Ratios and proportional relationships permeate our daily lives and underpin further study in mathematics and science. Understanding them and developing related problem-solving skills is essential and an important part of the Common Core Standards for grades six and seven.

It’s All Relative is a 98-page flipchart designed to engage students and develop deep conceptual understandings while correcting common misconceptions. Following the same practical format as the popular Xs and Whys of Algebra and Zeroing in on Number and Operations series, Anne Collins and Linda Dacey provide 30 modules that focus on key ideas with instructional strategies, activities, and reproducibles for:

• understanding the language of ratios;
• understanding the multiplicative relationships of ratios;
• using tables, tape diagrams, double number line diagrams, and graphs to represent ratios;
• using unit rates and scale factors to solve problems; and
• solving multi-step ratio and percent problems.

All middle-level math teachers should have this handy flipchart at their fingertips during planning and instruction. You can download three of the modules now in PDF format.

### An introduction to the Zeroing In series

The Zeroing In on Number and Operations series brings together the experience and insights of authors Anne Collins and Linda Dacey into easy-to-use tools for teaching key concepts in number and operations and for addressing common misconceptions. We asked Linda and Anne to introduce the series of flipcharts and share how the books came about and how teachers can make the most of them.

Between us, Linda and I have experience teaching mathematics at all grade levels. When we chat about the way our students talk about mathematics, we notice a general pattern of misconceptions, misuses of terminology, lack of number sense, and—above all—confusion about key concepts. This led us to ask, what are the key concepts and how should we teach them so that these common misconceptions don’t form? We decided to write a series, based on research that could be used by teachers regardless of their district’s adopted curriculum. In developing the Zeroing in on Number and Operations series we examined the National Council of Teachers of Mathematics Curriculum Focal Points and the recently developed Common Core State Standards for Mathematics to help classroom teachers as they align lessons to these sources. We targeted grade bands pre-K–K, 1–2, 3–4, 5–6, and 7–8 in such a way that teachers can tailor the mini-lessons and activities to the level of their students. That is, if students need more of a challenge in, say, fourth grade, the teacher may decide to use some of the material from the grades 5–6 flip books; conversely, if a third-grade student needs more intervention, practice, or scaffolding, a teacher may use the grades 1–2 book. Similarly, the forthcoming flip book, The Xs and Whys of Algebra:  Key Ideas and Common Misconceptions, can be used with students in grades 7, 8, or 9.

The Structure of the Flip Books
When you examine the Zeroing in on Number and Operations series you will see that the thirty modules in each resource include a mathematical focus, information on potential challenges and misconceptions, sections on the classroom and meeting individual needs, and references/further reading. As mentioned, the modules will support whichever text or program you use in your classroom. Each of the modules has game cards, problems, or graphic organizers in the appendix for you to reproduce.

As you flip through the Zeroing in on Number and Operations resources you will see plenty of suggestions for modeling specific numeric concepts geometrically or algebraically. For instance, many students may use the five-frame and ten-frame activities from the pre-K–K and grades 1–2 flip books. The strategies for posing problems are appropriate across the grades, and you will find specific problem-posing situations in each grade-level book. Because ratio and proportionality are so heavily emphasized in the Common Core State Standards for Mathematics, you will find a variety of lessons, activities, and differentiation tips in grades 5–6 and 7–8; you will notice too that the activities increase in sophistication from grades 5–6 to 7–8. Research shows that many students struggle with inverse operations such as subtraction and division, so you will find emphasis on these operations in the grades 3–4 and 5–6 flip books.

Meeting Individual Needs
When my son was in fifth grade, he announced that he did not need to go to school until after Thanksgiving. When pressed to explain why, he stated that all that happened from September through November was a review of what he had learned the previous year, and since he already knew that material it was a waste of time. Unfortunately, many students share my son’s experience, so we made sure that in our “Meeting Individual Needs” section we included challenges for students who need them as well as suggestions for scaffolding material for students who need additional support.

How to Use the Flip Books
We encourage you to have this resource handy on your desk so when you are working on a particular concept or topic and you need an activity, a suggestion to make the topic more engaging, or additional challenges or support for some of your students, you will have these ideas at your fingertips.