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Moving Math in the Write Direction

📝 Summary of "Moving Math in the Write Direction"

The article "Moving Math in the Write Direction" by Bostiga, Cantin, Fontana, and Casa discusses the implementation of math debate journals as an innovative way to encourage students to construct mathematical arguments and critique the reasoning of others. This approach aims to improve students' communication skills and mathematical competencies, aligning with the expectations of the Common Core State Standards for Mathematics (CCSSM).

Key Components of the Approach

The authors, who worked with small groups of third and fifth graders, share their experiences and offer considerations for other teachers. The core components of their implementation include:

Writing Meaningful Prompts: Prompts were based on common misconceptions to compel students to reason deeply about the concept and critique others' arguments. Prompts were structured to allow for various correct/incorrect scenarios (e.g., one correct argument, both correct, or both incorrect) to prevent students from relying on patterns.

Using a Consistent Debate Format: The prompts consistently asked students to compare two viewpoints and explain who they agreed with and why. This format directly addresses the CCSSM Standard for Mathematical Practice (SMP) 3, which emphasizes justifying conclusions and responding to others' arguments.

Ensuring Adequate Content Knowledge: Prompts must be deep enough for debate but not so challenging that students lack the necessary content understanding to write effectively.

Discussion Before Writing: Having students engage in a group discussion before writing was found to be crucial for furthering content knowledge, brainstorming ideas, and organizing their thinking. The discussion approach involved reading the question, solving the problem individually, comparing their answer to the presented arguments, and debating the positions.

Refining the Writing: To guide students toward quality responses that include detail, clear reasoning, and support for their claims, the authors used several methods:

Providing sample responses (good and underdeveloped).

Using a minilesson where students collectively revised a response to generate a list of characteristics for a high-quality answer.

Continuously giving feedback to encourage students to write with enough detail to convey their ideas to someone with no mathematical background.

Outcomes and Recommendations

The process led to significant improvements in students' reasoning and written communication, conveying a deeper understanding of mathematical concepts.

The authors also emphasized differentiating the writing instruction for individual needs, such as helping a student transfer verbal ideas onto paper by recording their thoughts.

Holding high expectations was a factor in students stepping up to the challenge.

Teacher Tips for Implementation

The article concludes with a list of tips for teachers implementing math debate journals:

Create prompts that require in-depth consideration of concepts.

Start slowly with plenty of modeling.

Revisit minilessons on written feedback, elements of a good response, and productive discussion.

Support student discussion of mathematical concepts before writing.

Recognize that this is a long-term process.

Allow time for peer review, feedback, rewriting, and editing.

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A teacher asked two students to calculate the sum of the fractions 1/4 + 2/3.

Student A says the answer is 3/7

Student B says the answer is 11/12

Who do you agree with, Student A or Student B? Explain your reasoning fully, including why the student you disagree with is incorrect. Use pictures or a diagram to support you thinking.

Student B is correct because you have to find a common denominator before you can add fractions. The least common multiple of 4 and 3 is 12. You have to convert 1/4 to an equivalent fraction with a denominator of 12: 1/4 x 3/3=3/12. Convert 2/3 to an equivalent fraction with a denominator of 12: 2/3 x 4/4= 8/12 and add the new fractions.

Student A is incorrect because they added the numerators and denominators without finding a common denominator first. The denominator tells you the size of the pieces, and you cannot add pieces of different sizes directly.

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Ms. Jones asked her class to simplify the fraction 12/18

Student A simplified the fraction to 6/9

Student B simplified the fraction to 2/3

Do you agree with Student A, Student B, or both? Explain why both students answers can be considered correct simplifications of the original fraction, even though they look different.

Student A is correct because they divided both the numerator and the denominator by a common factor of 2. Student B is also correct because they divided both the numerator and the denominator by the greatest common factor of 6. Both 6/9 and 2/3 are equivalent fractions to 12/18 and are therefore correct. 2/3 is the fraction in its simplest form, but 6/9 is a correct simplification.

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Two students were asked to find the answer to 4 x 5.

Student A says the answer is 4 + 5=9

Student B says the answer is 4x4x4x4x4=1024

Do you agree with Student A, Student B, or neither? Explain why each students reasoning is flawed and provide the correct answer and reasoning to both of them.

