Ages 5–12 math practice
How to get better at math
Build understanding, fluency, reasoning, and problem-solving range through connected practice that begins in a chosen school range and adapts as the learner responds.
AXIO connects number, operations, fractions, measurement, geometry, data, patterns, and logic through prerequisite-aware practice. The learner chooses where the math should begin, then each response helps shape the next useful task.
A connected math program
Mathematical ideas grow together. Counting supports place value, place value supports operations, equal shares lead into fractions, and operations connect with decimals, ratio, measurement, data, and algebraic thinking.
Mathematics curricula organize learning into connected domains and practices such as reasoning, representation, and problem solving123. AXIO turns those connections into practice routes across the Math Map.
What learners practise
The program spans seven connected learning areas. Each area draws on canonical Math Map skills and their prerequisite routes.
Number sense and place value
Connect quantities, counting, comparison, number lines, grouping, notation, and the base-ten structure of whole numbers.
Operations, fluency, and problem solving
Understand operation meaning, build flexible strategies and facts, estimate, calculate, and model one-step and multi-step situations with equations, arrays, and other representations.
Fractions, decimals, percent, and ratio
Connect equal parts, fractions as numbers, equivalence, operations, decimal place value, percentages, ratio, rate, and scaling.
Measurement, geometry, and space
Use units and tools, reason about shape properties, compose and transform figures, and explore coordinate planes, perimeter, area, angles, and volume.
Patterns and early algebra
Find and explain structure, generalize rules, work with unknowns and equality, and express relationships with diagrams, tables, words, and symbols.
Data and chance
Classify and collect information, build and interpret tables and graphs, compare displays, and reason about uncertainty.
Reasoning and problem solving
Organize cases, eliminate possibilities, compare strategies, justify conclusions, test conjectures, and transfer ideas to unfamiliar problems.
How practice builds stronger math
Make meaning
Connect objects, diagrams, number lines, tables, graphs, words, and symbols so each method has a mathematical reason behind it.
Build fluency
Develop accurate, flexible methods and retrieve useful knowledge while keeping the underlying relationships visible.
Reason and solve
Choose a representation, plan a route, monitor progress, explain decisions, and check whether the result fits the situation.
Revisit and transfer
Return to a prerequisite or clearer representation, then use the same idea independently in a changed context or form.
Fluency grows alongside understanding and flexible strategy choice6. Spaced retrieval, worked examples alongside problem solving, and links between visual, verbal, concrete, and abstract representations help organize practice7.
Where should the math begin?
The selected school range gives AXIO a curricular starting neighbourhood. Response evidence then adjusts task demand, representations, prerequisites, and challenge throughout practice.
Auto adapts as you solve
Let response evidence find the most useful entry point and keep adjusting from there.
Early primary foundations
Begin with early number, quantity, shape, pattern, and mathematical-language foundations.
Primary school skills
Begin across the main primary-school range, then move through prerequisites and connected skills as needed.
Primary-to-secondary bridge
Begin with upper-primary and bridge mathematics while keeping earlier prerequisites available.
Practice that responds to the learner
Correct independent work can bring a new form, a connected skill, or greater challenge. A difficult response can bring a clearer representation, a nearby prerequisite, or guided practice before the learner returns to the idea independently.
Sessions interleave connected skills and revisit important ideas after other work. This creates opportunities to retrieve knowledge, compare representations, repair misconceptions, and transfer a method to a changed situation. Systematic progressions, mathematical language, representations, number lines, and word-problem structures can support learners who need more help4.
See the kinds of thinking learners practise
These examples show different mathematical actions across the program: seeing structure, selecting operations, interpreting scale, navigating space, and reasoning from constraints.
Counting and Cardinality Foundations
Ten Frame Fill and Read
Teaching focus: See quantities in relation to five and ten.
How many triangle tiles are in the other part?
A structured quantity model makes number relationships visible and supports later place-value and calculation work.
Core · Reasoning · Fluency · Concept · Objects · Symbols · Enter a number
Explore this exampleWhole-Number Operations and Fact Fluency
Multi Digit Add Sub Fluent
Teaching focus: Use place value while adding and subtracting larger numbers.
