C. Algebra · Algebra: Patterns, Rational Numbers, Equations, Inequalities, Coding and Modelling

Patterns That Show How Rational Numbers Are Related

C1.4 A1About 45 minutes

Learning goals

I can create and describe patterns that show relationships among integers, fractions and decimals.
I can use a pattern to explain why a rule about rational numbers makes sense, such as why a negative times a negative is positive.
I can use patterns to predict the decimal form of a fraction.

Words to know

rational number
A number that can be written as one integer divided by another (not dividing by 0).

repeating decimal
A decimal with a digit or group of digits that repeats forever, like 0.333... or 0.1818...

terminating decimal
A decimal that stops, like 0.375.

product
The answer to a multiplication.

quotient
The answer to a division.

Remember from Grade 7

In Grade 7 you made patterns with integers on a number line and in tables, for example counting down by 3: 6, 3, 0, −3, −6.

In Grade 8 you'll use patterns to discover how all kinds of rational numbers are related: how integer products get their signs, how fractions turn into repeating decimals, and what happens when you keep adding halves. Patterns help you see why rules are true, not just memorize them.

Let's start with a mystery

You know that 3 × 4 = 12. But what is (−3) × (−4)? Is it 12 or −12? Instead of guessing, let's let a pattern tell us.

Product	Value
(−3) × 3	−9
(−3) × 2	−6
(−3) × 1	−3
(−3) × 0	0
(−3) × (−1)	?
(−3) × (−2)	?

As the second number goes down by 1, the product goes UP by 3.

Read down the table. Each time the second number goes down by 1, the product goes up by 3: −9, −6, −3, 0. To keep the pattern going, the next products must be 3, 6, 9, 12.

So (−3) × (−1) = 3 and (−3) × (−4) = 12. The pattern shows why a negative times a negative is positive. (You'll practise multiplying and dividing integers in the Operations unit.)

Plotting (−3) × the second number gives points on a straight line. To keep the pattern going, the products for −1, −2, −3 and −4 must be 3, 6, 9 and 12.

Pattern 2: Fractions that become repeating decimals

Use a calculator to find 1/9, 2/9 and 3/9 as decimals:

1/9 = 0.111...
2/9 = 0.222...
3/9 = 0.333... (which is also 1/3)

The pattern: n/9 = 0.nnn... for n from 1 to 8. Elevenths have a pattern too: 1/11 = 0.0909..., 2/11 = 0.1818..., 3/11 = 0.2727... The repeating pair is 9 times the numerator: 2 × 9 = 18, 3 × 9 = 27.

Pattern 3: Adding halves of halves

Start with 1/2. Add 1/4. Add 1/8. Keep adding half of the last piece:

1/2
1/2 + 1/4 = 3/4
3/4 + 1/8 = 7/8
7/8 + 1/16 = 15/16

The totals get closer and closer to 1, but never reach it. The amount still missing is always the last piece you added: 1/2, 1/4, 1/8, 1/16, ...

The totals 1/2, 3/4, 7/8 and 15/16 on a number line from 0 to 1. Each jump covers half of the distance that is left, so the totals creep closer to 1.

Patterns explain, then you check

A pattern gives you a strong reason to believe a rule. Then check it another way. For example, (−3) × (−4) = 12 also fits the rule that multiplying by −1 makes a number its opposite: (−3) × (−4) = 3 × 4 = 12.

Worked example

Example 1: The sign of a product

Use a pattern to find (−2) × (−5).

Start with (−2) × 2 = −4, (−2) × 1 = −2, (−2) × 0 = 0.
Each time the second number goes down by 1, the product goes up by 2.
(−2) × (−1) = 2, (−2) × (−2) = 4, (−2) × (−3) = 6, (−2) × (−4) = 8, (−2) × (−5) = 10.

(−2) × (−5) = 10

Worked example

Example 2: Predict a repeating decimal

Use the elevenths pattern to write 5/11 as a decimal.

The repeating pair for n/11 is 9 × n.
9 × 5 = 45.

5/11 = 0.4545... (Check with a calculator: 5 ÷ 11 = 0.454545...)

Worked example

Example 3: Dividing by smaller and smaller fractions

Find the pattern: 6 ÷ 2, 6 ÷ 1, 6 ÷ 1/2, 6 ÷ 1/4.

6 ÷ 2 = 3 and 6 ÷ 1 = 6.
6 ÷ 1/2 means "how many halves are in 6?" There are 12.
6 ÷ 1/4 means "how many quarters are in 6?" There are 24.

3, 6, 12, 24. Each time the divisor is cut in half, the quotient doubles. Dividing by a number less than 1 gives a quotient bigger than 6.

Worked example

Example 4: Fractions, decimals and percents together

Make a pattern with eighths as fractions, decimals and percents. What do you notice?

