C. Algebra · Patterns, Integers, Equations, Inequalities, Coding and ModellingPatterns with Integers
C1.4 A1About 45 minutes
Learning goals- I can create and describe growing and shrinking patterns that go past zero into the negative integers.
- I can use patterns to explain what happens when I add or subtract a negative integer.
- I can use patterns to show that opposite integers make zero.
- I can describe relationships among integers using patterns.
Words to know
integer
A whole number or its opposite: ..., −3, −2, −1, 0, 1, 2, 3, ...
positive integer
An integer greater than 0, like 4 or +4.
negative integer
An integer less than 0, like −4.
opposites
Two integers the same distance from 0 on opposite sides, like 5 and −5.
zero pair
A positive counter and a negative counter together. Their total is 0, just like 1 + (−1) = 0.
Remember from Grade 6
In Grade 6 you met integers: positive and negative whole numbers and zero. You placed them on horizontal and vertical number lines (temperatures, elevations, floors of a building), compared and ordered them, and found opposites, like 7 and −7.
Earlier this year, in the Operations unit, you added and subtracted integers with chips and number lines. In this lesson you will create patterns with integers and use them to see why those rules work, including what happens when you add or subtract a negative number.
Let's start with a cold evening
On a January evening it is 5°C at 4 p.m. The temperature drops 2°C every hour.
4 p.m.: 5°C, 5 p.m.: 3°C, 6 p.m.: 1°C, 7 p.m.: −1°C, 8 p.m.: −3°C, ...
The pattern doesn't stop at zero. It keeps going into the negative integers, still subtracting 2 each hour. Patterns like this help us understand how integers work.
Patterns that cross zero
Counting back by 3 from 9 gives 9, 6, 3, 0, −3, −6, −9. Look at the number line. The pattern is symmetric around 0: −3 is the opposite of 3, −6 is the opposite of 6, and so on.
Also notice: on the number line, numbers get smaller as you move left. So −9 < −6 < −3, even though 9 > 6 > 3. The farther a negative number is from 0, the smaller it is.
| Adding pattern | | Subtracting pattern | |
|---|
| 4 + 2 | 6 | 4 − 2 | 2 |
| 4 + 1 | 5 | 4 − 1 | 3 |
| 4 + 0 | 4 | 4 − 0 | 4 |
| 4 + (−1) | 3 | 4 − (−1) | 5 |
| 4 + (−2) | 2 | 4 − (−2) | 6 |
| 4 + (−3) | 1 | 4 − (−3) | 7 |
What do the patterns show?
Adding pattern: as the number we add goes down by 1, the answer goes down by 1. Keep the pattern going past 0, and 4 + (−1) = 3. That is the same as 4 − 1. Adding a negative integer gives the same answer as subtracting its opposite.
Subtracting pattern: as the number we subtract goes down by 1, the answer goes up by 1. So 4 − (−1) = 5, the same as 4 + 1. Subtracting a negative integer gives the same answer as adding its opposite.
Zero pairs: 1 + (−1) = 0, 2 + (−2) = 0, 3 + (−3) = 0. Opposites always add to 0. With counters, a + counter and a − counter make a zero pair.
Let the pattern be your guide
If you forget how adding or subtracting a negative works, build a quick pattern starting from something you know, like 4 + 2 and 4 + 1, and keep going. The pattern will tell you the answer.
Worked exampleExample 1: A temperature pattern
It is 5°C at 4 p.m. and the temperature drops 2°C each hour. At what time will it be −9°C?
- Extend the pattern: 5, 3, 1, −1, −3, −5, −7, −9.
- Count the jumps from 5 to −9: there are 7 jumps of 2. (From 5 down to −9 is 14 degrees, and 14 ÷ 2 = 7.)
- 7 hours after 4 p.m. is 11 p.m.
It will be −9°C at 11 p.m., if the temperature keeps dropping steadily.
Worked exampleExample 2: Use a pattern to find 2 + (−5)
Create a pattern to find 2 + (−5).
- Start with sums you know: 2 + 2 = 4, 2 + 1 = 3, 2 + 0 = 2.
- Continue: 2 + (−1) = 1, 2 + (−2) = 0, 2 + (−3) = −1, 2 + (−4) = −2, 2 + (−5) = −3.
- Check with counters: 2 positives and 5 negatives make 2 zero pairs, and 3 negatives are left: −3. ✓
Worked exampleExample 3: Use a pattern to find −3 − (−2)
Create a pattern to find −3 − (−2).
- Start with subtracting positive numbers: −3 − 2 = −5, −3 − 1 = −4, −3 − 0 = −3.
- Each time we subtract 1 less, the answer goes up by 1.
- Continue: −3 − (−1) = −2, −3 − (−2) = −1.
−3 − (−2) = −1. That's the same as −3 + 2.
Worked exampleExample 4: Create a pattern that crosses zero
Start at −20 and add 3 each time. What is the first positive term? Which term is it?
- −20, −17, −14, −11, −8, −5, −2, 1.
- Count: −20 is term 1, so 1 is term 8.
- The terms grow by 3. They get closer to 0, pass 0, then get farther from 0 on the positive side.