What is the sum of all ten whole numbers from 90 through 99?
Flexible calculation grows from the value of each digit and the relationships among units.
Core · Reasoning · Fluency · Concept · Objects · Symbols · Enter a number
Explore this exampleMeasurement Units, Tools, Time, and Money
Unit Conversion Word Problems
Teaching focus: Connect a measurement situation to compatible units and operations.
A jug holds 1.5 L. Add 750 mL. How many litres are in the jug now? The jug has room for all of it.
The learner interprets quantities and relationships before choosing a calculation.
Core · Reasoning · Fluency · Concept · Objects · Enter a number
Explore this exampleData Classification, Tables, and Graph Reasoning
Bar Graph Read and Interpret Scaled
Teaching focus: Read the scale before interpreting a graph.
In a community-center survey, each student chose exactly one weekend activity. The bar graph records each surveyed student’s choice exactly once. What fraction of the surveyed students chose Art or Chess?
A graph joins visual structure, numerical intervals, comparison, and evidence-based interpretation.
Core · Reasoning · Fluency · Concept · Objects · Table or graph · Symbols
Explore this exampleSpatial Composition, Transformations, Symmetry, and Coordinates
Coordinate paths
Teaching focus: Reason about position and movement on a coordinate grid.
Four cities are at the corners of a square with side 100 km. What is the least cost of a connected network of straight roads? New interior junctions may be anywhere inside; shared road segments count once. Roads cost 1 million dinars per km. Round to the nearest million dinars.
Coordinate paths connect spatial visualization, precise notation, and multi-step reasoning.
Core · Reasoning · Fluency · Concept · Objects · Diagram · Choose
Explore this exampleLogic, Constraint, Deduction, and Combinatorics
Constraints Elimination
Teaching focus: Remove impossible choices and justify what remains.
Which badge follows every rule?
Constraint tasks develop systematic search, deduction, checking, and explanation.
Core · Reasoning · Fluency · Concept · Objects · Choose · Draw or trace
Explore this exampleHow parents and teachers can help
Reflection, visual representations, comparison of strategies, and clear mathematical language can strengthen problem solving5.
Skills in this practice collection
These connected Math Map skills define the learning path across this practice collection.
Counting and Cardinality Foundations
Quantity Comparison and Order Reasoning
Base-Ten Place Value and Whole-Number Notation
- Compare Numbers to 1000
- Compare Numbers to 10000
- Expanded Form to 1000
- Large numbers to one million
- Large Numbers to Ten Million
- Place Value Hundreds
- Place Value Tens Ones
- Place Value Thousands
- Place Value to 100000
- Read Write Numbers to 1000
- Read Write Numbers to 10000
- Round Multi Place
- Round to Nearest 10
- Round to Nearest 100
- Round to Nearest 1000
- Whole Number Powers of 10
- Operations with positive and negative rational numbers
Whole-Number Operations and Fact Fluency
- Add Sub Within 1000 Algorithm
- Add Sub Within 1000 Models
- Add Two Digit No Regroup
- Add Two Digit Regroup
- Divide by 10 100
- Divide Multi Digit by One Digit Algorithm
- Divide Multi Digit by One Digit Partial Quotients
- Divide Multi Digit by One Digit Place Value
- Divide multi-digit whole numbers by two-digit divisors
- Multi Digit Add Sub Fluent
- Multiply by 10 100
- Multiply One Digit by Multi Digit Algorithm
- Multiply One Digit by Multi Digit Area Model
- Multiply One Digit by Multi Digit Partial Products
- Multiply Two Digit by Two Digit Algorithm
- Multiply Two Digit by Two Digit Area Model
- Multiply Two Digit by Two Digit Partial Products
- Sub Two Digit No Regroup
- Sub Two Digit Regroup
- Add Within 10 Concrete
- Add Within 20 Strategies
- Array Rows Columns
- Divide Equal Groups Basic
