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Do you have any online ratio games my students can engage with?

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Do you have any online ratio games my students can engage with?

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In Ratio Rumble, players choose from among 18 different characters as their avatar in the tournament.

Ratio Rumble is all about the player re-creating each ratio recipe given to them by the game. In this case, the game asks for two blue potions and one red potion. The player used double the original ratio. This is possible because the ratio is still the same even if the quantities are doubled.

Players can use health potions found on the gameboard to heal their characters.

Potions with lightning symbols, when played as part of a ratio, will weaken the opponent. Potions with numbers are time bombs that tick down each turn. It's important to use these potions in your ratios. If they are not used before they finish their countdown, they explode and weaken the character's health.

In Ratio Rumble, players learn to create ratios following numerical recipes. Ratios get more complicated as the game progresses and as the player masters new levels of difficulty.

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Students engaged in Math Snacks in class learned more math than those with access to just classroom instruction.

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What are common misconceptions when teaching proportions?

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What are common misconceptions when teaching proportions?

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One of the reasons many Year 7 and 8 students experience difficulty interpreting and using ratios, rates and percent, is that they have not yet acquired a capacity for proportional reasoning. This is a complex form of reasoning that depends on many interconnected ideas and strategies developed over a long period of time. These features are amply illustrated by Lamon's (1999) description of proportional reasoning as "the ability to recognise, to explain, to think about, to make conjectures about, to graph, to transform, to compare, to make judgements about, to represent, or to symbolise relationships of two simple types ... direct ... and inverse proportion" (p.8). At its core, proportional reasoning requires a capacity to identify and describe what is being compared with what. Essentially, there are two types of proportional reasoning problems, both of which require some form of comparison. The first typically involves a comparison of two rates, eg, Which car had a faster average speed, Car A which travelled 217 km in 1&frac34; hours or Car B which travelled 204 km in 1&frac12; hours? The second type is referred to as missing value problems. These problems typically provide 3 quantities and the fourth is missing, eg, If a supermarket worker can unpack 24 boxes in 1 hour, how many boxes could he unpack in 10 minutes?

Recognising what is being compared with what is not always straightforward. It can be confounded by the types of quantities used, how they are represented, and the number of variables involved. Also, not all problems in which 3 quantities are given and a fourth is missing require proportional reasoning. As Lamon notes, "There are no shortcuts available here! Thought, common sense, and experience must be used to determine whether a situation is proportional or not. You must always bring into play your knowledge about how things work in the real world" (P.225).

While this is undoubtedly true, proportional reasoning also requires a capacity to work flexibly and confidently with the quantities involved (that is, measures, rates and/or ratios expressed in terms of natural numbers, rational numbers, and/or integers), and an ability to recognise multiplicative relationships in a range of problem contexts including the idea of rational numbers as operators (eg, understanding ⅔ × $24 as ⅔ of 24, or 3.5 × 68 as 3 and a half times 68). Neither of which, according to recent research, can be assumed to be in place for all students at this level of schooling.

This could be due to/associated with:

a lack of access to a broader range of ideas for multiplication, in particular, the Cartesian product or the 'for each' idea which underpins rate (see Level 3);

an over-reliance on the quotition ('how many groups in') idea for division, and/or a failure to recognise the role of partition division in relation to fractions and decimals;

an inability to construct fraction models beyond those in the halving family and/or insufficient experience with making/representing, naming, comparing, and renaming mixed fractions and decimal fractions greater than 1 (see Level 4);

insufficient exposure to a broader range of continuous (part-whole) and discrete (subset-set) fraction representations where the unit varies (eg, "What is the unit if the brown Cuisenaire rod is 1 third? What is the unit if the brown rod is 2? ...);

a reliance on rule-based procedures to rename fractions (eg, "to change a fraction into a percentage multiply by 100 over 1") which suggests an inability to reconceptualise a quantity in terms of different sized chunks or 'units' (eg, &frac34; is 75% because we can think of 3 (25-units) out of 4 (25-units) or 75 out of 100);

a limited understanding of what it means to multiply a quantity by a rational number, for example, a belief that multiplication always mean bigger (whole number thinking)or that multiplication by a fraction always results in smaller (thinking limited to proper fractions), leading to a reluctance/inability to work with mixed fractions and rational numbers as factors/scalars (eg, recognise that the dimensions of a photo may be enlarged or reduced by rational factors such as 250% or 3/5 respectively);

a limited understanding of ratio (eg, assumes 1:4 is always the same as &frac14;) and/or a tendency to think about change in absolute rather than relative terms (eg, 2 is to 3 as 5 is to 6 because "3 was added to both sides")

