Integer Pond
A 6th grade lesson on the numbers above and below zero. Tap, drag, and watch the frog hop.
Meet the integers
Integers are the whole numbers, their opposites, and zero: …, −3, −2, −1, 0, 1, 2, 3, … Positive integers sit to the right of zero. Negative integers sit to the left. Zero is neither.
Tap any number on the line and the frog will hop there.
Opposites and absolute value
Opposites are the same distance from zero on different sides. The distance from zero is the absolute value, written |n|. Distance is never negative.
Comparing integers
On a number line, the number farther right is greater. Careful with negatives: −8 is deeper below zero than −2, so −8 is the smaller number.
Adding and subtracting
Start at the first number. Adding a positive hops right, adding a negative hops left. Subtracting means adding the opposite, so subtracting a negative hops right.
Why subtracting a negative adds
It feels strange that taking away a negative makes a number bigger. Here are three ways to see that it must be true.
1. Follow the pattern
Start with a number and subtract smaller and smaller numbers. Watch what happens to the answers.
Press Play pattern.
2. Measure the distance
Subtracting finds how far, and which way, you travel from the second number to the first. Watch the frog count the steps.
3. Rewrite it as adding
Multiplying and dividing
Multiplying is repeated hopping. a × b means make a hops of size b, starting at 0. A negative number of hops means hop the other way. Dividing asks how many hops of one size it takes to reach a total.
Sign rule: same signs give a positive answer. Different signs give a negative answer. So (+)(+) = +, (−)(−) = +, (+)(−) = −, (−)(+) = −. Division follows the same rule.
Practice
Integer Pond summary sheet
Key ideas
- Integers are whole numbers, their opposites, and 0: …, −2, −1, 0, 1, 2, …
- Positive is right of 0 (up, gain, deposit, above sea level). Negative is left of 0 (down, loss, owe, below).
- Opposites are the same distance from 0 on different sides: −5 and 5.
- Absolute value |n| is the distance from 0, so |−5| = 5.
- Comparing: farther right is greater, so −8 < −2.
- Adding a positive hops right. Adding a negative hops left.
- Subtracting means adding the opposite: 3 − (−4) = 3 + 4 = 7.
- Multiplying: a × b means a hops of size b. Same signs give a positive answer, different signs give a negative answer: −3 × 4 = −12.
- Dividing undoes multiplying and follows the same sign rule: −12 ÷ 3 = −4.
Try these
- What is the opposite of −7?
- What is |−9|?
- Compare −6 and −2 using < or >.
- −3 + 8 = ?
- 4 − 9 = ?
- −2 − (−6) = ?
- A submarine dives 12 metres. Which integer shows the change?
- A stock falls $4 in one day. Which integer shows the change?
- Order from least to greatest: 3, −5, 0, −1.
- Stock market day: A stock starts at $20 and changes by +6, −9, +4, −3. What is the final price?
- Stock market day: A stock starts at $25 and changes by −7, +2, −10, +5. What is the final price?
- Stock market day: A stock starts at $30 and changes by +8, −12, −6, +9. What is the final price, and after which change was the price lowest?
- −4 × 3 = ?
- −5 × (−6) = ?
- −18 ÷ 3 = ?
- −20 ÷ (−4) = ?
Answer key
1) 7 2) 9 3) −6 < −2 4) 5 5) −5 6) 4 7) −12 8) −4 9) −5, −1, 0, 3
10) 20 + 6 = 26, 26 − 9 = 17, 17 + 4 = 21, 21 − 3 = 18, so $18.
11) 25 − 7 = 18, 18 + 2 = 20, 20 − 10 = 10, 10 + 5 = 15, so $15.
12) 30 + 8 = 38, 38 − 12 = 26, 26 − 6 = 20, 20 + 9 = 29, so $29. The lowest price, $20, came after the third change.
13) −12 14) 30 15) −6 16) 5