Apply a defined symbol to compute with fractions

4.NF.B.3 · take · grade 4

Archetype: Apply a Newly Defined Operation · step in a 5-type progression

For any two numbers $a$ and $b$ , define a new operation $\odot$ as follows:

$a \odot b = \dfrac{a + b}{a - b}$

Find the difference between $8 \odot 3$ and $9 \odot 4$ .

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Understand

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.

Givens

a⊙b = (a+b)/(a-b).
We need 8⊙3 and 9⊙4.
We want the difference between the two results.

Unknowns

The value of 8⊙3 minus 9⊙4 (the difference).

Constraints

Substitute the given numbers into the definition exactly as written.

Plan

#7 Identify Subproblems · also uses: #6 Guess and Check

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.

Execute

#7 Identify Subproblems 4.OA.A.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\odot 3=\dfrac{8+3}{8-3}=\dfrac{11}{5}

Following a defined recipe is just doing the add and subtract it tells you to do.

#7 Identify Subproblems 4.OA.A.3

Substitute a=9, b=4: the top is 9+4 = 13 and the bottom is 9-4 = 5, so 9⊙4 = 13/5.

9\odot 4=\dfrac{9+4}{9-4}=\dfrac{13}{5}

Same recipe with different numbers — both results conveniently land over 5.

#7 Identify Subproblems 4.NF.B.3

Both fractions have denominator 5, so subtract the numerators: 13/5 - 11/5 = 2/5.

\dfrac{13}{5}-\dfrac{11}{5}=\dfrac{2}{5}

Same denominator means you just subtract the top numbers — a Grade 4 fraction subtraction.

Answer: 2/5

Review

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.

Guess and check the structure (tool 6): notice both bottoms are 5, so only the tops 11 and 13 matter; 13 - 11 = 2 over 5 gives 2/5 directly without writing the full subtraction.

Standards · min grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Evaluating each defined ⊙ expression by performing the whole-number add and subtract.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Subtracting the two like-denominator fractions 13/5 and 11/5.

💡 This only needs Grade 4 skills — just follow the rule, then subtract fractions that already share a denominator!