Guess and Check Heuristic withTables anf Bar Models

 Number Talk & Thinking Classroom Lesson Plan: Guess and Check Heuristic Using Tables 

 

This Number Talk and Thinking Classroom approach is designed for 4th-6th graders and also serves as a PD (Professional Development) session to introduce teachers to the power of:

• Guess and Check  "heuristics" tables
• Mathematical bar models
• Visual problem-solving using heuristics
• The Concrete-Pictorial-Abstract (CPA) approach

---

Guess and Check Heuristic Problem Solving with Bar Models FLIPPED PD PODCAST 

Lesson Flow for Students & PD Session Structure

1️⃣ Warm-Up: Quick Number Talk (5-10 minutes)

Objective: Activate prior knowledge and encourage mental math reasoning.

Prompt:
"If you bought a combination of two items, one costing $7 and another costing $4, and spent exactly $141, how might you figure out how many of each item you bought?"

Facilitation:

• Let students think independently for 1-2 minutes.
• Have them share strategies without solving it completely.
• Encourage multiple representations (e.g., breaking down numbers, estimation, or mental division).

Key Discussion Questions:

• How can you estimate a reasonable starting guess?
• If your guess is too high or too low, what should you do?
• How can a table or a visual math model help organize thinking?

---

2️⃣ Hands-On: Thinking Classroom Problem Solving (20-30 minutes)

Students work in groups using vertical whiteboards (or large chart paper) to solve the problem.

Step 1: Present the Word Problem Visually

Write or display:

💭 A box of caramel candy costs $7. A bag of lollipops costs $4. Dad bought 27 items in total and paid $141. How many bags of lollipops did he buy?

Ask:

• What do we know?
• What do we need to find?
• What might be a good first guess?

Step 2: Model with a Bar Diagram (Pictorial Stage)

Guide students to draw a bar model:

• One bar for boxes of candy ($7 per unit).
• One bar for bags of lollipops ($4 per unit).
• A total length of 27 items and a total cost of $141.

Encourage discussion:

• How does the model help visualize the problem?
• What if we start by assuming half-and-half (13 each)?
• What happens if we adjust the numbers?

Step 3: Use a Guess & Check Table

Introduce a table to organize their thinking:

Bags of Lollipops	Boxes of Candy	Lollipops (×$4)	Candy (× $7)	Total Cost
10	17	10 × $4 = $40	17 × $7 = $119	$159 (Too much)
5	22	5 × $4 = $20	22 × $7 = $154	$174 (Too much)
15	12	15 × $4 = $60	12 × $7 = $84	$144 (Too much)
11	16	11 × $4 = $	16 × $7 = $___	$141 ✅ (Correct Answer)

💡 Ask students: What patterns do you notice in the numbers?
💡 How does changing one variable affect the total?

---

3️⃣ Student Reflection & Discussion (10 minutes)

Whole-class discussion:

• How did the bar model help you think about the problem?
• How did the Guess & Check table organize your work?
• Why is visualization an important mathematical strategy?
• How could this method apply to real-world problems?

---

Professional Development (PD) Adaptation for Teachers

Focus:

• Demonstrate how heuristics like Guess & Check and bar models enhance problem-solving.
• Show the power of pictorial models (CPA) in deepening conceptual understanding.
• Highlight how vertical non-permanent surfaces (Thinking Classroom strategy) engage students in reasoning.

Interactive PD Activity:

1. Teachers solve the problem first, using bar models and Guess & Check.
2. Group discussion on why these heuristics work across grade levels.
3. Debrief on instructional strategies, emphasizing student-led discovery over direct instruction.

---

Final Takeaways for Teachers & Students

✅ Bar models make abstract problems visual.
✅ Guess & Check tables encourage systematic problem-solving.
✅ Thinking Classroom strategies (whiteboards, group work) engage students deeply.
✅ Mathematics is about reasoning, not just getting answers.

Please make any suggestions for refinements or additional resources for this PD session? 🚀

To solve this problem using the Guess and Check heuristic from Singapore Math, students would follow a structured approach to systematically test different possible solutions. Here's how it would be done using a Guess and Check table:

---

Step 1: Understand the Problem

We are given:

• Each box of caramel candy costs $7.00.
• Each bag of lollipops costs $4.00.
• Dad bought a total of 27 items (boxes + bags).
• Dad spent a total of $141.00.

The goal is to determine how many bags of lollipops Dad bought.

---

Step 2: Set Up the Guess and Check Table

A Guess and Check table allows us to systematically try different values for the number of lollipops and adjust based on whether the total cost is too high or too low.

Guess (Bags of Lollipops)	Boxes of Caramel Candies	Cost of Lollipops (× $4)	Cost of Candies (× $7)	Total Cost
10	17	10 × $4 = $40	17 × $7 = $119	$40 + $119 = $159 (Too high)
5	22	5 × $4 = $20	22 × $7 = $154	$20 + $154 = $174 (Too high)
15	12	15 × $4 = $60	12 × $7 = $84	$60 + $84 = $144 (Too high)
6	21	6 × $4 = $24	21 × $7 = $147	$24 + $147 = $171 (Too high)
9	18	9 × $4 = $36	18 × $7 = $126	$36 + $126 = $162 (Too high)
12	15	12 × $4 = $48	15 × $7 = $105	$48 + $105 = $153 (Too high)
8	19	8 × $4 = $32	19 × $7 = $133	$32 + $133 = $165 (Too high)
6	21	6 × $4 = $24	21 × $7 = $147	$24 + $147 = $171 (Too high)
16	11	16 × $4 = $___	11 × $7 = $___	$___ + $___ = $141 (Correct Answer)

---

Step 3: Verify the Answer

From the table, we see that when Dad buys ____ bags of lollipops and ____ boxes of caramel candy, the total cost is exactly $141, which matches the given information.

Thus, the correct answer is: Dad bought _____ bags of lollipops.

---

How Students Benefit from This Approach

1. Organized Thinking – The table helps students keep track of their guesses and the calculations.
2. Error Checking – They can immediately see if their guess is too high or too low and adjust accordingly.
3. Pattern Recognition – Students start noticing trends in how increasing or decreasing one variable affects the total cost.
4. Logical Deduction – Instead of randomly guessing, students make more informed choices based on previous results.

This method aligns perfectly with Singapore Math’s heuristic-based problem-solving and ensures students develop strong number sense and reasoning skills! 🚀

Alternative Word Problem & Solution

Problem Statement (Edited for Clarity):

A box of candy costs $7.00, and a bag of lollipops costs $4.00. Dad bought a total of 27 items (candies and lollipops combined) and paid $141.00 in total. How many bags of lollipops did Dad buy?

Solution Using the Guess & Check Heuristic:

We set up a table to systematically test different values for the number of lollipops and candies.

You're absolutely right! Let's carefully go through the correct calculation and then provide a revised solution.

---

10 More Word Problems for Students to Solve Using Guess & Check Tables

1️⃣ A bookstore sells notebooks for $3 each and pens for $2 each. A teacher buys 25 items for $62. How many notebooks did the teacher buy?

2️⃣ Tickets for a school event cost $8 for adults and $5 for students. If 30 people attended and the total revenue was $190, how many students bought tickets?

3️⃣ A pet store sells goldfish for $6 each and hamsters for $9 each. A customer buys 20 animals and spends $138. How many goldfish did they buy?

4️⃣ A farmer sells apples for $2 per pound and oranges for $3 per pound. He sells 50 pounds of fruit for a total of $120. How many pounds of apples did he sell?

5️⃣ A bakery sells cupcakes for $2 each and pies for $10 each. If a customer buys 15 items and spends $80, how many pies did they buy?

6️⃣ A game store sells board games for $12 each and video games for $25 each. If 20 games were sold for a total of $340, how many board games were sold?

7️⃣ A clothing store sells T-shirts for $8 each and hoodies for $25 each. If 35 items were sold for a total of $410, how many hoodies were sold?

8️⃣ A theater charges $12 for adult tickets and $7 for children’s tickets. If 40 tickets were sold for a total of $370, how many adult tickets were sold?

