Using Hands on Math Manipulatives to Build Understanding

🧮 The Power of Math Manipulatives: Why Hands-On Learning Builds Lasting Understanding

When many adults think back to their own math education, they remember worksheets, flashcards, and memorizing steps. What’s often missing from that picture is manipulatives—the hands-on tools that make math visible and concrete.

As a math interventionist and tutor, I know that manipulatives aren’t just for young learners. They are essential for building true mathematical understanding at every grade level. And the best way to think about how they fit into learning is through the CRA model.

🌟 What Is CRA?

CRA stands for Concrete–Representational–Abstract, a research-based teaching model that helps students truly understand math instead of memorizing procedures.

• Concrete: Students use physical objects (manipulatives) to explore the concept.

• Representational: Students move to drawings, diagrams, or visual models.

• Abstract: Finally, they use numbers and symbols alone.
  

This progression ensures that by the time students are working with abstract equations, they have already built meaning through hands-on experiences and visual representations.

✋ Manipulatives Aren’t Just for the Early Grades

One common misconception is that manipulatives are “babyish” or only belong in kindergarten or first grade. In reality, manipulatives are just as important in upper elementary school—and beyond. Even in 4th and 5th grade, tools like fraction bars, place value blocks, or arrays help students move from memorization to true number sense.

🔑 Manipulatives in Action: From Hands-On to Abstract

Let’s look at a few specific tools and how they support the CRA model.

1. 🟦 Ten Frames and Counters – Addition & Subtraction

Ten frames allow students to see numbers in relation to 10—a critical building block for addition, subtraction, and place value.

• Concrete: Students place counters on a ten frame to model problems like 8 + 5.

• Representational: They draw a ten frame with dots instead of using real counters.

• Abstract: Finally, they write 8 + 5 = 13, with a strong mental image of “making a ten.”

👉 Ten frames also support subtraction: 13 – 6 can be shown by filling a double ten frame with 13 counters, then removing 6.

2. 🟩 Connecting Cubes – Building Relationships

Connecting cubes are versatile for addition and subtraction. They help students explore number properties and relationships.

• Commutative Property: Build 3 + 5 and 5 + 3 to see that both equal 8.

• Number Combinations: Snap cubes into pairs to show all the ways to make 10.

• Doubles + 1: Build 7 + 8 by modeling 7 + 7, then adding one more cube.

Students move from cubes (concrete) → drawing towers (representational) → equations (abstract).

3. 🔢 Base Ten Blocks – Modeling Place Value

Base ten blocks give students a physical model of our number system. They show that numbers aren’t just digits—they represent ones, tens, hundreds, and beyond.

• Concrete: Build numbers like 243 with 2 hundreds, 4 tens, and 3 ones.

• Representational: Draw quick sketches of flats, rods, and units.

• Abstract: Transition to the written form: 200 + 40 + 3 = 243.

👉 This deepens place value understanding and prepares students for operations with larger numbers.

4. 🔄 Base Ten Blocks – Regrouping in Addition & Subtraction

Regrouping (or “borrowing” and “carrying”) can be confusing if it’s taught only as a procedure. Base ten blocks make the process clear and logical.

• Addition: When adding 27 + 36, students combine ones. If they have 13 ones, they physically trade 10 ones for 1 ten block.

• Subtraction: For 52 – 38, students may need to “break apart” a ten into 10 ones, making subtraction tangible.

• Representational: Students sketch blocks or use quick tens-and-ones drawings.

• Abstract: With practice, they apply the regrouping algorithm with confidence, because they understand what’s happening.
  
  

5. 🟨 Decimal Squares – Making Decimals Visual

Decimals can feel abstract and intimidating for many students—especially when they first encounter tenths and hundredths. Decimal squares make place value beyond the decimal point visible and easy to grasp.

• Concrete: Using decimal squares divided into 10 or 100 parts, students can shade to model numbers like 0.3 or 0.47. They literally see how decimals are parts of a whole.

