Problem
Rabbit pairs and the Fibonacci sequence
One pair of baby rabbits is born in January. A baby pair grows up in one month. A grown pair waits one more month, then has a baby pair every month. Count the pairs in July of the same year.
Reasoning · Grade 4-1 Working with Sequences
Your answer
Watch out Growing up looks like breeding time, but a pair waits one more month.
How to solve
Strategy Make a Systematic List — Only six steps separate January from July, so the safest route is simply to march month by month down a table that keeps baby pairs and adult pairs in separate columns — the two groups behave differently, and mixing them is exactly how this problem goes wrong. Once the table is filled in I look back along the totals for a pattern, both as a check on the arithmetic and because the pattern explains why the numbers grow the way they do.
1STEP 1
Set up two columns, babies and adults
Split the table into babies and adults.
Keeping babies and adults apart turns a wordy rule into two simple instructions you can apply without rereading the problem each time.
4.OA.C.5Make A Systematic List2STEP 2
January and February
January and February each hold 1 pair.
Jan: 1 + 0 = 1, Feb: 0 + 1 = 1
The count standing still for a month is the sign that the waiting rule has been applied properly — the new adults are not allowed to breed straight away.
4.OA.C.5Make A Systematic List3STEP 3
March and April
March gives 2 pairs and April 3 pairs.
Mar: 1 + 1 = 2, Apr: 1 + 2 = 3
Only the pairs that were already adult last month count as parents, so the number of new babies is copied from last month's adult column.
4.OA.C.5Make A Systematic ListOnly the pairs that were already adult last month become parents, so this month's babies copy last month's adult count.
Why?
Each adult pair produces exactly one new baby pair, so adults and new babies match up one for one.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
This month's total is last month's pairs together with the new babies, so the table grows by adding two known columns.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
4STEP 4
May and June
May gives 5 pairs and June 8 pairs.
May: 2 + 3 = 5, Jun: 3 + 5 = 8
Each new row is built from the row above by two easy additions, so no month can be skipped and no rabbit can be double-counted.
4.OA.C.5Make A Systematic List5STEP 5
July
July is 5 + 8 = 13 pairs.
Jul: 5 + 8 = 13
The last row is reached by the same two additions used on every earlier row, so if the table has been right so far it is right here too.
4.NBT.B.4Make A Systematic List6STEP 6
Read the pattern down the totals column
Each total is the two months before it added.
1, 1, 2, 3, 5, 8, 13
Finding a rule that connects the totals gives a second, independent way to reach 13 — a check that does not depend on the baby and adult columns.
4.OA.C.5Look For A Pattern7STEP 7
Picture the family tree as a check
Drawing the family tree also counts 13 pairs.
1 + 5 + 6 + 1 = 13
Seeing the same 13 come out of a drawing rather than a table is convincing because the two methods can only agree by being right.
4.OA.C.5Draw A DiagramAnswer
13 pairs
5 + 8 = 13
The totals 1, 1, 2, 3, 5, 8, 13 only ever grow, which they must, since nothing dies. They also grow by less than doubling each month (13 is less than 2 x 8), which is right, because only the older pairs breed while the youngest are still growing up. Counting the July rabbits the other way agrees too: 8 adult pairs plus 5 baby pairs is 13 pairs, and the 5 July babies match the 5 pairs that were adult in June. Finally, the answer is a whole number of pairs, as it must be.
Takeaway
Keep the babies and the grown-ups in separate columns, and this month's total is just the last two months added together!
- Set up two columns, babies and adults
- January and February
- March and April
- May and June
- July
- Read the pattern down the totals column
- Picture the family tree as a check