ALG1 – Algebraic Thinking 1: Foundational Knowledge
Welcome!
The aim of the ALG series is to support educators who are charged with advancing students’ development of algebraic thinking.
This is the first module in the algebraic thinking series, ALG1. In this module, we focus on ways you can support your students when they arrive to a course with some knowledge gaps.
Audience: Educators interested in enhancing their algebraic thinking, math content knowledge, and teaching strategies to support their students’ learning – particularly those students who experience and/or learn math differently. Upper elementary and middle school grade educators will find the modules most-aligned to the math standards they teach.Â
Module time estimate: 5.5 hours
Prerequisites: None
To enroll in module ALG1, click the button at the bottom of the page. 
ALG1 – Algebraic Thinking 1: Foundational Knowledge
Module Description
As an educator, you have likely seen the impact of foundational knowledge gaps. These gaps or overly simple understandings can make learning new yet related and more complex math content difficult for students.Â
In the ALG1 module, we help educators develop their own understanding about foundational algebraic topics and demonstrate ways to articulate and model math for students. We also discuss common misconceptions that students hold and how they can be addressed. Finally, we also draw on evidence-based strategies for teaching math that can be used within a Universal Design for Learning (UDL) framework.
Algebra is an Important Part of our Modern Lives
Math is an essential part of everyday life in the 21st century. Ohio’s Plan for K-12 Mathematics describes mathematics as, “the human activity of reasoning and problem solving with numbers, quantities, patterns, and shapes” (ODEW, 2025, p. 15). Perhaps you have had your students say, “Why do I even need to learn ‘hard’ math, like algebra?” Click the arrows on the slider below to see some of the ways algebra can impact our lives.Â
Optimize Relevance
What goals do your students have for their lives? Knowing your students’ life goals can help you to reinforce the value of algebra for the things they value and want to do!
Building strong math capabilities is important for many parts of life. Sometimes we think first of algebra as preparation for college and STEM careers; however, as Ohio’s Plan for K-12 Mathematics (2025) notes, our students will take many paths after high school. Algebra is also important for developing life skills (like budgeting or determining better values at the grocery story), making sense of the world (like reading infographics), and being informed members of a democratic society (understanding complexities of how people in power are using numbers).Â
As you work with your students, we can ask about the goals they have for their lives. How can you make connections between algebra learning and their goals? Motivation to work through the hard parts of algebra can be built when we help students to see how that hard work will pay off for the things they value most.
Unfortunately, in 2024Â only about half of students taking algebra scored proficient on their Ohio end-of-course assessment.Â
Algebraic Thinking Builds from Kindergarten
Did you know that there are Ohio math standards that build algebraic reasoning as early as kindergarten? The learning progression map below (ODE, 2020) shows the progression of number, operations, and algebraic thinking strands that are built on over time toward algebra 1. The green arrow and the boxes below detail the math content and learning strategies that we discuss in the ALG1 module.
This module focuses on some common difficulties students have as they work in middle school math or algebra 1. We will review a series of difficulties that you may notice your students exhibit. These can get in the way of their progress for on-grade level content. You can diagnose through early testing then remediate within the context of your course. The lessons focus on specific topics (listed below, left) that students have difficulty with and provide specific ways to address those difficulties (listed below, right).
ALG 1 Math Content Focus
- Review integers, fractions, decimals
- Review equal sign, equations, inequalities
- Word Problem structures for 4 operations
- Using models (CSA, number lines, coordinate grids)
- Properties and Order of Operations
ALG 1 Learning Strategies
- Teacher Collaboration
- Explicit and Systematic Instruction
- Understanding and applying UDL principles
- CSA: levels of math representations
- Graphic Organizers
- Talk- Alouds
- Visuals to support Remembering
Students can learn less than we hope for many reasons: perhaps when they learn in less-than-ideal conditions (such as during online COVID learning), have been sick or moved causing them to miss particular content instruction, or needed more practice to attain mastery. There are many reasons why students may fall behind. The question then is, “How do we ensure students catch up?”
In the video at right, Shanilee Sharma, founder of Zearn and author of Math Mind, describes how we can support students who miss some piece of learning in an orderly and non-stigmatizing manner. (Note: the “they” Sharma refers to are other countries’ teachers)
ALG1 focuses on foundational knowledge that may limit students’ ability to grasp upper elementary and middle school algebraic concepts. As you might imagine, teaching and supporting students on missing skills is vital for their algebraic readiness. We will break those topics down and provide you with ways to support your students in quickly filling those knowledge gaps.
Who is the Audience for this Module Series?
The ALG module series is designed for educators who are responsible for teaching algebraic thinking to upper elementary and middle school students and intervention specialists who are responsible for supporting upper elementary and middle school students with disabilities, but who may feel under-prepared to do so. What do we mean by under-prepared?
- Perhaps your background isn’t in mathematics, but you are charged with teaching math: this series will help you to both bolster your understanding of math concepts and think about how to teach them.
- Perhaps you have a strong math background but are unsure how to support students who struggle: this series will help you to think about how you can apply evidence-based practices aligned with Universal Design for Learning (UDL) to support your students who may experience/learn math differently.
Emphasis for this series is placed on educators of upper elementary and middle school students, as the lessons in the ALG series align most directly with grades 6, 7, and 8 mathematics standards. While ALG1 focuses on earlier foundations in algebraic thinking, educators in grades 6-8 will benefit from a review of the ways algebraic thinking is developed in earlier grades so they may a) build on that learning and b) intervene in topics not mastered during those earlier grades.Â
Teachers of grades 4 and 5 may also benefit from the modules to better understand how their instruction impacts the progression of algebraic thinking beyond their grade level.
