Elementary Pull-Out: Getting to the Root of A Problem

The Elementary Mathematician, COMAP's award-winning K-6 newsletter is available for the first time in electronic format. Each issue contains a collection of K-6 mathematic lessons that are fun to teach and exciting to learn.

Each issue of The Elementary Mathematician contains a Pull-Out section which includes thematic lessons based on the national census, wildlife, other cultures, technology, and provides a practical way to bring the concepts and pedagogy of the NCTM Standards into your classroom. 

Note: The information below was created with the assistance of AI.

Level of Mathematics
Target Grade Levels: Upper Elementary (Grades 4–6)

The activity is suited to this level based on:

Introduction of square roots—a concept typically explored in upper elementary or early middle school.

Use of concrete manipulatives and estimation to bridge to abstract calculator functions.

The requirement for students to understand fractions, estimation, and basic square numbers.

Application Areas
This lesson is highly applicable in the following educational domains:

1. Conceptual Mathematics
Introduces the abstract concept of square roots using hands-on, visual representations.

Strengthens understanding of area and side-length relationships in squares.

2. Numerical Estimation and Reasoning
Builds approximation and ratio reasoning using manipulatives and fractional comparisons.

Promotes understanding of the idea that square roots of non-perfect squares are irrational and must be estimated.

3. Technology Integration
Encourages appropriate and strategic use of calculators to confirm and refine manual estimates.

4. STEM Foundation
Lays groundwork for mathematical modeling and deeper exploration of square and root functions relevant to engineering and physical sciences.

Prerequisites
Students need to be familiar with:

Square Numbers: Recognizing perfect squares and their root values (e.g., 4 = 2²).

Fractions and Decimals: Ability to work with basic fractions (e.g., 3/5) and convert to decimals.

Area Concepts: Understanding that area = side × side in square configurations.

Calculator Use: Comfort using a square root key and interpreting calculator results.

Subject Matter

1. Square Roots
Introduced visually by constructing square arrays with colored tiles.

Explores estimation for square roots of non-square numbers using physical modeling.

2. Estimation Techniques
Combines fractional tile ratios (e.g., 3/5 ≈ 0.6) with visual models to estimate root values.

Students derive increasingly accurate estimates by refining their tile placement.

3. Mathematical Modeling
Comparison of three methods: visual tile model, ratio-based estimation, and calculator-derived values.

Students complete a chart comparing these to observe how each approach converges on a close approximation.

4. Rational vs. Irrational Numbers (Conceptual Intro)
Without using formal terminology, the lesson shows that many square roots are not whole numbers and must be approximated.

Correlation to Mathematics Standards
This lesson aligns well with Common Core State Standards (CCSS) for Grades 5–6:

Number and Operations – Fractions
CCSS.MATH.CONTENT.5.NF.B.7: Apply and extend previous understandings of division to divide unit fractions by whole numbers and vice versa.

Expressions and Equations
CCSS.MATH.CONTENT.6.EE.A.2: Write, read, and evaluate expressions involving whole-number exponents.

CCSS.MATH.CONTENT.6.EE.A.1: Understand square roots as a way to express numbers.

Mathematical Practices

MP2: Reason abstractly and quantitatively.

MP4: Model with mathematics.

MP5: Use appropriate tools strategically.

MP7: Look for and make use of structure.