Student A is incorrect because they did addition 4 + 5 instead of multiplication 4x5. Multiplication is repeated addition, not the sum of the factors.

Student B is incorrect because they did the exponents 4^5 instead of multiplication 4X5.

The correct answer is 20. Multiplication 4x5 means 4 groups of 5 or 5 groups of 4.

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Students will be able to independently apply measurement and problem-solving skills to estimate, measure, and reason about mass and liquid volume in real-world situations, selecting appropriate units and operations to make informed decisions in everyday contexts.

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Students will understand that measurements such as mass and liquid volume represent real quantities that help them make sense of everyday situations, and that selecting the correct unit and operation is essential for accurately solving problems involving these quantities.

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How do the units and tools we choose help us understand and solve real-world problems involving mass and liquid volume?

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Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).1 Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.)

The student must know and understand the following mathematical content:

Measurement Concepts: Understanding that liquid volume and mass are measurable attributes of objects.

Metric Units:

Mass: Identifying and understanding the standard units of grams (g) and kilograms (kg). Students must understand that kilograms are used for heavier objects, and grams for lighter ones.

Liquid Volume: Identifying and understanding the standard unit of liters (l).

Measurement Tools/Representations: Understanding how measurement tools (like a beaker with a scale) represent the quantities being measured and how to read the measurements.

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My Grades

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Sydney's Attendance and Learning Experience Sheet

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Fractions, Number Lines, and Third Graders

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Fractions, Number Lines, and Third Graders

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📚 Summary of "Fractions, Number Lines, & Third Graders"

This article, which discusses work with third-grade students in an urban district, explores ways to successfully introduce the challenging number line model for fractions and how it helps students develop robust mathematical understandings. The authors suggest that using the number line, a model required by the Common Core State Standards for Mathematics (CCSSM), is particularly complex for students.

The work was conducted in two phases:

Phase 1: Uncovering Student Misunderstandings

Initial fraction lessons included the use of fraction circles, paper-folding strips, chips, and the number line. This phase revealed several misunderstandings common among third graders:

Partitioning Misunderstanding (Sarah): Counting the number of tick marks instead of the spaces between them to determine the denominator.

Unit Misinterpretation (Shanya): Considering the distance between zero and two as the unit (whole) instead of the length between zero and one.

Incomplete Partitioning (AJ): Identifying the unit as the length between zero and one but failing to partition the distance completely or accurately.

These misunderstandings led the authors to identify key focus areas for instruction: making sense of the unit, understanding partitioning (dividing the unit length into equal parts), and interpreting the meaning of the point on the line as a measure of distance from zero.

Phase 2: Lesson Revision

The revised lessons employed a pedagogical strategy that involved using multiple representations and making connections among them to build meaning. The number line activities were integrated with previous part-whole models like fraction circles, paper-folding strips, and chips.

Key aspects of the revised lessons (Table 1) included:

Using Context (Lesson 1): Introducing number line tasks within contexts (e.g., distance from school to a library) helped students make sense of zero, one, and the meaning of a unit. Successful students recognized zero as the starting point and one mile as the unit length.

Connecting Representations (Lessons 2 & 3): Students explored connections between contexts, the paper-folding strip model (representing length), and the number line.

Multiple Representations and Partitioning (Lesson 4): Students examined how partitioning done with circles, chips, and paper folding could be reinterpreted onto the number line.

Outcomes for Successful Students

Interviews with successful students demonstrated a deeper understanding of the number line model:

Understanding the Unit: Successful students like Eva and Hanna consistently identified the unit first (zero to one) before locating fractions.

Partitioning Strategy: Successful students often partitioned the unit based on the denominator (e.g., cutting in half and then half again for fourths) before locating the fraction. They also mastered partitioning beyond halves, such as into thirds.

Fraction as Distance: Some students began to view fractions as distances from zero rather than just points on the line, extending their understanding beyond the part-whole model.

Conclusion

The work suggests that context plays an important role in helping third graders understand fraction concepts related to the number line, particularly in interpreting the distance from zero to one as the unit. The use of multiple representations and connections among them was a productive strategy for meeting CCSSM number line goals. The authors conclude that the number line should be sequenced after students have had experience with more concrete fraction representations.