- Distinguish group-size and number-of-groups unknowns
- Divide Equal Shares Basic
- Divide Relate to Multiplication
- Equal Groups Make
- Fact Families Multiply Divide
- Factors Basic
- Gcf Basic
- Lcm Basic
- Missing Addend Within 20
- Multiples Basic
- Multiply Commutative Property
- Multiply Distributive Property
- Multiply Equal Groups Basic
- Distinguish number of groups from group size
- Select and explain a multiplication fact strategy
- Multiply Facts 0 1
- Multiply Facts 2 5 10
- Multiply Facts 3 4 6
- Multiply Facts 7 8 9
- Use multiplication facts within 100 fluently
- Recognize and make equal sets
- Transfer multiplication across changed forms
- Translate among multiplication representations
- Prime Composite Basic
- Skip Count Repeated Addition
- Sub Within 10 Concrete
- Sub Within 20 Strategies
Operations Models, Bar Models, and Word Problems
- Word Problems Bar Model
- Add Sub Word Problems One Step
- Add Sub Word Problems Two Step
- Bar Model Comparison
- Bar Model Multiplicative Comparison
- Bar Model Part Whole
- Estimation Reasonableness
- Multi Step Word Problem Mixed Operations
- Multiply Divide Word Problems One Step
- Multiply Divide Word Problems Two Step
- Solve multiplicative comparison situations
- Remainder Interpret
Fractions as Parts, Wholes, and Fair Shares
- Fair Sharing Discrete Objects
- Fair Sharing Equal Shares
- Fraction of Quantity Unitary
- Fractions of a set
- Name Non Unit Fractions
- Name Unit Fractions
- Partition Shapes Equal Parts
- Represent Area Model
- Represent Bar Model
- Represent Number Line General
- Represent Number Line Unit
- Same Whole Constraint
- Whole As Equal Parts
Fraction Equivalence, Comparison, and Operations
- Add Same Denom Mixed
- Add Same Denom Unit
- Add fractions with general unlike denominators
- Add Unlike Denom Related
- Compare Benchmark Half
- Compare Same Denom
- Compare Same Numer
- Compare Unit Fractions
- Compare Unlike Denom
- Decompose
- Divide fractions in the general Grade 6 case
- Divide Unit Fraction by Whole Readiness
- Divide a whole number by a unit fraction
- Equivalent Basic Visual
- Equivalent Generate
- Equivalent Number Line
- Improper Convert Mixed
- Mixed Numbers Compose Decompose
- Multiply a fraction by a fraction
- Multiply Fraction by Whole
- Multiply Whole by Fraction
- Order Multiple
- Simplify Basic
- Sub Same Denom
- Subtract fractions with general unlike denominators
- Sub Unlike Denom Related
Decimal and Percent Reasoning
- Add Tenths Hundredths
- Compare Tenths Hundredths
- Compare to Thousandths
- Decimal Notation Tenths Hundredths
- Divide by 10 100
- Divide decimals to hundredths
- Equivalent Tenths Hundredths
- Fraction Convert Denominator 10 100
- Fraction Convert Denominator 1000
- Hundredths Concept
- Measurement Contexts
- Money Notation Connect
- Multiply by 10 100
- Multiply decimals to hundredths
- Number Line to Hundredths
- Place Value Tenths Hundredths
- Place Value Thousandths
- Round to Tenth Hundredth
- Round to Whole
- Sub Tenths Hundredths
- Tenths Concept
- Thousandths Concept
Measurement Units, Tools, Time, and Money
- Describe and compare measurable attributes
- Capacity and liquid measure
- Length Compare Direct
- Length Convert Customary Basic
- Length Convert Metric Basic
- Length Estimate
- Length Measure Ruler
- Length Nonstandard Units
- Length Standard Units
- Mass and weight
- Money Coin Values Uk
- Money Coin Values Us
- Money Count Mixed Coins
- Money Make Change
- Money Solve Word Problems
- Scale Read Intervals
- Temperature Read
- Elapsed time across hours
- Time Elapsed Same Hour
- Time Tell Five Minutes
- Time Tell Hour Half
- Tell and write time to the nearest minute
- Time Tell Quarter
- Unit Conversion Word Problems
Geometric Measurement
- Angle Sum Triangle Readiness
- Area Perimeter Distinguish
- Circle circumference and area
- Volume Surface Readiness
- Angles Additive Unknown