9️⃣ A music store sells guitars for $150 and keyboards for $100. If 12 instruments were sold for a total of $1,500, how many guitars were sold?

🔟 A bike shop sells helmets for $20 each and bicycles for $150 each. If 15 items were sold for a total of $1,000, how many helmets were sold?

---

Teaching Tip: Encourage students to create Guess & Check tables for each problem, adjusting their guesses systematically to find the correct answer. This will strengthen their problem-solving skills, numerical reasoning, and logical thinking. 🚀

Let's carefully go through the solution step by step and verify the correct answer.

---

Step 1: Define the Variables

• Let x = number of lollipops.
• Let y = number of boxes of candy.

We have two equations based on the problem:

1. Total items equation: $$x + y = 27$$
2. Total cost equation: $$4x + 7y = 141$$

---

Step 2: Solve the System of Equations

Step 2.1: Express $y$ in terms of $x$

From equation (1):

$$y = 27 - x$$

Step 2.2: Substitute into equation (2)

$$4x + 7(27 - x) = 141$$

Expand:

$$4x + 189 - 7x = 141$$

Combine like terms:

$$-3x + 189 = 141$$

Subtract 189 from both sides:

$$-3x = -48$$

Divide by -3:

$$x = 16$$

---

Step 3: Find $y$

$$y = 27 - 16 = 11$$

So, Dad bought 16 bags of lollipops and 11 boxes of candy. ✅

---

Step 4: Verify the Solution

1. Total items check:

   $$16 + 11 = 27 \quad \text{✅ Correct}$$

2. Total cost check:

   $$(16 \times 4) + (11 \times 7) = 64 + 77 = 141 \quad \text{✅ Correct}$$

---

Final Answer:

✅ Dad bought 16 bags of lollipops and 11 boxes of candy. 🎯

Thanks for your patience! You were right—16 is the correct answer. 🚀

Singapore Math Bar Model: Using Visualization to Solve Word Problems Grade 4-6

 Four-Week Singapore Math Bar Model Drawing Curriculum

 

Introduction to Mathematical Problem Solving: Visualization Through the Singapore Bar Model

Mathematical problem-solving is a fundamental skill that requires both logical reasoning and strategic thinking. One of the most powerful heuristics in mathematical problem-solving is visualization—specifically, using models to represent and break down complex problems. Among the various visualization techniques, the Singapore Bar Model has emerged as an essential tool for developing problem-solving proficiency, particularly in primary mathematics education.

The Power of Visualization in Mathematics

Visualization is one of the 13 recognized mathematical heuristics that support problem-solving. It allows students to convert abstract numerical and verbal problems into concrete visual representations, making it easier to analyze relationships, recognize patterns, and develop systematic solutions. Research has shown that students who use visual strategies to approach mathematical problems tend to develop deeper conceptual understanding and stronger problem-solving skills.

The Singapore Bar Model: A Structured Approach to Problem Solving

The Singapore Bar Model is a pictorial representation technique that helps students conceptualize and solve arithmetic and algebraic problems. It is a key component of the Concrete-Pictorial-Abstract (CPA) approach, which scaffolds learning from tangible, hands-on experiences to more symbolic representations.

1. Concrete Stage: Students use physical objects, such as counters or manipulatives, to model mathematical concepts.

2. Pictorial Stage: Students transition to drawing bar models, visually representing relationships between known and unknown quantities.

3. Abstract Stage: Students use equations and algebraic expressions to solve problems without the need for visual aids.

Explicit Instruction and Practice in Bar Model Representation

Although the Singapore Bar Model is highly effective, it must be explicitly taught and practiced. Students need structured guidance to:

• Identify problem types (e.g., part-whole relationships, comparison models, and proportion models).

• Determine the appropriate bar model to use.

• Draw accurate bar diagrams to reflect problem scenarios.

• Transition from visual models to algebraic representations.

Repeated exposure and deliberate practice are crucial for students to internalize the strategy and apply it fluently in diverse problem-solving situations. Without sufficient practice, students may struggle to recognize when and how to use bar models effectively.

Conclusion

The ability to visualize and represent mathematical problems is a cornerstone of effective problem-solving. The Singapore Bar Model provides a structured, intuitive way for students to break down problems, understand mathematical relationships, and develop algebraic thinking. By integrating the Concrete-Pictorial-Abstract approach, educators can equip students with the necessary skills to approach mathematical challenges with confidence and precision. Explicit instruction and ample practice ensure that students can harness the full potential of visualization in mathematics, fostering deeper comprehension and long-term success.

Singapore Math Word Problem Keywords Identification Guide

Introduction

This comprehensive guide identifies key signal words and phrases in Singapore Math word problems that help students determine:

1. Which type of bar model to use
2. Which operation(s) to apply
3. How to structure their solution approach

Understanding these keywords is essential for successfully translating word problems into visual bar models and mathematical operations.

Addition Keywords

Part-Whole Relationships (Finding the Total)

• altogether
• in all
• total
• sum
• combined
• put together
• join
• in total
• added to
• plus
• increased by
• both
• and (when combining quantities)
• all together
• in all

Examples:

• "John has 15 marbles and Mary has 12 marbles. How many marbles do they have altogether?"
• "The total number of students in the class is 28."
• "When the two amounts are combined, they sum to 56."

Bar Model Type:

Part-Whole (parts combining to form a whole)

Subtraction Keywords

Taking Away

• take away
• subtract
• minus
• less
• remove
• give away
• lose
• eat
• spend
• sell
• left
• remain
• remaining

Examples:

• "Sarah had 25 stickers. She gave away 12 stickers. How many stickers does she have left?"
• "After spending $35, Mark had $27 remaining."
• "The shopkeeper sold 18 apples from a box of 50 apples. How many apples remain?"

Bar Model Type:

Part-Whole (finding the remaining part)

Comparison (Finding the Difference)

• more than
• less than
• fewer than
• greater than
• difference between
• how many more
• how many less
• taller than
• shorter than
• younger than
• older than
• heavier than
• lighter than
• exceed
• compared to

Examples:

• "David is 5 years older than his sister. David is 12 years old. How old is his sister?"
• "The difference between the two numbers is 17."
• "Jane has 8 fewer stickers than Peter. Peter has 23 stickers. How many stickers does Jane have?"

Bar Model Type:

Comparison (showing the difference between two quantities)

Before-After (Finding the Starting Amount)

• now
• originally
• at first
• started with
• begin with
• initially
• before
• after
• then
• remaining
• left with

Examples:

• "After gaining 15 pounds, Tom weighs 178 pounds. How much did he weigh initially?"
• "Mary started with some money. After spending $12, she had $27 left. How much money did she have at first?"

Bar Model Type:

Before-After (showing change over time)

Multiplication Keywords

Equal Groups

• each
• every
• per
• for each
• for every
• at this rate
• times
• multiplied by
• product of
• twice
• triple
• double
• twice as many
• groups of
• rows of
• lots of

Examples:

• "There are 5 boxes with 12 crayons in each box. How many crayons are there altogether?"
• "The teacher gives 3 pencils to each student. There are 25 students. How many pencils does the teacher need?"
• "A car travels 55 miles per hour. How far will it travel in 4 hours?"
• "Sam has twice as many marbles as Jane. Jane has 15 marbles. How many marbles does Sam have?"

Bar Model Type:

Multiplication (equal groups)

Area/Array

• square
• rectangle
• dimensions
• length
• width
• area
• square units
• rows and columns
• grid

Examples:

• "A rectangle has a length of 7 cm and a width of 5 cm. What is its area?"
• "There are 6 rows of chairs with 8 chairs in each row. How many chairs are there in total?"

Bar Model Type:

Area model (rectangular array)

Division Keywords

Sharing Equally (Partitive Division)

• share equally
• divide equally
• distributed evenly
• equal parts
• equal groups
• same number
• each
• per
• average
• equal share

Examples:

• "48 candies are shared equally among 6 children. How many candies does each child get?"
• "A sum of $85 is divided equally among 5 people. How much does each person receive?"