By the end of Level 5 students need to be able to formally recognise and work meaningfully with a wider range of numbers, in particular, natural numbers, integers, and rational numbers in whatever form they appear (eg, proper fractions, mixed fractions, finite or recurring decimals, percentages, and rates). They also need to be able to work with a broader range of multiplicative situations including direct proportion and ratio.

A key indicator of the extent to which students have developed a broader range of ideas to support proportional reasoning is the extent to which they use multiplicative strategies such as partitioning to solve problems involving simple proportion (eg, find for 1 or a common composite unit such as 3 in a comparison involving 6 and 9, then multiply as appropriate). Another indicator is the extent to which students can work meaningfully with multiple representations of proportional relationships. For example, given a collection of 7 brown eggs and 5 white eggs, recognise that the following representations can all be used to refer to different relationships in this context depending on what is viewed as the unit:

A major issue here are the meanings students ascribe to the symbolic representation of ratio. Ratio involves the comparison of any two quantities, "it always makes a statement about one measurement in relation to another" (Lamon, 1999, p.13). For example, ratio can be used to compare the number of feral cats to the size of the national park. More specifically, ratios can be used to compare measures of the same type, for example, the number of feral cats compared to the number of feral dogs. In this case, two types of comparison are possible, for instance, one can compare the parts to the parts (eg, cats to dogs) or the parts to the whole (eg, cats to the total number of cats and dogs). Ratios can be also used to compare measures of different types, which is often but not exclusively referred to as rate (eg, the number of feral cats per square kilometre). And, although it is often assumed, ratios are not always rational numbers (eg, the ratio of the circumference of a circle to its diameter)

Tools

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Common Misunderstandings - Level 5 Proportional Reasoning

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One of the reasons many Year 7 and 8 students experience difficulty interpreting and using ratios, rates and percent, is that they have not yet acquired a capacity for proportional reasoning. This is a complex form of reasoning that depends on many interconnected ideas and strategies developed over a long period of time.

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Vertical Alignment Charts for Revised Mathematics TEKS

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This resource provides vertical alignment charts for the revised mathematics TEKS.

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STAAR Released Questions-Item Analysis Tool

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STAAR Released Questions-Item Analysis Tool

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lead4ward makes a difference in the lives of students by helping educators focus their work, creating structures that give teachers and students a sense of hope and confidence, and supporting leaders with systems they can trust.

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Mental Math - reducing fractions and ratio expressions
Problem Solving - identify equivalent fractions and ratios

Common Core Connection for Grades 6 and 7

Understand ratio concepts and use ratio reasoning to solve problems.

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Play Ratio Blaster at Math Playground! Find equivalent ratios in this space-themed math game.

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Do you have any real world examples of rate that I can use with my students?

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Do you have any real world examples of rate that I can use with my students?

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Pre-Algebra giving you a hard time? Shmoop's free Ratios & Percentages Guide has all the explanations, examples, and exercises you've been craving.

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AAA MATH - this site has over 200 pages of math lessons (K-8) which feature interactive practice, challenge games and an explanation of the topic

A+ MATH - a Math site for kids: game room, homework helper, flashcards, advanced problems; nice lay-out, easy to use.

The abacus: the art of calculating with beads - everything you ever wanted to know about the Abacus.

About today's date - a great site that gives history and facts about the numbers in today's date

AIMS - archives, puzzles, downloads, curriculum and ideas for Math and Science.

Global schoolnet foundation - Linking students around the world.

Googolplex - this site is about Googolplex. What is a Googolplex?

Harcourt multimedia math glossary - this site has definitions and visuals of math vocabulary for grades one through eight.

The MacTutor History of Mathematics archive - Contains the biographies of more than 1000 mathematicians, as well as histories of various mathematical topics

Hotel Infinity - When turn-of-the-century mathematician George Cantor approached the subject of infinity with mathematical rigor, he encountered many paradoxes. At the Hotel Infinity, these paradoxes come to life.