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Sydney

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Exploring Multiplication: Three In A Row Lucky Numbers

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Exploring Multiplication: Three In A Row Lucky Numbers

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Summary from Grok

The article, titled "Exploring Multiplication: Three-in-a-Row Lucky Numbers" by James A. Russo (published in Teaching Children Mathematics, April 2018), introduces an engaging game-based activity designed for second-grade students to build understanding of multiplication. Russo, a primary school teacher in Victoria, Australia, emphasizes using concrete and pictorial representations to help students grasp multiplication as "groups of" items, while fostering fluency, strategic thinking, and exploration of related concepts like the distributive property and prime numbers.

Game Setup and Rules

Materials: An eight-sided die (for the number of groups), a six-sided die with dots (for items per group), a 100 or 120 chart as the game board, markers or counters, and whiteboards for modeling.

Players: 2–4 students.

Objective: Be the first to score three different "three-in-a-rows" on the chart—consecutive numbers aligned horizontally, vertically, or diagonally (e.g., 2-3-4 or 6-16-26).

Gameplay:

Players roll the dice and calculate the product (e.g., 4 groups of 4 dots = 16, using skip counting or drawings).

Mark the resulting number on the chart.

If the number is already occupied, it's a "lucky number," allowing the player to mark any unoccupied spot—adding strategy and enabling access to harder numbers like primes.

Representations: Starts with "groups of" drawings, progresses to number sentences (e.g., 4 × 4 = 16 or repeated addition), and later introduces arrays for visual transitions to multiplicative thinking.

Classroom Implementation

Russo describes implementing the game over five 50-minute sessions in a two-week unit:

Session 1: Introduction and "groups of" focus, with mixed-ability pairs. Post-game discussions highlight patterns like frequent "lucky" (composite) numbers and rare ones (primes).

Sessions 2–3: Encourage predictions, recording number sentences, and exploring the distributive property (e.g., breaking 5 × 6 into 5 × 5 + 5 × 1 = 30). Student examples show strategies like splitting groups for easier counting (e.g., counting by fives instead of sixes).

Sessions 4–5: Introduce arrays to illustrate commutativity (e.g., 5 × 2 = 2 × 5 via rotation) and distributivity. Advanced groups use larger dice (e.g., ten-sided). Discussions lead to investigating primes (numbers only formable as 1 × n "skinny" rectangles) vs. composites.

Educational Benefits

Engagement and Fluency: The game balances fun with mathematical focus, as winning requires attention to multiplication structure.

Conceptual Depth: Exposes students to distributive and commutative properties organically through student-led strategies and discussions.

Prime/Composite Introduction: "Lucky" vs. "unlucky" numbers spark inquiries into why some products (e.g., 13, 43) rarely appear, leading to array-based explorations of factors.

Extensions: Adapt for addition/subtraction versions or larger dice for challenge. Russo notes it aligns with research on multiple representations for building multiplication understanding.

Examples from the Classroom

Gypsy uses distributivity intuitively (4 × 4 = 8 × 2) and wins via a lucky number.

Mikayla breaks 5 × 6 into easier parts (5 × 5 + 5 = 30).

Keaton and Rylan discover commutativity by rotating arrays and identify 13 as prime.

Concluding Insights

Russo concludes that the game provides hands-on experience with representations, encourages connections between ideas, and organically introduces advanced topics. He recommends it for early multiplication instruction and references related works on games, representations, and properties in math education. Additional resources are available at surfmaths.com.

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Standard:
Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, using drawings (such as beakers with a measurement scale) to represent the problem.

Learning Objective:

Students will accurately read measurements of liquid volume and mass from scales and containers (grams, kilograms, and liters) and use these measurements to solve one-step word problems.

Math Word Problem:

Sarah is baking muffins. She pours milk into a measuring cup.
The milk level lines up exactly with the 600-milliliter mark on the scale.
Then she adds more milk until the level reaches the 900-milliliter mark.

How much milk did Sarah add?

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Standard:
Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, using drawings (such as a beaker with a measurement scale) to represent the problem.

Learning Objective:

Students will be able to solve one-step mass and volume word problems by correctly selecting and applying the appropriate operation (addition, subtraction, multiplication, or division).

