4th Grade Equations

 A Pedagogical Guide to Teaching Equations and Algebraic Thinking in Grade 4

Beyond the Answer: A Pedagogical Guide to Teaching Equations and Algebraic Thinking in Grade 4


Section 1. Foundations of Early Algebraic Thinking in Grade 4: From Arithmetic to Modeling


1.1 Deconstructing the Standard: What "Equations" Mean in Grade 4

The introduction of "equations" in the 4th-grade curriculum frequently causes confusion, as the term evokes images of formal, abstract algebra (e.g., solving $2x + 5 = 15$). Analysis of 4th-grade standards, however, reveals a different and more foundational goal. The standards do not require students to learn the formal, procedural rules of solving for $x$.1 Instead, the objective is to build the cognitive foundations of early algebraic thinking: the ability to generalize arithmetic and model relationships.4
The core of this work is found in the Common Core State Standards (CCSS) for Operations and Algebraic Thinking, specifically standard 4.OA.A.3. This standard mandates that students "Solve multistep word problems... using the four operations" and, most critically, "Represent these problems using equations with a letter standing for the unknown quantity".6
This is a significant advancement from earlier grades, which typically use a box or question mark for an unknown. The introduction of a letter ($p$, $m$, $x$, etc.) as a variable is a deliberate step toward abstraction.11 This standard is paired with 4.OA.A.2, which specifies solving word problems involving "multiplicative comparison" (e.g., "3 times as many") and representing them with "equations with a symbol for the unknown".8
Therefore, in 4th grade, an "equation" is not a problem to be solved, but a model to be built.

1.2 The Core Pedagogical Shift: Modeling vs. Answering

The pairing of "solve" and "represent" in standard 4.OA.A.3 6 signals a crucial pedagogical shift from computation to modeling. The primary goal is no longer "answer-getting"; it is "model-building."
In a traditional K-3 arithmetic framework, a student reads a word problem, extracts the numbers (e.g., 5 and 3), selects an operation (e.g., addition), and computes an answer (8). In the 4th-grade algebraic thinking framework, the student must analyze the structure of a multi-step problem 16 and translate that structure into a single, abstract mathematical sentence. The equation itself—for example, $p = (2 \times 8) + 5$—is the "deliverable" that demonstrates a conceptual understanding of the problem's complete logic.
This shift moves students from being mathematical calculators to being mathematical architects. Instruction must be redesigned to value the representation (the equation) as much, if not more, than the final solution. The equation is the codification of the thinking process required to arrive at the answer.

Section 2. The Relational-Operational Chasm: Re-Teaching the Equal Sign as "The Same As"


2.1 The Primary Misconception: The "Operational" Equal Sign

The single greatest barrier to successfully implementing this 4th-grade standard is a deep-seated misconception about the equal sign ($=$).19 Research consistently shows that the "large majority" of elementary students 19 do not understand the equal sign as intended.
Operational View (Misconception): Students hold an "operational" view, believing the equal sign is a command or an operator symbol. They interpret it to mean "the answer comes next," "to do something," or "gives/makes".19
Relational View (Correct): The correct "relational" understanding, which is essential for algebra, defines the equal sign as a symbol of equivalence: "is the same as" or "is equivalent to".22
Without a proper relational understanding of the equal sign, it is "virtually impossible" for students to make sense of the symbolic language of mathematics or algebra.23

2.2 Diagnosing the Misconception: Why $8 + 4 = \_\_\_ + 5$ is the Key

This operational misconception is quickly exposed when students are presented with "non-standard" equations, such as $8 + 4 = \_\_\_ + 5$.22 A 4th-grade student with a deeply ingrained operational view will provide one of two predictable wrong answers:
"12": The student solves $8 + 4$, encounters the equal sign (which they read as "find the answer"), and places the answer "12" in the blank. They then ignore the $+ 5$.21
"17": The student views the equal sign as a command to find the total of all numbers present, computing $8 + 4 + 5$ to get "17".21
The correct answer, "7," is accessible only to a student with a relational understanding. This student thinks, "The left side, $8 + 4$, has a value of 12. The equal sign means the right side must also have a value of 12. To make the right side balance, the blank must be 7, because $7 + 5$ is 12." This type of problem is the primary diagnostic tool for assessing a student's readiness for algebraic thinking.

2.3 The Cause: An Incomplete but Entrenched Model

This operational view is not the result of student failure; it is a logical, developmentally appropriate, but incomplete model that has been reinforced by 3-4 years of arithmetic instruction.19
From Kindergarten through 3rd grade, almost all arithmetic is presented in the format $a + b = c$. The "running total" nature of early math activities 19 and the design of calculator buttons (where $=$ is pressed to get the answer) 29 reinforce this model. The student's conception that $=$ means "get the answer" has been 100% effective and correct in their experience.
The pedagogical failure is not that students develop this model, but that instruction often fails to intentionally disrupt and update it.19 The best practice, therefore, is not to simply add the relational definition as a vocabulary word. Instruction must force the operational model to fail in a safe, structured way—for example, by presenting "True/False" number sentences like $49 + 3 = 50 + 2$.5 This creates a cognitive conflict that compels the student to seek a new, more robust model: the relational one.