General Math Resources - Honor Roll Mathematics Hotlist. Lots of good math links in all areas, general math to computing. The Math Forum- this site has many math links listed by grade level or subject matter.

IlluminationsA site every math teacher should visit. Lesson Plans, simulations, and much more.

Interactive Mathematics Miscellany and Puzzles - games and puzzles, arithmetic and algebra, geometry, probability, fast reckoning, mathematics as a language, proofs in mathematics, the many ways to construct a triangle - and many teacher ideas and activities.

Interactive Mathematics Online - this site includes clear explanations of mathematical ideas with examples

Interactive Math Lessons

Algebra Sites

Equations

Purple Math - Purplemath contains practical algebra lessons demonstrating useful techniques and pointing out common errors.

Math forum: algebra lesson plans - This list contains some of the best algebra lesson plans or materials on which to base them.

Ask Dr. Math - a question and answer service for K-12 math students and their teachers. A searchable archive is available by level and topic, together with a FAQ.

Amusing Math Page - math jokes and more.

A Plus Student - This site has hundreds of interactive math lessons with great graphics, animation and audio. Math courses range from kindergarten to high school including pre-algebra, algebra, and geometry.

Awesome library's math lesson plans - The Awesome Library organizes 30000 carefully reviewed K-12 education resources, the top 5 percent for teachers, students, parents, and librarians.

Basket Math - clicking any of the questions in the Table of Contents will result in a mathematics question and animation. If you answer the question correctly you will make two points; correct answer shown if wrong.

Change Maker - a Funbrain simulation in which students can practice making change for a dollar or for one-hundred dollars

Chickscope and egg math - different mathematics topics related to eggs.

Click On Bricks - this is an interactive mathematics site for learning basic multiplication skills

Cool Math - an amusement park of mathematics!

Curious Math - Here you will find tricks and rules for quickly calculating certain types of math problems. There are also some entertaining trivia and math facts.

Currency Converter - You can view any exchange rates among the 164 currencies.

Math Tables - reference tables for general math, algebra, geometry, trigonometry, statistics, calculus and advanced topics.

The Internet Mathematics Library - from Math Forum, solid approaches to a variety of Math topics for elementary students.

Elementary teacher's place - wonderful site for the K-5 math teacher

Exercises in math readiness for university study - includes 3 levels, with hints and tutorials.

Fantastic Math - math tips and tricks.

Fibonacci puzzles - you want sophisticated patterns for your students to study? This site's has them: image and number puzzles based on Mathematics.

Figure This! - Mathematics Challenges for Families provides interesting math challenges that middle-school students can do at home with their families.

Flashcards for kids - this site has an interactive, online, elementary math problem solver

Frank Potter's Science Gems - this site has a useful, annotated list of math sites arranged by topic and grade level

Fun Brain will show you a number line. Students will click on the number line at the correct answer

Math Baseball - Kids can learn math and have fun playing baseball at the same time.

Calculus

Calculus Initiative - These modules emphasize geometric concepts of calculus while examining applications of mathematics to the physical and life sciences.

Math Archives - Calculus Resources On-line - contains information and links to numerous Internet resources, which could be used for teaching and learning of calculus.

Calculus and Mathematica - this site provides an exciting new way to learn calculus, differential equations, and matrix theory

Graphics for the calculus classroom - graphical demonstration developed for first year calculus

Visual Calculus - ideas on how technology can be used in the teaching of calculus

Fun Fractals - A Fractals Unit for Elementary and Middle School Students

Geometry

Gallery of Interactive Geometry - The Geometry Center at the University of Minnesota The Geometry Center has some great multimedia exhibits about geometric structures.

The Geometry Center - projects, Java applications, archives, downloads, materials

Geometry Junkyard - a number of excellent topics on geometry - neat, clean format.

Geometry Discussion Groups - eight geometry discussion groups, an online electronic community for all those interested in geometry

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Mathematics is the alphabet with which God has written the universe - Galileo Galilei

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Explore our growing collection of PBL resources to help you design and implement powerful projects for your students.

