Problem
Find an entry by row and column
The counting numbers are laid out in a triangle, row by row. Each row holds one more number than the row above. Row 1 is 1, Row 2 is 2 3, Row 3 is 4 5 6, Row 4 is 7 8 9 10. Find the 8th number in Row 10.
Reasoning · Grade 4-1 Number Arrangement Tables
Your answer
Watch out The 8th tempts you to add 8 to 46, but the first entry already counts — add only 7.
How to solve
Strategy Look for a Pattern — Rather than writing out all ten rows number by number, I track just one thing per row: the last number in it. Those end-numbers grow in an easy way, so a short systematic list carries me from Row 4 down to Row 9. Once I know where Row 9 ends, Row 10 starts on the very next number and I only have to step 7 places along a single row.
1STEP 1
Read the row lengths off the figure
Row r holds exactly r numbers.
Row r holds r numbers
The triangle shape is the rule: each new row is one entry wider than the one above, which a solver can see directly in the drawing.
4.OA.C.5Draw A Diagram2STEP 2
Notice how each row ends
The rows end on 1, 3, 6, 10.
1 → 3 → 6 → 10
Watching how much a total grows from one row to the next is easier than working with the totals themselves, and here the growth is just the row number.
4.OA.C.5Look For A Pattern3STEP 3
Extend the list of end-numbers down to Row 9
Carrying on, Row 9 ends at 45.
10 → 15 → 21 → 28 → 36 → 45
Only five small additions are needed to reach Row 9, far less work than writing out all 45 numbers in the triangle.
3.NBT.A.2Make A Systematic List4STEP 4
Find where Row 10 starts
So Row 10 starts at 46.
45 + 1 = 46
Nothing is skipped between rows, so the next row simply picks up where the last one stopped.
4.OA.C.5Look For A Pattern5STEP 5
Step along Row 10 to the 8th entry
Seven steps along gives 53.
46 + (8 - 1) = 46 + 7 = 53
Going from the 1st entry to the 8th is 7 moves, not 8 — the same fencepost care as counting gaps between posts.
4.OA.A.3Make A Systematic ListGoing from the 1st entry of a row to the 8th is 7 moves, not 8.
Why?
Each move sits between exactly one pair of neighbouring entries, so moves and the gaps between entries match one for one.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
The entries along a row climb by one each time, so the 8th entry is the 1st plus that step repeated seven times.
🧱A constant step reaches any termWhen a list climbs by the same amount every time, any term is the starting value plus that step repeated, so you never have to write out the ones between.
6STEP 6
Check from the other end of the row
Counting back from the row's end 55 confirms 53.
Row 10: 46,47,48,49,50,51,52,53,54,55
Checking a position from the far end catches an off-by-one slip without redoing any of the earlier row work.
3.NBT.A.2Make A Systematic ListAnswer
53
46 + 7 = 53
The answer must lie inside Row 10, which runs from 46 to 55, and 53 does. It also sits near the right-hand end of the row, which fits the 8th of 10 positions. A second route to the same place confirms it: the numbers used up through Row 9 total 1+2+3+4+5+6+7+8+9 = 45, so the 8th entry of Row 10 is the 45 + 8 = 53rd counting number, which is 53 because the entries are the counting numbers themselves.
Takeaway
Keep track of where each row ends — 1, 3, 6, 10, 15, ... — and then the number you want is just a short walk into the next row.
- Read the row lengths off the figure
- Notice how each row ends
- Extend the list of end-numbers down to Row 9
- Find where Row 10 starts
- Step along Row 10 to the 8th entry
- Check from the other end of the row