• Representational: Students draw their own grids or diagrams to represent decimals. For example, shading 47 out of 100 squares to show 0.47.

• Abstract: Finally, students work with numbers only, comparing, adding, or subtracting decimals with a clear mental picture of their size.

👉 Decimal squares are especially powerful for comparing decimals (which is larger, 0.3 or 0.29?), adding decimals, and later connecting decimals to fractions.

6. 🟥 Decimal Squares with Transparent Overlays – Multiplying Decimals

Multiplying decimals can feel counterintuitive. Students often wonder: “Why is 0.4 × 0.6 smaller than both 0.4 and 0.6?”

Decimal squares with transparent colored overlays provide the perfect visual.

• Concrete: Students shade one decimal (for example, 0.4 in yellow) on a 10×10 decimal grid. Then, using a transparent overlay, they shade another decimal (say, 0.6 in red). The overlapping shaded area shows the product 0.24.

• Representational: Students draw grids with overlapping shading to model decimal multiplication.

• Abstract: They then write and solve the equation: 0.4 × 0.6 = 0.24, but now it makes sense because they’ve seen it.

👉 This method clears up confusion and shows why multiplication doesn’t always mean “bigger”—it depends on the numbers being multiplied.

7. 🟦 Fraction Strips – Building Fraction Sense

Fractions are one of the trickiest concepts for many students because they can’t always “see” what a fraction represents. Fraction strips make fractions concrete, helping students understand their size and relationships.

• Concrete: Using a full set of fraction strips, students explore how parts make a whole. They see that 3 pieces of 1/3 equal the same length as one whole strip, or that 4 pieces of 1/4 also make one whole. By lining up different strips, students can compare fractions like 1/2 and 1/3 visually, noticing which is larger.

• Representational: Students sketch bars or number lines to represent fractions. For example, drawing three 1/3 bars to equal a whole

• Abstract: They then work with fraction notation confidently, writing equations like 3/3 = 1 or 4/4 = 1.

👉 Fraction strips are also excellent for modeling mixed numbers and improper fractions.

• Mixed to Improper: For example, with 1 2/3, students place one whole strip and two of the 1/3 strips. They can then count in thirds: 3/3 + 2/3 = 5/3.

• Improper to Mixed: With 7/4, students place seven of the 1/4 strips. They notice that 4/4 makes one whole, with 3/4 left over—so 7/4 = 1 3/4.

This hands-on approach allows students to see and build fractions, instead of just memorizing rules. Over time, they begin to visualize fractions mentally, making comparisons and operations much easier.

8. 📏 Fraction Benchmarking – Comparing Fractions to 0, 1/2, and 1

Once students are familiar with basic fractions, they can use benchmarking to make comparisons more efficient. Instead of always finding common denominators, they can build fractions with manipulatives to see where they fall relative to familiar benchmarks: 0, 1/2, and 1.

• Concrete: With fraction strips, students model a fraction like 3/8. They compare it side by side with the 1/2 strip and notice that 3/8 is just a little less.

• Representational: Students sketch bars or number lines, plotting fractions to see their relative sizes.

• Abstract: They move to reasoning strategies, such as recognizing that 5/12 is less than 1/2 because 6/12 equals 1/2 and 5/12 is smaller.

👉 Benchmarking helps students develop number sense with fractions, building the ability to judge size and make quick, logical comparisons.

9. 🟪 Fraction Multiplication with Fraction Squares and Overlays

Multiplying fractions can feel abstract if students jump straight to the algorithm. Fraction squares with transparent colored overlays provide a powerful way to show why multiplication works the way it does.

• Concrete: Start with a fraction square divided into fourths. Shade 1/2 vertically in yellow, then overlay 3/4 shaded horizontally in red. The overlapping region shows the product—3/8 of the whole.

• Representational: Students draw grids or rectangles, shading in two directions to model multiplication.

• Abstract: They then connect the visual to the procedure: 1/2 × 3/4 = 3/8.

