While preparing the rover to explore an asteroid, Nova and Luna programmed its communication safety zone. They represented positions along the planned route with real numbers, using the landing site as 0 kilometers. Negative values indicated one direction and positive values the other. Three sensor readings placed possible sampling sites at -3.8, 2.9, and 4.2 kilometers. Because distance from the landing site depends on magnitude rather than direction, Luna compared |-3.8|, |2.9|, and |4.2|, obtaining 3.8, 2.9, and 4.2 kilometers. The communication system allowed the rover to travel at most 4 kilometers from the landing site. Nova also simplified a safety calculation, 2(1.35 + 0.65), using the associative and distributive properties to confirm a 4-kilometer boundary. The sites at -3.8 and 2.9 were within range, but the site at 4.2 was too far away. They removed that site from the route and programmed the rover to remain between -4 and 4 kilometers throughout its exploration.

Challenges

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Challenge 1 of 5

Challenges

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Challenge 1 of 4

Hint

The sign tells the direction from 0, while absolute value tells the distance from 0.

A student says the sampling site at -3.8 kilometers is outside the safety zone because -3.8 is less than 0. Which statement best explains the student's error?

The student should change -3.8 to 3.8 because negative positions are not real numbers.The student should compare the absolute value of the position with 4 because distance from the landing site does not depend on direction.The student should compare -3.8 with -4 only because the positive boundary does not matter.The student should subtract 4 from -3.8 to determine whether the site is safe.

Correct. The distance is |-3.8| = 3.8 kilometers, which is within the 4-kilometer boundary.

Not quite. Use absolute value to compare the site's distance from 0 with the safety limit.

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Hint

A site is safe when its absolute value is no more than 4.

Nova adds four possible sampling sites at -4, 1.5, 3.9, and 4.6 kilometers to the three original sites at -3.8, 2.9, and 4.2 kilometers. The rover may travel at most 4 kilometers from the landing site. How many of all seven sites are within the safety zone?

Correct. Five of the seven sites are within the 4-kilometer safety zone.

Check the absolute value of each position and count those whose distance from 0 is at most 4 kilometers.

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Hint

When distributing 2, multiply both terms inside the parentheses by 2.

Nova checks the 4-kilometer boundary using 2(1.35 + 0.65). Which calculation correctly verifies the result using the distributive property?

2(1.35 + 0.65) = 2(1.35) + 0.65 = 3.35 2(1.35 + 0.65) = 1.35 + 2(0.65) = 2.65 2(1.35 + 0.65) = 2(1.35) + 2(0.65) = 2.7 + 1.3 = 4 2(1.35 + 0.65) = 2 + 1.35 + 0.65 = 4

Correct. Distributing 2 to both terms gives 2.7 + 1.3 = 4.

Not quite. The factor outside the parentheses must multiply each term inside.

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