Building Blocks for Math Mastery
Fraction Foundations

Ages 9-12

Teacher:   Mindy Seminario

 

Homework Requirement: 30 to 60 minutes per week.  Please do not sign up for this class if you do not have the time to complete homework.

 

Winter:  Advanced Fractions

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Prerequisite Information 

This class is designed for continuing students who participated in the fall trimester. In addition to learning foundational fraction concepts, these students have begun to internalize the relationships between Cuisenaire rods—an essential component of the learning approach described below. If your child was not enrolled in the fall, I recommend waiting to join the class when it is offered again from the beginning of the sequence. However, if you feel your child may be ready to join now, please contact me at seminariomindy@gmail.com and I can provide prerequisite readiness test to ensure they’re prepared to fully engage with the material.

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Class Description

I’m so proud of the progress the class has made this fall, and we have so much more in store as we continue our learning journey.

We began the year exploring fractions through hands-on work with Cuisenaire rods, covering a wide range of foundational topics: improper fractions, mixed numbers, fractional equivalence, comparing fractions, and adding and subtracting with both like and unlike denominators. We’ve practiced finding least common denominators and even started multiplying fractions—all through tactile, visual experiences that make abstract ideas feel concrete.

This winter, we’ll build on that foundation by diving deeper into multiplying and dividing fractions. We’ll continue reinforcing addition and subtraction, especially with mixed numbers and improper fractions. To stretch their thinking, we’ll introduce more complex equations using parentheses—an easy extension with the rods—which will help students understand the distributive property and why the “P” in PEMDAS matters.

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Underlying all of this is the power of the rods themselves. The magic of using Cuisenaire rods is that while their values are flexible, their relationships remain constant. Even as students build basic fractions, they’re internalizing these consistent relationships. For example, the purple rod is always—always—two-thirds the length of the dark green rod, no matter which rod is chosen as the whole or what the context is. It’s also equivalent to four white rods and two red rods. As students work with the rods, they begin to see a network of connections—relationships that stay true across different situations. This growing familiarity helps them link ideas and deepen their understanding of how fractions, proportions, and operations all relate. Fractions and proportions become intuitive, not just procedural. That kind of deep understanding simply isn’t possible with one-off models or through symbols and steps alone.

Here’s why this approach matters:

 

• Flexibility and Consistency: While the value of any rod can change depending on which rod is designated as the whole, the relationships (ratios) between the rods always stay the same. This is the magic!

• Internalizing Relationships: Students don't just memorize rules; they internalize proportional relationships. The fact that the purple rod is always 2/3 the dark green rod, regardless of the "whole," builds a deep, intuitive understanding.

• Network of Connections: The equivalence property (e.g., purple = ⅔ x dark green = 4 white = 2 red) helps students see how different fractional parts and operations relate to each other, creating a rich network of connections.

• Deep Conceptual Understanding: This model moves students beyond rote memorization of symbols and steps. It provides the physical, multi-sensory foundation necessary for an intuitive grasp of how fractions, proportions, and arithmetic operations are connected.

 

I’m deeply passionate about teaching fractions this way because I’ve seen how it transforms students’ thinking. It’s not just about getting answers—it’s about building a lasting understanding of how numbers work. I’m excited to continue this journey with your children and can’t wait to see how their mathematical thinking grows in the months ahead.

 

Homework Requirement

Homework is a required and essential part of our math program. It’s only through consistent practice that students begin to truly see and make the connections described. I want to sincerely thank you for supporting your child’s homework in the fall—it made a real difference. As we move into winter, homework will continue to be a mandatory and meaningful part of our learning journe

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Fall: Fraction Foundations:

Ready to conquer fractions and build a powerful math foundation? This engaging and comprehensive class is designed for students aged 9 to 12 to develop a deep and lasting understanding of fractions. We'll move beyond just memorizing rules and explore the exciting world of parts and wholes. Through hands-on activities, visual models, and real-world connections, students will gain confidence and fluency in comparing, finding equivalent fractions, and mastering the operations of multiplication and division.

 

Why are Fractions So Important for Math Success?

 

Fractions are not just a topic in elementary school; they are a fundamental building block for success in higher-level mathematics and many real-world applications. Understanding fractions is crucial because:

• They are the foundation for decimals, percentages, and ratios: Many essential mathematical concepts build directly upon a solid understanding of fractions. Without this foundation, students may struggle with these related topics.

• They are essential for algebra and beyond: Algebraic concepts often involve manipulating rational numbers, which are essentially fractions. A strong grasp of fractions makes the transition to algebra much smoother.

 

• They foster logical reasoning and problem-solving skills: Working with fractions requires students to think conceptually, reason proportionally, and develop flexible problem-solving strategies.

 

Research Highlights the Link Between Fraction Knowledge and Long-Term Math Success:

 

Numerous studies underscore the critical role of fraction understanding in predicting future mathematical achievement:

• Fraction proficiency predicts algebra success: Research has consistently shown a strong correlation between students' understanding of fractions in middle school and their success in algebra (Booth et al., 2014; Siegler et al., 2012). A solid foundation in fractions is a key predictor of who will succeed in more advanced mathematics.

• Early fraction knowledge impacts later math achievement: Studies have found that students' early understanding of fraction magnitudes is significantly related to their overall math achievement in high school (Bailey et al., 2015). This suggests that early intervention and strong instruction in fractions are crucial for long-term success.

• Difficulties with fractions are a barrier to higher math: Research indicates that students who struggle with fractions often face significant challenges in subsequent mathematics courses (National Mathematics Advisory Panel, 2008). Addressing these foundational difficulties early on is essential for preventing future struggles.

 

 

What Students Will Learn:

In this class, we will delve into the core concepts of fractions, including:

• Understanding the Meaning of Fractions: Exploring fractions as parts of a whole, ratios, and points on a number line.

• Comparing Fractions: Developing strategies for determining which fraction is larger or smaller using visual models and common denominators.

• Equivalent Fractions: Discovering how different fractions can represent the same value and learning how to find equivalent fractions.

• Multiplying and Dividing Fractions: Building a conceptual understanding of fraction multiplication division.

• Adding and Subtracting Fractions (with like and unlike denominators)

• Connecting Fractions to Decimals: Exploring the relationship between fractions and decimals.

• Problem-Solving with Fractions: Applying fraction concepts to solve real-world problems and develop critical thinking skills.

 

 

By the end of this class, students will have a strong conceptual understanding of fractions, enabling them to confidently tackle more advanced mathematical concepts and navigate the many ways fractions appear in the world around them!

 

• To reinforce learning 30 to 60 minutes of homework per week will be mandatory for this class. This dedicated time outside of the classroom is crucial for solidifying understanding and honing skills for long-term math success!