Section 3. A Pedagogical Framework for Abstraction: Implementing the Concrete-Representational-Abstract (CRA) Model


3.1 The CRA Framework Explained

The most effective, evidence-based pedagogical practice for teaching abstract mathematical concepts like equations is the Concrete-Representational-Abstract (CRA) instructional framework.31 This approach scaffolds learning by moving students through three distinct stages of understanding.
Concrete (The "Doing" Stage): Students first interact with and manipulate physical objects to model the concept.31 This could involve using balance scales, Cuisenaire rods, or counters to physically represent equality. This hands-on interaction "leads to more retention" 31 and builds a deep, conceptual understanding.34
Representational (The "Seeing" Stage): Students transition from physical objects to visual representations. They learn to "draw, tally, or use stamps" 31 to create pictorial models of the concept. Examples include drawing a balance scale or, more commonly, creating bar models (also known as tape diagrams).36
Abstract (The "Symbolic" Stage): Finally, the teacher explicitly connects the concrete and representational models to the abstract mathematical notation—the numbers and symbols of an equation.33

3.2 CRA as the Bridge for Algebraic Thinking

The CRA framework is the precise mechanism for solving the "relational-operational chasm" and bridging the gap between arithmetic and algebra.
The difficulty students face is not with the concept of equality itself, but with the symbolism. Research highlights this exact "disconnect": students can often solve an equality problem correctly using physical blocks (Concrete) but cannot transfer that understanding to the symbolic equation (Abstract). They remain "convinced the incorrect answer was right" when looking at the abstract problem, even after solving it with manipulatives.19
This is because the "Representational" stage—the bridge—is often skipped. The CRA model provides this critical intermediate step. By having students move from a physical pan balance (Concrete) to a drawing of a pan balance or a bar model (Representational), the conceptual structure is preserved. This visual model serves as a scaffold, allowing the student to map their concrete understanding directly onto the abstract letters and symbols of an equation.37

Section 4. A Toolkit of Concrete and Representational Models for Teaching Equations


4.1 The Balance Model (Concrete $\rightarrow$ Representational)

The balance model is the most direct and effective tool for physically re-teaching the equal sign as a relational symbol.
Concrete (Doing): A physical two-pan balance scale is used.38
Activity 1 (True/False): The teacher places cubes representing $6 + 2$ on the left side and $3 + 5$ on the right side. Students observe that the scale balances, providing physical, undeniable proof that the statement $6 + 2 = 3 + 5$ is true.40 This directly confronts and disproves the "one answer" operational model.
Activity 2 (Find the Unknown): The teacher models $8 + 4 = x + 5$. They place 12 cubes on the left side. On the right side, they place 5 cubes and a "mystery bag" or box labeled with a letter $x$.41 The question becomes physical: "How many cubes must we put in the mystery bag to make it balance?" Students can physically add cubes one by one until the scale is level, discovering the unknown value is 7.
Representational (Seeing): After working with the physical scales, students are given worksheets with drawings of balance scales.43 They must either write the abstract equation that matches the drawing or determine the missing value that would make the drawing "balanced."

4.2 The Bar Model / Tape Diagram (Representational $\rightarrow$ Abstract)

The bar model (or tape diagram) is the primary Representational tool for deconstructing and visualizing the structure of multi-step word problems (4.OA.A.3) and multiplicative comparisons (4.OA.A.2).36
Part-Whole Models (Addition/Subtraction): These models are used for problems where quantities are combined or separated. A "whole" bar is drawn, with two or more "part" bars underneath.36 This visually reinforces the relationships $Part + Part = Whole$ and $Whole - Part = Part$. This structure is ideal for representing problems with a missing part, such as $35 + m = 100$.47
Comparison Models (Multiplication/Division): This is where the bar model is most powerful for 4th-grade standards.
Additive Comparison ("more than"): To model "Sam has 3 more apples than Maria," two separate bars are drawn. The "Sam" bar is shown as the "Maria" bar plus a small, additional block labeled "3."
Multiplicative Comparison ("times as many"): To model "Sam has 3 times as many apples as Maria," the "Maria" bar is drawn (e.g., [ 8 ]), and the "Sam" bar is drawn as three identical repetitions of Maria's bar [ 8 | 8 | 8 ].11
This visual distinction between additive and multiplicative comparisons is a core requirement of standard 4.OA.A.2.11 The pedagogical best practice is to have students read the story, draw the bar model first, and then derive the abstract equation from their drawing.18 The "unknown" part of the model (an empty box) is simply labeled with a letter ($x$, $p$, etc.), directly linking the visual representation to the abstract variable.46

4.3 Alternative Models: Cuisenaire Rods and Number Lines

Cuisenaire Rods (Concrete): These colored, length-based rods are excellent for concrete modeling of equations.51 To model $3 + x = 8$, a student places the light green rod (length 3) next to the brown rod (length 8). They can physically see the "missing" length and find the rod that fills the gap—the yellow rod (length 5).53 This can be abstracted by assigning a value to a rod (e.g., "The dark green rod is worth 6") and asking for combinations (e.g., $p + p = 6$), which leads to $2p = 6$.54
Number Lines (Representational): A number line can represent an unknown quantity as a "jump" or a missing value on a scale.39 This is particularly useful for modeling problems involving measurement, elapsed time, or change.

Section 5. Comprehensive 4-Day Lesson Plan: "From Balanced Scales to Bar Models"

