How to Use Strip Diagrams to Solve Multi-step Word Problems

A Step-by-step Guide to Using Strip Diagrams to Solve Multi-step Word Problems

Let’s go through a multi-step word problem suitable:

Problem: Sam has 20 stickers. He gives 5 stickers to his friend Joe. Then, he buys a pack of stickers that contains 8 more. How many stickers does Sam have now?

Step 1: Understand the Problem

Read the problem carefully and identify the different events and operations taking place.

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Step 2: Draw the Initial Strip Diagram

Start by drawing a box (or strip) representing the total number of stickers Sam initially has. Label this box as 20 stickers.

|--------- 20 stickers ---------|

Step 3: Draw the Changes

Next, represent the changes (both increase and decrease) to the number of stickers Sam has.

When Sam gives 5 stickers to Joe, draw a smaller box to represent this subtraction:

|------ 15 stickers ------| - 5 stickers |

Then, when Sam buys more stickers, extend the box to represent this addition:

|------ 15 stickers ------| + | 8 stickers |

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Step 4: Calculate the Result

Add or subtract as indicated by the strip diagram. In this case, 15 stickers + 8 stickers equals 23 stickers.

So, Sam has 23 stickers now.

Using strip diagrams can help students visualize the problem and understand what operations to perform and in what order. It’s a very helpful strategy for solving multi-step word problems.

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What Are Strip Diagrams?

Strip diagrams, also called bar models or tape diagrams, are visual representations that break down multi-step word problems into manageable parts. They consist of rectangular bars representing known and unknown quantities, helping you organize information and plan your solution strategy before performing calculations.

Building a Strip Diagram Step by Step

Start by identifying all quantities in the problem. Draw a rectangle for each known quantity and label it with the value. For unknown quantities, draw unlabeled rectangles. Use the relationships described in the problem to arrange these strips: stack them for totals, place them in sequence for combinations, or position them side-by-side for comparisons.

Worked Example 1: Addition and Subtraction

Maria has 15 stickers. Her friend gave her 8 more. Then she gave 6 to her brother. How many does she have now?

Draw a strip for 15 (Maria’s initial stickers), add a segment for 8 (received), then remove 6 (given away). Visually, you can see: 15 + 8 – 6 = 17 stickers.

Worked Example 2: Multiplication and Division

A baker makes 4 trays of cookies. Each tray has 12 cookies. She sells 20 cookies. How many are left?

Draw 4 equal strips, each labeled 12, showing the total is 4 × 12 = 48 cookies. Then mark off 20 sold. Remaining: 48 – 20 = 28 cookies.

Worked Example 3: Ratio and Comparison

The ratio of boys to girls in a class is 3:2. If there are 15 boys, how many girls are there?

Draw 3 equal strips for boys (each part = 15 ÷ 3 = 5 students). Draw 2 equal strips for girls (each part = 5 students). So 2 × 5 = 10 girls.

Multi-Step Problems Involving Multiple Operations

For complex problems, draw separate diagrams for each step. First diagram shows the initial setup, second diagram shows what happens after the first operation, and so on. This sequential visualization makes it clear which operations to perform and in what order.

Complex Example: Mixed Operations

A store has 120 apples. They receive 30 more apples from a supplier. Then they sell half of the apples. Finally, 5 apples spoil. How many apples remain?

Step 1: Start with 120, add 30 → total 150. Step 2: Sell half → 150 ÷ 2 = 75 remain. Step 3: Subtract 5 spoiled → 75 – 5 = 70 apples left. The strip diagrams visually confirm each operation.

Advantages of Using Strip Diagrams

Strip diagrams transform abstract word problems into concrete visual representations. They reduce the likelihood of misinterpreting relationships, help identify which operations to use, and provide a clear strategy before you calculate. They’re especially valuable when problems involve multiple steps or complex relationships.

Common Pitfalls

Don’t rush to calculate before drawing the diagram. Ensure your diagram accurately reflects the problem relationships. If results don’t match your visual representation, check both your diagram and your calculations. Be consistent with the size of strips when they represent equal quantities.

