Problem
Build values from place-valued items
Liam throws 2 darts and both land on the board. Each dart scores 10, 50 or 100. Find every total the two darts can make.
Reasoning · Grade 2-1 Making Numbers
Your answer
Watch out Treating a swapped pair as new inflates the count. The total is the same, so count each pair once.
How to solve
Strategy Make a Systematic List — Only three scores are possible per dart, so every pair of darts can be listed completely. I list the unordered pairs in a fixed order (smallest value first) so I never repeat a pair, then add each pair. Adding the values is just two-digit/three-digit addition I can quickly check.
1STEP 1
List the value of one dart
One dart is 10, 50 or 100, so list pairs with the smaller score first.
one dart ∈ {10, 50, 100}
Treating the rings like coin values keeps the choices small and concrete — only three options per dart.
2.MD.C.8Make A Systematic List2STEP 2
Pair the first dart = 10 with the second dart
With 10 first the totals are 20, 60, 110.
10+10=20, 10+50=60, 10+100=110
Holding the first dart fixed at the smallest value and sweeping the second is the no-skip way to list pairs.
2.NBT.B.5Make A Systematic List3STEP 3
Pair the first dart = 50 with a second dart of 50 or more
With 50 first only two new pairs remain: 100, 150.
50+50=100, 50+100=150
Starting the second dart at 50 (not 10) avoids re-listing the 10+50 pair, so nothing is double-counted.
2.NBT.B.5Make A Systematic ListStarting the second dart at the first dart's value, never below it, lists every pair once and no pair twice.
Why?
Grouping by the first dart's value means a pair falls into exactly one group, so the groups can never overlap and none is forgotten.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
Why?
Two darts landing on 10 and 50 give the same score whichever was thrown first, so the pair is one thing and deserves one line in the list.
🧱Commutativity of additionPutting a with b or b with a piles up the same total — order does not change a sum.
4STEP 4
Pair the first dart = 100 with a second dart of 100
With 100 first just one new pair is left: 200.
100+100=200
By now the only fresh pair is the two biggest darts together, which matches the worked example of 200.
2.NBT.B.7Make A Systematic List5STEP 5
Collect every distinct total
Together, with no repeats: 20, 60, 100, 110, 150, 200.
20, 60, 100, 110, 150, 200
Putting the totals in order confirms each pair gave a different sum, so there are exactly six possible scores.
2.NBT.A.4Guess And CheckAnswer
20, 60, 100, 110, 150, 200 points (6 totals)
The smallest total should be both darts in the 10 ring (10 + 10 = 20) and the largest should be both in the center (100 + 100 = 200); the answers 20 and 200 match. All six totals fall between 20 and 200, and 200 matches the example. Listing pairs with the smaller value first guarantees no pair is missed or repeated.
Takeaway
List the dart pairs smallest-first so you never repeat one, then add — Grade 2 addition finds all six scores!
- List the value of one dart
- Pair the first dart = 10 with the second dart
- Pair the first dart = 50 with a second dart of 50 or more
- Pair the first dart = 100 with a second dart of 100
- Collect every distinct total