This lesson plan follows the I Do, We Do, You Do model of explicit instruction 57 and is structured to move students through the Concrete-Representational-Abstract (CRA) sequence.
Unit Overview & Learning Objectives
Standards:
4.OA.A.2: Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
4.OA.A.3: Solve multistep word problems posed with whole numbers... using the four operations... Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers..6
Content Objectives:
Students will be able to (SWBAT) define the equal sign as a symbol of relational equivalence ("is the same as").
SWBAT determine if equations are true or false by comparing the value of both sides.
SWBAT solve for an unknown in a non-standard equation (e.g., $10 + 2 = \_\_\_ + 5$).
SWBAT translate a multiplicative comparison word problem into a bar model and a symbolic equation (4.OA.A.2).
SWBAT deconstruct a multi-step word problem by drawing a bar model (4.OA.A.3).
SWBAT write a single, multi-step equation with a letter for the unknown to represent a word problem (4.OA.A.3).
Day 1: "Is it Balanced?" — Deconstructing the Equal Sign
Focus: Concrete & Abstract. Re-teaching the equal sign (relational equivalence).
Materials: Two-pan balance scale, uniform cubes, "True/False" equation cards, student whiteboards.
Warm-Up / Hook (5 min):
Teacher writes on the board: $49 + 3 = 50 + 2$.5
"Is this statement True or False? Turn and talk to your partner."
Discuss. Some students will use computation ($52 = 52$). Guide them to see the relational strategy: "To get from 49 to 50, you add 1. To keep it balanced, you must take 1 from the 3, which leaves 2. So, $50+2$ is the same. The statement is true.".5
I Do (I Do - Concrete Model) (10 min):
"We've been using this symbol ($=$) for years. Most people think it means 'here comes the answer.' Today, we learn its true meaning. The equal sign means 'is the same as' or 'is balanced.'".25
Show the two-pan balance.38 "This is our physical model for the equal sign."
Place 5 cubes on one side and 5 cubes on the other. "It balances. $5 = 5$ is a true statement.".25
Model $3 + 4$ on the left (7 cubes) and $7$ on the right. "It balances. $3 + 4 = 7$ is true."
Critically: Model $7$ on the left and $3 + 4$ on the right. "It still balances. $7 = 3 + 4$ is also true. The 'answer' doesn't have to come last." This explicitly challenges the operational view.19
We Do (We Do - Concrete/Representational) (15 min):
"Let's test some." Show the card $6 + 2 = 8$. Students confirm with cubes.
Show $6 + 2 = 3 + 5$. Students build $6+2$ (8 cubes) on the left and $3+5$ (8 cubes) on the right. "It balances! This is True.".40
Show $4 + 5 = 2 + 6$. Students build $4+5$ (9) on the left and $2+6$ (8) on the right. "It tips! This is False."
Connecting to Abstract: Hold up the problem $8 + 4 = \_\_\_ + 5$.
I Do (Concrete): "Let's build this." Place $8+4$ (12 cubes) on the left. Place 5 cubes on the right. "It's tipping! What must we add to the blank ($\_\_\_$) to make it balance?" Students will see they need to add 7 more cubes.
"The number 7 makes the equation $8 + 4 = 7 + 5$ true."
You Do (Independent Practice - Abstract) (10 min):
Worksheet with 10 "True/False" or "Find the Missing Number" problems.
Examples: $21 + 4 = 20 + 5$ (T/F?), $15 - 3 = 10 + 2$ (T/F?), $7 + 6 = \_\_\_ + 4$ 22, $9 + 9 = \_\_\_ \times 3$.
Exit Ticket (5 min): 61
What is one word that the equal sign means? (Expected: same, balanced)
Solve for the missing number: $14 + 5 = \_\_\_ + 8$
(This is the key diagnostic.22 Teacher sorts these: "11" = relational. "19" or "27" = operational.)
Day 2: "Seeing the Story" — Modeling with Bar Models (4.OA.A.2)
Focus: Representational & Abstract. Connecting word problems (multiplicative comparison) to bar models and equations.
Materials: Whiteboards, pre-printed "Problem/Model/Equation" graphic organizers 63, task cards.49
Warm-Up (5 min):
True/False: $4 \times 6 = 2 \times 12$.
Find the blank: $3 \times 8 = 4 \times \_\_\_$. (Use balance logic: 24 on left, so 24 on right).
I Do (I Do - Representational) (15 min):
"Yesterday, we saw equations balance. Today, we'll use drawings to model word problems before we write an equation.".36
Problem 1 (Additive Comparison): "Maria has 8 apples. Sam has 3 more apples than Maria. How many apples does Sam have?"
"I'll use a Bar Model." Draw one bar: "This is Maria: [ 8 ]".
"Sam has 3 more." Draw a second bar underneath: "This is Sam: [ 8 | +3 ]".
"My drawing shows the equation. $8 + 3 = s$. So, $s = 11$."
Problem 2 (Multiplicative Comparison - 4.OA.A.2): "Maria has 8 apples. Sam has 3 times as many apples as Maria. How many apples ($s$) does Sam have?".11
"Is this the same problem?" (No!) "How is my drawing different?"
"Maria: [ 8 ]"
"Sam has 3 times Maria's amount. That means he has 3 of her bars." Draw: "Sam: [ 8 | 8 | 8 ]"
"This drawing proves the correct equation. It's $3 \times 8 = s$. So, $s = 24$."
This explicit visual contrast between "more than" (additive) and "times as many" (multiplicative) is the core of 4.OA.A.2.11
We Do (We Do - Guided Practice) (15 min):
Problem: "A blue scarf costs $5. A red scarf costs 4 times as much. How much does the red scarf ($r$) cost?"
"Who/What are our bars?" (Blue scarf, red scarf).
"Which bar is smaller?" (Blue). "Let's draw it."`].
"How do we draw the red scarf?" (4 bars of 5).`].
