International Baccalaureate · PYP 3

Number Patterns

⌚ ~82 min ▢ 8 blocks ✓ 36 exercises ➣ Prerequisites : Times Tables to 10 × 10, Division, Patterns and Sequences

I Adding or Subtracting the Same Number

A sequence is a list of numbers in a special order that follows a rule. By finding the rule or pattern, we can figure out what number comes next.
In some patterns, we always add the same number, or we always subtract the same number. In others, we always multiply by the same number, or we always divide by the same number.

Report an error

Definition — The Step of a Pattern

In some patterns the same number is added each time, or the same number is subtracted each time: that number is the step. It is the difference between two terms that are next to each other in the pattern.
Such a pattern is also called an arithmetic sequence, and its step the common difference.

Example

What is the next number in the sequence?$$3, 6, 9, 12, ?$$

Answer

The next number is $15$, because each number increases by $3$.

Skills to practice

Finding the Next Term
Finding the First Term
Finding the Step

II Multiplying or Dividing by the Same Number

Example

What is the next number in the sequence?$$2, 4, 8, 16, ?$$

Answer

The next number is $32$. Each number doubles: the rule is “multiply by $2$”.

Example

In this pattern, each term is divided by the same number to give the next one. $80, 40, 20, 10, \ ?$ Find the rule and the next term. Then extend the pattern backwards with the same rule: which term would come just before $80$?

Answer

From $80$ to $40$, from $40$ to $20$, from $20$ to $10$: each term is half of the one before. The rule is divide by $2$ (halve).

$$\begin{aligned} 10 \div 2 &= 5 &&\text{(the rule)}\\ 80 \times 2 &= 160 &&\text{(the rule undone, to go back one term)} \end{aligned}$$ The next term is $5$. Extended backwards, the term just before $80$ would be $160$: halving $160$ gives $80$.

Skills to practice

Finding the Next Term
Finding the First Term

III Rules in a Table and in Two Steps

Example

In this table, one rule in a single step — adding, subtracting or multiplying by the same whole number — changes each number of the In row into the number below it, in the Out row, always in the same way.

In	$13$	$20$	$9$	$35$	?
Out	$11$	$18$	$7$	?	$40$

Find the rule, then complete the table.

Answer

Compare each In number with its Out number. $13$ becomes $11$: $2$ less. $20$ becomes $18$: $2$ less. $9$ becomes $7$: $2$ less. The rule is the same for every pair: subtract $2$.
Use the rule. For the In number $35$: $35 - 2 = 33$.
Undo the rule to go back. Which In number gives $40$ once $2$ is taken away? The one that is $2$ more: $40 + 2 = 42$. Check: $42 - 2 = 40$.

$$\begin{aligned} 35 - 2 &= 33 &&\text{(the rule)}\\ 40 + 2 &= 42 &&\text{(the rule undone)} \end{aligned}$$

In	$13$	$20$	$9$	$35$	$\boxed{42}$
Out	$11$	$18$	$7$	$\boxed{33}$	$40$

In a pattern, the rule goes from one term to the next. In a table, the same rule goes from each In number to its Out number, and the In numbers need not follow each other: the rule is checked on every pair.

Example

The rule of this pattern has two steps: double the number, then add $1$. The first term is $1$. $1, \ ?, \ ?, \ ?$ Write the next three terms. Then say whether one single step — adding, subtracting, multiplying or dividing by the same whole number each time — could give the same pattern.

Answer

Each new term needs the two steps, always in the same order: first double, then add $1$. $$\begin{aligned} 1 \times 2 &= 2 &\qquad 2 + 1 &= 3\\ 3 \times 2 &= 6 &\qquad 6 + 1 &= 7\\ 7 \times 2 &= 14 &\qquad 14 + 1 &= 15 \end{aligned}$$

The pattern is $1, 3, 7, 15$. A rule with one single step cannot give it:

adding the same number each time would make the step from one term to the next always the same, but it is $+2$, then $+4$, then $+8$;
multiplying by the same number: from $1$ to $3$ that number would be $3$, but $3 \times 3 = 9$, not $7$;
subtracting or dividing by a whole number cannot turn $1$ into the bigger number $3$.

The rule needs its two steps. Their order matters too: “add $1$, then double” gives $1, 4, 10, 22$, another pattern.

Skills to practice

Finding the Rule in a Table
Following a Rule in Two Steps

IV Going further

Skills to practice

Creating and Describing Patterns
Quiz

In	\(13\)	\(20\)	\(9\)	\(35\)	?
Out	\(11\)	\(18\)	\(7\)	?	\(40\)

In	\(13\)	\(20\)	\(9\)	\(35\)	\(\boxed{42}\)
Out	\(11\)	\(18\)	\(7\)	\(\boxed{33}\)	\(40\)