Standard 1: Solve problems involving rational numbers, including numbers in scientific notation, and extend the understanding of rational numbers to irrational numbers.

Students are asked to define a rational number and then explain why the sum of two rational numbers is rational.

Type: Formative Assessment

Students are asked to describe the difference between rational and irrational numbers and then explain why the sum of a rational and an irrational number is irrational.

Type: Formative Assessment

Students are asked to describe the difference between rational and irrational numbers, and then explain why the product of a non-zero rational and an irrational number is irrational.

Type: Formative Assessment

Students are given numerical expressions and asked to use properties of integer exponents to find equivalent expressions.

Type: Formative Assessment

Students are asked to complete a table of powers of three and provide an explanation of zero powers.

Type: Formative Assessment

Students are asked to evaluate perfect square roots and perfect cube roots.

Type: Formative Assessment

Students are asked to add and subtract numbers given in scientific notation in real-world contexts.

Type: Formative Assessment

Students are asked to multiply and divide numbers given in scientific notation in real-world contexts.

Type: Formative Assessment

Students are given word problems with numbers in both standard and scientific notation and asked to solve problems using various operations.

Type: Formative Assessment

Students are asked to apply the properties of integer exponents to generate equivalent numerical expressions.

Type: Formative Assessment

Students are asked to solve problems involving square roots and cube roots.

Type: Formative Assessment

Students are asked to estimate an extremely large and an extremely small number by writing it in the form a x .

Type: Formative Assessment

Students are asked to identify rational numbers from a list of real numbers, explain how to identify rational numbers, and to identify the number system that contains numbers that are not rational.

Type: Formative Assessment

Students are given pairs of numbers written in the form of an integer times a power of 10 and are asked to compare the numbers in each pair using the inequality symbols.

Type: Formative Assessment

Students are given pairs of numbers written in exponential form and are asked to compare them multiplicatively.

Type: Formative Assessment

Students are given pairs of numbers written in scientific notation and are asked to compare them multiplicatively.

Type: Formative Assessment

Students are given several terminating and repeating decimals and asked to convert them to fractions.

Type: Formative Assessment

Students are given expressions with negative exponents and are asked to identify those that are equivalent from given sets of expressions.

Type: Formative Assessment

Students are given examples of calculator displays and asked to convert the notation in the display to both scientific notation and standard form.

Type: Formative Assessment

Students are asked to find and interpret lower and upper bounds of an irrational expression using a calculator.

Type: Formative Assessment

Students are asked to plot the square root of eight on three number lines, scaled to progressively more precision.

Type: Formative Assessment

Students are asked to graph three different irrational numbers on number lines.

Type: Formative Assessment

Students are asked to estimate the value of several irrational numbers using a calculator and order them on a number line.

Type: Formative Assessment

Students will apply their knowledge of mathematical operations to provide inputs in a Scratch program to try and make a specific given output, in this lesson plan.

Type: Lesson Plan

In this lesson students will categorize a list of stars based on absolute brightness, size, and temperature. Students will analyze astronomical data presented in charts and plot their data on a special graph called a Hertzsprung-Russell Diagram (H-R Diagram). Using this diagram, they must determine the proper classification of individual stars. Using their data analysis, students completing this lesson will develop two short essay responses to a professional client indicating which stars are Main Sequence Stars and which ones are White Dwarfs, Giants, or Supergiants.

Type: Lesson Plan

In this discovery oriented lesson, students will explore the use of non-standard units of measurement. They will convert linear measurements within the metric system and convert measurements given in astronomical units (AU) into more familiar units, specifically meters and kilometers. The unit conversions will be completed with measurements that are expressed in scientific notation. Students will recall their prior knowledge of how to add and subtract numbers given in scientific notation. They will also use their knowledge of exponent rules to determine an efficient method for multiplying and dividing numbers expressed in scientific notation.

Type: Lesson Plan

Students will be introduced to a derivation of the Law of Sines and apply the Law of Sines to solve triangles.

Type: Lesson Plan

Use as a follow up lesson to solving systems of equations graphically. Students will explore graphs of systems to see how manipulating the equations affects the solutions (if at all).

Type: Lesson Plan

In this lesson, students model the orbit of a satellite and the trajectory of a missile with a system of equations. They solve the equations both graphically and algebraically.

Type: Lesson Plan

In this lesson, students determine when to use the Law of Cosine and/or the Law of Sine to calculate the measures of sides and angles of non-right triangles.

Type: Lesson Plan

This lesson serves as an introductory lesson on the Pythagorean Theorem and its converse. It has a hands-on discovery component. This lesson includes worksheets that are practical for individual or cooperative learning strategies. The worksheets contain prior knowledge exercises, practice exercises and a summative assignment.

Type: Lesson Plan

For a given circle m and point A in a coordinate plane, students will be able to show whether this given point A lies on the circumference of the given circle m using the Pythagorean Theorem. Subsequently, this can be used to prove that an airplane lies within or outside the radar's range with a given radius of detection.

