Operations.
Use materials to explain joining, separating, grouping, sharing, and comparing.
An operation describes a relationship or a change. A collection may be joined to another, shared into equal groups, or compared with a larger collection. Materials are useful when their arrangement makes that situation clear.
Use linking cubes for equal groups, math bars for parts and comparisons, or a number line for jumps and differences. Ask what each piece, group, and movement represents.
Record an equation alongside the model and connect each symbol to something visible. Different models can represent the same calculation, but some make a particular question easier to discuss. Choosing the model is part of the mathematical work.
Make it concrete.
Base-Ten Blocks
Units, rods, flats, and larger blocks that represent the grouping structure of base-ten numbers.
Explore Cuisenaire Rods
Colored rods with lengths in a one-to-ten relationship for comparing numbers, operations, and fractions.
Explore Dice
Objects with numbered or dotted faces for generating values, playing number games, and collecting chance data.
Explore Fraction Circles
Equal-sized circles divided into equal parts for comparing fractions of the same whole.
Explore Fraction Tiles
Equal-length wholes partitioned into fraction strips for comparing, composing, and ordering fractions.
Explore Linking Cubes
Equal-sized cubes that connect into trains, groups, arrays, and three-dimensional builds.
Explore Math Bars
Length-based bars for showing quantities, comparing parts, and drawing attention to numerical relationships.
Explore Number Lines
Lines with equal unit spacing for locating numbers, comparing distance, and representing operations.
Explore Play Money
Replica coins and notes for comparing values, making totals, and finding equivalent combinations.
Explore Rekenreks
Bead racks arranged in groups of five and ten for showing numbers and comparing addition strategies.
Explore Sketch Pad
An open drawing space for turning a math idea into a picture, revising it, and explaining your thinking.
Explore Ten Frames
Two rows of five spaces that organize counters around the benchmarks five and ten.
Explore Two-Color Counters
Small disks with contrasting sides for sorting, counting, and representing two parts of a quantity.
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