The first positive term is 1, at term 8.
Just for funWhy are negative temperatures so relaxed?
When things drop below zero, they just chill.
Hands-on activityTwo-colour counter patterns
You need: about 20 small items of two kinds, like coins showing heads and coins showing tails, or two kinds of dry pasta. One kind is +1 and the other kind is −1. Keep small items away from young children.
- Make 4 + (−1) with counters: 4 positives and 1 negative. Remove the zero pair. What's left? (3)
- Keep the 4 positives and add one more negative each time: 4 + (−2), 4 + (−3), 4 + (−4), 4 + (−5), 4 + (−6). Record each answer in a table. (2, 1, 0, −1, −2)
- Describe your pattern in words. (Each extra negative makes the answer 1 smaller.)
- Make up your own integer pattern with the counters and challenge a grown-up to describe it.
Your turn
1Give the next three terms: 8, 5, 2, −1, ...
Subtract 3 each time: −4, −7, −10.
2Describe the rule and give the next two terms: −15, −11, −7, −3, ...
3Continue the pattern: 6 + 1 = 7, 6 + 0 = 6, 6 + (−1) = ?, 6 + (−2) = ?
5 and 4. Each step, the answer goes down by 1.
4Continue the pattern: 3 − 1 = 2, 3 − 0 = 3, 3 − (−1) = ?, 3 − (−2) = ?
4 and 5. Each step, the answer goes up by 1.
0. 7 and −7 are opposites, so they make 7 zero pairs.
6Order from least to greatest: −2, 5, −8, 0
7It is −4°C at sunrise and the temperature rises 3°C each hour. Give the temperatures after 1, 2 and 3 hours.
More practice: warm-up
1What is the opposite of −6?
2Next term: 3, 2, 1, 0, ...
3Next term: −10, −20, −30, ...
5Which is greater: −3 or −7?
−3. It is closer to 0, to the right of −7 on the number line.
More practice: on your own
1Make a pattern to find −2 + (−3).
−2 + 1 = −1, −2 + 0 = −2, −2 + (−1) = −3, −2 + (−2) = −4, −2 + (−3) = −5.
2An elevator starts on level 6 and goes down 2 levels at each stop. The ground floor is level 0 and the parking levels are negative. List the levels for the first 5 stops. At which stop does it reach level −4?
Stops: 4, 2, 0, −2, −4. It reaches level −4 at stop 5.
3Use zero pairs to find 3 + (−5).
3 positives and 5 negatives make 3 zero pairs, and 2 negatives are left: −2.
4Continue the pattern: 7 − 9 = −2, 6 − 9 = −3, 5 − 9 = −4, ... Write the next two lines and describe the pattern.
4 − 9 = −5 and 3 − 9 = −6. When the first number goes down by 1, the answer goes down by 1.
5Create a shrinking integer pattern that starts at 12 and has −8 as its 5th term. Give the rule.
The change from term 1 to term 5 is −20 in 4 jumps, so subtract 5 each time: 12, 7, 2, −3, −8.
Challenge
1Make a table for t = 4n − 10 for n = 0 to 5. Which term number gives the first positive value? Where does the graph cross 0?
−10, −6, −2, 2, 6, 10. The first positive value is at n = 3. The values pass 0 between n = 2 (−2) and n = 3 (2), so the line crosses 0 halfway, at n = 2.5.
2Use a pattern to explain why 5 − (−3) = 8.
5 − 2 = 3, 5 − 1 = 4, 5 − 0 = 5, 5 − (−1) = 6, 5 − (−2) = 7, 5 − (−3) = 8. Each time we subtract one less, the answer goes up by 1. So subtracting −3 gives the same result as adding 3.
3Pattern A: 10, 7, 4, 1, ... Pattern B: −11, −7, −3, 1, ... Which term is the same in both? Will they ever meet again?
Both have 1 as term 4. After that A keeps going down by 3 and B keeps going up by 4, so they move apart and never meet again.
Quiz
1. What are the next two terms? 6, 2, −2, ...
2. Continue: 5 + 1 = 6, 5 + 0 = 5, 5 + (−1) = ?
3. Continue: 2 − 1 = 1, 2 − 0 = 2, 2 − (−1) = ?
5. Which list is ordered from least to greatest?
6. It is 3°C and the temperature drops 4°C each hour. What is the temperature after 2 hours?
Big idea: Grade 7 students create and describe patterns to illustrate relationships among integers (C1.4). Extending patterns past zero shows why the integer rules work: adding a negative is like subtracting, subtracting a negative is like adding, and opposites make zero. This builds on the Adding Integers and Subtracting Integers lessons in the Number strand (Operations unit), which come earlier in the year.
Common difficulties:
- Thinking −8 is greater than −2 because 8 > 2. Use a thermometer: −8°C is colder.
- Subtracting a negative. Many children think 4 − (−1) = 3. Build the subtraction pattern again and let the pattern decide.
- Losing track of the sign when crossing zero. Use a number line and count the jumps.
How to help: Look at outdoor temperatures in winter together and track how they change through a day. Use the counter activity with coins. This lesson builds A1 skills: noticing patterns, making conjectures and explaining them.