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Watch the design process in action with educational how-to videos of engineering projects and activities kids can try at home.

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"You see this with a young class of learners who've not yet learned classroom behavior. Little kindergarten children all raise their hands at once, and you're lucky if they raise their hands. Usually they just break out and say something when the teacher tells a story about her experience because they want to share their experiences. This is how life is much of the time. And it's not that when children are doing that they're not learning. They want to be able to apply their natural tendencies to the learning process. And I think by giving children project work, we open up areas during the school day when children can speak about what they already know, when they can ask questions, they can express interests that are different from [those of] other children. And, I believe, where they can work on their strengths."

Read the full Edutopia.org interview with Sylvia Chard.

Vicki Davis

Vicki Davis, a high school teacher from Georgia, is cofounder of the award-winning Flat Classroom Project and author of the popular Cool Cat Teacher blog.
Why should collaboration be built into a project?
"If one is teaching math, you can talk all day about it and look at problems on the board; however, students begin to understand math when they work on problems. Problems are a necessary part of math, and we shouldn't shy away from using that word."

"In order to understand collaboration, both face-to-face and global, students must also work on the problems they are having with one another. We must begin to view the interpersonal struggles between students not as problems in the sense of something to eliminate but rather as a potential teaching scenario. Having a real-life problem to coach students through will help them understand the skills they need to survive and thrive in the real world of collaborative online and off-line spaces."

"If a math teacher wants to have her students underachieve, then she will work out all the problems for her students. Doing problems for the students may result in neat, tidy papers, but it also results in a missed learning opportunity."

"Likewise, teachers often get caught up in the emotion of students' interpersonal conflicts when, in reality, we should learn to coach students through these problems. Working out problems for the students may result in neat, tidy classrooms, but it also results in a missed learning opportunity."

"Businesses ubiquitously request people skills, and now they're looking for online, collaborative project-management skills as well. This is not something that will happen tomorrow; it's going on today. Students can be the greatest textbooks for one another, but this kind of learning requires a change in how we, the teachers, view problems."

"Some of the "problem opportunities" that can occur when we use technology in the classroom include wiki wars, non-contributing students, difficulties in coordinating online projects with student teams around the world, troubles with creating videos for another student and delivering on time, and cultural misunderstandings."

"These things cannot be taught in a book or through a lecture but are, by definition, experiential skills that students can only learn by living through -- you guessed it -- problems."

Watch an Edutopia.org video about Vicki Davis' classroom, "Harness Your Students' Digital Smarts" and read an article she wrote, "Personal Learning Networks Are Virtual Lockers for Schoolkids".

Chris Lehmann

Chris Lehmann is the founding principal of the Science Leadership Academy, in Philadelphia. The Science Leadership Academy is a public magnet high school that emphasizes inquiry-driven, project-based learning. A partnership between the School District of Philadelphia and the Franklin Institute, SLA focuses on science, technology, mathematics, and entrepreneurship, and it emphasizes the core values of inquiry, research, collaboration, presentation, and reflection in all classes.

"Inquiry-driven and project-based learning makes sense for today's learners because it gives them the opportunity to connect the work they do in school with the larger world around them. We must stop saying that school is "preparation for real life" and acknowledge that school is real life for the kids while they are in it. Project-based learning allows that to happen in powerful, meaningful ways."

Read an Edutopia.org article about Chris Lehmann and the Science Leadership Academy, "Social Networking at Science Leadership Academy" and an additional interview with Lehmann,"How to Use Social-Networking Technology for Learning".

John Mergendoller

John Mergendoller is executive director of the Buck Institute for Education (BIE). BIE is dedicated to improving 21st-century teaching and learning throughout the world by creating and disseminating products, practices, and knowledge for effective project-based learning.
Why is PBL worth doing?
"Research has shown that students learn content as well or better using PBL than with traditional instruction. It also shows that students remember what they have learned longer and are better able to use what they have learned. PBL provides the opportunity to learn and practice skills that traditional instruction often ignores -- working in groups, making choices, monitoring progress, thinking deeply about a problem or challenge, and communicating what has been learned. In short, PBL helps students not only learn content but also the 21st-century skills they will need to thrive in a quickly changing, globally connected world."
How does one know if students are ready for PBL?
"I think the key point is that most, if not all students, have to be prepped to do PBL successfully. The teacher should think through what the project requires students to do, both in terms of academic tasks (such as writing an editorial, creating a poster, summarizing an argument) and in terms of the process of completing the project (discussing ideas and making choices in a group, giving constructive feedback on others' work, or making an oral presentation). And then teachers need to ask themselves whether students can do this right now or whether they need some scaffolding, guidance, and practice."