"What equation matches your drawing?".
Scaffolding Step: Give students 4 word problems and 4 pre-drawn bar models. Have them match the problem to its model.50 This builds their range of strategies.39
You Do (Independent Practice) (10 min):
Students get 3 word problems (4.OA.A.2). They must use the graphic organizer:
Problem: "A bike ride is 12 miles. A car ride is 6 times as long. How long is the car ride ($c$)?"
Bar Model:.
Equation:.
Exit Ticket (5 min):
"Draw a bar model and write an equation for this problem: A cat weighs 7 pounds. A dog weighs 3 times as much as the cat. What is the dog's weight ($d$)?"
Day 3: "Finding the Missing Piece" — Multi-Step Bar Models (4.OA.A.3)
Focus: Representational & Abstract. Deconstructing multi-step problems (all 4 ops).
Materials: Whiteboards, multi-step word problem task cards.65
Warm-Up (5 min):
"Draw the model: 'Ken has $18. This is 3 times as much as Lisa ($l$).' "
Analysis: This is a 4.OA.A.2 division problem. The total is known. Ken: [? |? |? ] = 18. This shows $3 \times l = 18$ or $18 / 3 = l$.
I Do (I Do - Representational) (15 min):
"Today, the problems have 2 or more steps. Our bar models will have 2 or more parts. The goal is to write one equation for the whole story.".17
Problem: "A baker had 2 trays of 12 cookies. He sold 15 cookies. How many cookies ($c$) are left?".60
"Step 1: What did he have at the start? 2 trays of 12."
"I'll model this first. The 'Whole' is the total number of cookies."
Model: Total (t): [ 12 | 12 ].
"Step 2: He sold 15. This means the total is made of two parts: the part that was sold (15) and the part that is left ($c$)."
Model: Total (t): [ 15 (sold) | c (left) ].
"Now let's put it all together in one equation, using our letter $c$."
$c = (Total) - 15$
"We know the Total is $2 \times 12$. Let's substitute that in."
$c = (2 \times 12) - 15$
"This single equation represents the entire story.".17
We Do (We Do - Guided Practice) (15 min):
Problem: "Sarah has $50. She buys 3 books that each cost $8. How much money ($m$) does she have left?"
"What's the whole amount?" ($50)..
"The $50 is made of two parts: the money she spent and the money she has left ($m$)."
Model: Whole (50):
"How do we find the 'Money Spent'?" (3 books at $8).
"Let's model that part." Money Spent: [ 8 | 8 | 8 ].
"What's our single equation for $m$?"
$m = 50 - (Money Spent)$
$m = 50 - (3 \times 8)$
Students solve on whiteboards. $m = 50 - 24$, so $m = 26$.
You Do (Independent Practice) (10 min):
Students get 3 multi-step problems. They must draw a bar model and write the single equation with a variable before solving.
Problem: "There are 4 buses. Each bus has 30 students. At the first stop, 10 students get off each bus. How many students ($s$) are left on all the buses combined?"
Exit Ticket (5 min):
"Write one equation using the letter $p$ to represent this problem: 'Jaime had 120 pennies. He gave 25 to his sister and 40 to his brother. How many pennies ($p$) does he have left?'"
(Expected: $p = 120 - 25 - 40$ or $p = 120 - (25 + 40)$).
Day 4: "Interpreting Remainders" & Assessment
Focus: Abstracting. Applying 4.OA.A.3 (remainders) and solidifying the equation.
Materials: Word problems with remainders, final assessment.
Warm-Up (5 min):
"Yesterday, you solved $m = 50 - (3 \times 8)$. What does the $(3 \times 8)$ part represent in the story?" (The total cost of the books). "Why are the parentheses important?"
I Do (I Do - Abstract) (10 min):
"The last part of 4.OA.A.3 is 'interpreting remainders'.".6 "The 'answer' from division isn't always the 'answer' to the problem."
Problem 1: "There are 30 students. 4 students can fit in a car. How many cars are needed?"
$30 / 4 = 7 \text{ R } 2$
"What is the answer? Is it 7? No! Those 2 'remainder' students need a ride! We must round up. The answer is 8 cars."
Problem 2: "You have 30 cookies. You want to give 4 cookies to each friend. How many friends get 4 cookies?"
$30 / 4 = 7 \text{ R } 2$
"What is the answer? It's 7 friends. The 2 'remainder' cookies are left over."
"The math is the same, but the context 60 changes the answer. You must interpret the remainder."
We Do (We Do - Abstract) (10 min):
"Showdown" activity.70 In groups, the captain reads a problem.
"125 students are going on a trip. A bus holds 50 students. How many buses are needed?"
$125 / 50 = 2 \text{ R } 25$..
"You have $20. A game costs $3. What is the most games you can buy?"
$20 / 3 = 6 \text{ R } 2$..
"I have 17 pencils. I want to put them in 3 boxes, with the same number in each box. How many pencils are left over?"
$17 / 3 = 5 \text{ R } 2$..
You Do (Summative Assessment) (20 min):
A 4-question quiz covering all unit objectives.
(4.OA.A.3/Relational): Find the missing number to make the equation true: $15 + 6 = 3 \times \_\_\_$.
(4.OA.A.2/Representational): A new video game costs $60. This is 5 times as much as an old game. Draw a bar model and write an equation to find the cost ($c$) of the old game.
(4.OA.A.3/Abstract): Write a single equation with a letter ($p$) to represent this problem: "Lisa bought 4 packs of stickers with 25 stickers in each. She then gave 30 stickers to her friend. How many stickers ($p$) does Lisa have left?"
(4.OA.A.3/Remainder): 53 people are waiting for an elevator that can hold 10 people. How many trips will the elevator need to make?

