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Pattern and Function - Basic Algebraic Thinking

Grade 3IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Arithmetic Patterns: A sequence of numbers where the difference between any two consecutive terms is constant. For example, in the pattern 3,6,9,123, 6, 9, 12, the rule is to add 33.

A number pattern showing an increase of 3 between each term.
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Input-Output Machines: A function can be thought of as a machine. You put a number (input) in, a rule is applied, and a new number (output) comes out. If the input is xx and the rule is +5+5, the output is x+5x + 5.

Flowchart of an input-output machine adding 5 to the number 10.
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Equality and Balance: An equation is like a balanced scale. Both sides of the equal sign (==) must have the same value. If 10+5=1510 + 5 = 15, then the scale is balanced.

A balanced scale with 8 + 2 on one side and 10 on the other.
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Commutative Property: In addition, changing the order of the numbers does not change the sum. For example, 4+3=3+44 + 3 = 3 + 4.

Illustration showing that 2 circles plus 1 circle is the same as 1 circle plus 2 circles.

📐Formulae

a+b=ca + b = c

Input+Rule=OutputInput + Rule = Output

a+b=b+aa + b = b + a

□=Output−Input\Box = Output - Input (when rule is addition)

Term+Difference=Next TermTerm + Difference = Next\ Term

💡Examples

Problem 1:

Find the missing numbers in the following sequence: 4,8,12,16,…,…4, 8, 12, 16, \dots, \dots

Solution:

Step 1: Identify the rule by finding the difference between the first two numbers: 8−4=48 - 4 = 4. Step 2: Check if the rule applies to the next pair: 12−8=412 - 8 = 4. The rule is 'add 44'. Step 3: Add 44 to the last known number to find the next one: 16+4=2016 + 4 = 20. Step 4: Add 44 to that result: 20+4=2420 + 4 = 24. The sequence is 4,8,12,16,20,244, 8, 12, 16, 20, 24.

Explanation:

To solve a growing pattern, we first determine the 'jump' or difference between numbers and then continue that same jump to find the unknown values.

Problem 2:

Solve for the missing value: 45−□=3245 - \Box = 32

Solution:

Step 1: Understand that 4545 is the total and 3232 is one part. Step 2: To find the missing part, subtract the known part from the total: 45−32=□45 - 32 = \Box. Step 3: Calculate the subtraction: 45−30=1545 - 30 = 15, and 15−2=1315 - 2 = 13. So, □=13\Box = 13. Step 4: Verify by checking if 45−13=3245 - 13 = 32.

Explanation:

This problem uses the concept of part-part-whole. If we know the whole (4545) and one part (3232), we subtract to find the missing part.

Problem 3:

Identify the rule and find the 5th term in the following geometric pattern represented by squares: Term 1 has 2 squares, Term 2 has 4 squares, Term 3 has 6 squares.

A visual pattern of squares increasing by 2 in each step.

Solution:

Rule:Term n×2Rule: Term\ n \times 2 Term 5:5×2=10Term\ 5: 5 \times 2 = 10

Explanation:

Each step adds 2 more squares than the previous one. To find any term, we multiply the term number by 2. For the 5th term, 5×2=105 \times 2 = 10.

Problem 4:

Determine the value of the unknown variable yy in the equation represented by the balanced hanger: y+7=15y + 7 = 15.

A balance hanger with y + 7 on one side and 15 on the other.

Solution:

y+7=15y + 7 = 15 y=15−7y = 15 - 7 y=8y = 8

Explanation:

The hanger is balanced, meaning the left side equals the right side. To find the unknown yy, we subtract the known amount (77) from the total (1515).