"If they do need practice or preparation -- either academic or process focused -- the teacher needs to provide that. Begin by having kids describe what it means to discuss things and learn from the discussion or talk about what a good presentation looks like (help them make a rubric!). Examine the steps in developing an argument. Sometimes the teacher will need to be direct and didactic: "A good argument is based on reasons that are logical and make sense; it is not based on what we feel or what we like." At other times, the teacher can draw the needed ideas out of the students in a discussion. But in all cases, the result should help students understand what they need to do and give them practice doing it in a supervised, structured situation before we turn them loose on a project."

"The teacher should also check in with students after the project starts: "Let's talk about the discussions you are having in your groups. How many students would say they are learning new things? What is your group doing to make that happen?" Good projects are done with preparation (for both teachers and students) and under supervision. If things are going swell, then the teacher can back off."
How can you make your first project a success?
"Start small. Keep the project short and the learning goals limited. Students need a chance to develop their PBL skills -- working productively in groups, monitoring their progress, making public presentations. This takes practice, teacher and peer feedback, and reflection time for both you and your students. You can tackle more complicated projects once your students have some initial success."

Read Edutopia.org blog posts about project-based learning by BIE consultant Andrew Miller and BIE's Director of Product Development John Larmer.

Seymour Papert

An expert on children and computing, Seymour Papert is a mathematician and one of the early pioneers of Artificial Intelligence. He is a distinguished professor at the Massachusetts Institute of Technology (MIT), co-founder of MIT's Media Lab, and author of major books on children and learning.
What must change in schools to better accommodate project-based learning?
"Well, first thing you have to do is to give up the idea of curriculum. Curriculum meaning you have to learn this on a given day. Replace it by a system where you learn this where you need it. So that means we're going to put kids in a position where they're going to use the knowledge that they're getting. So what I try to do is to develop kinds of activities that are rich in scientific, mathematical, and other contents like managerial skills and project skills, and which mesh with interests that particular kids might have."

"If, for example, this Lego stuff is why a lot of kids love building robotic kinds of things and programming -- and they love doing that -- and you can connect that work to all the powerful ideas that are important for kids to know. So I imagine the learning environment of the future as we've given up the idea of there being curriculum that says you have to learn this at the seventh of May in your eighth year. We've given up the age segregation which is just as, I think, wrong and harmful as any other kind of segregation. It's just as bad to segregate the seven-year-olds from the eight-year-olds, the eight-year-olds from the nine-year-olds, as it was to segregate people by color, religion, or whatever. That will go away. Kids will work in communities of common interest on rich projects that will connect with powerful ideas."

Read the full Edutopia.org interview with Seymour Papert.

Bob Pearlman

Bob Pearlman is a strategy consultant for 21st-century school development. Formerly the director of strategic planning for the New Technology Foundation, now the New Tech Network, he speaks and consults in the United States and abroad on 21st-century learning and is a longtime proponent of project-based learning.

"Students of today enter an increasingly globalized world in which technology plays a vital role. They must be good communicators as well as great collaborators. The new work environment requires responsibility and the ability to self-manage as well as the interpersonal and project-management skills needed for teamwork and leadership. Enter project-based learning, designed to put students into a students-as-workers setting in which they learn collaboration, critical-thinking, and written- and oral-communication skills along with the values of a strong work ethic, all while meeting state or national content standards. But don't confuse PBL with simply doing activities injected into traditional education to enliven things as a culminating event for a learning unit. Real PBL, in contrast, is deep, complex, rigorous, and integrated."

Read a 2002 Edutopia.org interview with Bob Pearlman on project-based learning.

Organizations and Resources that Support and Promote PBL

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Project-Based Learning: What Experts Say

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"You see this with a young class of learners who've not yet learned classroom behavior. Little kindergarten children all raise their hands at once, and you're lucky if they raise their hands. Usually they just break out and say something when the teacher tells a story about her experience because they want to share their experiences.

