Array Rods
Array Rods are a simple manipulative for exploring multiplication, division, and the properties of multiplication.
Uses:
- Modeling multiplication and division
- Illustrating the Distributive Property by rearranging facts (for example, 7x8 can be broken into 7x1 + 7x7, 7x2 + 7x6, 7x3 + 7x5, and so on)
- Building strategies for memorizing trickier facts, such as recognizing that 7x6 is the same as 5x6 plus 2x6
Numicons
Numicons are number frames built around 5 as a visual benchmark, paired with a pegboard that fills to 10.
Uses:
- Addition and subtraction
- General number sense
- Learning combinations that make 10
- Understanding regrouping when numbers exceed 10 (for example, seeing that 7 + 5 is the same as 7 + 3, plus 2 more)
Fraction Squares
Fraction Squares are a smaller version of a popular manipulative, designed for classroom use so every student can have their own set (or two).
Uses:
- Part-whole reasoning
- Comparing fractions
- Fraction addition and subtraction
- Fraction multiplication and division
- Mixed numbers and improper fractions
Fraction Tiles
Fraction Tiles support fraction comparison and arithmetic, with a hinged "whole" piece that folds for easy storage.
Uses:
- Comparing fractions
- Fraction addition, subtraction, multiplication, and division
- Placing and labeling fractions on a number line (especially when paired with a paper-based number line)
Classroom tip: Having more than one set per student (or 3 to 4 sets per small group) allows students to practice working beyond a single whole.
Decimal Rods
Decimal Rods let students represent decimals with a physical manipulative. Hundredths rods can be flipped rather than counted one by one, and a "whole" piece (1) is included for reference.
Uses:
- Representing tenths, hundredths, and wholes
- Comparing the size of decimal parts
- Regrouping in decimal addition and subtraction
- Area multiplication (with setups up to 3x3)
Classroom tip: Pairing the rods with decimal square paper helps students move from concrete to pictorial representation. For multiplication problems larger than 3x3, grid paper works well as a next step.
Place Value Rings
Place Value Rings are inspired by tools like arrow cards and homemade place value expanders, covering hundreds through hundredths for both whole numbers and decimals.
Uses:
- Building whole numbers and decimals across place value positions
- Combining and regrouping place value parts
- Engaging with "place value riddles," such as:
- I have 5 tens and 3 ones. What number am I?
- I have 5 tens and 13 ones. What number am I?
- I have 25 tens and 3 ones. What number am I?
- I have 25 tens and 13 ones. What number am I?
Classroom tip: These work especially well in pairs, letting students see and regroup digits across positions. Walking through the first one or two riddles together is often worthwhile when introducing a new type of regrouping.