Math Word Problem:

A recipe for fruit punch uses 250 mL of orange juice.
Aiden wants to make 4 batches of the recipe for his class party.

How much orange juice does Aiden need in total?

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Standard(s)
- Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division, or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

You may also connect:

- Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

These support the fluency needed to solve mass and volume word problems in

Learning Objective:

Students will fluently add, subtract, multiply, and divide whole numbers in order to accurately solve one-step mass and liquid volume word problems.

Math Word Problem:

A box contains 12 small bags of trail mix.
Each bag has a mass of 50 grams.

What is the total mass of all 12 bags?

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Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).1 Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.)

The student must be able to perform the following actions:

Measuring: Accurately measure liquid volumes and masses of objects using the standard units (g, kg, l). This implies the skill of reading a scale or container.

Estimating: Estimate the liquid volumes and masses of objects. This requires having a mental benchmark for the size of a gram, a kilogram, and a liter.

Computational Fluency: Add, subtract, multiply, or divide whole numbers fluently. This is a crucial prerequisite skill applied to the measurement context.

Problem Solving:

Solve one-step word problems involving mass or volume.

Determine which operation (addition, subtraction, multiplication, or division) is necessary to solve a given word problem context.

Representational Skills: Use drawings (such as a beaker with a measurement scale) to represent the problem, which aids in conceptual understanding and problem-solving.

Unit Recognition: Recognizing that the word problems will involve measurements that are already given in the same units (e.g., all grams or all liters), avoiding the need for unit conversion at this grade level.

📏 Measuring (Reading a Scale or Container)

Visual Perception: Discerning markings and lines accurately.

Spatial Reasoning: Mapping the physical quantity (mass/volume) to its numerical position on the scale.

Calibration/Mapping: Understanding how the space between markings represents specific units.

Working Memory: Holding and processing measurements to calculate amounts (if required).

🤔 Estimating (Mass and Volume)

Conceptual Benchmarking: Forming and recalling a mental model or schema for the size/weight of standard units (g, kg, l).

Proportional Reasoning: Comparing the target object's attributes to the recalled mental benchmark.

Inference/Judgment: Making an educated guess based on available conceptual knowledge.

➕ Computational Fluency (Arithmetic)

Automaticity: Accessing and performing basic arithmetic facts and procedures rapidly from long-term memory.

Retrieval: Quickly recalling appropriate algorithms for calculations.

Pattern Recognition: Identifying the structure of the math problem for efficient calculation.

❓ Problem Solving (Word Problems)

Reading Comprehension: Extracting necessary data and identifying the goal from the problem text.

Semantic Understanding: Interpreting the meaning of the text to understand the required relationship (e.g., combining, finding the difference).

Strategy Selection: Employing metacognition (thinking about thinking) to choose the correct mathematical operation.

Executive Function: Planning the solution steps and monitoring the overall process.

🖼️ Representational Skills (Using Drawings)

Translation/Encoding: Converting abstract information (the word problem) into a concrete, visual format.

Symbolic Reasoning: Understanding that lines, numbers, and drawings on a beaker/scale symbolize specific quantities and calculations.

Visualization: Creating a mental or physical image to clarify quantity relationships.

🏷️ Unit Recognition (g, kg, l)

Attention to Detail: Focusing on and identifying the unit labels (g, kg, l).

Rule Application: Applying the grade-level constraint of working only with measurements in the same units.

Classification: Categorizing units as belonging to either mass or volume measurements.

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1. What is the nature of a good life?

To me a good life is about being content with what you have while continuing to learn about yourself and grow as a person. It means building a life with a partner you choose, supporting each other, and working toward your goals and dreams. A large part of my happiness comes from the people I surround myself with friends and family, people I love, trust, and feel supported by. A good life is about doing things that bring me genuine happiness.

2. What makes a good society?

A good society is one where people are treated with respect, kindness, and fairness. I believe everyone should be treated equally, regardless of their differences. In a good society, people care about one another and are willing to help those around them. It also means taking responsibility for the communities and taking care of the places we share.

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1. Should education prepare young people to fit into society as it is, or to seek improvements in society?

