Problem
Skip-count by the changing place value
A chain of boxes skip-counts by a fixed step: 4,270,000; 4,273,000; 4,276,000; then two blanks; then the star (the 6th box). Find the rule and the star's value.
Base-ten numbersOperations
Your answer
Watch out It's easy to think the star is six jumps from the first box since it's the 6th box, but jumps happen between boxes, so it's really only 5 jumps from box 1.
How to solve
Strategy Look for a Pattern — Find the constant step from two known neighbors, confirm it with a second pair, then add that step repeatedly to reach the sixth box.
1STEP 1
Find the step
From 4,270,000 to 4,273,000 is +3,000, and the next jump is +3,000 too, so the step is +3,000 (the thousands place changes).
4,273,000 - 4,270,000 = 3,000
Equal jumps between known boxes reveal the repeating rule.
4.OA.C.5Look For A Pattern2STEP 2
Fill the two blanks
Continue adding 3,000: the 4th box is 4,276,000 + 3,000 = 4,279,000 and the 5th box is 4,279,000 + 3,000 = 4,282,000.
4,276,000 + 3,000 = 4,279,000, 4,279,000 + 3,000 = 4,282,000
Each new box is just one more hop of 3,000.
4.NBT.A.2Identify Subproblems3STEP 3
Reach the star
The 6th box is 4,282,000 + 3,000 = 4,285,000. (Equivalently, the star is the first box plus five steps: 4,270,000 + 5 × 3,000 = 4,285,000.)
4,270,000 + 5 × 3,000 = 4,285,000
Five equal hops of 3,000 from the start land on the star.
4.NBT.A.2Look For A PatternThe star, the sixth box, equals the first box plus five equal steps of the skip-count rule.
Why?
Each box is the box before it plus one step, so arriving at the sixth box means the first box has had every step in between added onto it.
Why?
Those separate step-additions can be gathered into a single total added to the first box, since regrouping additions never changes the sum.
🧱Associativity of additionAdding three piles, it does not matter which two you push together first.
Why?
There are five steps to add, because six boxes in a row leave five gaps and each later box pairs with exactly one arrow before it.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Every one of those steps is the same size, 3,000.
Why?
The size of a jump is found by subtracting one box from the next, which returns the amount that was added because subtraction undoes addition.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
That subtraction comes out to 3,000 because only the thousands place differs between neighboring boxes while every other place stays the same.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
Five equal steps of 3,000 pile up to five groups of 3,000, which is 5 times 3,000.
Why?
Five equal groups of one amount is exactly what multiplying that amount by five means.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Answer
4,285,000
4,270,000 + 5 × 3,000 = 4,285,000
From the first box to the sixth there are 5 steps of 3,000, totaling 15,000, and 4,270,000 + 15,000 = 4,285,000, matching the step-by-step result.
Takeaway
This only needs Grade 4 pattern sense: find the equal jump, then keep hopping by it!
- Find the step
- Fill the two blanks
- Reach the star
Where next?
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