Section 6. Instructional Supports, Assessment, and Differentiation


6.1 Comprehensive Materials List

Concrete:
Two-pan bucket balances 38
Uniform linking cubes (or other counters) 71
Cuisenaire rods 39
"Mystery bags" (small cloth bags for holding unknown quantities of cubes)
Representational:
Student whiteboards and markers
Poster paper for anchor charts
Pre-printed bar model graphic organizers 63
Templates for drawing pan balances
Abstract:
Pre-printed task cards (laminated) with word problems 49
"True/False" equation cards
"Showdown" 70 problem sets

6.2 Formative and Summative Assessment Strategy

Formative (Daily):
Day 1 Exit Ticket: The question $14 + 5 = \_\_\_ + 8$ 61 is a powerful diagnostic tool.22 The teacher must sort responses not as "right/wrong" but by type of misconception:
Answer "19": (from $14+5$) Indicates a pure Operational View.
Answer "27": (from $14+5+8$) Indicates an "add all numbers" operational view.21
Answer "11": Indicates the desired Relational View.
Day 2 Exit Ticket: The bar model drawing for the cat/dog problem assesses the student's ability to model 4.OA.A.2.
Day 3 Whiteboard Check: Having students hold up their "single equation" for the multi-step problem allows the teacher to instantly see 63 who is successfully abstracting (e.g., $50 - (3\times8) = m$) versus who is still solving procedurally in multiple steps.
Summative (Day 4):
The 4-question quiz detailed in the lesson plan serves as the final assessment, directly mapping to the key unit objectives and standards 4.OA.A.2 and 4.OA.A.3.

6.3 Differentiation and Enrichment Plan

Support (Scaffolding):
For students struggling with word problems, provide pre-drawn bar models.44 The student's task is simplified to labeling the knowns and the unknown ($p$).
Provide sentence stems 63 to help students articulate their reasoning: "First, I knew the whole was ___. The parts were ___ and ___. My bar model shows...".74
Allow students to remain at the Concrete stage longer. They can use Cuisenaire rods 53 or the pan balance 38 to model problems that other students are drawing.
Extension (Enrichment):
Use "low-floor, high-ceiling" tasks.76
Task 1 (Work in Reverse): "The answer to a multi-step word problem is $25. Write a story, draw the bar model, and create the single equation that matches your story.".77
Task 2 (Model $\rightarrow$ Story): "Create a real-world word problem that can only be modeled by the equation $(40 - 5) / 7 = n$."
Introduce two-step equations in a concrete puzzle format (e.g., "On the balance, 3 mystery bags and 2 cubes on the left is balanced by 14 cubes on the right. How many cubes are in each bag?").78
"Diagnose the Problem" 79: Present students with a common error (e.g., a word problem, an incorrect bar model, and an incorrect equation). Their task is to write a paragraph explaining the error in thinking and how to fix it.

Section 7. Appendix: Exemplar Word Problems and Visual Models

The following tables provide a set of exemplar problems aligned to the 4th-grade standards, categorized by their underlying mathematical structure.
Table 1: 4th Grade Word Problem Exemplars (4.OA.A.2 & 4.OA.A.3)
Standard
Problem Structure/Type
Exemplar Word Problem [14, 18, 65, 66, 69, 81]
4.OA.A.2
Multiplicative Comparison (Unknown Product)
A T-shirt costs $9. A sweatshirt costs 3 times as much. How much does the sweatshirt ($s$) cost?
4.OA.A.2
Multiplicative Comparison (Unknown Group Size)
A sweatshirt costs $27, which is 3 times as much as a T-shirt. How much does the T-shirt ($t$) cost?
4.OA.A.2
Multiplicative Comparison (Unknown Multiplier)
A sweatshirt costs $27 and a T-shirt costs $9. How many times as much does the sweatshirt cost than the T-shirt ($n$)?
4.OA.A.3
Multi-Step (Add/Subtract)
The library has 1,250 books. On Monday, they loaned out 120 books. On Tuesday, they received 85 new books. How many books ($b$) does the library have now?
4.OA.A.3
Multi-Step (Multiply/Subtract)
Mr. Jones bought 5 boxes of pencils with 12 pencils in each. He then gave 8 pencils to his students. How many pencils ($p$) does he have left?
4.OA.A.3
Multi-Step (Multiply/Add)
A family bought 4 movie tickets for $7 each and a large popcorn for $10. What was the total cost ($c$)?
4.OA.A.3
Multi-Step (Divide/Subtract)
Sarah had $45. She split the money evenly among her 5 friends. One friend, "Lee," took his share and then spent $3. How much money ($m$) does Lee have left?
4.OA.A.3
Multi-Step (All Ops)
Maria, Sam, and Lee wash cars. They charge $10 per car. They wash 8 cars. If they split the money evenly, how much does each person ($m$) get?
4.OA.A.3
Remainder Interpretation (Round Up)
140 students are going on the field trip. A bus can hold 40 students. How many buses ($b$) are needed?
4.OA.A.3
Remainder Interpretation (Ignore/Drop)
A farmer has 25 apples. He wants to pack them into bags of 4. How many full bags ($b$) can he make?
4.OA.A.3
Remainder Interpretation (Focus on Remainder)
20 toys are shared equally among 6 friends. How many toys are left over ($L$)?

Table 2: Visual Solution Key (Bar Models and Equations)
Exemplar Problem
Bar Model Diagram
Abstract Equation
...T-shirt costs $9. Sweatshirt costs 3 times as much... ($s$)?
T-shirt: [ 9 ] Sweatshirt: [ 9 | 9 | 9 ] = $ $
$3 \times 9 = s$
...Sweatshirt costs $27, which is 3 times as much... ($t$)?
T-shirt: [ $t$ ] Sweatshirt: [ $t$ | $$$ | $ $ ] = 27
$3 \times t = 27$
...5 boxes of 12 pencils... gave 8... ($p$)?
Step 1 (Find Total): Total (T): [ 12 | 12 | 12 | 12 | 12 ] Step 2 (Find Part): Total (T): [ 8 (gave) | $ $ (left) ]
$p = (5 \times 12) - 8$
...4 tickets for $7 each... popcorn for $10... ($c$)?
Whole ($c$): [ 7 | 7 | 7 | 7 | 10 ] (Tickets) (Popcorn)
$c = (4 \times 7) + 10$
...Wash 8 cars at $10 each... split by 3 people... ($m$)?
Step 1 (Total Money, $T$): $T$: [ 10 | 10 |... | 10 ] (8 times) Step 2 (Share, $,$): $*$: [ $[$ | $\$ | $ $ ]
$m = (8 \times 10) / 3$
...140 students... bus holds 40... ($b$)?
Whole (140): [ 40 | 40 | 40 | 20 ] Analysis: 3 full buses and 1 partial bus. Must round up.
$140 / 40 = 3 \text{ R } 20$ Answer: $b = 4$
...25 apples... bags of 4... full bags ($b$)?
Whole (25): [ 4 | 4 | 4 | 4 | 4 | 4 | 1 ] Analysis: 6 full bags. 1 apple left over. Q is "full bags".
$25 / 4 = 6 \text{ R } 1$ Answer: $b = 6$