1/8 = 0.125 = 12.5%
2/8 = 0.25 = 25%
3/8 = 0.375 = 37.5%
4/8 = 0.5 = 50%

Each step adds 1/8 = 0.125 = 12.5%. The three forms grow together, because they are the same numbers written in different ways.

Hands-on activity

Paper-strip halves

You need: a long strip of paper (about 30 cm), a pencil, and a ruler. (Ask an adult if you want to use scissors.)

The whole strip is 1. Fold it in half and shade one half. Write 1/2 on it.
The unshaded part is 1/2. Shade half of what's left (1/4 of the strip). Write the total shaded: 3/4.
Keep going: shade half of what's left each time. Record the total shaded and the amount unshaded in a table.
Stop when the pieces get too small to shade. How many steps did you manage?

What to look for: totals 1/2, 3/4, 7/8, 15/16, 31/32, ... The unshaded part halves each time: 1/2, 1/4, 1/8, ... The shaded total gets closer to 1 but never reaches it.

Bonus: on a calculator, find 1/9, 2/9, ... 8/9, then 1/11 to 5/11. Write each pattern in words.

Just for fun: recycled digits

The sevenths are the laziest fractions. 1/7 = 0.142857142857..., and 2/7, 3/7, 4/7, 5/7 and 6/7 all reuse the same six digits, just starting at a different spot. Try 3/7 on a calculator: 0.428571...

Your turn

Continue the pattern: (−4) × 3 = −12, (−4) × 2 = −8, (−4) × 1 = −4, (−4) × 0 = 0, (−4) × (−1) = ?, (−4) × (−2) = ?

Use the ninths pattern to write 4/9 as a decimal.

Use the elevenths pattern to write 6/11 as a decimal.

What is 1/2 + 1/4 + 1/8 + 1/16 + 1/32?

Continue: 12 ÷ 1 = 12, 12 ÷ 1/2 = 24, 12 ÷ 1/3 = ?

Continue: 0.7 × 10 = 7, 0.7 × 1 = 0.7, 0.7 × 0.1 = 0.07, 0.7 × 0.01 = ?

Describe the pattern 1/2, 1/3, 1/4, 1/5, ... as decimals.

More practice: warm-up

Write 2/9 as a decimal.

6 × 1 = 6, 6 × 0 = 0, 6 × (−1) = ?

1/2 + 1/4 = ?

5 × 0.1 = ?

Next term: 1/2, 1/4, 1/8, ...

More practice: on your own

Make a pattern that shows (−2) × (−6) = 12.

Write 7/9 and 8/9 as decimals. What does the pattern suggest 9/9 equals as a decimal? What do you know 9/9 equals?

Find the quotients: 20 ÷ 4, 20 ÷ 2, 20 ÷ 1, 20 ÷ 1/2, 20 ÷ 1/4. Describe the pattern.

Find a fraction halfway between 1/3 and 1/2.

Multiply: (−1) × (−1), then (−1) × (−1) × (−1), then four (−1)s. What is the product of ten (−1)s?

Challenge

What is 1/2 + 1/4 + 1/8 + ... + 1/1024 (10 terms)? How far is it from 1?

1/99 = 0.0101... and 2/99 = 0.0202... Predict 23/99 and 7/99 as decimals.

1/2 = 1/2. Then 1/2 + 1/6 = 2/3, and 2/3 + 1/12 = 3/4, and 3/4 + 1/20 = 4/5. Check one step, then predict the total after 9 terms.

Check your understanding

Quiz

(−5) × 2 = −10, (−5) × 1 = −5, (−5) × 0 = 0. So (−5) × (−1) = ?

Using the ninths pattern, 5/9 = ?

What is 1/2 + 1/4 + 1/8 + 1/16?

As the divisor goes 1, 1/2, 1/4, the quotients of 10 ÷ divisor are...

Which is true about the totals 1/2, 3/4, 7/8, 15/16, ...?

0.6 × 10 = 6, 0.6 × 1 = 0.6, 0.6 × 0.1 = ?

For parents and teachers

Big idea: C1.4 asks Grade 8 students to create and describe patterns that illustrate relationships among rational numbers. Patterns let students see why rules work, such as negative × negative = positive, or why 1/9 = 0.111...

Common difficulties:

Treating rules as magic. Ask your child to rebuild the multiplication pattern table rather than just quoting "two negatives make a positive."
Adding denominators (1/2 + 1/4 = 2/6). Use the paper strip to show quarters and halves.
Expecting division to always make numbers smaller. Dividing by 1/2 doubles a number.
Calculator rounding. A calculator may show 2/3 as 0.6666667. The last digit is rounded; the 6s really go on forever.

How to help: Explore the ninths and elevenths patterns on a phone calculator together and ask "What will 7/11 be?" before checking. This lesson builds A1 skills: curiosity, making conjectures and reasoning.