Bar Model Type:

Division (partitive - finding the size of each equal part)

Forming Groups (Quotative Division)

• how many groups
• batches of
• sets of
• teams of
• rows of
• packs of
• fit into
• how many times
• portions of
• bundles of

Examples:

• "There are 42 students. Each group needs 6 students. How many groups can be formed?"
• "Cookies are packed into boxes of 8. How many boxes are needed for 96 cookies?"
• "How many vans are needed to transport 52 people if each van can fit 8 people?"

Bar Model Type:

Division (quotative - finding the number of equal groups)

Fraction Keywords

Fractional Parts

• half of
• third of
• quarter of
• fifth of
• fraction of
• part of
• remainder
• out of
• percent of
• percentage of

Examples:

• "Three-quarters of the students are girls. There are 36 students. How many boys are there?"
• "Sam spent one-third of his money on a book. He spent $12. How much money did he have at first?"
• "40% of the marbles are red. There are 60 marbles in total. How many red marbles are there?"

Bar Model Type:

Fraction model (showing parts of a whole)

Ratio Keywords

Ratio Relationships

• ratio
• proportion
• for every
• corresponds to
• compared to
• to (as in "3 to 5")
• out of
• per
• for each
• for every
• scale

Examples:

• "The ratio of boys to girls is 3:5. There are 24 students. How many boys are there?"
• "For every 4 red beads, there are 7 blue beads. If there are 28 red beads, how many blue beads are there?"
• "The ingredients are mixed in the proportion 2:3:4. How much of each ingredient is needed to make 18 pounds of mixture?"

Bar Model Type:

Ratio model (showing proportion between quantities)

Rate Keywords

Rate Relationships

• per
• rate
• speed
• at this rate
• each
• for each
• for every
• miles per hour
• dollars per pound
• workers per task

Examples:

• "A car travels 55 miles per hour. How far will it travel in 4 hours?"
• "Oranges cost $1.50 per pound. How much will 3.5 pounds of oranges cost?"
• "At this rate, how many pages can be printed in 15 minutes?"

Bar Model Type:

Rate model (showing unit rates and conversions)

Multi-Step Problem Keywords

Sequential Events

• first, then
• initially, afterward
• begin, next, finally
• after that
• later
• subsequently
• at first, then

Examples:

• "John had 25 marbles. First, he lost 5 marbles. Then, he won 12 more. How many marbles does he have now?"
• "Initially, the tank contained 50 gallons of water. After 15 gallons were removed, how much water remained?"

Bar Model Type:

Multiple models in sequence or Before-After models

Conditional Statements

• if
• when
• suppose
• assuming that
• given that
• provided that

Examples:

• "If John has 3 times as many marbles as Peter, and together they have 48 marbles, how many marbles does each boy have?"
• "Given that the ratio of red to blue marbles is 3:4, and there are 21 red marbles, how many blue marbles are there?"

Bar Model Type:

Depends on the relationship described (often algebraic models)

Algebraic Problem Keywords

Unknown Quantities

• some
• a number
• a quantity
• as much as
• as many as
• how many
• how much
• what is
• find the

Examples:

• "Sam has some marbles. After giving away 12, he has 15 left. How many marbles did he have at first?"
• "A number multiplied by 5 gives 75. What is the number?"
• "Mary has as many books as John and Peter combined. John has 12 books and Peter has 15 books. How many books does Mary have?"

Bar Model Type:

Algebraic model (representing unknown quantities)

Multiple Unknowns (Related)

• times as many
• more than
• less than
• the same as
• together
• total
• difference
• sum
• product

Examples:

• "Jane has 5 more than twice the number of stickers Mary has. They have 53 stickers together. How many stickers does Mary have?"
• "The product of two consecutive numbers is 42. What are the numbers?"

Bar Model Type:

Algebraic model with multiple unknowns

Percentage Keywords

Percentage Relationships

• percent
• percentage
• discount
• increase
• decrease
• mark up
• mark down
• interest
• tax
• profit
• loss

Examples:

• "The shirt was discounted by 30%. Its original price was $45. What is the sale price?"
• "The price increased by 15%. If the new price is $69, what was the original price?"
• "A store makes a profit of 25% on each item. If an item costs the store $12, what is the selling price?"

Bar Model Type:

Percentage model or Before-After model

Systematic Strategy for Keyword Analysis

1. Identify relationship keywords to determine the type of bar model:

   • Part-whole: "altogether," "in all," "total"
   • Comparison: "more than," "less than," "difference"
   • Equal groups: "each," "every," "per"
   • Before-After: "initially," "after," "now"
   • Fraction: "half of," "third of," "portion of"
   • Ratio: "ratio," "proportion," "for every"

2. Identify operation keywords to determine the mathematical operation:

   • Addition: "sum," "total," "together"
   • Subtraction: "difference," "remain," "left"
   • Multiplication: "product," "times," "each"
   • Division: "share equally," "per," "quotient"

3. Identify quantity keywords to clarify what is known and unknown:

   • Known: actual numbers, specific quantities
   • Unknown: "some," "a number," "how many"

4. Identify sequencing keywords for multi-step problems:

   • "first," "then," "after that," "finally"

5. Identify condition keywords for algebraic relationships:

   • "if," "when," "suppose," "given that"

Common Misconceptions and Tricky Keywords

1. "Left" can indicate:

   • Subtraction: "How many are left?" (remaining after taking away)
   • Position: "The tree is on the left side" (no mathematical operation)

2. "More" can indicate:

   • Addition: "3 more candies" (add 3)
   • Comparison: "3 more than Jane" (comparison relationship)

3. "Times" can indicate:

   • Multiplication: "3 times as many" (multiply by 3)
   • Instances: "She visited 3 times" (frequency, not necessarily multiplication)

4. "Each" can indicate:

   • Multiplication: "3 cookies each" (multiply by number of people)
   • Division: "Share equally so each person gets the same" (divide by number of people)

5. "Average" typically indicates:

   • Division: "Find the average" (sum divided by count)

Strategies for Identifying Operations from Context

1. Looking for quantity changes:

   • Increase in quantity → Addition
   • Decrease in quantity → Subtraction
   • Repeated addition with same quantity → Multiplication
   • Breaking into equal parts → Division

2. Analyzing the question:

   • "How many altogether?" → Addition
   • "How many left?" → Subtraction
   • "How many in total?" → Addition or Multiplication
   • "How many in each?" → Division (partitive)
   • "How many groups?" → Division (quotative)

3. Examining relative values:

   • If answer should be larger than given values → Addition or Multiplication
   • If answer should be smaller than given values → Subtraction or Division

Practical Application Steps

1. Read the entire problem before focusing on keywords
2. Highlight or underline key signal words
3. Identify known and unknown quantities
4. Determine the relationship between quantities
5. Select the appropriate bar model type
6. Draw the bar model with proper labeling
7. Determine the operation(s) needed
8. Solve and check if the answer makes sense

Keyword Quick Reference Chart

Operation	Bar Model Type	Common Keywords
Addition	Part-Whole	altogether, in all, total, sum, combined
Subtraction (Take Away)	Part-Whole	take away, left, remain, less, fewer
Subtraction (Comparison)	Comparison	more than, less than, difference, taller than
Multiplication	Equal Groups	each, every, per, times, product
Division (Partitive)	Equal Groups	share equally, each gets, per person
Division (Quotative)	Equal Groups	how many groups, packs of, batches of
Fractions	Fraction Model	half of, third of, quarter of, percent of
Ratios	Ratio Model	ratio, proportion, for every, corresponds to
Algebraic	Algebraic Model	some, a number, how many, what is
Percentages	Percentage Model	percent, discount, increase, decrease
Before-After	Before-After Model	initially, now, originally, after, then

20 Comprehensive Lessons for Grades 2-3

In Singapore Math, bar models are used to visually represent several distinct types of mathematical relationships. Here are the main types of bar models used in Singapore Math:

1. Part-Whole Model

   • Purpose: Represents how parts combine to form a whole
   • Example: Sam has 12 red marbles and 15 blue marbles. How many marbles does he have altogether?
   • Representation: Two bars (representing 12 and 15) combine to form one larger bar (representing the total, 27)
   • Variation: Can include multiple parts making up a whole