Practice Problems

1. Tom spends 3 hours studying math and twice as much time studying English. How many total hours does he study?

2. A school has 250 students. One-third are in grade 3, and the rest are in grade 4. How many are in grade 4?

3. Sarah has some money. She spends 30 dollars, then earns 50 dollars. Now she has 120 dollars. How much did she start with?

Master this essential technique in our word problems resources and strengthen your skills with the SAT Math Course.

What Are Strip Diagrams?

Strip diagrams, also called bar models or tape diagrams, are visual representations that break down multi-step word problems into manageable parts. They consist of rectangular bars representing known and unknown quantities, helping you organize information and plan your solution strategy before performing calculations.

Building a Strip Diagram Step by Step

Start by identifying all quantities in the problem. Draw a rectangle for each known quantity and label it with the value. For unknown quantities, draw unlabeled rectangles. Use the relationships described in the problem to arrange these strips: stack them for totals, place them in sequence for combinations, or position them side-by-side for comparisons.

Worked Example 1: Addition and Subtraction

Maria has 15 stickers. Her friend gave her 8 more. Then she gave 6 to her brother. How many does she have now?

Draw a strip for 15 (Maria’s initial stickers), add a segment for 8 (received), then remove 6 (given away). Visually, you can see: 15 + 8 – 6 = 17 stickers.

Worked Example 2: Multiplication and Division

A baker makes 4 trays of cookies. Each tray has 12 cookies. She sells 20 cookies. How many are left?

Draw 4 equal strips, each labeled 12, showing the total is 4 × 12 = 48 cookies. Then mark off 20 sold. Remaining: 48 – 20 = 28 cookies.

Worked Example 3: Ratio and Comparison

The ratio of boys to girls in a class is 3:2. If there are 15 boys, how many girls are there?

Draw 3 equal strips for boys (each part = 15 ÷ 3 = 5 students). Draw 2 equal strips for girls (each part = 5 students). So 2 × 5 = 10 girls.

Multi-Step Problems Involving Multiple Operations

For complex problems, draw separate diagrams for each step. First diagram shows the initial setup, second diagram shows what happens after the first operation, and so on. This sequential visualization makes it clear which operations to perform and in what order.

Complex Example: Mixed Operations

A store has 120 apples. They receive 30 more apples from a supplier. Then they sell half of the apples. Finally, 5 apples spoil. How many apples remain?

Step 1: Start with 120, add 30 → total 150. Step 2: Sell half → 150 ÷ 2 = 75 remain. Step 3: Subtract 5 spoiled → 75 – 5 = 70 apples left. The strip diagrams visually confirm each operation.

Advantages of Using Strip Diagrams

Strip diagrams transform abstract word problems into concrete visual representations. They reduce the likelihood of misinterpreting relationships, help identify which operations to use, and provide a clear strategy before you calculate. They’re especially valuable when problems involve multiple steps or complex relationships.

Common Pitfalls

Don’t rush to calculate before drawing the diagram. Ensure your diagram accurately reflects the problem relationships. If results don’t match your visual representation, check both your diagram and your calculations. Be consistent with the size of strips when they represent equal quantities.

Practice Problems

1. Tom spends 3 hours studying math and twice as much time studying English. How many total hours does he study?

2. A school has 250 students. One-third are in grade 3, and the rest are in grade 4. How many are in grade 4?

3. Sarah has some money. She spends 30 dollars, then earns 50 dollars. Now she has 120 dollars. How much did she start with?

Master this essential technique in our word problems resources and strengthen your skills with the SAT Math Course.

Fundamentals of Strip Diagrams

Strip diagrams (also called bar models, tape diagrams, or pictorial representations) are visual problem-solving tools that represent quantities as rectangular bars. Each bar’s length represents the magnitude of a quantity. This visual approach makes abstract quantities concrete and relationships clear. Students can see at a glance whether quantities are being combined, divided, or compared. The method originated in Singapore mathematics education and has proven remarkably effective internationally.