Section 8. Conclusion

Teaching "equations" in the 4th grade is a critical, foundational practice for future mathematical success. The pedagogical focus must shift away from rote computation and toward the development of algebraic thinking. This is achieved by recasting equations as models that represent the structure of a problem.
The primary obstacle to this goal is the pervasive "operational" misconception of the equal sign. Effective, evidence-based instruction must directly confront and remediate this misconception. The Concrete-Representational-Abstract (CRA) framework provides the necessary scaffolding to do so. By using balance scales (Concrete) to reteach the equal sign as "the same as," and bar models (Representational) to help students deconstruct and visualize word problems, teachers can successfully bridge the gap to the Abstract symbolic equation.
By focusing on modeling, not just solving, 4th-grade educators can demystify abstraction and equip students with the relational understanding of mathematics necessary to succeed in algebra and beyond.
Works cited
Common Core Essential Elements/Common Core State Standards side by side for Mathematics - OSPI, accessed November 5, 2025, https://ospi.k12.wa.us/sites/default/files/2022-12/ccee-ccss-math.pdf
Common Core State Standards for Mathematics, accessed November 5, 2025, https://learning.ccsso.org/wp-content/uploads/2022/11/Math_Standards1.pdf
California Common Core State Standards: Mathematics, accessed November 5, 2025, https://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf
Transformative teaching strategies for algebraic thinking: A systematic review of cognitive, pedagogical, and curricular advances - Eurasia Journal of Mathematics, Science and Technology Education, accessed November 5, 2025, https://www.ejmste.com/article/transformative-teaching-strategies-for-algebraic-thinking-a-systematic-review-of-cognitive-17250
Algebra for 3rd Graders? - Education Week, accessed November 5, 2025, https://www.edweek.org/teaching-learning/algebra-for-3rd-graders/2019/04
Grade 4 » Operations & Algebraic Thinking | Common Core State Standards Initiative, accessed November 5, 2025, https://www.thecorestandards.org/Math/Content/4/OA/
Mathematics > Grade 4 > Operations and Algebraic Thinking - Standards Navigator - Curriculum Map, accessed November 5, 2025, https://www.doe.mass.edu/frameworks/search/Map.aspx?ST_CODE=Mathematics.4.OA.A.03
MCCRSM Grade 4 - Maryland State Department of Education, accessed November 5, 2025, https://www.marylandpublicschools.org/about/Documents/DCAA/Math/MCCRSM/MCCRSMGrade4-A.pdf
Grade 4 - Claim 1 - Target A | Smarter Content Explorer, accessed November 5, 2025, https://contentexplorer.smarterbalanced.org/target/m-g4-c1-ta
Fourth Grade Math Student Learning Objective Template - Illinois State Board of Education, accessed November 5, 2025, https://www.isbe.net/Documents/slo-4th-grade-math.pdf
Mississippi College and Career Readiness Standards for Mathematics Scaffolding Document Grade 4, accessed November 5, 2025, https://www.mdek12.org/sites/default/files/Offices/Secondary%20Ed/ELA/ccr/Math/04.Grade-4-Math-Scaffolding-Doc.pdf
Write variable expressions: word problems | 4th grade math - IXL, accessed November 5, 2025, https://www.ixl.com/math/grade-4/write-variable-expressions-word-problems
Unknown Variable Equations - TPT, accessed November 5, 2025, https://www.teacherspayteachers.com/browse/free?search=unknown%20variable%20equations
4.OA.A.2 - Illustrative Mathematics, accessed November 5, 2025, https://tasks.illustrativemathematics.org/content-standards/4/OA/A/2
4.OA.A.2 Worksheets - Common Core Math - Education.com, accessed November 5, 2025, https://www.education.com/common-core/CCSS.MATH.CONTENT.4.OA.A.2/worksheets/
Washington fourth-grade math standards - IXL, accessed November 5, 2025, https://www.ixl.com/standards/washington/math/grade-4
4th Grade Math 2.12, Solve Multi-step Word Problems Using Equations - YouTube, accessed November 5, 2025, https://www.youtube.com/watch?v=lwdLI0DJeJs
4th Grade Math 4.12, Word Problem Solving, Multi-Step Division Problems - YouTube, accessed November 5, 2025, https://www.youtube.com/watch?v=Kq4Gyoh7lGg
Children's Understanding of the Equal Sign Sue P. Prestwood Georgia College and State University, accessed November 5, 2025, https://www.gcsu.edu/sites/files/page-assets/node-808/attachments/prestwood.pdf
accessed November 5, 2025, https://pmc.ncbi.nlm.nih.gov/articles/PMC3374577/#:~:text=Some%20students%20believe%20the%20equal,sign%20is%20an%20operator%20symbol.
EQUATIONS AND THE EQUAL SIGN IN ELEMENTARY MATHEMATICS TEXTBOOKS - NIH, accessed November 5, 2025, https://pmc.ncbi.nlm.nih.gov/articles/PMC3374577/
Student Conceptions of the Equal Sign : Knowledge Trajectories ..., accessed November 5, 2025, https://www.journals.uchicago.edu/doi/10.1086/717999
The meaning of “equals” - New Zealand Council for Educational Research, accessed November 5, 2025, https://www.nzcer.org.nz/nzcerpress/set/articles/meaning-equals%E2%80%9D