School should teach young people to seek improvements in society. Just like in our personal lives, there is always room for growth and change in the world around us. I think its important for students to learn about what is happening in their communities and the world, but also to understand how they can contribute and make a difference on their own. Education should help young people grow as individuals and encourage them to think critically and independently.

2. Should there be different educational paths for different groups, or a unified approach?

I do not think there is one clear answer to this question. I believe there should be a balance of both. Everyone has different learning styles and interests, and students should have the opportunity to choose what work's best for them. At the same time, there is value in having a unified education system because it makes sure that everyone learns the same essential information and skills.

3. Should public education focus on general citizenship or specific vocational training?

I think public education should focus on both. Schools should teach practical skills that students will use throughout their lives, while also incorporating lessons about citizenship, responsibility, and how to interact with other people in our society. This balance helps students prepare for both their future careers and their own community.

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Lesson One: Solving and Creating One-Step Mass & Liquid Volume Word Problems

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Lesson One: Solving and Creating One-Step Mass & Liquid Volume Word Problems

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Grade Level
3rd Grade

Time
30–40 minutes

Evidence of Learning

Students create and solve a one-step word problem involving mass or liquid volume using the correct operation (addition, subtraction, multiplication, or division), appropriate units, and clear work (numbers, words, or drawings).

Standard

NY-3.MD.2
Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (L). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes given in the same units.

Learning Objective

Students will solve and create one-step word problems involving mass or liquid volume by choosing and correctly using addition, subtraction, multiplication, or division.

UDL Support (Multiple Means of Representation & Expression)

To support diverse learners, students are provided with visual supports and flexible response options. During modeling and independent work, students may use drawings, diagrams, number lines, measuring visuals, or written explanations to represent their thinking. Sentence starters and problem frames are available for students who need language support, while all students may choose how to show their work (numbers, words, or pictures).

Materials

Whiteboard or chart paper

Measuring visuals (pictures of measuring cups, scales, beakers, or containers)

Student worksheet or notebook paper

Pencils

Optional: Real or toy measuring tools for demonstration

UDL Support Materials: sentence starters (e.g., “I chose ___ because…,” “The total mass/volume is…”) and a visual anchor chart showing operations and when to use them

Procedure

Introduction (5 minutes)

Quickly review units of mass (grams, kilograms) and liquid volume (milliliters, liters).

Display measuring visuals and ask:
“When we measure things like juice or trail mix, when might we need to add, subtract, multiply, or divide?”

Briefly discuss real-life situations for each operation (e.g., adding more liquid, finding how much was added, making multiple batches, finding total mass of groups).

Point to the anchor chart to support students who benefit from visual cues.

Support for NLL

Simple Strategy: Use picture–word matching with gestures

As you review units and operations, point to each measuring visual and say the word while gesturing (e.g., pretend to pour for milliliters, show grouping with hands for multiplication). Have new language learners repeat the word and match it to the picture on the anchor chart. This supports understanding by pairing spoken language, visuals, and movement without requiring students to produce full sentences right away.

Teacher Modeling (10 minutes)

Present a sample word problem on the board:

“A recipe calls for 250 mL of orange juice. Aiden wants to make 4 batches for his party. How much orange juice does he need in total?”

Think aloud while solving:

Identify the unit (mL).

Decide on the operation (multiplication—making batches means repeated addition).

Draw a simple beaker or number line to visually represent the problem.

Solve step-by-step:
250 × 4 = 1,000 mL

Write a clear math sentence and label the answer.

Explicitly remind students that they may choose to draw, write, or use numbers when they solve their own problems.

Guided Practice (10 minutes)

Solve two problems together as a class (one mass, one volume):

Mass: “Each bag of trail mix has a mass of 50 grams. What is the total mass of 12 bags?”

Volume: “Sarah poured 600 mL of milk into a bowl and then added more until she had 900 mL. How much milk did she add?”

Have students turn and talk to justify the operation chosen, using sentence starters if helpful.

Record solutions on the board using both numbers and visuals, emphasizing correct units and a math sentence.

Creating the Evidence of Learning (10 minutes)

Students work independently to:

Create their own original one-step word problem involving mass or liquid volume (using correct units: g, kg, mL, or L).

Solve their problem, showing work using numbers, words, drawings, or models.

Write a clear math sentence with the labeled answer.

