Mathematics & Reasoning

D05 P3: Intellectual & Cognitive Awareness

Number, pattern, shape, and data — and the habit of reasoning clearly and honestly about quantities and relationships.

Mathematics is a way of seeing structure in the world, not a gatekeeping ritual. We teach it as sense-making: children count real things, find real patterns, and build real shapes before they ever meet an abstract symbol. Reasoning here is for everyone, drawn from many mathematical traditions, not one.

K–12 objective ladder

D05.S1 Number & Quantity

Counting, comparing, and operating with numbers and quantities in meaningful contexts.

68 lessons

K D05.S1.K.01 Count objects up to 20 and say how many there are.
1 D05.S1.01.01 Count, read, and write whole numbers up to 120 and use them to say how many objects are in a group.
1 D05.S1.01.02 Add and subtract within 20 using objects, drawings, or counting strategies, and explain what the operation means.
2 D05.S1.02.01 Use place value of tens and ones to read, write, compare, and order two-digit numbers.
2 D05.S1.02.02 Add and subtract within 100 using models, drawings, or written methods, and explain the strategy used.
3 D05.S1.03.01 Use place value to 1,000 to read, write, round, and compare whole numbers.
3 D05.S1.03.02 Multiply and divide within 100 and explain the meaning of each operation using objects, drawings, and known facts.
4 D05.S1.04.01 Use place value to 1,000,000 to read, write, round, and compare multi-digit whole numbers.
4 D05.S1.04.02 Add, subtract, multiply, and divide multi-digit whole numbers, and use estimation to check that answers are reasonable.
5 D05.S1.05.01 Read, write, compare, and round decimals to the thousandths and connect them to fractions with denominators of 10 and 100.
5 D05.S1.05.02 Add, subtract, multiply, and divide fractions and decimals in real contexts such as sharing, cooking, and money.
6 D05.S1.06.01 Use ratios and rates to describe and compare real situations such as fair shares, recipes, and unit prices.
6 D05.S1.06.02 Use positive and negative numbers to represent real situations such as temperature and elevation, and locate and order them on a number line.
7 D05.S1.07.01 Use proportional relationships, including unit rates, scale, and percentages, to solve multi-step problems in real contexts.
7 D05.S1.07.02 Perform all four operations with rational numbers such as fractions, decimals, and integers, and explain each step of the reasoning.
8 D05.S1.08.01 Distinguish rational and irrational numbers and approximate square roots by locating them on the number line.
8 D05.S1.08.02 Use integer exponents and scientific notation to represent and compare very large and very small quantities, such as distances in space or the size of cells.
9 D05.S1.09.01 Work with real numbers, roots, and radicals in expressions and equations, explaining which properties justify each step.
9 D05.S1.09.02 Choose appropriate units and levels of precision when measuring and calculating, and interpret the results in context.
10 D05.S1.10.01 Use properties of exponents and roots to simplify expressions and solve problems involving exponential growth and decay.
10 D05.S1.10.02 Extend number reasoning to complex numbers and explain why new kinds of numbers arise from real problems.
11 D05.S1.11.01 Model and solve real problems with sequences and series, and interpret their meaning in financial or environmental contexts.
11 D05.S1.11.02 Use vectors and matrices to represent and manipulate quantities with both size and direction or with several dimensions.
12 D05.S1.12.01 Apply the ideas of limits and continuity informally to describe change, approximation, and the behavior of quantities.
12 D05.S1.12.02 Reason quantitatively about large systems such as populations, energy use, or carbon, using orders of magnitude and estimation, and critique the numbers used in public claims.

D05.S2 Patterns & Relationships

Seeing, making, and generalizing patterns and relationships, and reasoning about change.

