Learn on PengiSaxon Math, Intermediate 4Chapter 6: Lessons 51–60, Investigation 6

Lesson 56: Using Models and Pictures to Compare Fractions, Activity Comparing Fractions

In this Grade 4 Saxon Math Intermediate 4 lesson, students learn to compare fractions such as 1/2, 1/3, and 1/4 by using fraction manipulatives and drawing congruent figures like circles and rectangles that are shaded to represent each fraction. The lesson emphasizes that comparison figures must be congruent — the same shape and size — to make accurate comparisons. Students practice writing comparisons using greater than and less than symbols and extend their skills to ordering decimals from greatest to least.

Section 1

📘 Using Models and Pictures to Compare Fractions

New Concept

When we draw figures to compare fractions, the figures should be congruent. Congruent figures have the same shape and size.

Why it matters

Visualizing fractions with congruent shapes builds the essential skill of representing abstract numbers in the real world. Mastering this prepares you to intuitively grasp more complex relationships in geometry and algebraic functions later on.

What’s next

Next, you’ll apply this concept by drawing and shading your own congruent figures to compare different fractions.

Section 2

Comparing fractions with models

Property

To compare fractions, draw and shade parts of two congruent (identical) figures. The fraction representing the larger shaded area is the greater fraction. This visual method helps you see the difference in value between fractions like 12\frac{1}{2} and 13\frac{1}{3} without needing to find a common denominator.

Example

To compare 12◯13\frac{1}{2} \bigcirc \frac{1}{3}, draw two identical circles. Shading half of one is more than a third of the other, so 12>13\frac{1}{2} > \frac{1}{3}.
To compare 14◯13\frac{1}{4} \bigcirc \frac{1}{3}, draw two congruent rectangles. Shading 14\frac{1}{4} covers less area than shading 13\frac{1}{3}, so 14<13\frac{1}{4} < \frac{1}{3}.
To compare 23◯35\frac{2}{3} \bigcirc \frac{3}{5}, drawing two identical bars shows that the shaded area for 23\frac{2}{3} is larger, so 23>35\frac{2}{3} > \frac{3}{5}.

Explanation

Think of it like sharing two identical candy bars! If you shade 12\frac{1}{2} of one bar and your friend shades 13\frac{1}{3} of the other, your drawing will clearly show that your piece is bigger. This method turns abstract fractions into a simple visual contest where the bigger shaded part wins!

Section 3

Congruent

Property

When we draw figures to compare fractions, the figures should be congruent. Congruent figures have the same shape and size. Using congruent shapes ensures that the comparison is fair and accurate, focusing only on the fractional parts rather than differences in the wholes.

Example

To compare 34\frac{3}{4} and 45\frac{4}{5}, you must use two rectangles of the exact same length and width.
When comparing 23\frac{2}{3} and 12\frac{1}{2} with circles, both circles must have the same radius.
A 4-inch by 6-inch rectangle is congruent to another 4-inch by 6-inch rectangle, but not to a 4-inch by 7-inch one.

Explanation

Imagine comparing who drank more juice, but you have a giant cup and your friend has a tiny one. It's not a fair test! Congruent figures are like using two identical cups. This ensures you're only comparing the fractions themselves, not the size of the shapes. It’s the key to an honest, apples-to-apples fraction showdown.

Section 4

Ordering decimals

Property

To arrange decimals in order, compare the digits in the same place value from left to right, starting with the ones place. If digits are equal, move to the tenths place, then the hundredths. You can add placeholder zeros to the end of a decimal (like changing 0.70.7 to 0.700.70) to make comparisons easier.

Example

To order 0.5,0.2,0.250.5, 0.2, 0.25 from greatest to least, compare the tenths place: 5>25 > 2. To compare 0.20.2 and 0.250.25, use 0.200.20. In the hundredths, 5>05 > 0. The order is 0.5,0.25,0.20.5, 0.25, 0.2.
To order 0.75,0.9,0.70.75, 0.9, 0.7 from greatest to least, compare the tenths place: 9>79 > 7. To compare 0.750.75 and 0.70.7, use 0.700.70. In the hundredths, 5>05 > 0. The order is 0.9,0.75,0.70.9, 0.75, 0.7.

Explanation

It's like judging a talent show! First, look at the whole numbers. If they're tied, check the tenths place. Still tied? Move on to the hundredths! Adding zeros to make the decimals the same length helps you see the comparison clearly, just like lining everyone up side-by-side to see who is taller.

Book overview

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Chapter 6: Lessons 51–60, Investigation 6

  1. Lesson 1

    Lesson 51: Adding Numbers with More Than Three Digits, Checking One-Digit Division

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  2. Lesson 2

    Lesson 52: Subtracting Numbers with More Than Three Digits, Word Problems About Equal Groups, Part 2

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  3. Lesson 3

    Lesson 53: One-Digit Division with a Remainder, Activity Finding Equal Groups with Remainders

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  4. Lesson 4

    Lesson 54: The Calendar, Rounding Numbers to the Nearest Thousand

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  5. Lesson 5

    Lesson 55: Prime and Composite Numbers, Activity Using Arrays to Find Factors

    LearnPractice
  6. Lesson 6Current

    Lesson 56: Using Models and Pictures to Compare Fractions, Activity Comparing Fractions

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  7. Lesson 7

    Lesson 57: Rate Word Problems

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  8. Lesson 8

    Lesson 58: Multiplying Three-Digit Numbers

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  9. Lesson 9

    Lesson 59: Estimating Arithmetic Answers

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  10. Lesson 10

    Lesson 60: Rate Problems with a Given Total

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  11. Lesson 11

    Investigation 6: Displaying Data Using Graphs, Activity Displaying Information on Graphs

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Lesson overview

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Section 1

📘 Using Models and Pictures to Compare Fractions

New Concept

When we draw figures to compare fractions, the figures should be congruent. Congruent figures have the same shape and size.

