This resource links to both the Fractions progression document published by the Common Core Writing Teams in June 2011 and the Module posted on the Illustrative Mathematics website.

Students learn about the Foucault pendulum an engineering tool used to demonstrate and measure the Earth's rotation. Student groups create small experimental versions, each comprised of a pendulum and a video camera mounted on a rotating platform actuated by a LEGO MINDSTORMS(TM) NXT motor. When the platform is fixed, the pendulum motion forms a line, as observed in the recorded video. When the rotating, the pendulum's motion is observed as a set of spirals with a common center. Observing the patterns that the pendulum bob makes when the platform is rotating provides insight as to how a full-size Foucault pendulum operates. It helps students understand some of the physical phenomena induced by the Earth's rotation, as well as the tricky concept of how the perception of movement varies, depending on one's frame of reference.

This lesson will be the first of six lessons guiding students in constructing the abstract concepts necessary to understand adding positive and negative integers.
It is designed for adult learners (or middle school or older) who are not fluent with using numbers.

Rational Numbers

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Solve problems with positive rational numbers.
Plot positive rational numbers on a number line.
Understand the equal sign.
Use the greater than and less than symbols with positive numbers (not variables) and understand their relative positions on a number line.
Recognize the first quadrant of the coordinate plane.

Lesson Flow

The first part of this unit builds on the prerequisite skills needed to develop the concept of negative numbers, the opposites of numbers, and absolute value. The unit starts with a real-world application that uses negative numbers so that students understand the need for them. The unit then introduces the idea of the opposite of a number and its absolute value and compares the difference in the definitions. The number line and positions of numbers on the number line is at the heart of the unit, including comparing positions with less than or greater than symbols.

The second part of the unit deals with the coordinate plane and extends student knowledge to all four quadrants. Students graph geometric figures on the coordinate plane and do initial calculations of distances that are a straight line. Students conclude the unit by investigating the reflections of figures across the x- and y-axes on the coordinate plane.

Students revise their work on the Self Check based on feedback from the teacher and their peers.Key ConceptsConcepts from previous lessons are integrated into this assessment task: integers, absolute value, and comparing numbers. Students apply their knowledge, review their work, and make revisions based on feedback from the teacher and their peers. This process creates a deeper understanding of the concepts.Goals and Learning ObjectivesApply your knowledge of integers, absolute value, and comparing numbers to solve problems.Track and review your choice of strategy when problem solving.

Students draw a figure on the coordinate plane that matches a written description.Key ConceptsOrdered pairs name locations on the coordinate plane. The first coordinate tells how many units to go left or right of the origin (0,0) along the x-axis. The second coordinate tells how many units to go up or down from the origin along the y-axis.Goals and Learning ObjectivesDraw a figure that matches a description of a figure on the coordinate plane.Give coordinates of points on the coordinate plane.Write descriptions of figures on the coordinate plane.

Students identify whether an inequality statement is true or false using a number line to support their reasoning.Key ConceptsThe meaning of mThe meaning of n > m is that n is located to the right of m on a number line. The inequality statement n > m is read “n is greater than m.”To decide on the order of two numbers m and n, locate the numbers on a number line. If m is to the left of n, then m < n. If m is to the right of n, then m > n.Goals and Learning ObjectivesState whether an inequality is true or false.Use a number line to prove that an inequality is true or false.

Students watch a dot get tossed from one number on a number line to the opposite of the number. Students predict where the dot will land each time based on its starting location.Key ConceptsThe opposite of a number is the same distance from 0 as the number itself, but on the other side of 0 on a number line.In the diagram, m is the opposite of n, and n is the opposite of m. The distance from m to 0 is d, and the distance from n to 0 is d; this distance to 0 is the same for both n and m. The absolute value of a number is its distance from 0 on a number line.Positive numbers are numbers that are greater than 0.Negative numbers are numbers that are less than 0.The opposite of a positive number is negative, and the opposite of a negative number is positive.Since the opposite of 0 is 0 (which is neither positive nor negative), then 0 = 0. The number 0 is the only number that is its own opposite.Whole numbers and the opposites of those numbers are all integers.Rational numbers are numbers that can be expressed as ab, where a and b are integers and b ≠ 0.Goals and Learning ObjectivesIdentify a number and its oppositeLocate the opposite of a number on a number lineDefine the opposite of a number, negative numbers, rational numbers, and integers

Students analyze whether given statements are possible or impossible using their definitions of absolute value and the opposite of a number. If the statements are possible, students give an example of a pair of numbers that fit the statement. If the statements are impossible, students explain why.Key ConceptsA number and the opposite of the number always have the same absolute value.In general, taking the opposite of n changes the sign of n. For example, the opposite of 3 is −3.In general, taking the absolute value of n gives a number |n|, which is always positive. For example, |3| = 3 and |−3| = 3.Since the opposite of 0 is 0 (which is neither positive nor negative), therefore −0 = 0. The number 0 is the only number which is its own opposite.Goals and Learning ObjectivesFind pairs of numbers that satisfy different statements about absolute values and/or the opposites of numbers.State when it is impossible to find a pair of numbers that satisfies the statement and explain why.

Students reflect a figure across one of the axes on the coordinate plane and name the vertices of the reflection. As they are working, students look for and make use of structure to identify a convention for naming the coordinates of the reflected figure.Key ConceptsWhen point (m,n) is reflected across the y-axis, the reflected point is (−m,n).When point (m,n) is reflected across the x-axis, the reflected point is (m,−n).When point (m,n) is reflected across the origin (0,0), the reflected point is (−m,−n).Goals and Learning ObjectivesReflect a figure across one of the axes on the coordinate plane.Name the vertices of the reflected figure.Discern a pattern in the coordinates of the reflected figure.

Students answer questions about low temperatures recorded in Barrow, Alaska, to understand when to use negative numbers and when to use the absolute values of numbers.Key ConceptsThe absolute value of a number is its distance from 0 on a number line.The absolute value of a number n is written |n| and is read as “the absolute value of n.”A number and the opposite of the number always have the same absolute value. As shown in the diagram, |3| = 3 and |−3| = 3.In general, taking the opposite of n changes the sign of n. For example, the opposite of 3 is –3.In general, taking the absolute value of n gives a number, |n|, that is always positive unless n = 0. For example, |3| = 3 and |−3| = 3.The absolute value of 0 is 0, which is neither positive nor negative: |0| = 0.Goals and Learning ObjectivesUnderstand when to talk about a number as negative and when to talk about the absolute value of a number.Locate the absolute value of a and the absolute value of b on a number line that shows the location of a and b in different places in relation to 0.

Whole numbers are no better than any others! Practice plotting values on the number line as a passionate activist rises up and demands equity for all numbers, including fractions and decimals.

This supplemental resource provides problems and activities related to Numerical and Algebraic Operations & Analytical Thinking in Middle School Mathematics.

The intent of clarifying statements is to provide additional guidance for educators to communicate the intent of the standard to support the future development of curricular resources and assessments aligned to the 2021 math standards.  Clarifying statements can be in the form of succinct sentences or paragraphs that attend to one of four types of clarifications: (1) Student Experiences; (2) Examples; (3) Boundaries; and (4) Connection to Math Practices.

Navigate the number line while diving amidst shipwrecks and sunken ruins. Will you find a pearl, or an old boot? Watch out for the electric eel! Pearl Diver teaches properties of numbers, how to plot numbers, how to visualize quantity on the number line, how to order numbers, and how to use the number line as a visual model for mathematical operations.

Students explore the concepts of positive and negative numbers through Newton's Third Law and rocket thrust, and other examples.