Type: Lesson Plan

Students will explore the relationship between vertical angles and prove the Vertical Angle Theorem. They will use vertical angle relationships to calculate other angle measurements.

Type: Lesson Plan

Students will use similar triangles to determine inaccessible measurements. Examples include exploring dangerous caves and discovering craters on Mars.

Type: Lesson Plan

Students will explore triangle inequalities that exist between side lengths with physical models of segments. They will determine when a triangle can/cannot be created with given side lengths and a range of lengths that can create a triangle.

Type: Lesson Plan

Students will algebraically find the missing coordinates to create a specified quadrilateral using theorems that represent them, and then algebraically prove their coordinates are correct.

Note: This is not an introductory lesson for this standard.

Type: Lesson Plan

Students will prove that alternate interior angles and corresponding angles are congruent given two parallel lines and a traversal. Students will use GeoGebra to explore real-world images to prove their line segments are parallel.

Type: Lesson Plan

This lesson introduces the law of sine. It is designed to give students practice in using the law to guide understanding. The summative assessment requires students to use the law of sine to plan a city project.

Type: Lesson Plan

Students will compare the cost of pre-made solar car kits to cars made from a 3-D printer. In the second part of the activity, students will research other available 3-D printers and determine what attributes are important to consider. There is also an optional solar panel car race for day 3.

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. Click here to learn more about MEAs and how they can transform your classroom.

Type: Lesson Plan

Students will use number lines to approximate the square root value of non-perfect square numbers to the tenth place. This lesson supports plotting, comparing, and ordering irrational numbers as well as graphing them on a number line, specifically those in the form of nonperfect square roots.

Type: Lesson Plan

Students explore numerical examples involving multiplying exponential terms that have the same base. They generalize the property of exponents where, when multiplying terms with the same base, the base stays the same and the exponents are added together.

Type: Lesson Plan

Students will participate in a gallery walk in which they observe patterns in algebraic expressions. Students will apply the properties of integer exponents to simplify expressions.

Type: Lesson Plan

Students will use their knowledge of perfect squares and square roots to determine a rational number to approximate an irrational number and find their locations on a number line. They will complete an activity that guides them to zoom further into a number line to find more accurate approximations for irrational numbers. They will conclude that between two rational numbers is another rational number and therefore the further the place value in the approximation, the more accurate the location on the number line.

Type: Lesson Plan

In this lesson, students will learn how irrational numbers differ from rational numbers. The students will complete a graphic organizer that categorizes rational and irrational numbers. Students will also be able to identify irrational numbers found in the real world.

Type: Lesson Plan

This low-tech lesson will have students stand up holding different exponent cards. This will help them write and justify an equivalent expression and see the pattern for expressions with the same base and descending exponents. What happens as you change from 2 to the fourth power to 2 to the third power; 2 to the second power; and so forth? This is an introductory lesson to two of the properties of exponents: and

Type: Lesson Plan

Get your students up and moving and interested in simplifying expressions with whole integer powers. After getting your students to figure out what it takes to multiply and divide powers with whole number exponents, have your students scurry about the room to find the questions and answers for scavenger hunt exercise. The lesson includes questions and answers for the hunt, directions for the hunt, printable cards for the hunt, and step by step directions on how to get your students to figure out what they need to do when multiplying and dividing powers with whole number exponents.

Type: Lesson Plan

The students will informally learn the rules for exponents: product of powers, powers of powers, zero and negative exponents. The activities provide the teacher with a progression of steps that help lead students to determine results without knowing the rules formally. The closing activity is hands-on to help reinforce all rules.

Type: Lesson Plan

Students use hands-on materials to understand that only certain combinations of lengths will create closed triangles.

Type: Lesson Plan

Students will organize the set of real numbers and be able to identify when a number is rational or irrational. They will also learn the process of how to change a repeating decimal to its equivalent fraction.

Type: Lesson Plan

In this lesson, students will complete proof tables to justify the steps taken to solve multi-step equations. Justifications include mathematical properties and definitions..

Type: Lesson Plan

Students must decide the destination of a multi-billion dollar space flight to an unexplored world. The location must be selected based on its potential for valuable research opportunities. Some locations may have life, while others could hold the answers to global warming or our energy crisis. Students must choose the destination that they feel will be most helpful to human-kind.

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 process. 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 MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

Type: Lesson Plan

Students will practice and create problems solving linear equations that involve one solution, no solution, infinitely many solutions. There will be class discussion so students can verbalize their thoughts. In addition, students will create their own real-world problems that can be used for the next day’s extension exercise.

Type: Lesson Plan

Students will learn to approximate non-perfect square roots as rational numbers. Understanding that irrational numbers can be approximated by rational numbers can assist students and their understanding of the real number system.

Type: Lesson Plan

Using different activities, students will find real life uses for the Pythagorean Theorem.

Type: Lesson Plan

In this lesson students will learn the properties of integer exponents and how to apply them to multiplication and division. Students will have the opportunity to work with concrete manipulatives to create an understanding of these properties and then apply them abstractly. The students will also develop the understanding of the value of any integer with a zero exponent.