- Angles Compare
- Angles Draw
- Angles Identify
- Angles Measure Protractor
- Area Composite Rectilinear
- Area Count Square Units
- Area Rectangles Array
- Area Triangle Readiness
- Perimeter Composite Rectilinear
- Perimeter Count Units
- Perimeter Rectangles
- Find surface area from polygonal nets
- Volume Composite
- Volume Cubic Units Count
- Volume Formula
- Volume Rectangular Prism Pack
Data Classification, Tables, and Graph Reasoning
- Average Readiness Mean Median Mode
- Bar Graph Draw Basic
- Bar Graph Draw Scaled
- Bar Graph Read and Interpret Scaled
- Bar Graph Read Basic
- Construct a line graph
- Line Graph Read Basic
- Line Plot Create
- Line Plot Fractions
- Line Plot Read
- Picture Graph Draw
- Picture Graph Read
- Compound Events, Sample Spaces, and Tree Diagrams
- Experimental probability and relative frequency
- Probability Models and Probability Scale
- Simple chance and likelihood language
- Sort Classify One Attribute
- Display numerical data with dot plots histograms and box plots
- Center, spread, and distribution shape
- Statistical questions and variability
- Sampling and informal inference
- Reading tables
- Tables Two Way Readiness
- Tally and Frequency
- Variation and distribution readiness
- Sort Venn One Set
- Sort Venn Two Set
Geometry Shape Attributes and Classification
Spatial Composition, Transformations, Symmetry, and Coordinates
- Circular Turns Basic
- Coordinate Grid First Quadrant
- Multi-quadrant coordinate readiness
- Coordinate Plot Points
- Coordinate paths
- Slices and dimensions
- Nets Cube Cuboid
- Spatial Compose Shapes
- Spatial Decompose Shapes
- Describe relative position of objects and shapes
- Symmetry Complete Figure
- Symmetry Identify Line
- Symmetry Rotational Readiness
- Tessellation Basic
- Transform Compose
- Transform Reflect
- Transform Rotate
- Transform Translate
- Spatial Visualization
Patterns, Functions, and Generalization
Equations, Variables, Expressions, and Inequalities
- Truth Value Equations
- Order of Operations Basic
- Equations Unknown Add Sub
- Equations Unknown Mult Div
- Operations Properties Apply
- Order of Operations Parentheses
- Algebraic Thinking Word to Expression
- Balancing equations
- Generate and identify equivalent expressions
- Exponents Squares Cubes
- Expression Evaluate
- Inequality Compare
- Represent simple inequality constraints on a number line
- Linear equations in one variable
- Negative Numbers Contexts
- Negative number order and absolute value readiness
- Variable As Unknown
Logic, Constraint, Deduction, and Combinatorics
Ratio, Rate, Scaling, and Proportional Readiness
- Similarity Scale Readiness
- Ratio Readiness Tables
- Double number line missing values
- Percent As Fraction 100
- Percent Applications, Increase, Decrease, and Reverse Percent
- Find a simple percent of a quantity
- Proportional relationships across tables, graphs, diagrams, equations
- Rate Unit Rate Readiness
- Ratio Table Basic
- Scale drawing and map scale readiness
- Ratio notation and equivalent ratios
Evidence behind the approach
- Common Core State Standards Initiative. Mathematics Standards. 2010.
- Department for Education. National curriculum in England: mathematics programmes of study. Updated 28 September 2021.
- Singapore Ministry of Education. 2021 Primary Mathematics Syllabus, Primary 1 to 6. Updated December 2024.
- U.S. Institute of Education Sciences, What Works Clearinghouse. Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades. March 2021.
- U.S. Institute of Education Sciences, What Works Clearinghouse. Improving Mathematical Problem Solving in Grades 4 Through 8. Revised October 2018.
- National Council of Teachers of Mathematics. Procedural Fluency in Mathematics. Position statement.
- U.S. Institute of Education Sciences, What Works Clearinghouse. Organizing Instruction and Study to Improve Student Learning. September 2007.