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Rigor, Relevance, and Relationships ,edited by Enrique Galindo and Jean Lee, supplies teachers with a challenging and rewarding new way to deeply engage their students in secondary mathematics. In project-based learning (PBL), students investigate and solve real-world issues through mathematics. This book describes the nuts and bolts of designing an effective PBL unit, outlines how this method can help prepare students for future challenges in life and learning, and showcases a number of classroom-tested examples of PBL lessons.

Each unit provided here kicks off with an Entry Event (a letter, video, or presentation of a real-life problem) followed by a Driving Question, an open-ended challenge that students explore together. Mathematics instruction is integrated throughout their investigation. All of these lessons work to support the three pillars of project-based learning:

RIGOR

Projects teach students the rigorous mathematics necessary for college and career readiness. These units address the Common Core State Standards for Mathematics-in particular its Standards for Mathematical Practice-and serve to develop the 21st-century skills that students will need.

RELEVANCE

As they work on mathematics projects situated in the real world, students discover connections to their own lives and communities. Units in this book include ones on buying a car, building a solar cooker, modeling quarterback performance in the Super Bowl, and more. Students immersed in PBL-taught classes have higher levels of engagement and emerge with more useful content knowledge.

RELATIONSHIPS

PBL promotes collaborative learning. To complete these projects, students work in close cooperation with their peers, and they develop positive relationships with their teachers, parents, and other members of their communities.

To help teachers on their journey to implementing project-based learning in their own classrooms, this book includes planning forms and project calendars, real examples of student work, teaching rubrics, and reflections and tips from educators experienced in the PBL method. Classroom-ready forms are available to download and print at NCTM's More4U website.

When students are engaged in both the process and the content of their learning, their educational experience is more enjoyable, effective, and lasting. Incorporating these project-based lessons into instruction increases engagement for students, teachers, and the entire learning community.

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Rigor, Relevance, and Relationships: Making Mathematics Come Alive with Project-Based Learning - National Council of Teachers of Mathematics

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Rigor, Relevance, and Relationships: Making Mathematics Come Alive with Project-Based Learning, edited by Enrique Galindo and Jean Lee, introduces project-based learning (PBL), an exciting new teaching methodology. PBL units that were designed and implemented by high school mathematics teachers are showcased throughout the book, which concludes with tips from mathematics educators who have taught and researched in PBL settings.

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Scaffolding and Differentiating for Students

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Scaffolding and Differentiating for Students

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Plan for success before the lesson

There are differences between differentiation and scaffolding. Differentiation refers to the idea of modifying instruction to meet a student's individual needs and learning styles. Scaffolding refers to modifications you make while designing and teaching lessons that allow all students to be successful in learning the same content.

Differentiation is often directed at individual students while scaffolding is done for the entire class. For example, when an individual student can't answer a Checking for Understanding question and you rephrase your question from open-ended to multiple choice, you have differentiated for that one student. When you pre-read a Learning Objective before having the entire class read chorally, you are scaffolding.

In reality, differentiation and scaffolding strategies overlap, but they have the same ultimate goal: increasing student success.

Differentiating and Scaffolding to increase student success

All educators want student success. A common goal at the lesson level is to have 80%-100% of students successful through whole class instruction. Student success can be improved by making strategic decisions before the lesson, during the lesson, and after the lesson.

Design for success before teaching

Differentiation and scaffolding start with the of the lesson. Explicit Direct Instruction (EDI) lessons are purposefully crafted to generate student success. The Learning Objective and Independent Practice are exactly synchronized. Each lesson contains written definitions of the concepts you are teaching. Labeled examples illustrate for the students what the concept definitions mean. Lessons include clear steps for you and your students to use during Skill Development and Guided Practice. Over the course of the lesson, you teach your students exactly what they need so they can successfully complete the matching Independent Practice.

Much of the success of EDI lessons, including 1,200 online EDI lessons at educeri.com , is due to the use of the "Rule of Two." Each lesson provides matched pairs of problems. The first problem is for you to solve while modeling your thinking out loud so students can hear you solve it. The second problem is for the students to solve.