2. Comparison Model

   • Purpose: Shows the difference between two quantities
   • Example: Jane has 35 stickers. John has 22 stickers. How many more stickers does Jane have than John?
   • Representation: Two bars of different lengths are aligned at the left, with a bracket showing the difference
   • Variations: Can show "more than" or "less than" relationships

3. Multiplication Model

   • Purpose: Represents equal groups or units
   • Example: There are 6 baskets with 8 oranges in each basket. How many oranges are there altogether?
   • Representation: One bar divided into equal segments, with each segment representing the same quantity
   • Variation: Can represent area (length × width) with a rectangular model

4. Division Model

   • Purpose: Shows dividing a quantity into equal parts
   • Types:
     • Partitive Division: Finding the size of each equal part
       • Example: 24 candies are shared equally among 6 children. How many candies does each child get?
     • Quotative Division: Finding the number of equal parts
       • Example: 24 candies are packed into bags of 6 candies each. How many bags are needed?
   • Representation: One bar representing the total, divided into equal parts

5. Fraction Model

   • Purpose: Represents fractional parts of a whole
   • Example: 3/4 of the students in a class are girls. There are 36 students. How many boys are in the class?
   • Representation: A bar divided into equal parts with some parts shaded or labeled differently
   • Variation: Can represent equivalent fractions with equally divided bars

6. Ratio Model

   • Purpose: Shows the relationship between quantities in a ratio
   • Example: The ratio of boys to girls is 3:5. There are 24 students in total. How many boys are there?
   • Representation: Multiple bars of equal units representing each quantity in the ratio
   • Variation: Can display complex ratios with multiple quantities

7. Before-After Model

   • Purpose: Represents changes in quantity over time or after an action
   • Example: Sarah had some money. After spending $15, she had $27 left. How much money did she have at first?
   • Representation: Two bars (often stacked) showing quantities before and after a change
   • Variation: Can show increases or decreases in quantity

8. Rate Model

   • Purpose: Represents unit rates or conversions
   • Example: If 5 pencils cost $3, how much do 8 pencils cost?
   • Representation: Bars showing the relationship between quantity and cost/rate
   • Variation: Can be used for speed, pricing, or other rate problems

9. Percentage Model

   • Purpose: Represents percentage relationships
   • Example: 40% of a number is 60. What is the number?
   • Representation: A bar divided into 100 units or proportional sections
   • Variation: Can show percentage increase/decrease

10. Algebraic Model

    • Purpose: Represents unknown quantities and relationships
    • Example: Peter has 5 more marbles than Mary. Together they have 37 marbles. How many marbles does Mary have?
    • Representation: Uses units to represent unknown quantities, often with one unit representing the variable
    • Variation: Can represent systems of equations with multiple unknowns

These models provide a visual framework for students to understand and solve a wide range of mathematical problems. The power of Singapore Math lies in how these models help students visualize abstract relationships and develop algebraic thinking without formal algebraic notation.

Overview

This four-week curriculum is designed to explicitly teach students how to draw and use bar models to solve word problems using Singapore Math strategies. Students will learn to identify signal words, break down problems, create proportional divisions, and use multiple representations including bar models and a 120-bead number line.

Week 1: Introduction to Bar Models & Part-Whole Relationships

Lesson 1: Understanding Part-Whole Relationships

Objective: Introduce the concept of part-whole relationships and basic bar model representation.

Materials:

• Connecting cubes
• Blank paper
• Rulers
• 120-bead number line
• Drawing templates

Lesson Flow:

1. Concrete Exploration (10 min)

   • Provide students with connecting cubes to create a set of 10
   • Have students break 10 into different combinations (7+3, 6+4, etc.)
   • Record findings as number sentences

2. Introduce Bar Model Concept (15 min)

   • Show how to represent the whole (10) as a rectangle
   • Demonstrate drawing two parts within the whole
   • Model proper use of rulers to create straight lines
   • Label each part with numbers

3. Signal Words (10 min)

   • Introduce part-whole signal words: "altogether," "in all," "total"
   • Show how these words indicate the whole amount

4. Guided Practice (20 min)

   • Read simple word problems together
   • Highlight signal words
   • Model drawing a bar to represent the whole
   • Divide and label parts
   • Example: "Sam has 6 red marbles and 8 blue marbles. How many marbles does he have altogether?"

5. 120-Bead Number Line Connection (10 min)

   • Demonstrate how the 120-bead number line can represent the same problem
   • Show how to count combined quantities on the number line

6. Independent Practice (15 min)

   • Students practice drawing bars for 2-3 simple part-whole problems
   • Teacher circulates to provide feedback on drawing technique and proportions

Lesson 2: Drawing Proportional Parts

Objective: Teach students to draw parts with appropriate proportional sizes.

Materials:

• Graph paper
• Rulers
• Colored pencils
• 120-bead number line

Lesson Flow:

1. Review (5 min)

   • Recap part-whole concept from previous lesson

2. Proportional Drawing Instruction (15 min)

   • Demonstrate how larger numbers should have longer bars
   • Show examples of proportional vs. non-proportional drawings
   • Teach students to count graph paper squares to ensure proportionality

3. Drawing Technique Mini-Lesson (15 min)

   • Model how to hold and use rulers effectively
   • Demonstrate drawing straight edges and even divisions
   • Show how to label clearly outside the bars

4. Guided Practice (20 min)

   • Work through problems with numbers of different magnitudes
   • Example: "Max read 15 pages on Monday and 5 pages on Tuesday. How many pages did he read in all?"
   • Emphasize drawing the 15-unit part three times longer than the 5-unit part

5. 120-Bead Visualization (10 min)

   • Use the bead number line to physically show proportional differences
   • Connect physical proportion to drawn proportion

6. Independent Practice (15 min)

   • Students draw proportional bar models for given number pairs
   • Self-check using graph paper squares to verify proportions

Lesson 3: Multiple Parts in a Whole

Objective: Extend bar modeling to represent problems with more than two parts.

Materials:

• Connecting cubes in different colors
• Graph paper
• Rulers
• Drawing templates
• 120-bead number line

Lesson Flow:

1. Concrete Exploration (10 min)

   • Use different colored connecting cubes to represent multiple parts
   • Create a set of 24 with 3-4 different colored groups

2. Bar Model Extension (15 min)

   • Demonstrate drawing a bar divided into 3-4 sections
   • Show how to maintain proportional sizing with multiple parts
   • Label each part and the whole

3. Signal Words (10 min)

   • Review part-whole signal words
   • Introduce sequence words: "first," "then," "after that"

4. Guided Practice (20 min)

   • Read multi-part word problems together
   • Example: "Sara picked 12 apples, 8 pears, and 5 oranges. How many pieces of fruit did she pick altogether?"
   • Model drawing a bar with three proportional sections

5. 120-Bead Representation (10 min)

   • Show how the bead number line can represent combining multiple quantities
   • Demonstrate using different colored beads or markers for each part

6. Independent Practice (15 min)

   • Students practice drawing bar models for problems with 3+ parts
   • Focus on maintaining proportionality across all sections

Lesson 4: Finding the Missing Part

Objective: Use bar models to find unknown parts in part-whole relationships.

Materials:

• Graph paper
• Rulers
• Colored pencils
• 120-bead number line

Lesson Flow:

1. Review (5 min)

   • Recap multi-part bar models from previous lesson

2. Missing Part Introduction (15 min)

   • Introduce problems where the whole and one part are known
   • Demonstrate drawing a bar with a question mark for the unknown part
   • Show how to label known values and the unknown

3. Signal Words (10 min)

   • Introduce missing part signal words: "how many more," "how many left," "still need"

4. Guided Practice (20 min)

   • Example: "Jamal has 15 stickers in all. He used 6 stickers. How many stickers does he have left?"
   • Model drawing the whole bar (15)
   • Draw and label the known part (6)
   • Use "?" for the unknown part

5. 120-Bead Strategy (10 min)

   • Demonstrate finding the missing part using the bead number line
   • Show how to count from the known part to the whole

6. Independent Practice (15 min)

   • Students solve missing part problems using bar models
   • Focus on clearly labeling known values and the unknown

Lesson 5: Part-Whole Assessment and Review

Objective: Assess student understanding of part-whole bar models and address misconceptions.

