AMC 8 · 2006 · #4

Grade 4 geometry-2d

Archetype Arithmetic Expression Decomposition

fraction-arithmeticmodular-arithmeticspatial-visualization identify-subproblemsmodular-arithmetic ↑ Prerequisites: fraction-arithmeticmodular-arithmetic

📏 Short solution 💡 3 insights 📊 Diagram

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Problem

Initially, a spinner points west. Chenille moves it clockwise $2 \dfrac{1}{4}$ revolutions and then counterclockwise $3 \dfrac{3}{4}$ revolutions. In what direction does the spinner point after the two moves?

Pick an answer.

(A)

north

(B)

east

(C)

south

(D)

west

(E)

northwest

View mode:

Toolkit + CCSS Solution

Understand

Restated: A spinner starts pointing west. It is rotated clockwise $2\dfrac{1}{4}$ revolutions, then counterclockwise $3\dfrac{3}{4}$ revolutions. Which direction does it point at the end?

Givens: Starting direction: west; First move: $2\dfrac{1}{4}$ revolutions clockwise; Second move: $3\dfrac{3}{4}$ revolutions counterclockwise; Answer choices: (A) north, (B) east, (C) south, (D) west, (E) northwest

Unknowns: The final direction of the spinner

Understand

Restated: A spinner starts pointing west. It is rotated clockwise $2\dfrac{1}{4}$ revolutions, then counterclockwise $3\dfrac{3}{4}$ revolutions. Which direction does it point at the end?

Plan

Primary tool: #1 Draw a Diagram

Secondary: #7 Identify Subproblems

A compass rose is the natural picture for this problem — tool #1 (Draw a Diagram) lets us walk around the four cardinal points and watch quarter-turns happen. Tool #7 (Identify Subproblems) splits the work into two cleaner pieces: first combine the two rotations into a single net revolution, then apply only its fractional part to the starting direction. Whole revolutions can be thrown away because they bring the spinner back to where it was.

Execute — Answer: B

#7 Identify Subproblems 4.NF.B.3 Step 1

Combine the two moves into one net rotation.
Treat counterclockwise as positive and clockwise as negative, then subtract whole and fractional parts.

$$\text{Net} = +3\dfrac{3}{4} - 2\dfrac{1}{4} = (3-2) + \left(\dfrac{3}{4}-\dfrac{1}{4}\right) = 1\dfrac{1}{2} \text{ revolutions CCW}$$

💡 Subtracting mixed numbers with the same denominator is a Grade 4 move: handle the whole parts and the fraction parts separately.

#7 Identify Subproblems 4.MD.C.5 Step 2

Throw away the whole-number part.
One full revolution brings the spinner back to where it started, so $1\dfrac{1}{2}$ revolutions has the same effect as just $\dfrac{1}{2}$ revolution counterclockwise.

$$1\dfrac{1}{2} \text{ rev} \;\equiv\; \dfrac{1}{2} \text{ rev CCW}$$

💡 Each full revolution is a $360^\circ$ turn — same direction in, same direction out. Only the leftover $\dfrac{1}{2}$ changes anything.

#1 Draw a Diagram 4.MD.C.5 Step 3

Draw the compass and walk a half-revolution counterclockwise from West.
Counterclockwise on a standard N-up compass goes W $\rightarrow$ S $\rightarrow$ E $\rightarrow$ N $\rightarrow$ W.
Two quarter-turns from West land on East.

$$\text{West} \xrightarrow{\frac{1}{4}\,\text{CCW}} \text{South} \xrightarrow{\frac{1}{4}\,\text{CCW}} \text{East} \;\Rightarrow\; \textbf{(B)}$$

💡 A half revolution is $180^\circ$ — the direction directly opposite West, which is East.

[1] #7 4.NF.B.3 Combine the two moves into one net rotation. Treat counterclockwise as positive

[2] #7 4.MD.C.5 Throw away the whole-number part. One full revolution brings the spinner back to

[3] #1 4.MD.C.5 Draw the compass and walk a half-revolution counterclockwise from West. Counterc

Review

Reasonableness: Check by adding the fractional parts only. The clockwise $2\dfrac{1}{4}$ is the same as $\dfrac{1}{4}$ CW (drop the $2$ full spins): West $\to$ North. The counterclockwise $3\dfrac{3}{4}$ is the same as $\dfrac{3}{4}$ CCW (drop the $3$): from North, CCW goes North $\to$ West $\to$ South $\to$ East. Final direction is East, matching answer (B). The two methods agree, which is the cross-check we want.

Alternative: Tool #3 (Eliminate Possibilities) on the choices. The net rotation $1\dfrac{1}{2}$ revolutions is exactly $\dfrac{1}{2}$ turn after the full spin cancels, i.e. $180^\circ$. So the final direction must be the one directly opposite West, which immediately rules out (A) north, (C) south, (D) west, and (E) northwest, leaving only (B) east.

CCSS standards used (min grade 4)

4.NF.B.3 Understand a fraction $a/b$ with $a>1$ as a sum of fractions $1/b$; add and subtract mixed numbers with like denominators (Subtracting $3\dfrac{3}{4} - 2\dfrac{1}{4} = 1\dfrac{1}{2}$ by handling whole and fraction parts separately.)
4.MD.C.5 Recognize angles as geometric shapes formed by two rays sharing a common endpoint, and understand angle measurement as fractions of a circular arc (Reading one revolution as a $360^\circ$ turn, a half revolution as $180^\circ$, and quarter-turns as the steps between adjacent cardinal directions.)

⭐ Cancel the full spins first, then walk the leftover fraction around the compass. Half a revolution counterclockwise from West lands on East — a Grade 4 mixed-number subtraction plus a compass walk.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 4)

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