Students may use problem frames or visual supports as needed. Circulate to provide support and ensure problems are one-step and realistic.

Closure (5 minutes)

Invite 3–4 students to share their created problems and solutions with the class (oral sharing or displaying work).

As a group, discuss:
“How did you decide which operation to use?”

Collect student work for assessment.

Assessment / Evidence of Learning

Rubric Focus:

Student-created word problem is clear, realistic, and involves mass or liquid volume.

Correct operation is chosen and justified (verbally, in writing, or through visuals).

Solution is accurate with proper units (g, kg, mL, L).

Work is shown clearly using a method of the student’s choice (numbers, words, drawings, or models).

Collected student problems serve as the primary evidence of mastery.

Technology Integration (SAMR)

SAMR – Augmentation (simple use of technology)

Have students create and solve their word problems in a shared digital slide or document instead of on paper. Students can type the word problem, insert a simple drawing or digital image (e.g., a beaker or scale), and label units using text boxes. This slightly enhances the task by making it easier to revise work, add visuals, and clearly label thinking, while keeping the learning goal the same.

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Lesson Two: Choosing the Right Operation for Volume Word Problems

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Lesson Two: Choosing the Right Operation for Volume Word Problems

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Grade Level: 3rd Grade
Time:

30–40 minutes

Evidence of Learning
Students will solve a set of one-step mass and volume word problems by correctly selecting and applying the appropriate operation (addition, subtraction, multiplication, or division) using numbers, words, or drawings. Students justify their choice of operation and clearly label units in their solutions.

Standard
NY-3.MD.2
Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (L). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes given in the same units.

Learning Objective
Students will be able to solve one-step mass and volume word problems by analyzing the problem, selecting the correct operation, and representing their solution accurately with proper units.

UDL Support (Multiple Means of Representation & Expression)

Students are provided with an Operation Choice Visual that pairs each operation with a simple icon and sentence frame (e.g., multiplication = equal groups icon + “I see ___ equal groups of ___”).

Students may use drawings, diagrams, number lines, measuring visuals, or written explanations to represent their thinking.

Sentence starters and problem frames are available for language support.

Visuals and manipulatives remain available throughout the lesson to support engagement and understanding.

Materials

Whiteboard or chart paper

Operation Choice Visual with icons and sentence frames

Chart paper or large graphic organizer showing keywords for operations

Student worksheets with a mix of one-step mass and volume problems

Visual representation of measuring cups, beakers, or scales

Pencils

Procedure

Introduction / Review (5 minutes)

Quickly review units of mass (grams, kilograms) and liquid volume (milliliters, liters).

Display measuring visuals and ask:
“When we measure things like juice or trail mix, when might we need to add, subtract, multiply, or divide?”

Briefly discuss real-life situations for each operation.

Point to the Operation Choice Visual and keyword chart for visual support.

Teacher Modeling (15 minutes)

Present a sample word problem on the board:
“A recipe for fruit punch uses 250 mL of orange juice. Aiden wants to make 4 batches for his class party. How much orange juice does he need in total?”

Think aloud while solving:

Identify the unit (mL).

Underline key information: “4 batches” and “250 mL”.

Refer to the Operation Choice Visual to select the operation.

Use the sentence frame: “I see ___ equal groups, so I use ___.”

Solve step-by-step: 250 × 4 = 1,000 mL.

Extend understanding: 1,000 mL = 1 L, using a measuring cup visual.

Guided Practice (10 minutes)

Solve two to four problems together (covering addition, subtraction, multiplication, division).

For each problem:

Identify keywords.

Select the operation using the Operation Choice Visual and keyword chart.

Solve using numbers, words, or drawings.

Students turn and talk to justify operation choice using sentence frames.

Support for NLL
Think–Pair–Share with Sentence Frames:

Students identify the operation for a problem, then explain their reasoning to a partner:
“I chose ___ because the problem says ___.”

This reinforces academic language and comprehension in a low-pressure setting.

Independent Practice / Evidence of Learning (10 minutes)

Students complete a set of one-step mass and volume word problems independently.

For each problem:

Circle keywords

Write the operation name

Solve the problem and label units

Optionally, use drawings, words, or numbers to show work

Closure / Assessment (5 minutes)

Invite 2–3 students to share their solutions and reasoning.