Materials:

• Assessment worksheets
• Rulers
• Graph paper
• 120-bead number line

Lesson Flow:

1. Review Game (15 min)

   • Play a quick game matching word problems to appropriate bar model representations

2. Common Mistakes Discussion (10 min)

   • Show examples of common drawing errors
   • Discuss proportionality issues and labeling confusion
   • Model correction strategies

3. Guided Review (15 min)

   • Work through mixed part-whole problems as a class
   • Reinforce proper drawing techniques and proportions

4. Assessment (20 min)

   • Students complete an assessment with varied part-whole problems
   • Include at least one problem requiring them to draw a multi-part model
   • Include finding missing parts and total problems

5. 120-Bead Practice (10 min)

   • While students finish assessments, work with small groups on the bead number line
   • Reinforce the connection between physical beads and drawn models

6. Self-Reflection (10 min)

   • Students identify their strengths and one area for improvement in bar modeling

Week 2: Comparison Bar Models

Lesson 6: Introduction to Comparison Relationships

Objective: Introduce comparison bar models and teach drawing techniques for representing differences.

Materials:

• Connecting cubes
• Rulers
• Graph paper
• 120-bead number line

Lesson Flow:

1. Concrete Exploration (10 min)

   • Create two towers of different heights with connecting cubes
   • Physically compare the difference between them

2. Comparison Bar Model Introduction (15 min)

   • Demonstrate drawing two separate bars of different lengths
   • Show how to align bars for visual comparison
   • Teach how to label the difference between bars

3. Signal Words (10 min)

   • Introduce comparison signal words: "more than," "fewer than," "taller than," "shorter than"
   • Show how these words indicate a comparison relationship

4. Guided Practice (20 min)

   • Example: "Maya has 12 stickers. Leo has 7 stickers. How many more stickers does Maya have than Leo?"
   • Model drawing two bars aligned at left
   • Show how to label the difference
   • Demonstrate using brackets to highlight the difference

5. 120-Bead Comparison (10 min)

   • Show how to represent two quantities on the bead number line
   • Demonstrate measuring the difference between positions

6. Independent Practice (15 min)

   • Students practice drawing comparison bar models
   • Focus on proper alignment and labeling differences

Lesson 7: "More Than" Comparison Problems

Objective: Focus on "more than" comparison problems and drawing techniques.

Materials:

• Rulers
• Graph paper
• Colored pencils
• 120-bead number line

Lesson Flow:

1. Review (5 min)

   • Recap comparison bar models from the previous lesson

2. "More Than" Language Focus (15 min)

   • Analyze "more than" phrases in word problems
   • Show how the language indicates which bar should be longer
   • Demonstrate labeling the bars to match problem language

3. Drawing Technique (15 min)

   • Teach strategies for drawing accurate comparison bars
   • Demonstrate using graph paper to maintain proportions
   • Show how to bracket and label the difference clearly

4. Guided Practice (20 min)

   • Example: "Tina is 7 years older than her brother. Tina is 15 years old. How old is her brother?"
   • Model how to draw and label both bars
   • Emphasize aligning bars for clear visual comparison

5. 120-Bead Strategy (10 min)

   • Use the bead number line to physically show "more than" relationship
   • Demonstrate counting backward to find the smaller quantity

6. Independent Practice (15 min)

   • Students solve various "more than" problems
   • Focus on proper labeling of known and unknown values

Lesson 8: "Less Than" Comparison Problems

Objective: Extend comparison modeling to "less than" relationships.

Materials:

• Rulers
• Graph paper
• Colored pencils
• 120-bead number line

Lesson Flow:

1. Contrast with "More Than" (10 min)

   • Compare "more than" vs. "less than" language
   • Show how the language affects which bar is longer/shorter

2. "Less Than" Language Focus (15 min)

   • Analyze "less than," "fewer than," "shorter than" phrases
   • Demonstrate accurate drawing and labeling based on language

3. Drawing Strategy (15 min)

   • Show how to draw the shorter bar first, then the longer one
   • Demonstrate clear labeling of the difference
   • Emphasize starting bars at the same baseline

4. Guided Practice (20 min)

   • Example: "Tim has 8 fewer marbles than Sasha. Tim has 12 marbles. How many marbles does Sasha have?"
   • Model drawing both bars with proper alignment
   • Show how to solve for the unknown quantity

5. 120-Bead Visualization (10 min)

   • Demonstrate "less than" relationships on the bead number line
   • Show counting forward to find the larger quantity

6. Independent Practice (15 min)

   • Students solve "less than" comparison problems
   • Focus on correctly interpreting the relationship direction

Lesson 9: Unknown Difference Problems

Objective: Work with comparison problems where the difference is unknown.

Materials:

• Rulers
• Graph paper
• Colored pencils
• 120-bead number line

Lesson Flow:

1. Review (5 min)

   • Review previous comparison problems

2. Unknown Difference Introduction (15 min)

   • Introduce problems where both quantities are known but the difference is unknown
   • Show how to draw two bars with known lengths
   • Demonstrate labeling the unknown difference with "?"

3. Drawing Technique (15 min)

   • Model drawing accurately proportioned bars
   • Show how to create a bracket to highlight the difference
   • Demonstrate labeling with "?" and appropriate units

4. Guided Practice (20 min)

   • Example: "Jill has 17 books. Mike has 9 books. How many more books does Jill have than Mike?"
   • Model drawing both bars accurately
   • Show how to label the unknown difference

5. 120-Bead Method (10 min)

   • Use the bead number line to visually show the gap between quantities
   • Demonstrate counting the beads between positions to find difference

6. Independent Practice (15 min)

   • Students solve unknown difference problems
   • Focus on accurate drawing and difference labeling

Lesson 10: Comparison Assessment and Review

Objective: Assess student understanding of comparison bar models and address misconceptions.

Materials:

• Assessment worksheets
• Rulers
• Graph paper
• 120-bead number line

Lesson Flow:

1. Review Activity (15 min)

   • Sort word problems into "more than," "less than," and "unknown difference" categories
   • Discuss how to identify the relationship type from the problem language

2. Common Mistakes Discussion (10 min)

   • Address typical comparison drawing errors
   • Show examples of misaligned bars and incorrect labeling
   • Model correction strategies

3. Mixed Comparison Problems (15 min)

   • Work through various comparison problems as a class
   • Reinforce correct drawing and labeling techniques

4. Assessment (20 min)

   • Students complete an assessment with different types of comparison problems
   • Include "more than," "less than," and "unknown difference" scenarios

5. 120-Bead Reinforcement (10 min)

   • Small group work with the bead number line while others finish assessment
   • Practice finding differences using physical beads

6. Self-Reflection (10 min)

   • Students identify their strengths and challenges with comparison bar models

Week 3: Multiplication and Division Bar Models

Lesson 11: Introduction to Equal Groups (Multiplication)

Objective: Introduce bar models for equal groups multiplication problems.

Materials:

• Connecting cubes
• Graph paper
• Rulers
• 120-bead number line

Lesson Flow:

1. Concrete Exploration (10 min)

   • Create multiple equal groups with connecting cubes
   • Discuss how to represent repeated addition visually

2. Equal Groups Bar Model Introduction (15 min)

   • Demonstrate drawing a bar divided into equal sections
   • Show how to label each section and the total
   • Connect to multiplication notation

3. Signal Words (10 min)

   • Introduce multiplication signal words: "each," "every," "times," "groups of"
   • Show how these words indicate equal groups situations

4. Guided Practice (20 min)

   • Example: "There are 4 vases. Each vase has 3 flowers. How many flowers are there in total?"
   • Model drawing one long bar divided into 4 equal sections
   • Label each section with 3 and the total as 12

5. 120-Bead Representation (10 min)

   • Use the bead number line to show equal jumps (repeated addition)
   • Connect to the bar model representation

6. Independent Practice (15 min)

   • Students practice drawing equal groups bar models
   • Focus on making equal divisions and clear labeling

Lesson 12: Unknown Factor Problems

Objective: Use bar models to solve problems with unknown factors.