Type: Lesson Plan

This lesson allows students to explore and estimate the values of imperfect squares, using perfect square anchors and number lines as resources. The conversations throughout the lesson will also emphasize that imperfect squares are irrational numbers that must be estimated to compare.

Type: Lesson Plan

In this lesson students will gain an understanding of how repeating decimals are converted into a ratio in the form of by completing an exploration worksheet. They will conclude that any number which can be written in this form is called a rational number.

Type: Lesson Plan

Use scientific notation to compare the distances of planets and other objects from the Sun in this interactive tutorial.

Type: Original Student Tutorial

Use astronomical units to compare distances betweeen objects in our solar system in this interactive tutorial.

Type: Original Student Tutorial

Learn how to simplify radicals in this interactive tutorial.

Type: Original Student Tutorial

Explore how to express large quantities using scientific notation in this interactive tutorial.

Type: Original Student Tutorial

Apply the Pythagorean Theorem to solve mathematical and real-rorld problems in this interactive tutorial.

Type: Original Student Tutorial

How are fluency and automaticity defined? Dr. Lawrence Gray explains fluency and automaticity in the B.E.S.T. mathematics benchmarks in this Expert Perspectives video.

Type: Perspectives Video: Expert

What roles do exploration, procedural reliability, automaticity, and procedural fluency play in a student's journey through the B.E.S.T. benchmarks? Dr. Lawrence Gray explains the path through the B.E.S.T. mathematics benchmarks in this Expert Perspectives video.

Type: Perspectives Video: Expert

What is fluency? What are the ingredients required to become procedurally fluent in mathematics? Dr. Lawrence Gray explores what it means for students to be fluent in mathematics in this Expert Perspectives video.

Type: Perspectives Video: Expert

Why is it important to look beyond whether a student gets the right answer? Dr. Lawrence Gray explores the importance of understanding why we perform certain steps or what those steps mean, and the impact this understanding can have on our ability to solve more complex problems and address them in the context of real life in this Expert Perspectives video.

Type: Perspectives Video: Expert

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

Type: Problem-Solving Task

In this problem students are comparing a very small quantity with a very large quantity using the metric system. The metric system is especially convenient when comparing measurements using scientific notations since different units within the system are related by powers of ten.

Type: Problem-Solving Task

This task requires students to work with very large and small values expressed both in scientific notation and in decimal notation (standard form). In addition, students need to convert units of mass. The solution below converts the mass of humans into grams; however, we could just as easily converted the mass of ants into kilograms. Students are unable to go directly to a calculator without taking into account all of the considerations mentioned above. Even after converting units and decimals to scientific notation, students should be encouraged to use the structure of scientific notation to regroup the products by extending the properties of operations and then use the properties of exponents to more fluently perform the calculations involved rather than rely heavily on a calculator.

Type: Problem-Solving Task

This task is intended for instructional purposes so that students can become familiar and confident with using a calculator and understanding what it can and cannot do. This task gives an opportunity to work on the notion of place value (in parts [b] and [c]) and also to understand part of an argument for why the square root of 2 is not a rational number.

Type: Problem-Solving Task

The task assumes that students can express a given repeating decimal as a fraction. Teachers looking for a task to fill in this background knowledge could consider the related task "Converting Decimal Representations of Rational Numbers to Fraction Representations".

Type: Problem-Solving Task

When students plot irrational numbers on the number line, it helps reinforce the idea that they fit into a number system that includes the more familiar integer and rational numbers. This is a good time for teachers to start using the term "real number line" to emphasize the fact that the number system represented by the number line is the real numbers. When students begin to study complex numbers in high school, they will encounter numbers that are not on the real number line (and are, in fact, on a "number plane"). This task could be used for assessment, or if elaborated a bit, could be used in an instructional setting.

Type: Problem-Solving Task

In this task, students explore some important mathematical implications of using a calculator. Specifically, they experiment with the approximation of common irrational numbers such as pi (p) and the square root of 2 (v2) and discover how to properly use the calculator for best accuracy. Other related activities involve converting fractions to decimal form and a concrete example where rounding and then multiplying does not yield the same answer as multiplying and then rounding.

Type: Problem-Solving Task

Students are given a pair of numbers. They are asked to determine which is larger, and then justify their answer. The numbers involved are rational numbers and common irrational numbers, such p and square roots. This task can be used to either build or assess initial understandings related to rational approximations of irrational numbers.

Type: Problem-Solving Task

The student is asked to perform operations with numbers expressed in scientific notation to decide whether 7% of Americans really do eat at Giantburger every day.

Type: Problem-Solving Task

This is an instructional task meant to generate a conversation around the meaning of negative integer exponents. It is good for students to learn the convention that negative time is simply any time before t=0.

Type: Problem-Solving Task

Exponentiation can be understood in terms of repeated multiplication much like multiplication can be understood in terms of repeated addition. Properties of multiplication and division of exponential expressions with the same base are derived.

Type: Video/Audio/Animation