Problem
Maximize or minimize a mixed expression
The numbers 5, 4, 3, 2, 1 sit in a row in that order. Each of the four circles between them takes a plus or a times. Signs may repeat. Fill them so the value comes out as small as possible.
Reasoning · Grade 5-1 Mixed Operations with Whole Numbers
Your answer
Watch out All-plus looks smallest, but times beats plus next to the 1.
How to solve
Strategy Look for a Pattern — Rather than evaluating sixteen expressions blindly, I first ask a single small question about one circle at a time: for a neighbouring pair, does + or x give the smaller contribution? Testing that on tiny pairs shows a clean pattern — for two numbers that are both 2 or more, multiplying gives at least as much as adding, so + keeps things small; but 1 breaks the pattern, because multiplying by 1 changes nothing while adding 1 makes the total grow. That pattern decides all four circles at once. Finally I write out the full systematic list of all 16 expressions to make sure nothing smaller was missed.
1STEP 1
Fix the order of operations first
Fix the rule: times before plus.
5 + 4 × 3 + 2 + 1 = 5 + 12 + 2 + 1 = 20
Trying the rule on one short expression, where both readings can be worked out in a second, settles the reading rule before any comparing starts — a wrong reading here would spoil every later step.
5.OA.A.1Solve An Easier Related Problem2STEP 2
Compare + and x on one pair of neighbours
For numbers above 2, times makes it bigger.
2+3 = 5 ＜ 6 = 2 × 3, 3+4 = 7 ＜ 12 = 3 × 4, 4+5 = 9 ＜ 20 = 4 × 5
Multiplying by a number that is 2 or more means taking two or more copies, which overshoots simply putting the two numbers together — that is why addition is the small-making sign here.
3.OA.D.9Look For A PatternFor neighbours of 2 or more, multiplying always overshoots adding, so a plus sign is what keeps the value small.
Why?
Multiplying by a number that is 2 or more means taking two or more copies, which is more than simply putting the two numbers together.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
Because the comparison comes out the same way at every circle, the separate verdicts chain into one choice for the whole expression.
🧱Transitivity of orderIf the first is smaller than the second and the second smaller than the third, the first is smaller than the third, so pairwise comparisons chain into one full order.
3STEP 3
Notice that 1 breaks the pattern
But next to 1, times comes out smaller.
2 × 1 = 2 ＜ 3 = 2 + 1
One copy of a number is just the number itself, so multiplying by 1 is a free move; that is exactly what makes 1 the one place where x, not +, keeps the total down.
3.OA.A.1Look For A Pattern4STEP 4
Put the four signs in and compute
So 5 + 4 + 3 + 2 × 1 = 14.
5 + 4 + 3 + 2 × 1 = 5 + 4 + 3 + 2 = 14
Each circle was decided on its own and each decision only made the total smaller, so putting all four best choices together must give the smallest expression of all.
5.OA.A.1Look For A Pattern5STEP 5
Check every possibility with a systematic list
Listing all sixteen confirms 14 is smallest.
14 ＜ 15 ＜ 16 ＜ 19 ＜ 20 ＜ 25 ＜ 26 ＜ 27 ＜ 29 ＜ 30 ＜ 62 ＜ 63 ＜ 120 ＜ 121
Sixteen is a small enough number of cases to write out completely, so the list turns a clever argument into a certainty — no expression was skipped.
5.OA.A.2Make A Systematic ListAnswer
14
5 + 4 + 3 + 2 × 1 = 14
The value is a plain number with no units, and it has to be a whole number since only whole numbers and the signs + and x are involved. Putting plus in all four circles gives 5 + 4 + 3 + 2 + 1 = 15, so the smallest possible value cannot be more than 15 — and 14 is just under it, exactly one less, which is the single unit saved by turning + 1 into x 1. It also cannot be smaller than 5 + 4 + 3 + 2 = 14, because 5, 4, 3 and 2 each contribute at least themselves however the signs fall. The answer sits precisely at that floor, so 14 is exactly right, and it is far below the largest value 121.
Takeaway
To keep a chain of + and x small, use + everywhere among the big numbers, but multiply by 1 instead of adding it — one copy of a number costs nothing.
- Fix the order of operations first
- Compare + and x on one pair of neighbours
- Notice that 1 breaks the pattern
- Put the four signs in and compute
- Check every possibility with a systematic list