51 lessons

K D05.S2.K.01 Continue or copy a simple repeating pattern of colors, shapes, or sounds.
1 D05.S2.01.01 Identify, extend, and create repeating patterns using objects, sounds, or colors, and tell the rule of the pattern.
1 D05.S2.01.02 Find the missing number in a simple number sentence such as 3 + ? = 7 and explain how they know.
2 D05.S2.02.01 Identify and extend growing patterns and describe how the pattern changes from one step to the next.
2 D05.S2.02.02 Solve one-step addition and subtraction problems with a missing number in a real context, such as sharing or spending.
3 D05.S2.03.01 Identify arithmetic patterns in tables and number grids and explain them using the properties of operations.
3 D05.S2.03.02 Solve two-step word problems using the four operations and explain the relationship between addition and subtraction or multiplication and division.
4 D05.S2.04.01 Generate and analyze a number or shape pattern that follows a given rule and describe its features.
4 D05.S2.04.02 Use a letter or symbol to stand for an unknown number in a simple equation and find its value.
5 D05.S2.05.01 Plot points and graph relationships in the first quadrant of a coordinate grid to represent real situations.
5 D05.S2.05.02 Write and evaluate simple expressions and interpret what they mean in a real situation.
6 D05.S2.06.01 Use variables to write expressions and equations that represent real situations, and solve one-step equations.
6 D05.S2.06.02 Represent the relationship between two quantities with a table and graph, and tell when one quantity depends on the other.
7 D05.S2.07.01 Write and solve two-step equations and inequalities to represent and solve real problems.
7 D05.S2.07.02 Identify proportional relationships in tables, graphs, and equations, and express them in the form y = kx.
8 D05.S2.08.01 Define, evaluate, and compare functions and use function notation to describe how one quantity depends on another.
8 D05.S2.08.02 Solve linear equations in one variable and analyze a pair of simultaneous linear equations by graphing.
9 D05.S2.09.01 Model real situations with linear and exponential functions and interpret slope and intercept in context.
9 D05.S2.09.02 Solve quadratic equations using several methods and interpret what their solutions mean in real situations.
10 D05.S2.10.01 Build and interpret polynomial and rational functions that model real relationships and describe their key features.
10 D05.S2.10.02 Model exponential growth and decay in population, finance, or environmental systems and explain the effect of the rate.
11 D05.S2.11.01 Model periodic phenomena with trigonometric functions and interpret amplitude, frequency, and shift in context.
11 D05.S2.11.02 Use logarithmic and exponential relationships to solve problems involving growth, decay, and large ranges of scale.
12 D05.S2.12.01 Model systems of relationships with equations or matrices and interpret the meaning of a solution in a real context.
12 D05.S2.12.02 Describe an algorithm as a step-by-step procedure for a real problem and analyze whether it works correctly and fairly.

D05.S3 Geometry & Spatial Reasoning

Understanding shape, space, position, and how objects fit and move in the world.

48 lessons

K D05.S3.K.01 Name and match common shapes and use position words (over, under, beside).
1 D05.S3.01.01 Compose and take apart two- and three-dimensional shapes and describe their sides, corners, and faces.
1 D05.S3.01.02 Measure and compare lengths using non-standard units such as hand spans or paper clips.
2 D05.S3.02.01 Identify and draw shapes with given attributes, such as a certain number of angles or equal sides.
2 D05.S3.02.02 Partition circles and rectangles into equal shares and describe the shares as halves, thirds, and fourths.
3 D05.S3.03.01 Understand the area and perimeter of rectangles and solve problems using them in real contexts.
3 D05.S3.03.02 Sort shapes into categories by their attributes and recognize that shapes in a category share common properties.
4 D05.S3.04.01 Draw and identify points, lines, line segments, rays, and angles, and classify shapes by their lines and angles.
4 D05.S3.04.02 Apply formulas for the area and perimeter of rectangles to solve real-world and mathematical problems.
5 D05.S3.05.01 Graph points in the first quadrant of a coordinate grid and use them to solve problems about location and distance.
5 D05.S3.05.02 Find the volume of right rectangular prisms using unit cubes and formulas, and explain what volume measures.
6 D05.S3.06.01 Find area, surface area, and volume of common figures and use nets to build and analyze three-dimensional shapes.
6 D05.S3.06.02 Draw and use polygons on the coordinate plane to solve real problems involving position and length.
7 D05.S3.07.01 Construct and describe triangles and other shapes from given conditions, and tell when such a shape is impossible.
7 D05.S3.07.02 Solve problems with scale drawings and circles, using the relationship between circumference and diameter.
8 D05.S3.08.01 Describe translations, rotations, reflections, and dilations, and use them to show congruence and similarity.
8 D05.S3.08.02 Use the Pythagorean theorem to solve problems about right triangles and distances.
9 D05.S3.09.01 Use rigid motions and coordinates to prove and reason about geometric relationships such as congruence.
9 D05.S3.09.02 Prove and apply theorems about lines, angles, triangles, and parallelograms.
10 D05.S3.10.01 Use trigonometric ratios and the Pythagorean theorem to solve right-triangle problems in real contexts such as surveying or navigation.
10 D05.S3.10.02 Prove and apply theorems about circles and use coordinate geometry to describe and analyze figures.
11 D05.S3.11.01 Use the unit circle and trigonometry of general triangles to model periodic relationships in real phenomena.
11 D05.S3.11.02 Reason about three-dimensional shapes, finding the volume and surface area of composite solids in real contexts.
12 D05.S3.12.01 Use coordinates and vectors to analyze geometric situations and prove results.
12 D05.S3.12.02 Apply geometric reasoning to real-world design, such as architecture, mapping, or the layout of shared spaces, considering access and sustainability.