accessed November 5, 2025, https://pmc.ncbi.nlm.nih.gov/articles/PMC3374577/#:~:text=Students%20in%20elementary%20school%20often,Sherman%20%26%20Bisanz%2C%202009).
What does research suggest about teaching and learning the equal sign? - Cambridge Mathematics, accessed November 5, 2025, https://www.cambridgemaths.org/Images/espresso_34_the_equal_sign.pdf
Full article: Differences in grade 7 students' understanding of the equal sign, accessed November 5, 2025, https://www.tandfonline.com/doi/full/10.1080/10986065.2022.2058160
Middle-School Students' Understanding of the Equal Sign: The Books They Read Can't Help - Department of Psychology, accessed November 5, 2025, https://psych.wisc.edu/alibali/home/Publications_files/McNeil_Grandau_etal_06.pdf
A Study of Mathematical Equivalence: The Importance of the Equal Sign - LSU Scholarly Repository, accessed November 5, 2025, https://repository.lsu.edu/cgi/viewcontent.cgi?article=1397&context=gradschool_theses
How to help your young child understand the equals sign (plus free printable), accessed November 5, 2025, https://www.mathkidsandchaos.com/understanding-equals-sign/
Lesson 4. 1 Understanding the Equal Sign - Didax, accessed November 5, 2025, https://www.didax.com/pub/media/fileuploader/custom/upload/File-1576689484.pdf
All About The Concrete Representational Abstract (CRA) Method - Accelerate Learning, accessed November 5, 2025, https://blog.acceleratelearning.com/concrete-representational-and-abstract
Using the CRA Framework in Elementary Math - Edutopia, accessed November 5, 2025, https://www.edutopia.org/article/using-cra-framework-elementary-math/
ALGEBRA TILES | Calculate, accessed November 5, 2025, https://calculate.org.au/wp-content/uploads/sites/15/2019/03/lesson-sequence-for-algebra-tiles.pdf
Concrete–Representational–Abstract (CRA) Instructional Approach in an Algebra I Inclusion Class: Knowledge Retention Versus Students' Perception - MDPI, accessed November 5, 2025, https://www.mdpi.com/2227-7102/13/10/1061
Concrete-Representational-Abstract: Instructional Sequence for Mathematics - PaTTAN, accessed November 5, 2025, https://www.pattan.net/getmedia/9059e5f0-7edc-4391-8c8e-ebaf8c3c95d6/CRA_Methods0117
What Is a Bar Model? How to Use This Math Problem-Solving Method in Your Classroom, accessed November 5, 2025, https://www.teachstarter.com/us/blog/the-bar-model-for-problem-solving-us/
Concrete, Representational, Abstract (CRA) - Mathematics Hub, accessed November 5, 2025, https://www.mathematicshub.edu.au/plan-teach-and-assess/teaching/teaching-strategies/concrete-representational-abstract-cra/
Tips for Teaching Balancing Equations: Activities, Games & More, accessed November 5, 2025, https://mollylynch.com/2024/11/tips-for-teaching-balancing-equations.html
Find unknown values: Year 4: Planning tool - Mathematics Hub, accessed November 5, 2025, https://www.mathematicshub.edu.au/planning-tool/4/algebra/find-unknown-values/
Activities to Help Your Students Understand the Equal Sign - The Chocolate Teacher, accessed November 5, 2025, https://www.thechocolateteacher.com/2022/01/teaching-the-equal-sign.html
Using Models to Solve Equations | Texas Gateway, accessed November 5, 2025, https://texasgateway.org/resource/using-models-solve-equations
1.4 Activity: Solving Algebraic Equations-Making Connections to Balance Act, accessed November 5, 2025, https://rotel.pressbooks.pub/algebra4elementaryteachers/chapter/1-4-activity-solving-algebraic-equations-making-connections-to-balance-act/
Equations With a Balance Scale | TPT, accessed November 5, 2025, https://www.teacherspayteachers.com/browse?search=equations%20with%20a%20balance%20scale
What Is A Bar Model & How Is It Used In Elementary School Math? - Third Space Learning, accessed November 5, 2025, https://thirdspacelearning.com/us/blog/what-is-bar-model/
Our guide to teaching with a bar model - White Rose, accessed November 5, 2025, https://whiteroseeducation.com/latest-news/guide-teaching-bar-model
The Ultimate Elementary School Guide To The Bar Model, accessed November 5, 2025, https://thirdspacelearning.com/us/blog/teach-bar-model-method/
Bar models in addition and subtraction problems - 4th grade math - YouTube, accessed November 5, 2025, https://www.youtube.com/watch?v=cAGpjInd6iw
Bar Models Worksheets, accessed November 5, 2025, https://www.easyteacherworksheets.com/math/integers-barmodels.html
Unknown Numbers Multiplication Word Problems - TPT, accessed November 5, 2025, https://www.teacherspayteachers.com/browse?search=unknown%20numbers%20multiplication%20word%20problems
Modeling Equations with Tape Diagrams - Solvent Learning, accessed November 5, 2025, https://solventlearning.com/modeling-equations-with-tape-diagrams/
Cuisenaire Rod Manipulatives: Learning Key Math Concepts - hand2mind, accessed November 5, 2025, https://www.hand2mind.com/glossary-of-hands-on-manipulatives/cuisenaire-rods
Cuisenaire rods - Wikipedia, accessed November 5, 2025, https://en.wikipedia.org/wiki/Cuisenaire_rods
