Problem
Insert plus or minus to make an equation true
Write only + or − in each circle so the equation comes out true. Compute left to right. The example is 54 − 21 + 14 = 47. Complete all five equations.
Reasoning · Grade 2-1 Calculating to Meet Conditions
Your answer
Watch out A flip looks like it costs the number itself, but it stops adding and starts taking away — twice the number.
How to solve
Strategy Guess and Check — Each equation has only a few circles, so the number of sign choices is small enough to test by guess-and-check. Comparing 'all plus' against the target tells me how much I must lower the total, and each minus sign lowers it by twice that number — that 'how much do I need to subtract' reasoning is working backwards from the target.
1STEP 1
A handy idea: each minus lowers the all-plus total by twice that number
One flipped number lowers the all-plus total by twice that number.
(all-plus sum) - 2×(numbers made minus) = target
Working backwards from the target turns 'where do the minus signs go?' into the easier question 'which numbers, doubled, fill the gap?'
1.OA.D.8Work BackwardsTurning one plus into a minus lowers the all-plus total by exactly twice that number.
Why?
The all-plus total already counts that number once as an addition, so it must first be removed and then taken away again.
Why?
Subtracting a number reverses exactly what adding it did, so undoing an addition and then subtracting costs two copies of it.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
Every term contributes its own amount to the running total, so changing one term's sign changes only that term's contribution.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
2STEP 2
(1) 10 ○ 4 ○ 6 ○ 3 = 9
(1) 23 overshoots 9 by 14, so flip 4 and 3: 10 − 4 + 6 − 3.
10 - 4 + 6 - 3 = 9
Knowing the gap is 14 (so flipped numbers must total 7) points straight to 4 and 3 — quick guess-and-check within 20.
2.OA.B.2Guess And Check3STEP 3
(2) 14 ○ 5 ○ 4 ○ 3 = 18
(2) 26 overshoots 18 by 8, so flip just the 4: 14 + 5 − 4 + 3.
14 + 5 - 4 + 3 = 18
The gap of 8 means flip numbers totalling 4 — exactly the single card 4.
2.OA.B.2Guess And Check4STEP 4
(3) 5 ○ 4 ○ 3 ○ 2 ○ 1 = 7
(3) 15 overshoots 7 by 8, so flip 3 and 1: 5 + 4 − 3 + 2 − 1.
5 + 4 - 3 + 2 - 1 = 7
Flipped numbers must total 4; both {3,1} and {4} work, so there is more than one correct equation.
2.OA.B.2Make A Systematic List5STEP 5
(4) 9 ○ 5 ○ 4 ○ 3 ○ 2 = 9
(4) The same move flips 5 and 2: 9 − 5 + 4 + 3 − 2.
9 - 5 + 4 + 3 - 2 = 9
Gap 14 means flip numbers totalling 7; both {5,2} and {4,3} reach 7, giving two correct equations.
2.OA.B.2Guess And Check6STEP 6
(5) 10 ○ 9 ○ 8 ○ 7 ○ 6 ○ 5 = 19
(5) 45 overshoots 19 by 26, so flip 8 and 5: 10 + 9 − 8 + 7 + 6 − 5.
10 + 9 - 8 + 7 + 6 - 5 = 19
Gap 26 means flip numbers totalling 13; both {8,5} and {7,6} reach 13, so again more than one answer works.
2.NBT.B.5Guess And CheckAnswer
(1) 10 − 4 + 6 − 3 (2) 14 + 5 − 4 + 3 (3) 5 + 4 − 3 + 2 − 1 (4) 9 − 5 + 4 + 3 − 2 (5) 10 + 9 − 8 + 7 + 6 − 5
Every equation was re-computed left to right and lands exactly on its target (9, 18, 7, 9, 19). The 'gap = twice the flipped numbers' rule is consistent each time: the all-plus sum minus the given target is always even, and half of it equals the sum of the numbers given a minus sign.
Takeaway
Add everything first, then see how much too big it is — each minus sign chops off twice that number, so just pick numbers to flip that fill the gap!
- A handy idea: each minus lowers the all-plus total by twice that number
- (1) 10 ○ 4 ○ 6 ○ 3 = 9
- (2) 14 ○ 5 ○ 4 ○ 3 = 18
- (3) 5 ○ 4 ○ 3 ○ 2 ○ 1 = 7
- (4) 9 ○ 5 ○ 4 ○ 3 ○ 2 = 9
- (5) 10 ○ 9 ○ 8 ○ 7 ○ 6 ○ 5 = 19