At DataWORKS Educational Research we have found that when students are not successful, it's not because they can't learn. It's usually because of defects in the lesson itself. For example, a teacher uses a strategy such as "sounds right" during a grammar lesson instead of providing clear definitions and rules for the students to apply.

Sometimes, teachers solve problems but cannot describe in words exactly how they determined the answer. Often, a teacher provides lots of information and then tells the students to "learn it" without providing any strategies or techniques for students to use to remember or organize the information. When students are unsuccessful in these lessons, it's not the students. It's the lesson.

EDI and lessons are designed to prevent these problems by using clearly written definitions, labeled examples, matched problems, steps to use while solving problems, and integrated Checking for Understanding questions.

In a future blog, we'll talk about differentiating and scaffolding the lesson.

These strategies are based on DataWORKS' Explicit Direct Instruction.

Explicit Direct Instruction helps all students. Download this document to learn more about our EDI Professional Development.

All of our EDI lessons have built-in differentiation and scaffolding. Create your account now and access over 1,200 resources in seconds.

Author: John Hollingsworth

John is the president and co-founder of DataWORKS. With his varied background as a systems analyst, engineer, and musician, John has applied his analytical and presenting skills to the field of education, transforming lives with his insights into the teaching process. He and his wife Silvia Ybarra have pioneered research-based teaching strategies, known as Explicit Direct Instruction, which have received praise from teachers worldwide. John has published numerous articles, trained thousands of educators, and presented at many conferences. His mastery of lesson delivery is inspiring to educators. He and Silvia are co-authors of two books on teaching.

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How to Differentiate and Scaffold - Before the Lesson

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Plan for success before the lesson There are differences between differentiation and scaffolding. Differentiation refers to the idea of modifying instruction to meet a student's individual needs and learning styles. Scaffolding refers to modifications you make while designing and teaching lessons that allow all students to be successful in learning the same content.

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Pre-testing and post-testing is the easy part; but how do you TEACH differently with Standards based grading? Proficiency Scales are all about helping students show growth in small steps instead of being overwhelmed by a year-end goal that seems way over their head. This video will walk you through one way to get your students understanding a scale. In this sixth grade standard, students are going to be asked to show that they understand the concept of a ratio and ratio language to describe ratios, which are relationships between two quantities. We’re going to visually build on last year’s content for student’s to make lasting connections that facilitate progress in the standard. Let’s see how to differentiate sixth grade ratios with proficiency scales!

Level 1

Now when you introduce new vocabulary to students, sometimes they can really get thrown off guard. If you work with a lot of lower level students, title one students, students who don’t feel confident in math, it’s really helpful to identify the things that they’ve already learned. Fifth Grade, fourth grade previously that’s going to help them to understand and master this standard. One thing I like to do is compare ratios to multiplication and division or fractions because what students don’t know is all the time they spent in fourth and fifth grade studying fractions parts to wholes groups of objects in different arrangements is a form of ratios. Once they understand that they already know something about ratios and they’re a little more open to learning something new. I started level one and two, to help students access their background knowledge from fourth and fifth grades.

Level three describes the grade level expectation, which explains that students will be able to use ratio language to describe ratio relationships. They’ll have to understand how to identify the ratio and then put that into a sentence. Level four is a challenge for students who have already mastered grade level skills. Not all of your students will need to go this far, but we’ll get to that later.

At level one I asked students to think about multiplication and division and how to compare amounts in that way. They were asked to do this in fourth grade, so hopefully they’ll feel like this is something pretty manageable. Here’s a story problem to help them demonstrate what they already know.

Pool admission costs a $1.50 per day. A pool pass costs $13.50. How many times more expensive is the pass?

First students have to remember what’s important for them to know. They’re going to have to remember how they worked with money and decimals in the past. They’re going to use multiplication or division and they can make the decision of how to compare these two quantities, $1.50 to $13.50. If you notice they’re struggling, you can tell them how to do it. Say use multiplication, use division, whatever they think is easier. You might find students scribbling pictures on their paper trying to add $1.50 multiple times, whatever they need to do to get through the problem. Seeing how they work out the problem is a great part of the assessment because you can use this as a pre test to determine how much instruction you’ll need to do for them to understand ratios and what kind of background knowledge they may be missing.

