Support
Answer and teaching response
1488 · 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.
Find the place-value error in a product
Audit a whole-number multiplication algorithm by naming each partial product, repairing its place value and checking the result against a useful bound.
Get student and teacher materialsPrepare the lesson
Learning goal: Produce a corrected two-row multiplication with a written explanation of the tens row and an independent reasonableness check.
You will need: Paper, pencil and squared paper if available. Prepare columns headed thousands, hundreds, tens and ones. Label quantities and do the initial calculations without a calculator; all pack counts are invented.
Start with a diagnostic
Calculate 46 × 20. Enter the product and explain what the digit 2 means in the multiplier 20.
Check the reasoning and choose a starting point
920 · The 2 means two tens, or twenty groups: 46 × 2 = 92, then 92 × 10 = 920. Forty-six times two would count only two groups.
Start with support: 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 for core: 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.
Work through an example
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?
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.
- 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.
- 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.
- 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.
- 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.
Run the 40-minute lesson
- 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.
- 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.
- 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.
- 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.
A useful error to discuss
A learner might say: The second digit is 3, so its row is simply 246 × 3 = 738 in the same columns as the first row.
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.
Practise at your level, then check independently
Use the diagnostic to choose support or core practice, then try the challenge. Complete the independent check without help. Enter a number, fraction or mixed number. The number check does not assess your drawing or explanation.
Core
Answer and teaching response
7368 · 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.
Challenge
Answer and teaching response
243 · 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.
Independent check
Answer and teaching response
9374 · 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 the idea
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?
Discuss a possible response
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.
Choose the next teaching step
- 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.
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.
Materials for your group
Student pages and teacher answers are separate. Keep the HTML files or print them in your browser.
Revisit a prerequisite
Use Repeated Fraction Groups when the prerequisite gap is understanding multiplication as a number of equal groups. Its examples use fractions; this investigation's 46 × 20 diagnostic and place-value columns supply the separate whole-number tens work needed for the algorithm.