Materials:

• Graph paper
• Rulers
• Colored pencils
• 120-bead number line

Lesson Flow:

1. Review (5 min)

   • Recap equal groups bar models from previous lesson

2. Unknown Factor Introduction (15 min)

   • Introduce problems where the product and one factor are known
   • Demonstrate drawing a bar with known number of equal parts
   • Show how to label the unknown value in each part

3. Signal Words and Analysis (10 min)

   • Revisit multiplication signal words
   • Show how to identify which factor is unknown from problem language

4. Drawing Technique (15 min)

• Model drawing equal divisions precisely
• Demonstrate using graph paper to

4. Drawing Technique (15 min)

   • Model drawing equal divisions precisely
   • Demonstrate using graph paper to create equal sections
   • Show clear labeling of the unknown value

5. Guided Practice (20 min)

   • Example: "Kim has 24 stickers arranged in 6 equal rows. How many stickers are in each row?"
   • Model drawing one bar representing 24
   • Divide into 6 equal sections
   • Label each section with "?" and the total as 24

6. 120-Bead Division Approach (10 min)

   • Use the bead number line to demonstrate dividing a quantity into equal groups
   • Show how to find how many in each group

7. Independent Practice (15 min)

   • Students solve unknown factor problems using bar models
   • Focus on equal divisions and proper labeling

Lesson 13: Division as Equal Groups

Objective: Teach students to model division problems using equal groups bar models.

Materials:

• Connecting cubes
• Graph paper
• Rulers
• 120-bead number line

Lesson Flow:

1. Division Concept Review (10 min)

   • Review division as sharing equally or finding group size
   • Connect to previous multiplication bar models

2. Division Bar Model Introduction (15 min)

   • Demonstrate two types of division problems:
     • How many in each group? (partition division)
     • How many groups? (quotative division)
   • Show how to draw each type differently

3. Signal Words (10 min)

   • Introduce division signal words: "share equally," "distributed evenly," "each gets," "how many groups"
   • Show how language determines the type of division

4. Guided Practice (20 min)

   • Example 1: "18 candies are shared equally among 6 children. How many candies does each child get?"
   • Example 2: "18 candies are distributed so each child gets 3 candies. How many children receive candy?"
   • Model drawing both types with appropriate labeling

5. 120-Bead Division Demonstration (10 min)

   • Use the bead number line to physically show division
   • Demonstrate both types of division problems

6. Independent Practice (15 min)

   • Students practice drawing bar models for different division situations
   • Focus on identifying the type of division and appropriate representation

Lesson 14: Unknown Group Size and Number of Groups

Objective: Develop proficiency with division bar models for different unknown values.

Materials:

• Graph paper
• Rulers
• Colored pencils
• 120-bead number line

Lesson Flow:

1. Review (5 min)

   • Recap division bar models from previous lesson

2. Problem Type Analysis (15 min)

   • Analyze language that indicates unknown group size vs. unknown number of groups
   • Show how to determine what the question is asking

3. Drawing Strategies (15 min)

   • Demonstrate different drawing approaches for each problem type
   • Show how to label unknown values appropriately
   • Emphasize proportional representation

4. Guided Practice (20 min)

   • Model solving problems with unknown group size:
     • "36 marbles are shared equally among 4 children. How many marbles does each child get?"
   • Model solving problems with unknown number of groups:
     • "36 marbles are packed into bags of 9 marbles each. How many bags are needed?"

5. 120-Bead Division Strategies (10 min)

   • Show how to use the bead number line for both types of division problems
   • Demonstrate grouping and partitioning approaches

6. Independent Practice (15 min)

   • Students solve a mix of division problems
   • Focus on identifying the problem type and appropriate modeling

Lesson 15: Multi-Step Problems with Multiplication and Division

Objective: Apply bar models to multi-step problems involving multiplication and division.

Materials:

• Graph paper
• Rulers
• Colored pencils
• 120-bead number line

Lesson Flow:

1. Multi-Step Problem Introduction (10 min)

   • Discuss how some problems require multiple operations
   • Show how to break down complex problems into steps

2. Bar Model Strategy (15 min)

   • Demonstrate drawing separate bar models for each step
   • Show how to connect models in a logical sequence
   • Teach labeling intermediate results

3. Process Focus (15 min)

   • Introduce systematic approach:
     1. Identify known and unknown values
     2. Determine operations needed
     3. Draw models for each step
     4. Connect steps logically

4. Guided Practice (20 min)

   • Example: "A baker made 48 cookies. He packed them equally into 6 boxes. Then he sold each box for $12. How much money did he earn?"
   • Model drawing the division step (48 ÷ 6)
   • Model drawing the multiplication step (6 × $12)
   • Show how the answers connect

5. 120-Bead Representation (10 min)

   • Use the bead number line to work through multi-step problems
   • Show how to track intermediate results

6. Independent Practice (15 min)

   • Students solve multi-step problems using bar models
   • Focus on clear step representation and connecting steps

Week 4: Advanced Applications and Mixed Problem Types

Lesson 16: Before and After Problems

Objective: Use bar models to represent and solve before/after quantity change problems.

Materials:

• Graph paper
• Rulers
• Colored pencils
• 120-bead number line

Lesson Flow:

1. Before/After Concept Introduction (10 min)

   • Discuss situations where quantities change over time
   • Introduce drawing two bars showing before and after states

2. Signal Words (10 min)

   • Introduce temporal signal words: "now," "before," "after," "increased by," "decreased by"
   • Show how language indicates sequence and change direction

3. Drawing Technique (15 min)

   • Demonstrate stacking before/after bars vertically for comparison
   • Show how to indicate increase/decrease with brackets
   • Model proper labeling of initial, final, and change amounts

4. Guided Practice (20 min)

   • Example: "Sarah had some marbles. After she won 15 more marbles in a game, she had 42 marbles. How many marbles did Sarah have at first?"
   • Model drawing two bars (before and after)
   • Show how to label known values and the unknown

5. 120-Bead Representation (10 min)

   • Use the bead number line to show before/after positions
   • Demonstrate counting forward/backward to show changes

6. Independent Practice (15 min)

   • Students solve before/after problems using bar models
   • Focus on clearly showing the change and proper labeling

Lesson 17: Fraction Bar Models

Objective: Extend bar modeling to represent fractional parts and relationships.

Materials:

• Graph paper
• Rulers
• Colored pencils
• Fraction circles/bars
• 120-bead number line

Lesson Flow:

1. Fraction Review (10 min)

   • Review basic fraction concepts using physical models
   • Discuss how fractions represent parts of a whole

2. Fraction Bar Model Introduction (15 min)

   • Demonstrate drawing a bar and dividing it into equal fractional parts
   • Show how to label each part with appropriate fractions
   • Introduce finding fractional amounts of quantities

3. Signal Words (10 min)

   • Introduce fraction signal words: "half of," "third of," "quarter of," "fraction of"
   • Show how these indicate fractional relationships

4. Guided Practice (20 min)

   • Example: "3/4 of the students in the class are girls. There are 28 students in the class. How many boys are in the class?"
   • Model drawing a bar representing the whole class
   • Show dividing into fourths and labeling appropriately

5. 120-Bead Fraction Connection (10 min)

   • Use the bead number line to represent fractional quantities
   • Show dividing the beads into equal groups

6. Independent Practice (15 min)

   • Students practice drawing and solving fraction bar models
   • Focus on equal divisions and proper labeling

Lesson 18: Mixed Problem Types Practice

Objective: Build fluency with selecting appropriate bar model types for different word problems.

Materials:

• Graph paper
• Rulers
• Colored pencils
• 120-bead number line
• Problem type cards

Lesson Flow:

1. Problem Classification Review (15 min)

   • Review all bar model types learned:
     • Part-whole
     • Comparison
     • Multiplication/Division
     • Before/After

2. Signal Word Analysis (15 min)

   • Compile a reference chart of signal words for each problem type
   • Practice identifying problem types from language cues

3. Selection Strategy (10 min)

   • Teach systematic approach to problem analysis:
     1. Read problem completely
     2. Identify key information and relationships
     3. Look for signal words
     4. Select appropriate model type

4. Guided Practice (20 min)

   • Work through varied problem types
   • Model the thought process for selecting appropriate models
   • Demonstrate drawing and solving each type

5. 120-Bead Integration (10 min)

   • Show how the bead number line can represent all problem types
   • Connect physical representation to drawn models

6. Independent Practice (20 min)

   • Students solve mixed problems, first identifying type, then drawing appropriate models
   • Focus on correct model selection and drawing technique

Lesson 19: Solving Complex Word Problems

Objective: Apply bar modeling to solve multi-step problems with mixed operations.

