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Examples
Jasmine is on a cruise and is going on a scuba diving excursion. Her elevations of 10 feet above sea level and 8 feet below sea level can be compared on a number line, where 0 represents sea level.Clarifications
Clarification 1: Instruction includes vertical and horizontal number lines, context referring to distances, temperatures and finances and using informal verbal comparisons, such as, lower, warmer or more in debt.Clarification 2: Within this benchmark, the expectation is to compare positive and negative rational numbers when given in the same form.
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Integers
- Number Line
- Rational Numbers
- Whole Numbers
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In elementary grades, students plotted, ordered and compared positive numbers. In grade 6, students plot, order and compare on both sides of zero on the number line with all forms of rational numbers. This benchmark focuses on the comparison of quantities with opposite directions (one positive and one negative) and understanding what the zero represents within the provided context. In grade 7, students will use the concept of opposites when solving problems involving order of operations and absolute values.- A strong foundation in this skill can help students more efficiently solve problems involving absolute value, finding the distance between points on a coordinate plane, combining rational numbers as well as assessing the reasonableness of solutions.
- Within this benchmark, it is the expectation that students make informal verbal comparisons of rational numbers using “is greater than” or “is less than” or another comparison within context.
- For example, when comparing the temperature in Chipley, Florida, which is 34°F, to Minneapolis, Minnesota, which is −4°F, one could say that it is warmer in Florida than Minnesota or that it is colder in Minnesota than Florida.
- Students can describe the relative relationships between quantities and include the distance between the values in their description, as this is a way of connecting and reinforcing the MA.6.GR.1 standard (MTR.2.1, MTR.7.1).
- For example, if a shark is 7.5 meters below sea level and a cliff diver is standing on a cliff at a point that is 22 meters from the surface of the water, the shark is 29.5 meters below the cliff diver. The shark’s position can also be described as– 7.5 meters from the surface of the water if the surface of the water represents 0.
Common Misconceptions or Errors
- Students may incorrectly assign double negatives when describing relationships that are below zero.
- For example, they may say the temperature is – 3 degrees below zero when they mean it is – 3 degrees from zero or it is 3 degrees below zero. In this same context, it is important for students to understand that identifying a value as negative indicates the position as being lower or colder than the neutral position of 0.
- Instruction provides many opportunities for students to share verbal descriptions (in written and oral forms) to better develop this skill (MTR.4.1, MTR.7.1).
- Students may experience difficulty when trying to create a number line and determining the meaning of 0 within the context. Instruction showcases drawing a picture of the context first and then connect it to the building of a number line.
Strategies to Support Tiered Instruction
- Instruction includes the use of a physical or digital number line to plot various values and to use their location on the number line to assist in comparing them. Instruction focuses on the generalization that the farther to the left a value is on the number line, the lesser the number.
- Teacher assists students with representing rational values in a real-world context, by first drawing a visual representation of the situation and labeling key features of the picture such as the “neutral” point of the visual (sea level, zero degrees, ground level, etc.), direction (above or below sea level, forward or backwards movement, etc.) and magnitude (the absolute value of the given quantity) of the given situation. Once a visual is drawn, the teacher or students create a corresponding horizontal or vertical number line to match the picture and the key features and then use the number line to write the correct rational number, paying close attention to the location of the point relative to zero.
- For example, Jasmine is on a cruise and is going on a scuba diving excursion. Her elevations of 10 feet above sea level and 8 feet below sea level can be compared on a number line, where 0 represents sea level.
- Teacher provides opportunities for students to represent verbal descriptions of situations pictorially and on a number line. Students then restate the same situation using different vocabulary and check the reasonableness of the new situations by making connections back to the visual representation, number line, and meaning of zero in the context.
- For example, on Saturday,Thomas’s football team lost 3 yards in their first drive while on Thursday, Derrek’s football team gained 5 yards in their first drive. The first drives for each team can be compared on a number line, where 0 represents the line of scrimmage for each play.
- Instruction includes providing students with opportunities to practice placing rational number cards on a provided number line. While placing the cards, students should share their thinking out loud while the teacher listens for reasonable estimations and assumptions for correct placement.
- For example, if a student is placing a number card with −4 on the number line, the student needs to know that is more than half, so to place the card, the student should be between −4 and −5, with the point closer to the −5. As more cards are added to the number line, the students may adjust the placement of other cards, if necessary, for more accurate estimates.
- Teacher and students co-create a list of common terms from contextual situations that may be used to describe positive or negative values.
- Examples of this could include:
| Positive | Negative |
| Deposit Gain Above | Withdrawal Loss Below |
Instructional Tasks
Instructional Task 1 (MTR.4.1, MTR.7.1)Part B. What role does 0 play in this context?
Part C. Using this reasoning, describe how any number x and any number y could be positioned relative to each other on a number line and the role played by 0.
Instructional Items
Instructional Item 1*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Original Student Tutorial
Perspectives Video: Professional/Enthusiast
Perspectives Video: Teaching Idea
Problem-Solving Tasks
Tutorials
Video/Audio/Animation
STEM Lessons - Model Eliciting Activity
Rangoli is a traditional Indian art that is used in decorating the entrance of the house to welcome guests. In this activity students will use the concept of lines of symmetry to select the best rangoli design for a school event.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx
MFAS Formative Assessments
Students are asked to graph -4, 0, and 4 on a number line and to explain the relationship between a number and its opposite in terms of the number line.
Students are asked to interpret values given in a chart that represent positive and negative deviations from average rainfall.
Students are asked to explain the meaning of positive and negative decimals within the context of a problem.
Students are given positive and negative fractions and asked to explain their meanings within the context of a problem.
Students are asked to use numbers to represent gains/losses and to interpret the meaning of zero in the context of football.
Original Student Tutorials Mathematics - Grades 6-8
Follow Matteo as he explores opposite numbers, positive and negative rational numbers, and zero in real-world contexts while planning and going on a cruise in Alaska in this interactive tutorial.
Student Resources
Original Student Tutorial
Follow Matteo as he explores opposite numbers, positive and negative rational numbers, and zero in real-world contexts while planning and going on a cruise in Alaska in this interactive tutorial.
Type: Original Student Tutorial
Problem-Solving Tasks
Students are asked to reason about and explain the position of two locations relative to sea level.
Type: Problem-Solving Task
The purpose of this task is meant to reinforce students' understanding of rational numbers as points on the number line and to provide them with a visual way of understanding that the sum of a number and its additive inverse (usually called its "opposite") is zero.
Type: Problem-Solving Task
The purpose of this task is for students to apply their knowledge of integers in a real-world context.
Type: Problem-Solving Task
Tutorials
This video discusses the negative sign as meaning "opposite."
Type: Tutorial
This video guides you through comparisons of values, including opposites.
Type: Tutorial
This video uses a number line to describe the opposite of a number.
Type: Tutorial
Parent Resources
Problem-Solving Tasks
Students are asked to reason about and explain the position of two locations relative to sea level.
Type: Problem-Solving Task
The purpose of this task is meant to reinforce students' understanding of rational numbers as points on the number line and to provide them with a visual way of understanding that the sum of a number and its additive inverse (usually called its "opposite") is zero.
Type: Problem-Solving Task
The purpose of this task is for students to apply their knowledge of integers in a real-world context.
Type: Problem-Solving Task