Building Blocks: Single-Step Problems

A single-step problem like “Sarah has 12 apples and gets 5 more” uses one diagram. Draw a rectangle labeled 12 for her original apples. Draw another labeled 5 for the new apples. Connect them to show 12+5=17 total. For “A book costs 20 dollars and is 25% off,” draw a rectangle for the original price, then show that 25% of it is the discount, leaving 75% as the new price. The visual immediately clarifies the relationship.

Multi-Step Problem Strategy

Complex problems require sequential diagrams. “A store starts with 100 items. They receive 30 more, then sell 40. How many remain?” Draw diagram 1: 100+30=130. Draw diagram 2: 130-40=90. Each diagram focuses on one operation, and the sequence shows the logical flow. Students see exactly which operations are needed and in what order, reducing errors from misunderstanding the problem structure.

Strip Diagram Example: Ratio Problems

When boys outnumber girls 3:2 and there are 60 boys total, create 3 equal-sized boxes for boys and 2 for girls. Since 3 boxes=60 boys, each box=20. So girls=2 boxes=40. Strip diagrams make ratio reasoning visual and intuitive, eliminating confusion about which quantity gets which multiplier.

Advanced Application: Fractional Relationships

For “One-third of the apples are red, and red apples number 8,” draw a bar divided into 3 equal parts. Mark that one part represents 8 red apples. Then the whole bar (3 parts) represents 24 total apples. This visualization clarifies that one-third of something means dividing into three equal parts, with each part having the same size.

Comparison Problems with Strip Diagrams

When comparing quantities like “Tom reads 5 more books than Jerry,” draw Jerry’s bar, then draw Tom’s bar slightly longer by 5 units. This immediately shows the relationship visually. If Jerry reads x books and Tom reads y books, then y=x+5 becomes obvious from the diagram. Extending this, if together they read 27 books, you can see x+(x+5)=27, so 2x+5=27, giving x=11 for Jerry and y=16 for Tom.

Handling Unknowns in Complex Scenarios

When a problem has multiple unknowns, label them systematically. “Alex has twice as many coins as Bailey, and together they have 15” uses one bar for Bailey with label B, another bar (twice as long) for Alex with label 2B, and shows both equal 3B=15 in total, so B=5 and Alex has 10. The diagram manages multiple unknowns simultaneously without requiring algebraic setup before visualization.

Common Student Pitfalls and Solutions

Many students draw bars the wrong size, making proportions unclear. Teach them to use grid paper for consistent scaling. Others forget to label what each bar represents, losing track of quantities. Always label the bar and its value. Some struggle with operations appearing in different orders in the problem statement. The diagram helps by showing the actual mathematical relationships rather than following word order. Emphasize that the diagram represents the mathematics, not the reading order.

Transitioning from Diagrams to Equations

Once students master visual strip diagrams, help them connect to algebraic equations. A diagram showing x+2x=15 naturally leads to the equation. The bar model serves as a bridge from concrete visualization to abstract symbolic math, supporting mathematical development across learning styles. Students who benefit from visuals gain confidence, while all students deepen understanding through multiple representations.

Extended Practice Problems

1. Sarah spends 40 dollars on books and 15 dollars on pens, leaving her with 25 dollars. How much did she start with? (Answer: 80 dollars, worked via strip diagrams showing three quantities). 2. A recipe uses flour and sugar in ratio 5:2. If sugar is 40 grams, how much flour? (Five equal parts for flour, two for sugar; 2 parts=40g, so 1 part=20g, flour=100g). 3. Three students’ test scores: Alex scored 10 points more than Bailey. Bailey scored 5 more than Charlie. Total: 195 points. Find each score. (Diagram shows this reduces to 3C+15=195, so C=60, B=65, A=75).

Resources: Word Problems Guide, SAT Math Course, GED Math.