Why it matters

Visualizing fractions with congruent shapes builds the essential skill of representing abstract numbers in the real world. Mastering this prepares you to intuitively grasp more complex relationships in geometry and algebraic functions later on.

What’s next

Next, you’ll apply this concept by drawing and shading your own congruent figures to compare different fractions.

Section 2

Comparing fractions with models

Property

To compare fractions, draw and shade parts of two congruent (identical) figures. The fraction representing the larger shaded area is the greater fraction. This visual method helps you see the difference in value between fractions like 12\frac{1}{2} and 13\frac{1}{3} without needing to find a common denominator.

Example

To compare 12◯13\frac{1}{2} \bigcirc \frac{1}{3}, draw two identical circles. Shading half of one is more than a third of the other, so 12>13\frac{1}{2} > \frac{1}{3}.
To compare 14◯13\frac{1}{4} \bigcirc \frac{1}{3}, draw two congruent rectangles. Shading 14\frac{1}{4} covers less area than shading 13\frac{1}{3}, so 14<13\frac{1}{4} < \frac{1}{3}.
To compare 23◯35\frac{2}{3} \bigcirc \frac{3}{5}, drawing two identical bars shows that the shaded area for 23\frac{2}{3} is larger, so 23>35\frac{2}{3} > \frac{3}{5}.

Explanation

Think of it like sharing two identical candy bars! If you shade 12\frac{1}{2} of one bar and your friend shades 13\frac{1}{3} of the other, your drawing will clearly show that your piece is bigger. This method turns abstract fractions into a simple visual contest where the bigger shaded part wins!

Section 3

Congruent

Property

When we draw figures to compare fractions, the figures should be congruent. Congruent figures have the same shape and size. Using congruent shapes ensures that the comparison is fair and accurate, focusing only on the fractional parts rather than differences in the wholes.

Example

To compare 34\frac{3}{4} and 45\frac{4}{5}, you must use two rectangles of the exact same length and width.
When comparing 23\frac{2}{3} and 12\frac{1}{2} with circles, both circles must have the same radius.
A 4-inch by 6-inch rectangle is congruent to another 4-inch by 6-inch rectangle, but not to a 4-inch by 7-inch one.

Explanation

Imagine comparing who drank more juice, but you have a giant cup and your friend has a tiny one. It's not a fair test! Congruent figures are like using two identical cups. This ensures you're only comparing the fractions themselves, not the size of the shapes. It’s the key to an honest, apples-to-apples fraction showdown.

Section 4

Ordering decimals

Property

To arrange decimals in order, compare the digits in the same place value from left to right, starting with the ones place. If digits are equal, move to the tenths place, then the hundredths. You can add placeholder zeros to the end of a decimal (like changing 0.70.7 to 0.700.70) to make comparisons easier.

Example

To order 0.5,0.2,0.250.5, 0.2, 0.25 from greatest to least, compare the tenths place: 5>25 > 2. To compare 0.20.2 and 0.250.25, use 0.200.20. In the hundredths, 5>05 > 0. The order is 0.5,0.25,0.20.5, 0.25, 0.2.
To order 0.75,0.9,0.70.75, 0.9, 0.7 from greatest to least, compare the tenths place: 9>79 > 7. To compare 0.750.75 and 0.70.7, use 0.700.70. In the hundredths, 5>05 > 0. The order is 0.9,0.75,0.70.9, 0.75, 0.7.

Explanation

It's like judging a talent show! First, look at the whole numbers. If they're tied, check the tenths place. Still tied? Move on to the hundredths! Adding zeros to make the decimals the same length helps you see the comparison clearly, just like lining everyone up side-by-side to see who is taller.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 6: Lessons 51–60, Investigation 6

  1. Lesson 1

    Lesson 51: Adding Numbers with More Than Three Digits, Checking One-Digit Division

    LearnPractice
  2. Lesson 2

    Lesson 52: Subtracting Numbers with More Than Three Digits, Word Problems About Equal Groups, Part 2

    LearnPractice
  3. Lesson 3

    Lesson 53: One-Digit Division with a Remainder, Activity Finding Equal Groups with Remainders

    LearnPractice
  4. Lesson 4

    Lesson 54: The Calendar, Rounding Numbers to the Nearest Thousand

    LearnPractice
  5. Lesson 5

    Lesson 55: Prime and Composite Numbers, Activity Using Arrays to Find Factors

    LearnPractice
  6. Lesson 6Current

    Lesson 56: Using Models and Pictures to Compare Fractions, Activity Comparing Fractions

    LearnPractice
  7. Lesson 7

    Lesson 57: Rate Word Problems

    LearnPractice
  8. Lesson 8

    Lesson 58: Multiplying Three-Digit Numbers

    LearnPractice
  9. Lesson 9

    Lesson 59: Estimating Arithmetic Answers

    LearnPractice
  10. Lesson 10

    Lesson 60: Rate Problems with a Given Total

    LearnPractice
  11. Lesson 11

    Investigation 6: Displaying Data Using Graphs, Activity Displaying Information on Graphs

    LearnPractice