Learning Algebra with Cuisenaire Rods - Highhill Homeschool, accessed November 5, 2025, http://highhillhomeschool.blogspot.com/2014/07/learning-algebra-with-cuisenaire-rods.html
Inspire me! Cuisenaire Rods: The CPA approach to teaching algebra - HFL Education, accessed November 5, 2025, https://www.hfleducation.org/blog/inspire-me-cuisenaire-rods-cpa-approach-teaching-algebra
Number Line. Mathematics Worksheets and Study Guides Fourth Grade. - NewPathWorksheets.com, accessed November 5, 2025, https://newpathworksheets.com/math/grade-4/number-line
How to Work Out Unknown Values on a Number Line or Scale - YouTube, accessed November 5, 2025, https://www.youtube.com/watch?v=Sf7ObeTVTUk
I Do We Do You Do Lesson Plans | TPT, accessed November 5, 2025, https://www.teacherspayteachers.com/browse/free?search=i%20do%20we%20do%20you%20do%20lesson%20plans
Classroom Practice | What is 'I Do, We Do, You Do' Structural Learning, accessed November 5, 2025, https://www.structural-learning.com/post/i-do-we-do-you-do
Special Education Math Curriculum - enCORE Lesson Plans - TeachTown, accessed November 5, 2025, https://web.teachtown.com/solutions/encore/lesson-plans-math/
Content Standard CCSS.Math.Content.4.OA Operations and Algebraic Thinking Math.Content.4.OA.A Use the four operations with whole, accessed November 5, 2025, https://www.in.gov/doe/files/ILEARN-Grade-4-ELA-Math-Item-Specifications.pdf
Exit Ticket - The Teacher Toolkit, accessed November 5, 2025, https://www.theteachertoolkit.com/index.php/tool/exit-ticket
The Equal Sign Exit Slips Exit Tickets Assessment Quick Check Balancing Equation - TPT, accessed November 5, 2025, https://www.teacherspayteachers.com/Product/The-Equal-Sign-Exit-Slips-Exit-Tickets-Assessment-Quick-Check-Balancing-Equation-4853404
16 Ways to Differentiate Math Instruction in the Classroom - What I Have Learned, accessed November 5, 2025, https://whatihavelearnedteaching.com/ways-to-differentiate-math-instruction/
Equations Bar Model - TPT, accessed November 5, 2025, https://www.teacherspayteachers.com/browse?search=equations%20bar%20model
4th Grade Math Word Problems Worksheets - K5 Learning, accessed November 5, 2025, https://www.k5learning.com/free-math-worksheets/fourth-grade-4/word-problems
Multiplication & division word problems - Grade 4 - K5 Learning, accessed November 5, 2025, https://www.k5learning.com/free-math-worksheets/fourth-grade-4/word-problems/multiplication-division
Multi-Step Word Problems - 4th Grade Math - Print & Digital - 4.OA.A.3 - MagiCore, accessed November 5, 2025, https://magicorelearning.com/shop/grade/4th-grade/multistep-word-problems-google-slides-distance-learning-4-oa-a-3
4.OA.A.3 Worksheets, Workbooks, Lesson Plans, and Games - Education.com, accessed November 5, 2025, https://www.education.com/common-core/CCSS.MATH.CONTENT.4.OA.A.3/
Free Math Common Core 4.OA.A.3 Worksheets - TPT, accessed November 5, 2025, https://www.teacherspayteachers.com/browse/independent-work/worksheets/math/ccss-4-OA-A-3/free
Multiplicative Comparison Worksheets 4th Grade Math | Teaching 4.OA.A.1 and 4.OA.A.2, accessed November 5, 2025, https://theteacherscafe.com/fouth-grade-math-4-oa-a-1-and-4-oa-a-2/
Using a Balance Scale | TPT, accessed November 5, 2025, https://www.teacherspayteachers.com/browse/free?search=using%20a%20balance%20scale
One Step Word Problems 4 Operations - TPT, accessed November 5, 2025, https://www.teacherspayteachers.com/browse?search=one%20step%20word%20problems%204%20operations
Sentence Stems - Third Space Learning, accessed November 5, 2025, https://thirdspacelearning.com/us/math-resources/sentence-stems/
Start the Conversation in Math: Sentence Starters, Stems & Frames - HMH, accessed November 5, 2025, https://www.hmhco.com/blog/math-sentence-starters-stems-frames
Math Sentence Stems | TPT, accessed November 5, 2025, https://www.teacherspayteachers.com/browse?search=math%20sentence%20stems
Differentiation in Math: Strategies to Support ALL Students in the Math Classroom, accessed November 5, 2025, https://lightbulbmomentsinmath.com/differentiated-instruction-strategies-how-to-support-all-students-in-the-math-classroom/
Multistep word problems - 4th Grade Math 4.OA.A.3 - YouTube, accessed November 5, 2025, https://www.youtube.com/watch?v=Dmagiho6fME
Two Step Equations Activities - TPT, accessed November 5, 2025, https://www.teacherspayteachers.com/browse/free?search=two%20step%20equations%20activities
12 Activities that Make Practicing Two-Step Equations Pop - Idea Galaxy, accessed November 5, 2025, https://ideagalaxyteacher.com/two-step-equations-activities/
Two-step equations | Algebra (practice) - Khan Academy, accessed November 5, 2025, https://www.khanacademy.org/math/get-ready-for-algebra-i/x127ac35e11aba30e:get-ready-for-equations-inequalities/x127ac35e11aba30e:two-step-equations/e/linear_equations_2

https://docs.google.com/document/d/1u2oerY4dmesHkGCaaUezav2OYFM5u8YNt9iCfA168eo/edit?usp=sharing