Problem
Apply a defined symbol to compute with fractions
A made-up operation a⊙b means (a+b) divided by (a-b). I must compute 8⊙3 and 9⊙4 and find how much bigger one result is than the other.
Fractions
Your answer
Watch out It's easy to flip the fraction and compute (a-b)/(a+b) instead of the defined (a+b)/(a-b).
How to solve
Strategy Identify Subproblems — Break the task into two subproblems — evaluate each ⊙ expression by plugging into the rule — then subtract the two fractions. The rule is a recipe to follow step by step.
1STEP 1
Compute 8⊙3
Substitute a=8, b=3 into (a+b)/(a-b): the top is 8+3 = 11 and the bottom is 8-3 = 5, so 8⊙3 = 11/5.
8⊙3=(8+3)/(8-3)=11/5
Following a defined recipe is just doing the add and subtract it tells you to do.
4.OA.A.3Identify Subproblems2STEP 2
Compute 9⊙4
Substitute a=9, b=4: the top is 9+4 = 13 and the bottom is 9-4 = 5, so 9⊙4 = 13/5.
9⊙4=(9+4)/(9-4)=13/5
Same recipe with different numbers — both results conveniently land over 5.
4.OA.A.3Identify Subproblems3STEP 3
Find the difference
Both fractions have denominator 5, so subtract the numerators: 13/5 - 11/5 = 2/5.
13/5-11/5=2/5
Same denominator means you just subtract the top numbers — a Grade 4 fraction subtraction.
4.NF.B.3Identify SubproblemsBecause 13/5 and 11/5 share the denominator 5, their difference is found by subtracting the numerators and keeping that same 5 underneath.
Why?
Two fractions with the same bottom number are measured in the same unit piece, so subtracting one from the other is just counting how many of that unit are left.
Why?
A fraction like 13/5 is 13 copies of the unit piece 1/5, and 11/5 is 11 copies of that very same 1/5.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Taking 11 of those identical fifth-pieces away from 13 of them leaves 13 minus 11 fifth-pieces, and each leftover piece is still one-fifth.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
Answer
2/5
13/5 - 11/5 = 2/5
Both results are a little over 2 (11/5 = 2.2 and 13/5 = 2.6), and their difference 2/5 = 0.4 matches 2.6 - 2.2. The denominators happened to be equal (both 5), which makes the small difference reasonable.
Takeaway
This only needs Grade 4 skills — just follow the rule, then subtract fractions that already share a denominator!
- Compute 8⊙3
- Compute 9⊙4
- Find the difference
Where next?
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Same look, different logicsuggested
Step back and solidifysuggested