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Math, Grade 7, Working With Rational Numbers, Understanding Rational Numbers

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Lesson OverviewStudents learn the definition of rational number, and they write rational numbers as ratios of integers and as repeating or terminating decimals.Key ConceptsStudents have been working with rational numbers throughout this unit, but the term rational number is formally defined in this lesson. A rational number is a number that can be written in the form pq, where p and q are integers. All the integers, fractions, decimals, and percents students have worked with so far in their math classes are rational numbers. Following are some rational numbers written as ratios of integers:36=361−1.2=−12105%=5100 −12=−12Any rational number can also be written as a decimal that terminates or that repeats forever in a regular pattern. For example, 35 = 0.6 and 711 = 0.63636363… Repeating decimals are often written with a bar over the digits that repeat. For example, 0.63636363… can be written as 0.63¯.There are numbers that are irrational. These numbers include π and the square root of any whole number that is not a perfect square, such as 2. The decimal form of an irrational number does not terminate, and the digits do not follow a repeating pattern. Students will study irrational numbers in Grade 8.Goals and Learning ObjectivesUnderstand the definition of rational number.Write rational numbers as ratios of integers.Write rational numbers as terminating or repeating decimals.SWD: Students with disabilities may have difficulty working with decimals and fractions, especially moving between the two. If students demonstrate difficulty to the point of frustration, provide direct instruction on the basics for finding equivalent fractions and decimals.ELL: Target and model key language and vocabulary. Specifically, focus on the term rational, as well as terms such as terminate. As you’re discussing the key points, write the words on the board or on large sheets of paper and explain/demonstrate what the words mean. Since these are important points that students will be using throughout the module, write them on large poster board so that students can use them as a reference. Have students record new terms, definitions, and examples in their Notebook. 

Material Type: Lesson Plan

Math, Grade 7, Algebraic Reasoning, Matching Equations To Problems

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Students match equations such as 3x − 50 = 90 and 3(x − 50) = 90 to real-world and mathematical situations. They identify the steps needed to solve these equations.Key ConceptsStudents solve equations such as 3x − 50 = 90 by using first the addition property and then the multiplication property of equality.Students also solve equations such as 3(x − 50) = 90. Equations with parentheses were introduced in the Challenge Problem of Lesson 6. Now, in this lesson, students use two methods to solve the equation. First method: use the multiplication property of equality and then the addition property of equality; second method: use the distributive property to eliminate the parentheses, then use the addition property of equality, and then the multiplication property of equality.Goals and Learning ObjectivesMatch equations to problems.Solve two-step equations.

Material Type: Lesson Plan

Solving Linear Equations in One Variable

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This lesson unit is intended to help teachers assess how well students are able to: solve linear equations in one variable with rational number coefficients; collect like terms; expand expressions using the distributive property; and categorize linear equations in one variable as having one, none, or infinitely many solutions. It also aims to encourage discussion on some common misconceptions about algebra.

Material Type: Assessment, Lesson Plan

Classifying Solutions to Systems of Equations

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This lesson unit is intended to help teachers assess how well students are able to classify solutions to a pair of linear equations by considering their graphical representations. In particular, this unit aims to help teachers identify and assist students who have difficulties in: using substitution to complete a table of values for a linear equation; identifying a linear equation from a given table of values; and graphing and solving linear equations.

Material Type: Assessment, Lesson Plan

Introduction to Linear Functions

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This task lets students explore the differences between linear and non-linear functions. By contrasting the two, it reinforces properties of linear functions. The task lends itself to an extended discussion comparing the differences that students have found and relating them back to the equation and the graph of the two functions.

Material Type: Activity/Lab

Author: Illustrative Mathematics

The Pythagorean Theorem: Square Areas

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This lesson unit is intended to help teachers assess how well students are able to: use the area of right triangles to deduce the areas of other shapes; use dissection methods for finding areas; organize an investigation systematically and collect data; deduce a generalizable method for finding lengths and areas (The Pythagorean Theorem.)

Material Type: Assessment, Lesson Plan

Equivalent fractions approach to non-repeating decimals

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The purpose of the task is to get students to reflect on the definition of decimals as fractions (or sums of fractions), at a time when they are seeing them primarily as an extension of the base-ten number system and may have lost contact with the basic fraction meaning. Students also have their understanding of equivalent fractions and factors reinforced.

Material Type: Activity/Lab

Author: Illustrative Mathematics

8.NS Placing a square root on the number line

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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 asects of the task and its potential use. Here are the first few lines of the commentary for this task: Place $\sqrt{28}$ on a number line, accurate to one decimal point....

Material Type: Activity/Lab

Author: Illustrative Mathematics

8.NS Estimating Square Roots

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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 asects of the task and its potential use. Here are the first few lines of the commentary for this task: Without using the square root button on your calculator, estimate $\sqrt{800}$ as accurately as possible to $2$ decimal places. (Hint: It is worth noti...

Material Type: Activity/Lab

Author: Illustrative Mathematics

Repeating Decimals

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This lesson unit is intended to help teachers assess how well students are able to: translate between decimal and fraction notation, particularly when the decimals are repeating; create and solve simple linear equations to find the fractional equivalent of a repeating decimal; and understand the effect of multiplying a decimal by a power of 10.

Material Type: Assessment, Lesson Plan

Grade 7 ELA Module 1

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In this 8 eight-week module, students explore the experiences of people of Southern Sudan during and after the Second Sudanese Civil War. They build proficiency in using textual evidence to support ideas in their writing, both in shorter responses and in an extended essay. In Unit 1, students begin the novel A Long Walk to Water (720L) by Linda Sue Park. Students will read closely to practice citing evidence and drawing inferences from this compelling text as they begin to analyze and contrast the points of view of the two central characters, Salva and Nya. They also will read informational text to gather evidence on the perspectives of the Dinka and Nuer tribes of Southern Sudan. In Unit 2, students will read the remainder of the novel, focusing on the commonalities between Salva and Nya in relation to the novel’s theme: how individuals survive in challenging environments. (The main characters’ journeys are fraught with challenges imposed by the environment, including the lack of safe drinking water, threats posed by animals, and the constant scarcity of food. They are also challenged by political and social environments.). As in Unit 1, students will read this literature closely alongside complex informational texts (focusing on background on Sudan and factual accounts of the experiences of refugees from the Second Sudanese Civil War). Unit 2 culminates with a literary analysis essay about the theme of survival. Unit 3 brings students back to a deep exploration of character and point of view: students will combine their research about Sudan with specific quotes from A Long Walk to Water as they craft a two-voice poem, comparing and contrasting the points of view of the two main characters, Salva and Nya,. The two-voice poem gives students an opportunity to use both their analysis of the characters and theme in the novel and their research about the experiences of the people of Southern Sudan during the Second Sudanese Civil War. Find the rest of the EngageNY ELA resources at https://archive.org/details/engageny-ela-archive .

Material Type: Module

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