This answer key gives a full worked solution for each of the 10 problems on integers, opposites, ordering, and absolute value. Number line figures show the position of each answer, and the word problems explain the reasoning in plain language.
Problem 1: Integers in context
Below zero and below sea level are negative; deposits and gains are positive.
a) \(-8\) b) \(25\) c) \(-40\) d) \(6\)
Problem 2: Find the opposite
The opposite is the same distance from 0 on the other side. The opposite of 0 is 0.
a) \(-7\) b) \(12\) c) \(0\) d) \(3.5\)
Problem 3: Graph on a number line
Start at 0. \(-4\) is 4 units to the left, \(2\) is 2 units to the right, and \(-1\) is 1 unit to the left.
Points at \(-4\), \(-1\), and \(2\).
Problem 4: Compare numbers
a) \(-3\) is left of 2. b) \(-7\) is left of \(-4\). c) 0 is right of \(-5\). d) \(-2.5\) is left of \(-2.25\).
a) \(-3 < 2\) b) \(-7 < -4\) c) \(0 > -5\) d) \(-2.5 < -2.25\)
Problem 5: Absolute value
Absolute value is the distance from 0.
a) \(|-9| = 9\) b) \(|6| = 6\) c) \(|0| = 0\) d) \(|-4.5| = 4.5\)
Problem 6: Order integers
Negative numbers come first, with the farthest left being least: \(-7 < -2\). Then 0, then the positive numbers \(4 < 5\).
\(-7,\ -2,\ 0,\ 4,\ 5\)
Problem 7: Order rational numbers
Write each number as a decimal: \(-\frac{1}{2} = -0.5\) and \(\frac{1}{4} = 0.25\). The decimals are \(-0.5,\ 0.75,\ -1.5,\ 0.25\). In order: \(-1.5 < -0.5 < 0.25 < 0.75\).
\(-1.5,\ -\frac{1}{2},\ \frac{1}{4},\ 0.75\)
Problem 8: Hiking elevations
Order by position: \(-120 < -45 < 30\), so the order is Lake B, Lake A, town.
Distances from sea level: \(|-120| = 120\), \(|-45| = 45\), \(|30| = 30\). The greatest distance is 120 feet.
Lake B, Lake A, town; Lake B is farthest from sea level.
Problem 9: Winter temperatures
a) The lowest temperature is \(-11\), so Thursday is coldest.
b) \(-11 < -8 < -3 < 2\). \(-11,\ -8,\ -3,\ 2\)
c) \(-11\)°F is 11 degrees below zero and \(-8\)°F is only 8 degrees below zero. Since \(-11\) is farther to the left on the number line, it is the lesser number and the colder temperature.
Problem 10: Account balances
a) \(-35 < -20 < 15\). \(-35,\ -20,\ 15\)
b) \(|-35| = 35\) and \(|-20| = 20\). Ethan owes more, 35 dollars compared with 20 dollars.
c) \(-35\) is farther to the left on the number line, so it is less than \(-20\). Its absolute value is greater because it is farther from 0, which means a larger debt.
See every chapter on the Grade 6 math page.