Build a deeper understanding of mathematical concepts that are relevant across different grades.
See the relationships between topics, making math more meaningful and applicable.
Develop problem-solving skills in their early years that provide them with strategies to tackle more complex problems later.
In module 5, lesson 8 in TK, students explore different ways to represent addition by using tools (e.g., objects, drawings, and fingers). When students have various tools available to directly model the action, they can confidently approach more challenging story problems and eventually use more sophisticated counting strategies. 
Students explore different ways to represent subtraction by using concrete objects and drawings. Most kindergarten students will use a counting-all strategy. This strategy may involve up to three steps: counting the total, counting the part that is removed, and counting the part that remains. Certain math tools, such as fingers, 5-groups, or the rekenrek, may nudge students toward subitizing or using the structure of 5 to make counting more efficient and accurate.
Students represent and solve all of the K–2 problem types. They make sense of word problems by drawing and labeling a tape diagram. Drawing and analyzing the resulting tape diagram clarifies the relationships in the problem. This helps students identify the meaning of the unknown and write an addition or subtraction equation to solve.
Students use part–whole thinking and the relationship between addition and subtraction to solve take from with start unknown and add to with start unknown word problems. Start unknown problems are new for grade 2 students and present a greater cognitive challenge because it is more difficult to begin with an unknown, such as ___ – 25 = 20. These problems use only numbers within 100 so that students can focus on the relationships presented. As students read carefully and draw what they know, they recognize the value in using a tape diagram to represent problems.
Students iterate unit fractions to create non-unit fractions concretely, pictorially, and numerically in unit form and fraction form. They compose and decompose wholes with unit and non-unit fractions by using number bonds and tape diagrams. After establishing that the fractions refer to the same whole, students compare unit fractions and fractions with the same numerator concretely and pictorially by reasoning about the size of the parts.
Within the familiar context of money, students use decimal points to record amounts of money as decimal numbers for the first time. Students see that numbers can be represented in different ways. They use tape diagrams, number lines, and area models to represent the fractional unit of tenths.
Students begin by using arrays to find fractions of a set and then use number lines and tape diagrams to find fractions of sets. Students continue to use number lines and tape diagrams to find products of fractions and whole numbers. When students begin multiplying a whole number by a fraction greater than 1, they rely on tape diagrams because tape diagrams can be more efficient for finding larger products.
This topic introduces students to ratios and ratio notation. Students use tape diagrams to model ratios and solve problems. They explore different ways to group and compare objects to develop an understanding of equivalent ratios by the end of the topic.
Students use familiar and new angle relationships to write and solve equations that help determine unknown angle measures. Students continue to use properties of operations and visual models to solve equations. They are introduced to a new strategy for solving equations: if–then moves.
Students use their knowledge of tape diagrams to provide a visual support to solving systems of equations. Students use the substitution method to solve systems of equations that gradually increase in difficulty. They are first exposed to systems in which each equation shares the same expression isolated on one side, thus allowing for very simple cases of substitution. Then students move to solving more complex systems of equations that do not have the same expression on one side of the equation. 
In grades 6, 7, and 8, students solve one-variable equations and inequalities of increasing complexity. Topic B builds on that foundation by formalizing the properties of equality and inequality. When solving equations, students justify each step of the solution path. Their work with equations is further extended to include equations with coefficients represented by letters and the concept of rearranging a formula to highlight a particular quantity. Students solve multi-step linear inequalities, write the solution sets using set-builder notation, and graph the solution sets on the number line. They are introduced to solving problems in real-world and mathematical contexts by creating equations and inequalities in one variable, helping them understand how equations and inequalities can represent the constraints of a given situation.
