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: A penny is about $\frac{1}{16}$ of an inch thick. In 2011 there were approximately 5 billion pennies minted. If all of these pennies were placed in a s...

This task provides a good entry point for students into representing quantities in contexts with variables and expressions and building equations that reflect the relationships presented in the context.

CSDE Model Curricula Quick Start GuideEquitable and Inclusive Curriculum  The CSDE believes in providing a set of conditions where learners are repositioned at the center of curricula planning and design. Curricula, from a culturally responsive perspective, require intentional planning for diversity, equity, and inclusion in the development of units and implementation of lessons. It is critical to develop a learning environment that is relevant to and reflective of students’ social, cultural, and linguistic experiences to effectively connect their culturally and community-based knowledge to the class. Begin by connecting what is known about students’ cognitive and interdisciplinary diversity to the learning of the unit. Opposed to starting instructional planning with gaps in students’ knowledge, plan from an asset-based perspective by starting from students’ strengths. In doing so, curricula’s implementation will be grounded in instruction that engages, motivates, and supports the intellectual capacity of all students. Course Description:  In Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking.  Upon completion of this course students will have the ability to:Understand ratio concepts and use ratio reasoning to solve problems. Apply and extend previous understandings of multiplication and division to divide fractions by fractions. Multiply and divide multi-digit numbers and find common factors and multiples. Apply and extend previous understandings of numbers to the system of rational numbers. Apply and extend previous understandings of arithmetic to algebraic expressions. Reason about and solve one-variable equations and inequalities. Represent and analyze quantitative relationships between dependent and independent variables. Solve real-world and mathematical problems involving area, surface area, and volume. Develop understanding of statistical variability. Summarize and describe distributions.Aligned Core Resources:  It is critical that curriculum be implemented using high quality instructional materials to ensure all students meet Connecticut’s standards. Ensuring alignment of resources to the standards is critical for success. There are tools that are available to districts to assist in evaluating alignment of resources, such as CCSSO’s Mathematics Curriculum Analysis Project and Student Achievement Partner’s Instructional Materials Evaluation Tool.   In addition, there exist compilations of completed reviews from a variety of resources. Some of these include but are not limited to EdReports, Louisiana Believes, CURATE, and Oregon Adopted Instructional Materials. Aligned Core Programs:  The CSDE in partnership with SERC has engaged with providers of high-quality vetted resources to provide additional alignment guidance to the CSDE model curriculum.  High-quality instructional resources are critical for improving student outcomes. The alignment guidance is intended to clarify content and support understanding for clear implementation and coherence. Materials selection is a local control decision and these documents have been provided from participating publishers to assist districts in implementation. Use of the materials from these publishers is not required. These aligned core programs meet expectations as reported by EdReports. If your resource is not listed below, you are encouraged to review EdReports to ensure the alignment of your resource to the Connecticut Core Standards. Strong alignment of curricula and instructional materials have the potential to support student engagement of meaningful grade level content daily and teacher growth.   Carnegie Learning Math Grade 6EdGems Math Grade 6enVisions Grade 6Eureka Math Grade 6Fishtank Plus Math Grade 6HMH Into Math Grade 6Imagine Learning Illustrative Mathematics Grade 6i-Ready Math Grade 6MidSchoolMath Grade 6Open Up Resouces Math Grade 6Reveal Math Grade 6Additional Course Information:  Major work of Grade 6 mathematics focuses on ratios and proportional relationships and early expressions and equations.  Fluencies required upon completion of grade 6 include multi-digit division and multi-digit decimal operations. Habits of Mind/SEIH/Transferable Skills Addressed in the Course: The Standards for Mathematical Practice describe the thinking processes, habits of mind, and dispositions that students need to develop a deep, flexible, and enduring understanding of mathematics. They describe student behaviors, ensure an understanding of math, and focus on developing reasoning and building mathematical communication. Therefore, the following should be addressed throughout the course: Make sense of problems & persevere in solving them Reason abstractly & quantitatively Construct viable arguments & critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for & make use of structure Look for & express regularity in repeated reasoning  

Unit Overview/Summary - FOCUS:  This unit focuses on the Equations and Expressions domain.  Learning in this unit will enable students to: • Reason about and solve one-variable equations and inequalities. • Represent and analyze quantitative relationships between dependent and independent variables. All of the learning in this unit is major work of the grade. 

Unit Overview/Summary - FOCUS:  This unit focuses on the Expressions and Equations domain.  Learning this unit will enable students to: Apply and extend previous understandings of arithmetic to algebraic expressions. Reason about and solve one-variable equations and inequalities. Represent and analyze quantitative relationships between dependent and independent variables.  

In this task students are asked to write an equation to solve a real-world problem. There are two natural approaches to this task. In the first approach, students have to notice that even though there is one variable, namely the number of firefighters, it is used in two different places. In the other approach, students can find the total cost per firefighter and then write the equation.

In Module 4, Expressions and Equations, students extend their arithmetic work to include using letters to represent numbers in order to understand that letters are simply "stand-ins" for numbers and that arithmetic is carried out exactly as it is with numbers. Students explore operations in terms of verbal expressions and determine that arithmetic properties hold true with expressions because nothing has changed—they are still doing arithmetic with numbers. Students determine that letters are used to represent specific but unknown numbers and are used to make statements or identities that are true for all numbers or a range of numbers. They understand the relationships of operations and use them to generate equivalent expressions, ultimately extending arithmetic properties from manipulating numbers to manipulating expressions. Students read, write and evaluate expressions in order to develop and evaluate formulas. From there, they move to the study of true and false number sentences, where students conclude that solving an equation is the process of determining the number(s) that, when substituted for the variable, result in a true sentence. They conclude the module using arithmetic properties, identities, bar models, and finally algebra to solve one-step, two-step, and multi-step equations.

**NOTE: The New York State Education Department shut down the EngageNY website in 2022. In order to maintain educators' access, nearly all resources have been uploaded to archive.org and the resource links above have been updated to reflect their new locations.**

This task provides a good opportunity for group work and class discussions where students generate and compare equivalent expressions. In class discussion, students should be asked to connect the terms of an expression with quantities shown in the diagram.