GEMS Anchor Chart .
OPERATION GEMS
Order of Operations Reference Guide
Anchor Chart
The GEMS Schema is a streamlined approach to solving multi-step math problems. It reduces the traditional six steps into four clear, sequential stages. Always solve from left to right within each step!
G Step 1
Grouping Symbols
Solve inside parentheses, brackets, or braces first. Work from the innermost grouping outward.
\( ( \quad ) \) \( [ \quad ] \) Innermost first!
E Step 2
Exponents
Simplify terms with powers and roots. Exponents tell you how many times to multiply a base by itself.
\( x^2 \) \( 5^3 \) \( 5 \times 5 \times 5 \)
M Step 3
Multiply & Divide
Perform multiplication and division. These have equal priority—always solve from left to right !
\( \times \) \( \div \) Left to Right
S Step 4
Subtract & Add
Perform subtraction and addition. These have equal priority—always solve from left to right !
\( - \) \( + \) Left to Right
GEMS in Action: Step-by-Step Walkthrough
Simplify: \( 15 - 3 \times (4 + 1) + 2^3 \)
G
Grouping Symbols First \( 4 + 1 = 5 \)
Problem becomes: \( 15 - 3 \times \mathbf{5} + 2^3 \)
E
Evaluate Exponents \( 2^3 = 8 \)
Problem becomes: \( 15 - 3 \times 5 + \mathbf{8} \)
M
Multiply & Divide (Left to Right) \( 3 \times 5 = 15 \)
Problem becomes: \( 15 - \mathbf{15} + 8 \)
S
Subtract & Add (Left to Right) \( 15 - 15 = 0 \), then \( 0 + 8 = 8 \)
Final Answer: \( \mathbf{8} \)
Pro-Tip: Multiplication and Division do not "come first" over each other. Solve whichever appears first when reading from left to right. The same rule applies to Subtraction and Addition!
Gems Level 1 Worksheet .
OPERATION GEMS: LEVEL 1
Scaffolded Two-Step Practice
Two-Step Problems
Name:
Date:
Score:
Instructions: Solve each two-step expression using **GEMS**. Identify which operation comes first (G, E, M, or S), complete that step first, then solve the remaining operation.
1
\( 12 + 4 \times 3 \)
Step 1 (M/D):
\( \times \) \( = \)
Step 2 (S/A):
\( + \) \( = \)
Final Gem:
2
\( (15 - 7) \times 2 \)
Step 1 (G):
\( - \) \( = \)
Step 2 (M/D):
\( \times \) \( = \)
Final Gem:
3
\( 5 + 3^2 \)
Step 1 (E):
\( 3^2 \to \) \( \times \) \( = \)
Step 2 (S/A):
\( + \) \( = \)
Final Gem:
4
\( 20 \div (2 + 3) \)
Step 1 (G):
\( + \) \( = \)
Step 2 (M/D):
\( \div \) \( = \)
Final Gem:
5
\( 14 - 15 \div 3 \)
Step 1 (M/D):
\( \div \) \( = \)
Step 2 (S/A):
\( - \) \( = \)
Final Gem:
6
\( 4^2 - 6 \)
Step 1 (E):
\( 4^2 \to \) \( \times \) \( = \)
Step 2 (S/A):
\( - \) \( = \)
Final Gem:
Self-Check Question
In Problem 5, why did you perform Step 1 (M/D) before Step 2 (S/A)? Write your answer below:
Gems Level 2 Worksheet .
OPERATION GEMS: LEVEL 2
Multi-Step Funnel Challenge
Multi-Step Problems
Name:
Date:
Score:
Instructions: Take your GEMS skills to the next level! Simplify each multi-step expression by showing your work in the funnel blocks. Write down the simplified equation after each calculation until you reach the final **Gem Answer**.
1
Start Expression
\( (3 + 5) \times 2 - 2^3 \)
Step 1:
Step 2:
Step 3:
Gem Answer:
2
Start Expression
\( 5 \times (10 - 2^3) + 4 \)
Step 1:
Step 2:
Step 3:
Gem Answer:
3
Start Expression
\( (12 - 4) \div 2 + 3^2 \)
Step 1:
Step 2:
Step 3:
Gem Answer:
4
Start Expression
\( 3 \times 4^2 - (15 + 5) \)
Step 1:
Step 2:
Step 3:
Gem Answer:
Critical Thinking Question
In Problem 2, why did we evaluate the exponent \(2^3\) BEFORE evaluating the subtraction inside the parenthesis?
Gems Teacher Key .
TEACHER ANSWER KEY
Level 1: Scaffolded Two-Step Solutions
Page 1 of 2
Note to Educator: Solutions are marked in bold blue font to simulate handwriting. Key sub-steps of GEMS are pre-tallied below.
Level 1 Reference
1
\( 12 + 4 \times 3 \)
Step 1 (M/D):
4 \( \times \) 3 \( = \) 12
Step 2 (S/A):
12 \( + \) 12 \( = \) 24
Final Gem:
24
2
\( (15 - 7) \times 2 \)
Step 1 (G):
15 \( - \) 7 \( = \) 8
Step 2 (M/D):
8 \( \times \) 2 \( = \) 16
Final Gem:
16
3
\( 5 + 3^2 \)
Step 1 (E):
\( 3^2 \to \) 3 \( \times \) 3 \( = \) 9
Step 2 (S/A):
5 \( + \) 9 \( = \) 14
Final Gem:
14
4
\( 20 \div (2 + 3) \)
Step 1 (G):
2 \( + \) 3 \( = \) 5
Step 2 (M/D):
20 \( \div \) 5 \( = \) 4
Final Gem:
4
5
\( 14 - 15 \div 3 \)
Step 1 (M/D):
15 \( \div \) 3 \( = \) 5
Step 2 (S/A):
14 \( - \) 5 \( = \) 9
Final Gem:
9
6
\( 4^2 - 6 \)
Step 1 (E):
\( 4^2 \to \) 4 \( \times \) 4 \( = \) 16
Step 2 (S/A):
16 \( - \) 6 \( = \) 10
Final Gem:
10
Self-Check Answer (Sample)
According to the GEMS rules, Multiply/Divide comes before Subtract/Add. Therefore, we must divide 15 by 3 first (obtaining 5) before subtracting it from 14.
.
TEACHER ANSWER KEY
Level 2: Multi-Step Funnel Challenge Solutions
Page 2 of 2
Note to Educator: Detailed step-by-step math reductions are clearly demonstrated inside each funnel block.
Level 2 Reference
1
Start Expression
\( (3 + 5) \times 2 - 2^3 \)
Step 1:
\( \mathbf{8} \times 2 - 2^3 \)
Step 2:
\( 8 \times 2 - \mathbf{8} \)
Step 3:
\( \mathbf{16} - 8 \)
Gem Answer:
8
