rows 492 and 7,380 but gives their sum as 7,772. What should be repaired, and why would shifting a row again make matters worse?

Answer: Repair the addition: 492 + 7,380 = 7,872, a change of 100. Both partial products already have the correct place value. In the tens column, 9 tens + 8 tens = 17 tens, so 1 hundred must be carried. Shifting a correct row would change the number of packs it represents.

teach-1. 5 minutes: Collect the 46 × 20 diagnostic and ask for the meaning of its tens digit. Route learners to a tens model or to labelling the faulty rows. Record the estimate range before revealing the corrected total.

teach-2. 10 minutes: Model 246 × 32 in place-value columns. Have partners point to the row for two packs and the row for thirty, then explain why the draft's addition can be correct while the product is wrong. Each learner writes the corrected algorithm.

teach-3. 15 minutes: Use five minutes for support or core according to the diagnostic, six minutes for a partner audit of the other product, and four minutes for the challenge. The checker must identify whether the error is in a row's value, a row's arithmetic or the final addition; switch roles once.

teach-4. 10 minutes: Spend six minutes on the independent product without partner help and four on the transfer. Assess three exit criteria: correct partial products, aligned total and an explanation or bound that rejects 1,526. Return a missing tens explanation to the place-value model, then recheck with a fresh product.

misconception. The second digit is 3, so its row is simply 246 × 3 = 738 in the same columns as the first row.

Teacher response: Cover the ones digit of 32 and ask how many packs the remaining 3 represents. Write 30 beside the row. Connect shifting the row to multiplying by thirty, and ask what quantity each column records instead of treating a written zero as a rule to memorise.

support. Keep the multiplier expanded as 10 + 2 while completing 124 × 12. Put a label above each row before using the compact algorithm. Use a place-value grid to make the empty ones column in the tens row visible.

extend. Audit 246 × 302 using 300 + 0 + 2. The partial products are 73,800, 0 and 492, giving 74,292. Explain why omitting a zero-valued row is acceptable only when the remaining rows keep their correct place values.

support-route. If the learner gives 92 or cannot name twenty groups, write 20 = 2 × 10 and combine two groups of 46 tens. Use 14 × 20 = 280 as a recheck before beginning the support product.

ready-route. If the learner gives 920 and explains the tens, ask them to label both partial products in the faulty worked example before doing any addition.