Find one handspan from measured counts

2.MD.A.12.MD.B.5 · adapt · grade 2

Archetype: Length as Sum of Parts with Unit Matching · step in a 7-type progression

Mia and Liam measured the length of a baseball bat using handspans. Starting from the left end, Mia counted off $5$ handspans, and starting from the right end, Liam counted off $3$ handspans. Their fingertips met exactly, with no overlap and no gap.

The bat is $3\ \text{ft}\ 9\ \text{in}$ long, and one of Mia's handspans is $6\ \text{in}$ . How many inches long is one of Liam's handspans?

Show solution

Understand

A baseball bat is measured in handspans from both ends. Mia counts 5 handspans from the left, Liam counts 3 from the right, and their fingertips meet exactly. Given the bat is 3 ft 9 in long and Mia's handspan is 6 in, find the length of one of Liam's handspans.

Givens

The bat is 3 ft 9 in long.
Mia measures 5 handspans from the left end, each 6 in.
Liam measures 3 handspans from the right end, all equal.
Their fingertips meet exactly, with no overlap and no gap.

Unknowns

The length of one of Liam's handspans, in inches.

Constraints

Mia's part plus Liam's part equals the whole bat (no overlap, no gap).
1 ft = 12 in.

Plan

#11 Work Backwards · also uses: #8 Analyze the Units

Convert everything to inches, find Mia's covered length, subtract from the whole to get Liam's covered length, then divide by his 3 handspans to find one. Working backwards from the total isolates Liam's part.

Execute

#8 Analyze the Units 2.MD.A.1

Convert 3 ft 9 in to inches: 3 feet is 36 inches, plus 9 inches.

3 \times 12 + 9 = 45 \text{ in}

Using one unit (inches) lets us add and subtract lengths directly.

#8 Analyze the Units 2.MD.B.5

Mia's 5 handspans are each 6 in, so multiply.

5 \times 6 = 30 \text{ in}

Repeated equal handspans add up to a total length.

#11 Work Backwards 2.MD.B.5

Since the two parts meet exactly with no overlap or gap, Liam covers the rest of the bat.

45 - 30 = 15 \text{ in}

The whole minus Mia's part leaves exactly Liam's part.

#11 Work Backwards 2.MD.B.5

Liam's 15 in is 3 equal handspans, so divide.

15 \div 3 = 5 \text{ in}

Splitting his total evenly among 3 spans gives one span.

Answer: 5 in

Review

Check: Mia 30 in + Liam 3x5 = 15 in totals 45 in = 3 ft 9 in, the full bat. A 5 in handspan is a bit smaller than Mia's 6 in, which is reasonable for a smaller hand.

Draw a Diagram (tool 1): mark the 45 in bat, shade Mia's 5 x 6 in from the left, and the remaining segment split into 3 equal pieces is clearly 5 in each.

Standards · min grade 2

2.MD.A.1 Measure the length of an object by selecting and using appropriate tools — Converting feet and inches into a single unit (inches) for measuring.
2.MD.B.5 Solve word problems involving lengths using same units — Adding, subtracting, and splitting lengths to isolate one of Liam's handspans.

💡 Turn everything into inches, take away Mia's part, then share what's left into 3 — pure Grade 2 length sense!