Build your number sense
Count, compare quantities, investigate shapes, and make sense of addition and subtraction through visual models.
Explore number foundations ↓Free interactive learning
Try it. See it. Understand it.
Explore free interactive explorations inspired by Virginia's 2023 Mathematics Standards of Learning. No sign-in needed. Choose a topic and start discovering.
An independent learning journey
Choose what you want to strengthen.
Start with your grade, revisit an earlier foundation when helpful, and move forward by exploring, checking, and practicing. The interactive labs below are available to try right now.
Make connections with fractions
Explore fraction parts, visual equivalence, multiplication ideas, and connections to real quantities.
Explore fraction models ↓Reason with patterns and relationships
Build algebra readiness through equations, unit rates, proportional thinking, and linear models.
Explore middle school labs ↓Explore algebra and geometric reasoning
Use polynomial structure, geometry, graphs, and mathematical explanations to develop deeper understanding.
Explore Algebra I ↓Early Number Foundations · Kindergarten–Grade 2
Make ten. See how numbers fit together.
One way to understand addition is to see a number as two parts. A ten-frame shows how many counters you already have and how many more would make a full set of ten.
- Choose how many counters to show.
- Count the open places.
- Explain how the two parts make ten.
When you see 6 counters and 4 open spaces, you know 6 + 4 = 10. Try another number and discover a new pair!
Visual number explorer
Make-Ten Lab
6 + ? = 10
Move the slider and find the missing part.
Fraction Foundations · Grades 3–5
See why two fractions can be equal.
A fraction describes equal parts of a whole. Dividing each part into two new equal pieces changes the number of pieces, but not the amount shaded.
- Choose the total number of equal parts.
- Change how many parts are shaded.
- Compare each fraction with an equivalent fraction.
For example, 2/3 = 4/6. The top and bottom numbers both double, and the shaded portion stays the same.
Interactive fraction bars
Fraction Equivalence Lab
Your fraction
2/4
The same fraction with twice as many parts
4/8
Both bars show the same part of a whole.
Multiplication Foundations · Grades 3–5
Build multiplication with rows and columns.
An array turns multiplication into something you can see. The total number of objects is the number of rows multiplied by the number of columns.
- Change the number of rows.
- Change the number of columns.
- Notice that rotating the array does not change the product.
A 4-by-6 array and a 6-by-4 array both contain 24 objects. This is the commutative property of multiplication.
Visual multiplication explorer
Multiplication Array Lab
4 × 6 = 24
4 equal rows with 6 objects in each row make 24 objects.
Grade 6 · SOL 6.PFA.3
Solve equations by keeping them balanced.
Both sides of an equation represent the same amount. To find the value of an unknown, undo an operation on both sides.
- Identify the operation applied to the unknown.
- Apply its inverse to both sides.
- Substitute your result to check the equation.
Example: x + 5 = 14. Subtract 5 from both sides to get x = 9, because 9 + 5 = 14.
Equation Balance Lab
x + 5 = 14
An equal amount on each side.
Which operation helps you solve for x?
Challenge 1 of 4Try, check, and grow.
Grade 7 · SOL 7.CE.2
Use unit rates to solve everyday problems.
Proportional relationships are all around you. Find the cost of one item, then scale it to a different quantity.
- Divide the total cost by the number of items.
- Multiply the unit price by the new number of items.
- Explain why the relationship is proportional.
A constant unit price means cost = (price per item) × quantity.
Try a real-world challenge
Unit Rate Lab
3 granola bars cost $6. What is the cost of 7 granola bars?
Unit price = $6 ÷ 3Think per 1 item
Find the unit price first, then multiply.
Percent & Proportional Reasoning · Middle School Foundation
See percent as “per hundred.”
A percent describes a part of 100. Converting a percent to a decimal makes it easier to calculate a part of another quantity.
- Use the 100-square model to see the percent.
- Convert the percent to a decimal.
- Multiply the decimal by the whole quantity.
35% of 80 means 0.35 × 80 = 28. The 100-square shows what 35% looks like; the multiplication finds 35% of a different whole.
Percent model
Percent of a Quantity Lab
35% of 80 = 28
35% = 0.35, and 0.35 × 80 = 28.
Grade 8 · SOL 8.PFA.3
See how slope changes a line.
A linear relationship has a constant rate of change. Move the controls and watch the graph respond instantly.
- Slope tells you the vertical change for one horizontal step.
- The y-intercept is where the graph crosses the vertical axis.
- The equation y = mx + b connects both ideas.
When the slope is positive, the line rises from left to right. When it is negative, the line falls.
Manipulate the mathematics
Linear Relationships Explorer
y = x + 0
Geometry Foundation · Middle School
A triangle always has 180° inside.
If you know two interior angles of a triangle, the third angle is determined. Add the known angles, then subtract from 180°.
- Change angle A.
- Change angle B.
- Watch angle C adjust so the total remains 180°.
If A = 52° and B = 71°, then C = 180° − 52° − 71° = 57°.
Angle relationship explorer
Triangle Angle Lab
Diagram not drawn to scale.
52° + 71° + 57° = 180°
The third angle is 57°.
Data Reasoning · Middle School Foundation
Compare mean, median, and range.
Different statistics answer different questions. The mean describes a balance point, the median identifies the middle of ordered data, and the range describes overall spread.
- Change any of the five data values.
- Compare how mean and median respond.
- Watch the range change as the smallest or largest value moves.
An extreme value can pull the mean more strongly than the median. That is why choosing a summary statistic depends on the shape of the data.
Data center
Mean · Median · Range Lab
Mean6
Median5
Range7
Ordered data: 3, 5, 5, 7, 10.
Algebra I · SOL A.EO.2
Discover the structure of a quadratic expression.
Factoring connects an expression to the values that make it equal to zero. Look for two numbers with the correct product and sum.
- For x² − 4x + 3, the constant product is +3.
- The two numbers must add to −4.
- Multiply the factors to check your choice.
Factoring and expanding are inverse processes: (x + a)(x + b) = x² + (a + b)x + ab.
Think like an algebraist
Factoring Explorer
Which expression is equivalent to this quadratic?
x² − 4x + 3
Choose an equivalent factored expression.
Geometry Foundation · High School
Connect the sides of a right triangle.
For a right triangle, the Pythagorean Theorem states that the square of the hypotenuse equals the sum of the squares of the legs.
- Choose the lengths of the two legs.
- Square each leg and add.
- Take the square root to find the hypotenuse.
For legs 3 and 4: 3² + 4² = 9 + 16 = 25, so the hypotenuse is √25 = 5.
Right-triangle explorer
Pythagorean Theorem Lab
3² + 4² = 25 → c = 5
The hypotenuse is 5.
Continue exploring
Mathematics can be entered from more than one direction.
After trying the English Math Labs, explore examples from the broader Brazilian HUBMAT portfolio. These linked projects are in Portuguese and are presented as part of the author's professional work.
See a mathematical idea
Visual Mathematics uses representations and direct manipulation to make relationships easier to inspect.
Explore Visual Mathematics ↗Think through a challenge
Mathematics games create situations involving logic, strategy, space, precision, and decision-making.
Explore Mathematics Games ↗Build a daily mathematics habit
HUBMAT 5min organizes short learning sessions designed to make regular practice easier to sustain.
Explore HUBMAT 5min ↗Meet mathematics inside a narrative
Mathematical stories use characters and situations to place ideas inside a broader context.
Explore Mathematical Stories ↗Learning continues
More understanding, one discovery at a time.
Explore an idea, check your reasoning, and practice again with a new example. Each step can strengthen your mathematical understanding.