D05.S4 Data, Measurement & Probability

Measuring, collecting and making sense of data, and reasoning about chance and uncertainty.

37 lessons

K D05.S4.K.01 Sort objects into groups by one feature and tell which group has more.
1 D05.S4.01.01 Collect and organize data into a simple chart or picture graph and answer questions about what it shows.
1 D05.S4.01.02 Tell and write time and identify the value of coins, using them in simple real situations such as buying a snack.
2 D05.S4.02.01 Measure length using standard units such as centimeters, meters, and inches, choosing the right tool for the job.
2 D05.S4.02.02 Draw and interpret picture graphs and bar graphs and answer questions about how many more or less.
3 D05.S4.03.01 Measure and estimate liquid volume and mass using standard units and solve real problems with them.
3 D05.S4.03.02 Draw scaled picture and bar graphs and use them to solve problems and make comparisons.
4 D05.S4.04.01 Convert measurements within a system and solve problems involving distance, time, volume, and money.
4 D05.S4.04.02 Make and interpret line plots with fractional measurements and answer questions about the data.
5 D05.S4.05.01 Convert among different-sized standard measurement units and use the conversions to solve real problems.
5 D05.S4.05.02 Represent and interpret data on line plots with fractions and use the data to solve problems.
6 D05.S4.06.01 Recognize and describe statistical variability, including the center, spread, and shape of a data set.
6 D05.S4.06.02 Summarize a data set with mean, median, mode, and range, and choose the measure that best answers a question.
7 D05.S4.07.01 Use random sampling to draw conclusions about a population and recognize bias in how data is collected.
7 D05.S4.07.02 Investigate chance processes and compute probabilities of simple and compound events.
8 D05.S4.08.01 Construct and interpret scatter plots for two related variables and describe patterns of association.
8 D05.S4.08.02 Fit a line informally to data to model a relationship, evaluate the fit, and explain that association does not prove causation.
9 D05.S4.09.01 Summarize and compare distributions of a single variable using plots and summary statistics.
9 D05.S4.09.02 Use two-way tables and conditional relative frequencies to explore relationships between categories, including questions of fairness.
10 D05.S4.10.01 Understand independence and conditional probability and use them to interpret real situations such as screening tests and risk.
10 D05.S4.10.02 Use probability rules to compute probabilities of compound events and interpret expected value in everyday decisions.
11 D05.S4.11.01 Use the normal distribution and standard deviation to describe data and reason about variability and outliers.
11 D05.S4.11.02 Design and carry out a survey or experiment, and critique the sampling, bias, and conclusions it produces.
12 D05.S4.12.01 Use inferential reasoning, including margin of error, confidence, and significance, to evaluate claims made from data.
12 D05.S4.12.02 Detect and critique misleading statistics, data visualizations, and algorithmic bias in media and public life, and defend a conclusion with honest data.

Mathematics & Reasoning

This domain file is part of the standards tree. It declares the strands and the full vertical ladder of grade-level objectives (K–12) for Mathematics & Reasoning, so every lesson can trace to a standard (see docs/roadmap.md).

Strands

D05.S1 — Number & Quantity

Counting, comparing, and operating with numbers in meaningful contexts.

D05.S2 — Patterns & Relationships

Seeing, making, and generalizing patterns and relationships.

D05.S3 — Geometry & Spatial Reasoning

Shape, space, position, and how objects fit and move.

D05.S4 — Data, Measurement & Probability

Measuring, making sense of data, and reasoning about chance.

Progression (spiral)

Each strand spirals from Kindergarten to Grade 12 with no gaps or leaps. Number & Quantity moves from counting and comparing real objects (K) through place value and the four operations (1–4), fractions, decimals, ratios, and rational and irrational numbers (5–8), to real and complex numbers, precision, and quantitative reasoning about large systems (9–12). Patterns & Relationships grows from copying repeating patterns (K) through growing patterns, simple equations, variables, and functions (1–8) to modeling linear, exponential, periodic, and algorithmic relationships (9–12). Geometry & Spatial Reasoning begins with naming shapes and position words (K), then builds through attributes, area, perimeter, volume, and the coordinate plane (1–6) to transformations, the Pythagorean theorem, trigonometry, and design reasoning (7–12). Data, Measurement & Probability starts with sorting and grouping (K), then measures, graphs, and summarizes data (1–5), and rises through variability, sampling, probability, and inference (6–12) to weighing uncertainty and detecting misleading statistics and algorithmic bias. Every grade deepens the sense-making that began in the sandbox and the circle, never jumping ahead.