Materials:

• Graph paper
• Rulers
• Colored pencils
• Highlighters
• 120-bead number line

Lesson Flow:

1. Complex Problem Analysis (15 min)

   • Introduce strategy for breaking down complex problems:
     1. Identify what's asked
     2. Identify given information
     3. Plan solution steps
     4. Select appropriate models for each step

2. Breaking Down Process (15 min)

   • Demonstrate reading and highlighting key information
   • Show how to identify relationships between quantities
   • Model planning multiple steps before drawing

3. Guided Practice (25 min)

   • Example: "A toy store had 120 action figures. They sold 3/4 of them during a sale. Each action figure costs $12. How much money did they make from selling the action figures?"
   • Work through step by step:
     1. Find 3/4 of 120 (fraction bar model)
     2. Calculate total value (multiplication bar model)

4. 120-Bead Complex Problem Solving (10 min)

   • Use the bead number line to work through complex problems
   • Show tracking multiple operations

5. Independent Practice (15 min)

   • Students solve complex multi-step problems
   • Focus on breaking down problems and connecting steps

Lesson 20: Comprehensive Review and Assessment

Objective: Review all bar model types and assess student proficiency in bar model drawing and application.

Materials:

• Assessment packet
• Graph paper
• Rulers
• Colored pencils
• 120-bead number line

Lesson Flow:

1. Comprehensive Review (20 min)

   • Review all bar model types and their applications
   • Discuss when each type is most useful
   • Review drawing techniques and common pitfalls

2. Bar Model Strategy Summary (15 min)

   • Create a class reference chart of problem types, signal words, and model examples
   • Discuss key drawing and labeling requirements for each type

3. 120-Bead Number Line Connection (10 min)

   • Review how the bead number line connects to each bar model type
   • Demonstrate the parallel between physical and drawn models

4. Final Assessment (30 min)

   • Students complete a comprehensive assessment including:
     • Problem type identification
     • Bar model drawing for various problem types
     • Multi-step problem solving
     • Self-explanation of solution process

5. Self-Reflection (15 min)

   • Students complete reflection on their bar model learning journey
   • Identify strengths and areas for continued practice
   • Set goals for applying bar modeling in future problem solving

Assessment and Differentiation Strategies

Formative Assessment

• Daily exit tickets checking specific skills
• Bar model drawing checks using rubrics
• Problem-type identification quizzes
• Peer review of drawing techniques

Summative Assessment

• Weekly skill checks (Lessons 5, 10, 15, 20)
• Bar model drawing portfolio
• Application assessment with varied problem types
• Problem-solving explanation (written or verbal)

Differentiation Strategies

For Students Needing Additional Support:

• Provide partially completed bar model templates
• Use grid paper with larger squares
• Offer step-by-step drawing guides
• Work with smaller numbers initially
• Create visual word problem analysis cards
• Provide additional concrete experience with manipulatives

For Students Needing Extension:

• Present problems with larger numbers requiring scale considerations
• Include problems with extraneous information to analyze
• Challenge with multi-step problems requiring multiple bar models
• Have students create their own word problems for specific bar model types
• Introduce algebraic thinking through bar models with multiple unknowns

120-Bead Number Line Integration

Setup and Use

• Position the 120-bead number line where all students can access it
• Use different colored beads or markers to distinguish quantities
• Demonstrate parallel modeling between beads and drawn bar models

Specific Applications

• Part-Whole: Use different colored sections to show parts making up a whole
• Comparison: Show two quantities and the gap between them
• Multiplication: Show equal jumps (e.g., counting by 4s)
• Division: Show grouping into equal sections
• Fractions: Divide total beads into equal sections

Connection to Bar Models

• Always draw the corresponding bar model after demonstrating on the bead line
• Have students explain how the physical model connects to the drawn model
• Use consistent coloring between bead line demonstrations and drawn models

Teacher Notes and Resources

Common Student Difficulties

• Maintaining proportionality in bar lengths
• Aligning bars properly for comparison
• Dividing bars into truly equal sections
• Labeling consistently and clearly
• Transferring between concrete and pictorial representations

Key Vocabulary Focus

• Part-whole
• Comparison
• Equal groups
• Before/after
• Proportion
• Division (partitive vs. quotative)

Materials Organization

• Provide drawing tool kits (ruler, pencil, eraser, colored pencils)
• Use consistent graph paper throughout
• Create bar model templates for scaffolding
• Prepare word problem cards sorted by type
• Make signal word reference cards available

Parent Communication

• Send home explanations of bar modeling approach
• Provide sample problems and drawing techniques
• Include glossary of signal words and meanings
• Share online resources for additional practice

Implementation Recommendations

Classroom Setup

• Create a math word wall with signal words and corresponding bar model examples
• Set up a dedicated drawing station with rulers, graph paper, and sample problems
• Display anchor charts showing the different bar model types
• Position the 120-bead number line prominently for daily demonstrations
• Create a gallery space to showcase exemplary student work

Pacing Considerations

• Each lesson is designed for approximately 60 minutes
• Consider breaking lessons into smaller segments for younger students
• Allow additional time for students who need more practice with drawing techniques
• The four-week timeline assumes daily math lessons; extend as needed if teaching math fewer times per week

Pre-Assessment

Before beginning the unit, assess students' current abilities with:

• Basic drawing skills and ruler use
• Understanding of part-whole relationships
• Knowledge of basic operations
• Reading comprehension of simple word problems

Ongoing Practice Opportunities

• Morning work problems focused on bar model drawing
• Math center activities with self-checking bar model tasks
• Homework assignments with simple drawing practice
• Word problem of the day requiring appropriate model selection

Extension Activities

Cross-Curricular Connections

• Science: Use bar models to compare measurements and experimental results
• Social Studies: Create bar models to represent population data or timeline comparisons
• Language Arts: Have students write their own word problems for specific bar model types
• Art: Explore proportional drawing in artistic contexts

Technology Integration

• Document camera demonstrations of drawing techniques
• Digital drawing tools for additional practice
• Creating video tutorials explaining drawing processes
• Virtual manipulatives that parallel physical models

Project-Based Learning

• Data Collection Project: Students collect real-world data and represent it using appropriate bar models
• Math Story Project: Create illustrated math story books with bar model representations
• Real-World Application: Solve authentic problems from school context using bar modeling

Detailed Materials List

Teacher Materials

• Large demonstration graph paper
• Document camera
• Colored markers for modeling
• 120-bead number line (teacher demonstration size)
• Word problem cards sorted by type
• Answer keys with exemplar drawings
• Assessment rubrics for bar model drawings
• Signal word posters for each problem type

Student Materials (per student)

• Math notebook with graph paper
• Ruler (with centimeter markings)
• Pencils and erasers
• Set of colored pencils
• Individual reference sheet of bar model types
• Signal word glossary
• Individual or small group 120-bead number lines
• Self-assessment checklists

Additional Resources

Singapore Math Problem Types Reference

Part-Whole Problems:

• Finding the whole when parts are known
• Finding a part when the whole and other part(s) are known
• Signal words: altogether, in all, total, sum, combined

Comparison Problems:

• Finding the difference between two quantities
• Finding one quantity when the other and the difference are known
• Signal words: more than, less than, fewer than, taller than, difference between

Multiplication Problems:

• Equal groups with known quantity per group
• Equal groups with unknown quantity per group
• Signal words: each, every, per, times, groups of

Division Problems:

• Partitive division (finding amount in each group)
• Quotative division (finding number of groups)
• Signal words: share equally, distributed evenly, each gets, how many groups

Before/After Problems:

• Finding quantity before a change
• Finding quantity after a change
• Finding the amount of change
• Signal words: now, before, after, increased by, decreased by

Fraction Problems:

• Finding fractional parts of wholes
• Finding wholes from fractional parts
• Signal words: fraction of, part of, half of, third of

Sample Word Problems by Type

Part-Whole:

1. Emma has 24 red beads and 18 blue beads. How many beads does she have altogether?
2. James has 35 marbles in total. He has 17 glass marbles. How many clay marbles does he have?
3. A bookshelf has 3 shelves. The top shelf has 12 books, the middle shelf has 15 books, and the bottom shelf has 9 books. How many books are on the bookshelf in total?