Fundamentals of Strip Diagrams

Strip diagrams (also called bar models, tape diagrams, or pictorial representations) are visual problem-solving tools that represent quantities as rectangular bars. Each bar’s length represents the magnitude of a quantity. This visual approach makes abstract quantities concrete and relationships clear. Students can see at a glance whether quantities are being combined, divided, or compared. The method originated in Singapore mathematics education and has proven remarkably effective internationally.

Building Blocks: Single-Step Problems

A single-step problem like “Sarah has 12 apples and gets 5 more” uses one diagram. Draw a rectangle labeled 12 for her original apples. Draw another labeled 5 for the new apples. Connect them to show 12+5=17 total. For “A book costs 20 dollars and is 25% off,” draw a rectangle for the original price, then show that 25% of it is the discount, leaving 75% as the new price. The visual immediately clarifies the relationship.

Multi-Step Problem Strategy

Complex problems require sequential diagrams. “A store starts with 100 items. They receive 30 more, then sell 40. How many remain?” Draw diagram 1: 100+30=130. Draw diagram 2: 130-40=90. Each diagram focuses on one operation, and the sequence shows the logical flow. Students see exactly which operations are needed and in what order, reducing errors from misunderstanding the problem structure.

Strip Diagram Example: Ratio Problems

When boys outnumber girls 3:2 and there are 60 boys total, create 3 equal-sized boxes for boys and 2 for girls. Since 3 boxes=60 boys, each box=20. So girls=2 boxes=40. Strip diagrams make ratio reasoning visual and intuitive, eliminating confusion about which quantity gets which multiplier.

Advanced Application: Fractional Relationships

For “One-third of the apples are red, and red apples number 8,” draw a bar divided into 3 equal parts. Mark that one part represents 8 red apples. Then the whole bar (3 parts) represents 24 total apples. This visualization clarifies that one-third of something means dividing into three equal parts, with each part having the same size.

Comparison Problems with Strip Diagrams

When comparing quantities like “Tom reads 5 more books than Jerry,” draw Jerry’s bar, then draw Tom’s bar slightly longer by 5 units. This immediately shows the relationship visually. If Jerry reads x books and Tom reads y books, then y=x+5 becomes obvious from the diagram. Extending this, if together they read 27 books, you can see x+(x+5)=27, so 2x+5=27, giving x=11 for Jerry and y=16 for Tom.

Handling Unknowns in Complex Scenarios

When a problem has multiple unknowns, label them systematically. “Alex has twice as many coins as Bailey, and together they have 15” uses one bar for Bailey with label B, another bar (twice as long) for Alex with label 2B, and shows both equal 3B=15 in total, so B=5 and Alex has 10. The diagram manages multiple unknowns simultaneously without requiring algebraic setup before visualization.

Common Student Pitfalls and Solutions

Many students draw bars the wrong size, making proportions unclear. Teach them to use grid paper for consistent scaling. Others forget to label what each bar represents, losing track of quantities. Always label the bar and its value. Some struggle with operations appearing in different orders in the problem statement. The diagram helps by showing the actual mathematical relationships rather than following word order. Emphasize that the diagram represents the mathematics, not the reading order.

Transitioning from Diagrams to Equations

Once students master visual strip diagrams, help them connect to algebraic equations. A diagram showing x+2x=15 naturally leads to the equation. The bar model serves as a bridge from concrete visualization to abstract symbolic math, supporting mathematical development across learning styles. Students who benefit from visuals gain confidence, while all students deepen understanding through multiple representations.

Extended Practice Problems

1. Sarah spends 40 dollars on books and 15 dollars on pens, leaving her with 25 dollars. How much did she start with? (Answer: 80 dollars, worked via strip diagrams showing three quantities). 2. A recipe uses flour and sugar in ratio 5:2. If sugar is 40 grams, how much flour? (Five equal parts for flour, two for sugar; 2 parts=40g, so 1 part=20g, flour=100g). 3. Three students’ test scores: Alex scored 10 points more than Bailey. Bailey scored 5 more than Charlie. Total: 195 points. Find each score. (Diagram shows this reduces to 3C+15=195, so C=60, B=65, A=75).

Resources: Word Problems Guide, SAT Math Course, GED Math.