In this final section, students have the opportunity to apply their thinking from throughout the unit. The lesson in this section is optional because it offers additional opportunities to practice standards that are not a focus of the grade. As this is a short section followed by an End-of-Unit Assessment, there are no section goals or checkpoint questions.

Work in this section introduces students to writing inequalities and representing solutions to inequalities on a number line. Students must consider reasonable solutions for situations in context. For each situation, they consider possible minimum and maximum values, whether boundary values should be included, whether negative numbers make sense, and whether solutions are discrete or continuous. They compare the different ways each of these are represented on a number line: open or closed circles and shading to the right or left of a number. Students find solutions to an inequality by determining if given values make the inequality true and observe that a value is a solution to an inequality if it lies within the shaded region on a number line.

Work in this section expands students’ understanding of the number system to include both positive and negative numbers. Students begin by examining situations involving temperature and elevation to understand the need for numbers less than 0. They work with both vertical and horizontal number lines to represent positive and negative values, observing the symmetry present in the number line. Students use the number line to compare values and practice ordering rational numbers from least to greatest. They recognize that points to the right on a horizontal number line (or farther up on a vertical number line) have a greater value than points to the left (or farther down). They use the inequality symbols < and > to write comparison statements.

Next, students examine more situations that can be represented using both positive and negative numbers, such as money and inventory. The end of this section introduces students to the idea of absolute value, which is the distance of a number from zero. Students return to working with temperature and elevation. They differentiate between objects’ elevations and their distances from sea level

In this section, students explore ways to represent pairs of numbers that include negative values. They do so by extending the horizontal axis to the left of and the vertical axis below and plotting ordered pairs in all four quadrants of the coordinate plane. They reason about how to appropriately scale and label a set of axes in order to most effectively display a given set of coordinate pairs. Students consider situations that include points plotted in all four quadrants and interpret those points in the context of the situation.

Students observe patterns between the location of a point and the signs of the numbers in its coordinates. They notice that when the numbers in two ordered pairs differ only by signs, those two points are located on opposite sides of the - or -axis. Students also develop strategies for determining the distance between two points on the same horizontal or vertical line. While use of the distance formula or arithmetic with negative numbers is not expected in this course, students may use strategies such as counting grid squares or calculating distances from an axis and adding them together.

In this section, students expand on their knowledge of factors and multiples and use the greatest common factor and the least common multiple of two whole numbers to solve problems. Students begin by finding factors of numbers in the context of splitting items up into groups. They relate the greatest common factor as the largest number of groups that can be created where each group is identical and every item belongs to one and only one group. 

• Factors of 18: 1, 2, 3, 6, 9, 18
• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Students also find multiples of numbers in the context of repeating events or purchasing items that come in certain-sized groups. They relate the least common multiple as the first instance of two events occurring at the same time or as the smallest number of items that can be collected where there are equal numbers of both items. 

• Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, . . .
• Multiples of 10: 10, 20, 30, 40, 50, . . .

Lastly, students investigate multiple situations and discern whether the problem involves common multiples or common factors before solving.