A diver at −12 m, a magpie at +5 m — which is farther from the surface?
Grade 7
In this lesson▾
What is absolute value?
A scuba diver hangs at −12 m in Lake Minnewanka, down near the drowned village. A magpie circles above the water at +5 m. On the number line, 5 beats −12 every time — and yet the diver is the one farther from the surface. How does the smaller number win?
Because those are two different questions. Which is greater asks where a number sits on the line. Which is farther asks how far it is from zero, direction ignored — and that is exactly what absolute value measures. It's the distance from zero, written |x|.
Before you drag, predict: what's |−7|? Then take the challenge in the panel — there are exactly two ways to win it.
Try it
Drag the marker — see its distance from zero.
Challenge: park the marker where the distance bar reads exactly the target. More than one spot works — find them all. |x| = 4.5
x3
same x, as a fraction3
|x|3
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
Click anywhere on the line to place the marker.
In symbols, the rule reads the same for any number — integers, decimals, fractions:
∣3∣=3∣−3∣=3∣0∣=0∣−2.5∣=2.543=43
Worksheet
These aren't graded. Get them right, get them wrong. The goal is to feel out where the idea works.
Practice · Not graded
MA.7.NUM.1
Practice the idea
01 / 10
01
What is ∣−7∣?
Multiple choice: what is the absolute value of negative seven?
02
A January morning in Calgary starts at −25°C. By mid-afternoon a chinook pushes the temperature to +12°C. What's the size of the swing, in °C?
Word problem: a January morning in Calgary starts at minus 25 degrees Celsius. By mid-afternoon a chinook has pushed the temperature to plus 12 degrees. What is the size of the swing in degrees?
03
Back at the lake: the magpie dips to +3.5 m while the diver sinks to −9.5 m. How far apart are they vertically?
Word problem: at Lake Minnewanka the magpie dips to plus 3.5 metres while the diver descends to minus 9.5 metres. How far apart are they vertically, in metres?
04
Find every x with ∣x∣=43. (More than one works.)
Multi-correct: which numbers x satisfy the absolute value of x equals three quarters? Two correct answers.
05
Use the number line above: you found both winning spots for ∣x∣=4.5. Add those two positions together. What's the sum?
Uses the number line slider's challenge: the two positions whose distance from zero is 4.5 are added together. What is their sum?
06
Is −21 greater than, less than, or equal to 43?
Compare magnitudes of two fractions: is the absolute value of negative one half greater than, less than, or equal to the absolute value of three quarters?
07
True or false: for every pair of numbers a and b, ∣a−b∣=∣b−a∣.
True or false: for every pair of numbers a and b, the absolute value of a minus b equals the absolute value of b minus a.
08
Back at Lake Minnewanka: a student claims |−12| < |5| because −12 < 5. Which side has the larger absolute value?
Two cards comparing absolute values: the absolute value of negative twelve, and the absolute value of five. Pick the larger.Try it
09
Try it
Drag the cards in order — smallest on the left, largest on the right.
Drag with the mouse or finger, or focus a card and use ←/→.
10
Reflection
▸Show common mistakes
Student says
“|−5| = −5. I just removed the bars.”
What it reveals
Treating the bars as decoration.
Targeted response
Number Line Slider: place −5 and read the orange arc. The distance is 5, not −5.
Student says
“|−8| < |3| because −8 < 3.”
What it reveals
Conflating number-line order with magnitude.
Targeted response
Compute each magnitude first. |−8| = 8 and |3| = 3. Magnitude 8 is bigger than magnitude 3, independent of where the numbers sit on the line.
Student says
“|x| = 4 means x = 4.”
What it reveals
Missing the two-case nature of absolute value.
Targeted response
Number Line: find ALL points 4 units from zero. Two answers: +4 and −4.
Going further
|a − b| is the distance between two points, regardless of order. The next lesson, negative numbers in context, is where this matters: some questions want a signed answer, some only a magnitude.