Trail Maps: Elevation and Route Reversals

Use signed products to describe climbs, descents, and reversed routes.

Read the Trail Map

Elevation gains are positive; elevation losses are negative. Each map entry describes one forward traversal. In this worksheet, a positive multiplier means forward traversals, and a negative multiplier means reverse traversals. For example, −3 means three reverse traversals. Count only the named traversals, not any travel between them.

Elevation Change per Forward Traversal
Trail section	Change (m)
Granite Steps	+18
Fern Hollow	−12
Lookout Spur	+7
Creek Link	−5

Match each calculation using the map to its total elevation change.

The trip log shows (−3) × (−12) = 36 for Fern Hollow. What does this describe?

A. Three reverse traversals produce a total gain of 36 m.B. Three forward traversals produce a total loss of 36 m.C. Three reverse traversals produce a total loss of 36 m.D. Three forward traversals produce a total gain of 36 m.

A hiking group skips Granite Steps. Its log uses 0 × 18 for that section.

What elevation change does this product record? m

Fern Hollow descends evenly. Lena walks forward through exactly half of the section.

Find her signed elevation change: 1/2 × (−12) = m.

Which expression records two reverse traversals of Granite Steps?

A. (−2) × 18B. 2 × 18C. (−2) × (−18)D. 2 + (−18)

Check New Route Plans

Use the same forward-and-reverse convention. Give signed elevation changes.

On a different trail, each forward switchback changes elevation by −7.5 m. Amir walks four forward switchbacks.

What is his total elevation change? m

A trail map measures horizontal distance along a straight path. Forward distance is positive and backward distance is negative. The path's elevation changes by −3/4 m per meter of forward distance. Nia walks a signed horizontal distance of −8 m.

Find her signed elevation change: (−8) × (−3/4) = m.

One forward route descends 12 m and then climbs 7 m. Which expression correctly distributes the multiplier in 3 × [(−12) + 7]?

A. [3 × (−12)] − [3 × 7]B. [3 × 12] + [3 × 7]C. [3 × (−12)] + [3 × 7]D. [3 × (−12)] + 7

A map entry is smudged. Four reverse traversals of that section produce a total elevation gain of 28 m.

What signed elevation change belongs on the map for one forward traversal? m

10.

One forward traversal of a short trail gains 2.5 m. Which statement correctly interprets (−2) × 2.5 = −5?

A. Two forward traversals lose a total of 5 m.B. Two reverse traversals lose a total of 5 m.C. Two reverse traversals gain a total of 5 m.D. Two forward traversals gain a total of 5 m.

Explain the Sign Rules

Connect the hiking model to the distributive property. Check both the sign and the size of each product.

11.

A one-meter descent has signed change −1 m. Reversing it models (−1) × (−1). To justify this product, start with (−1) × [1 + (−1)] = 0. Which equation correctly uses the distributive property and makes the sum zero?

A. 0 = 1 + [(−1) × (−1)] = 1 + (−1)B. 0 = −1 + [(−1) × (−1)] = −1 + (−1)C. 0 = 1 − [(−1) × (−1)] = 1 − (−1)D. 0 = −1 + [(−1) × (−1)] = −1 + 1

12.

A forward route climbs 6 m and then descends 2 m. Three reverse traversals are modeled by (−3) × [6 + (−2)]. Complete the distributed calculation.

(−3) × [6 + (−2)] = −18 + = −12.

13.

A forward route climbs 10 m and then descends 4 m. Tomas records three reverse traversals of the entire route. The product is (−3) × [10 + (−4)].

What is the total signed elevation change? m

14.

A forward trail section descends 9 m. Maya says, “Walking it in reverse makes the elevation change positive, but it also makes the size of the change smaller.” Explain what she gets right and what she gets wrong. Use (−1) × (−9) in your reasoning.