ALG1 Module Objectives
Objective 1.Â
List and review key foundational topics and skills that students need to promote greater success in algebra.
Objective 2.
Describe common misconceptions and difficulties students have related to those foundational math topics and skills.
Objective 3.
Apply content-specific tactics to intervene when students have gaps in knowledge or misconceptions that are likely to impact their ability to be successful in algebra.
What the Module Series is Not:
- The modules are not a curriculum; instead, the ideas shared in the modules are meant to help you effectively teach the curriculum your school has adopted.
- The modules are not meant for your students to use; the modules have been developed for educators.
Use of People-First Language
The BRICK Science and Mathematics modules have been created to support intervention specialists (aka special education teachers) and teams of general and special education personnel in more effectively delivering science and mathematics content to students with disabilities and learning difficulties who need additional support. In these modules, we have opted to follow the general guideline of using person-first language. While we are sensitive to the evolving nature of preferred terms for the various communities impacted by naming conventions (e.g., Woodridge, 2023), we have opted to use the term Students with Disabilities, or SWD, throughout the modules when referring to students who receive special education services, regardless of their specific disability condition.Â
Policies and Navigating the Course
The BRICK (Building Robust Intervention Specialist Content Knowledge) project’s primary goal is to enhance the content knowledge and teaching practices of Ohio educators with a particular emphasis on educators whose primary role is to teach and support students with disabilities and/or learning difficulties. Increased content knowledge in evidence-based math practices will support intervention specialists and other educators (general education teachers, paraprofessionals, school-based tutors, instructional specialists) in making such content more accessible for students with disabilities and learning challenges. BRICK online learning modules aim to develop educators’ content knowledge as well as content-specific pedagogical knowledge, particularly in the use of UDL 3.0, to support student learning. The project focuses on supporting educators to use explicit, systematic instruction, emphasizes the connections between achievement in literacy and achievement in math, offers a virtual community of practice for educators, and provides family materials that students and their family members can use to talk about math content at home.
Throughout each course, you will see a Course Outline with a green ribbon and the title of the course at the top right corner of the page. The outline displays every section of the course. Click any section to jump right to it.
As you go through the course, you will have the opportunity to click the Mark Complete box at the end of each section. When you do this, the sections you have completed will be crossed off in your outline, and you will be taken to the next section of the course.
Your progress will automatically be saved within the learning management system, and you can view your course progress through the progress bar located near the top of most sections.
Exiting and Resuming a Module
Know that your progress will automatically be saved within the learning management system. If you exit the website, once you log back in you can easily access your courses by clicking on the My Dashboard tab under the My ALL Ohio menu at the top of every page.
Note Feature within the Modules
Throughout each module, you will notice some Reflection boxes that provide various prompts. Each reflection will be in a grey box with an image of a pen/lightbulb. A built in feature of each module is the ability to take notes and save them for your future use. This feature can be used within any part of the module, but is especially helpful for the reflection activities.
To access the note feature, simply click on the purple Take Notes button fixed in the lower right-hand corner of the screen. When you click on this button, a Take Notes box will appear. This box can be moved anywhere on the screen. Within the box, there is a SAVE button as well as a Print Button.
The easiest way to print all the notes you take from the entire module is to click View All Notes. When you click View All Notes, you will be taken to a page like the one below.
Click the boxes of the Notes you would like to print, and then click Download Selected. All your notes will be downloaded into one document separated by headers.
ALL Ohio’s comprehensive learning modules are relevant to any educator working to support students’ literacy development. In each module, you will find in-depth narrative text, linked resources, realistic scenarios, reflection activities, and embedded audio and videos.
Each module is self-paced. The approximate time for completion of the module is listed in the Welcome text box on the home page of the module.Â
Certificates
Professional Development Certificates
are available for each comprehensive learning module. To earn a Professional Development Certificate, you must obtain a passing score of 85% or more on the module post-test. Please note that you are only allowed up to two attempts to obtain a passing score to earn your certificate.Badges
When completing professional development modules within All Ohio, you can earn badges based on your engagement in and mastery of content within each comprehensive learning module. You earn
Achievement Badges when you demonstrate proficiency with module content (i.e., obtaining a score of at least 85%), whereas Engagement Badges are awarded based on your level of participation on the site. Your earned Certificates and Badges will be displayed on your “All Ohio Dashboard.”
High standards of honesty and integrity are expected when completing professional development within this learning management system. Academic dishonesty includes, but is not limited to, dishonesty in the completion of activities and/or claiming credit for work not done or done by others. Any violation of these expectations can result in removal from this learning management system.
If you have any issues with accessing the content or functionality of this course, please email: allohioprojectteam@uc.edu
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Please contact Rachel Evans, Research Associate, Systems Development & Improvement Center, CECH, UC, evans2rm@ucmail.uc.edu
Helpful Resources Related to this Module:
Welcome and Navigation
Step-by-step guide to assist you in navigating the professional learning modules.
Click below to enroll in the ALG1 module
Please note:Â If you have not registered for an account yet, when you click the enroll button below you will be redirected to the registration page.
Once registered for a FREE account, you will be able to enroll in this and other modules. All enrolled modules will then appear on “My Dashboard” and your work will be saved so that you can easily stop work and begin again at any point. All of your certificates and badges will be accessible from My Dashboard too.
Resources mentioned on this page:
Ohio Department of Education (ODE). (2020). Ohio’s 6-8 Learning Progressions by Topic.Â
Ohio Department of Education & Workforce (ODEW). (2025). Ohio’s Plan for K-12 Mathematics.Â
Sharma, S. (2025, March 25). Meeting of the Maryland State Board of Education.Â
Thomas B. Fordham Institute. (2025). Ohio Education by The Numbers.