Comparison:

1. Tyler is 142 cm tall. His sister is 124 cm tall. How much taller is Tyler than his sister?
2. Maria has 16 more stickers than Pedro. Maria has 42 stickers. How many stickers does Pedro have?
3. Omar scored 75 points on his test. Jasmine scored 89 points. How many fewer points did Omar score than Jasmine?

Multiplication:

1. There are 4 vases. Each vase has 7 flowers. How many flowers are there in total?
2. A store sells eggs in cartons of 12. How many eggs are in 8 cartons?
3. Ms. Lee gives 5 pencils to each student. There are 23 students. How many pencils does she need?

Division:

1. 48 cupcakes are shared equally among 6 children. How many cupcakes does each child get?
2. 36 marbles are packed into bags of 9 marbles each. How many bags are needed?
3. A teacher has 56 stickers to distribute equally to 8 groups. How many stickers will each group receive?

Before/After:

1. Sarah had some marbles. After she won 15 more marbles in a game, she had 42 marbles. How many marbles did Sarah have at first?
2. David had 34 cards. After giving some to his friend, he had 19 cards left. How many cards did he give away?
3. A bottle contained 750 ml of water. After drinking some water, 320 ml remained. How much water was drunk?

Fraction:

1. 3/4 of the students in the class are girls. There are 28 students in the class. How many boys are in the class?
2. Mark spent 1/5 of his money on a book. He spent $12 on the book. How much money did he have at first?
3. 2/3 of the marbles in a bag are blue. There are 18 blue marbles. How many marbles are in the bag altogether?

Assessment Rubrics

Bar Model Drawing Rubric

Criteria	Beginning (1)	Developing (2)	Proficient (3)	Advanced (4)
Proportionality	Bars are not proportional to the values they represent	Bars show some proportionality but with significant errors	Bars are mostly proportional with minor errors	Bars are accurately proportional to the values they represent
Layout	Bars are not aligned properly; comparison is difficult	Bars are partially aligned with some alignment issues	Bars are well-aligned with minor placement issues	Bars are perfectly aligned for clear visual comparison
Labeling	Missing most labels or labels are incorrect	Some correct labels but with inconsistencies	Most values correctly labeled with minor errors	All values accurately and clearly labeled
Division of Parts	Unequal divisions when equal divisions are needed	Somewhat equal divisions with noticeable errors	Mostly equal divisions with minor imprecisions	Precise equal divisions where required
Problem Representation	Bar model does not represent the problem structure	Bar model partially represents the problem structure	Bar model accurately represents most aspects of the problem	Bar model perfectly represents all aspects of the problem

Problem-Solving Process Rubric

Criteria	Beginning (1)	Developing (2)	Proficient (3)	Advanced (4)
Problem Analysis	Unable to identify key information and relationships	Identifies some key information but misses critical elements	Identifies most key information and relationships	Accurately identifies all key information and relationships
Model Selection	Selects inappropriate model type for the problem	Selects partially appropriate model with significant flaws	Selects appropriate model with minor application issues	Selects optimal model type for the problem
Drawing Execution	Poor execution with major drawing errors	Basic execution with several drawing mistakes	Good execution with minor drawing issues	Excellent execution with precise drawing technique
Solution Process	Unable to use model to find solution	Partially uses model with logical errors	Uses model effectively with minor computational errors	Uses model optimally with accurate computation
Explanation	Cannot explain solution approach	Limited explanation of solution approach	Clear explanation of most solution steps	Comprehensive explanation of all solution steps

Reflection Questions for Students

End of Week 1:

1. What helps you decide when to use a part-whole bar model?
2. What is challenging about drawing parts with the correct size?
3. How does using the 120-bead number line help you understand part-whole relationships?

End of Week 2:

1. How do you know whether to draw the longer or shorter bar first in a comparison problem?
2. What signal words help you identify comparison situations?
3. How do you make sure your comparison bars are aligned correctly?

End of Week 3:

1. What's the difference between drawing a bar model for multiplication versus division?
2. How do you make sure you divide your bar into equal parts?
3. How does the 120-bead number line help you understand equal groups?

End of Week 4:

1. Which type of bar model do you find easiest to draw? Why?
2. Which type of bar model do you find most challenging? Why?
3. How has learning to draw bar models helped you solve word problems?
4. What drawing techniques have you improved the most?

Final Thoughts on Implementation

This comprehensive curriculum aims to systematically develop students' ability to represent mathematical relationships visually through bar modeling. By explicitly teaching drawing techniques alongside mathematical concepts, students develop both the visualization skills and problem-solving strategies needed for success with complex word problems.

The integration of the 120-bead number line provides a concrete foundation that bridges to the pictorial representation, supporting students who need multiple access points to abstract concepts. The careful attention to proportionality and drawing technique helps students create accurate visual models that truly represent the mathematical relationships in each problem.

Remember that developing proficiency with bar modeling is a progressive journey. Some students will need additional practice and scaffolding, while others will quickly move to applying these strategies independently. The curriculum is designed to be flexible enough to accommodate these different learning paths while maintaining high expectations for all students.

With consistent practice and explicit instruction, students will develop not only drawing skills but also a deeper conceptual understanding of mathematical relationships and operations.

1

3:04

How to solve age problems easily | P5 math Singapore

Practicle Math

2

4:35

Length word problems | Adding fractions with different denominators | P4 Maths

Practicle Math

3

8:09

PSLE Math 2024 Question [Answered!] | Paper 2

Practicle Math

4

4:19

Fraction word problems | Primary 4 Maths word problems

Practicle Math

5

2:32

Ratio proportion word problems | Primary 5 math

Practicle Math

6

3:19

Primary 5 ratio problem sum | Singapore math bar models

Practicle Math

7

2:46

How to solve fraction word problems | Primary 2 Mathematics

Practicle Math

8

3:54

Math problem sum made easy | Model drawing

Practicle Math

9

5:26

Remainder Concept Problem [A Quick 5 Mins Guide!]

Practicle Math

10

3:36

Fraction Word Problem [Made Easy!] | Primary 4 Model Drawing

Practicle Math

11

6:47

P4 Decimal word problem made easy | Shortage and excess question | Singapore Math

Practicle Math

12

4:43

Part Part Whole Model for Addition Word Problems Made Easy | Primary 2 Maths Singapore

Practicle Math

13

3:04

Primary 2 Maths Multiplication Word Problem for kids | Singapore Math

Practicle Math

14

3:50

Before and After Math Word Problems Made Easy

Practicle Math

15

3:29

Draw Part-Whole Model Examples - Singapore Math

Practicle Math

16

3:23

Primary 3 Maths Division Word Problem Explained

Practicle Math

17

5:07

Primary 4 Math Before and After Math Word Problem Made Easy

Practicle Math

18

6:03

P3 Maths Word Problems [Model Drawing Made Easy!]

Practicle Math

19

6:01

Solve Wordy Fraction Word Problems Easily!

Practicle Math

20

8:36

Learn Primary 3 Maths Comparison Models in 10 mins [QUICK & EASY!]

Practicle Math

▶

10:33

Bar Models Math Made Easy

Practicle Math

22

8:20

Learn Singapore bar model drawing easily | P2 Maths

Practicle Math

23

8:59

Learn the Excess and Shortage Concept [Fast!] | Practicle Math Made Easy 2

Practicle Math

24

9:27

Learn the Gap and Difference Concept [Fast!] | Practicle Math Made Easy 1

Practicle Math

25

4:24

Singapore Math Model Drawing Basics - Part-Whole Model

Practicle Math

26

5:11

Solve Primary 5 Math Fraction Problem Sums Easily! (Part 1)

Practicle Math