This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

Students are introduced to some basic civil engineering concepts in an exciting and interactive manner. Bridges and skyscrapers, the two most visible structures designed by civil engineers, are discussed in depth, including the design principles behind them. To help students visualize in three dimensions, one hands-on activity presents three-dimensional coordinate systems and gives students practice finding and describing points in space. After learning about skyscrapers, tower design principles and how materials absorb different types of forces, students compete to build their own newspaper towers to meet specific design criteria.The unit concludes with student groups using balsa wood and glue to design and build tower structures to withstand vertical and lateral forces.

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.

Have you ever wondered why it takes such a long period of time for NASA to build space exploration equipment? What is involved in manufacturing and building a rover for the Red Planet? During this lesson, students will discover the journey that a Mars rover embarks upon after being designed by engineers and before being prepared for launch. Students will investigate the fabrication techniques, tolerance concepts, assembly and field-testing associated with a Mars exploratory rover.

In this video from Cyberchase, Harry must score a total of 50 points in various events to qualify for the Northern Hemisphere games in New Orleans.

Game Over Gopher is an exciting tower defense game that guides students in plotting coordinate pairs, differentiating negative coordinates from positive coordinates, and identifying the four quadrants. Hungry space gophers are marching towards a prize carrot, and to defend it players place tools around the coordinate grid to “feed” gophers and make them lose interest. Ruby mines (which must be placed at designated x, y coordinates) yield currency that players spend to strategically place carrot launchers, garlic rays, corn silos, and beet traps – gopher feeding tools. To introduce students to the coordinate grid, Game Over Gopher introduces new skills via interactive tutorials and uses consistent visual clues (e.g., x red, y blue) to guide players in plotting coordinates. This means it’s not necessary to teach students coordinate plotting ahead of time. As the levels progress, the number of tools available increases, the level of math vocabulary increases, the scale of the grid changes, and players are asked to expand their mastery of the grid by reflecting points across axes. The game lowers intimidation about the coordinate grid, helps students understand how positive and negative numbers reflect each other across the axes, and helps students get comfortable with the four quadrants.

In this Cyberchase video segment, the CyberSquad locates the Cyberchase Council by using negative 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 play a game in which they try to find dinosaur bones in an archaeological dig simulator. The players guess where the bones are on the coordinate plane using hints and reasoning.Key ConceptsThe coordinate plane consists of a horizontal number line and a vertical number line that intersect at right angles. The point of intersection is the origin, or (0,0). The horizontal number line is often called the x-axis. The vertical number line is often called the y-axis.A point’s location on the coordinate plane can be described using words or numbers. Ordered pairs name locations on the coordinate plane. To find the location of the ordered pair (m,n), first locate m on the x-axis and draw a vertical line through this point. Then locate n on the y-axis and draw a horizontal line through this point. The intersection of these lines is the location of (m,n).The coordinate plane is divided into four quadrants:Quadrant I: (+,+)Quadrant II: (−,+)Quadrant III: (−,−)Quadrant IV: (+,−)Goals and Learning ObjectivesName locations on the coordinate plane.

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 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.

The purpose of this lesson is to teach students about the three dimensional Cartesian coordinate system. It is important for structural engineers to be confident graphing in 3D in order to be able to describe locations in space to fellow engineers.

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