Discuss as a class: “How did you decide which operation to use?”

Collect student worksheets as primary evidence of mastery, focusing on:

Correct operation selection and justification

Accurate solution with proper units

Clear representation of work (numbers, words, drawings)

Technology Integration (SAMR)

Substitution: Use a digital whiteboard or interactive slide where students can highlight key information, drag operation icons, and type solutions instead of using paper and pencil. This allows immediate feedback and maintains the same learning goal.

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Lesson Three: Using Drawings to Solve Liquid Volume Addition Problems

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Lesson Three: Using Drawings to Solve Liquid Volume Addition Problems

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Grade Level: 3rd Grade
Time:

30–40 minutes

Evidence of Learning
Students will solve one-step liquid volume addition problems by accurately representing the volumes using drawings and correctly calculating the total. Students may use numbers, words, or drawings to show their work and clearly label units (mL or L).

Standard
NY-3.MD.2
Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (L). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes given in the same units, using drawings (such as beakers with a measurement scale) to represent the problem.

Learning Objective
Students will accurately read and draw liquid volumes using a measurement scale and use the drawing to solve one-step addition word problems involving liquid volume.

UDL Support (Multiple Means of Representation & Expression)

Pre-labeled measuring cup visuals and partially completed beaker diagrams reduce cognitive load and provide visual scaffolds.

Students can shade, label, or annotate the drawing digitally or physically.

Students can represent their solution using numbers, words, drawings, or a combination, giving multiple ways to express understanding.

Visual vocabulary and hand gestures are used during modeling to support comprehension for new language learners.

Materials

Whiteboard or projector

Large diagram of a measuring cup or beaker with clear mL markings

Student grid paper or worksheet with pre-drawn blank measuring cup templates

Rulers for drawing straight lines

Optional: colored pencils for shading volumes

Pencils

Procedure

Introduction / Review (5–10 minutes)

Review units of volume: milliliters (mL) and liters (L) and the relationship: 1,000 mL = 1 L.

Display a measuring cup diagram and discuss interval values (e.g., if 5 lines between 100 mL and 200 mL, each line = 20 mL).

Practice reading volumes on the scale (e.g., “Show me 450 mL” or “Which line shows 350 mL?”).

Use visuals, gestures, and repeated verbal cues to support understanding, especially for new language learners.

Teacher Modeling (15 minutes)

Present a word problem on the board:
“Maya is making lemonade. She pours 400 mL of water into a large pitcher, then adds 250 mL of lemon juice. What is the total volume?”

Think aloud while solving:

Identify the operation (addition).

Draw a beaker on the board, mark 400 mL, and shade to that line.

Add the second volume: draw a dashed line representing 250 mL above the first shading.

Calculate total: 400 + 250 = 650 mL.

Label the final volume clearly.

Reference pre-labeled or partially completed templates to scaffold student understanding.

Pair vocabulary with gestures: “add/combine” = bring two groups together; “volume” = point to shaded area.

Guided Practice (10 minutes)

Present a similar word problem:
“A chemist mixes 150 mL of Solution A with 80 mL of Solution B. Draw a diagram to find the total volume.”

Students create their own visual on the pre-drawn template.

Teacher circulates to check:

Correct starting value reading

Accurate marking of total volume

Proper labeling of units

Support for NLL
Visual Vocabulary with Gestures:

As students follow the modeling, they mimic gestures paired with key terms.

Encourages students to connect spoken language, visuals, and actions to the math problem.

Independent Practice / Evidence of Learning (10 minutes)

Students complete an independent problem, choosing to represent their solution using a drawing, numbers, or both.

For each problem, students:

Read and mark starting volume correctly

Draw or annotate the added volume

Solve and label the total volume

Collect student work as primary evidence of mastery.

Closure / Assessment (5 minutes)

Invite 2–3 students to share their visual representation and solution with the class.

Discuss strategies used to determine the total volume and how drawings helped.

Ensure that students clearly label units and show work visually or numerically.

Technology Integration (SAMR)

Augmentation: Students may use a digital drawing tool or app to create the measuring container, shade volumes, and label totals. This allows easy editing, adding colors or text, and sharing work digitally while keeping the task essentially the same.