👉 This method helps students understand why multiplying fractions results in a smaller product, and it lays the foundation for reasoning about area models and proportional thinking later on.

10. 🔲 Area with Square Tiles – Building Area Understanding

Area is often taught as “length × width,” but if students only memorize the formula, they may not understand why it works. Using square tiles makes the concept concrete and visual.

• Concrete: Students cover a rectangle with unit square tiles, counting how many tiles fit inside. For example, a 3-by-4 rectangle is filled with 12 unit squares.

• Representational: Students draw rectangles on grid paper, shading the squares inside to show the total area.

• Abstract: Finally, students connect the model to the formula: 3 × 4 = 12 square units.
  
  

👉 This approach helps students see area as covering space rather than just plugging numbers into a formula. It also prepares them for more advanced topics like the distributive property, irregular areas, and even polynomial multiplication later on.

11. 🧊 Volume with Inch Cubes – Building 3D Understanding

Just like area, volume makes the most sense when students can see and build it. Using one-inch cubes gives learners a tangible way to understand volume as the amount of space an object occupies.

• Concrete: Students physically build rectangular prisms with inch cubes. For example, they might stack cubes to make a prism that measures 3 inches by 2 inches by 4 inches. Then, they count the total cubes to find the volume—24 cubic inches.

• Representational: Students draw prisms on grid paper or use isometric dot paper, sketching layers to represent the cubes inside.

• Abstract: Finally, they connect the model to the formula: Volume = length × width × height. In this case, 3 × 2 × 4 = 24
  👉 This process helps students see why the volume formula works instead of memorizing it as another rule. They can even extend this to irregular prisms by building and counting layers, which builds a strong foundation for middle school geometry.

12. 🔷 Exploring 2-D and 3-D Shapes – Understanding Attributes

Geometry can feel abstract when students are only shown pictures in a textbook. By physically handling 2-D and 3-D shapes, students can explore and compare their attributes in meaningful ways.

• Concrete: Students handle cut-outs of 2-D shapes (triangles, quadrilaterals, hexagons, etc.) or use solid 3-D figures (cubes, rectangular prisms, spheres, cones, pyramids). They can count sides, vertices, faces, and edges, and begin noticing patterns.

• Representational: Students sketch these shapes, label attributes, or build them on dot paper and graph paper. With 3-D solids, they can also draw nets to see how flat 2-D faces fold into a 3-D object.

• Abstract: Students use correct mathematical vocabulary and properties to classify shapes (e.g., “all squares are rectangles, but not all rectangles are squares”). They begin using formulas for perimeter, area, or surface area once they’ve built a solid understanding of the attributes.

👉 By handling and building shapes first, students don’t just memorize terms like edge or face—they experience them. This hands-on exploration leads to a deeper, longer-lasting understanding of geometry.

13. 📐 Angle Legs – Building Polygons and Angles

When students are only shown angles on paper, they may not fully grasp how angles work or how shapes are formed. Angle legs (hinged, movable sticks) allow students to physically create and adjust angles and polygons.

• Concrete: Students connect angle legs and move them to form different angles—acute, right, obtuse, and straight. They can also link multiple angle legs to build polygons like triangles, quadrilaterals, pentagons, and beyond. This hands-on exploration helps them see how side lengths and angles combine to form shapes.

• Representational: Students sketch the shapes they’ve built, label the angles, and begin to compare and classify polygons. They can also measure angles they’ve created with a protractor to connect physical movement to precise measurement.

• Abstract: Students apply angle rules (like the sum of angles in a triangle is 180°) and classify polygons by their attributes. They begin reasoning about geometry instead of simply memorizing properties.

👉 Angle legs give students a tactile and visual way to see how angles connect and how polygons are built, making geometry both interactive and memorable.

As you can see, there are so many places where hands-on manipulatives can help students make sense of concepts and truly visualize their learning.  At Math With Miss Malloy, I incorporate these tools to support student growth and confidence.