Level 2

A level two problem might look like this, I asked students to convert measurements using rates. This was something they were also asked to do in fifth grade. I’m using a similar problem so they can build on what they just did and I can see if they can take the problem one step further. In this problem, pool admission costs a $1.50 a day, a pool pass worth five days cost $6.50.

The students have two questions. They may already be comparing the amounts using multiplication division, which is what we just asked them to do at level one. But now I’m going to ask them to do a little more with it. If you buy a pool pass, how much are you paying per day? We know they’d want to use division to find that out, but they might work backwards and use multiplication. Which option is cheaper? So now they’re comparing amounts and making an evaluation to answer a question with the information.

Why is this a rate? You’re going to want to make sure that your students know the difference between ratios, which is comparing two quantities and rates. See if your students know what the key word is in the problem that identifies that they’re finding a rate. There is a special secret word, a key word, that will tip them off that they’re finding a rate. That word is PER,  “Per Day”.

In fifth grade, students were introduced to the concept of rates, that they’re comparing two quantities, but that the denominator’s always one thing. A $1.50 for every one day, and $6.50 for every one pass. So if the denominators, one item that’s a rate they may or may not remember that. So again, this could be a great pretest to see exactly what vocabulary and details you will have to reteach to make sure that your students can describe a ratio and ratio language accurately at the sixth grade level.

Level 3

Here’s some example of a problem that they would have to do in sixth grade to show mastery of the standard. There’s three different kinds of chocolates in a box, milk chocolate, mint chocolate, and dark chocolate. There’s different quantities of each one in one box. Boxes of chocolate candy come with the three flavors.

Write one sentence describing the ratio of milk to dark chocolate, Write One sentence describing the ratio of mint to dark chocolate. Now this is also a great pre-assessment to see if your students remember how to reduce fractions and ratios. You’re going want to make sure that they have done that in order to get a completely correct answer. If they don’t, you’ll know something else that you want to include in your instruction. Have them do this on a piece of paper. They can do it alone or with a partner, depending how much information you want for your planning and have them turn it in. It’ll be a quick way for you to assess how many students need some background instruction before they’re ready to master this standard.

Level 4

Now if you have students doing really well, there’s a lot of fun ways that you can extend this standard. Have them create their own problems, have them create their own ratios, have them do more complicated ratios in this problem. I’m using the same setup, but I’m going to ask my level four students to create a ratio table listing all of the possibilities.

This is going to be a lot more thinking for them and they’re going to have to make sure that they’re very thorough at finding all the possible combinations of comparison between the three kinds of chocolate and the total amount in each box. Because we can’t have a real ratio without including parts and wholes. Now create three more box sizes with different total amounts using the same ratios. Once your students reduced the ratios, all possible ratios in their table, they can use the same ratios to expand again and make different amounts. They can’t use 1812 and nine again in the same combination, but they might think of a different combination with bigger boxes of chocolates. This could keep your gifted kids busy for a little while in a small group or on their own while you attend to the students who really need your attention in class.

I love planning this way. I love having my scale ready so that if I need to make a quick modification or double check a concept or vocab with a student, I have it right in front of me on a paper on the wall posted in my room and it makes a great pretest post test or a quick reference for me to modify classroom assignments and help my students who really need to fill in some holes.

I hope you found this example helpful. If you would like to see more math scales for sixth grade, please visit my website, Mrslsleveledlearning.com. I have a freebies tab where you can download examples of math and ELA Scales and assessments that are already done for you. The assessments have problems attached to each level of this scale. Just like our examples here, except many more so that you can assess whether or not your students have mastered the grade level standard and which ones need some extra work. I hope you found this example helpful. I’d love to hear any comments or ideas you have. Questions or suggestions. Criticisms are welcome too, and have a great year. Thank you for your time. Yeah.

You can find this resource in my TpT store. Follow the links to download a FREE Leveled Assessment and FREE Scales for this math standard and more!

https://www.teacherspayteachers.com/Product/6th-Grade-Math-Posters-with-Learning-Goals-Scales-RP1-3-Editable-Levels-FREE-3290750?aref=6369kafd

https://www.teacherspayteachers.com/Product/6th-Grade-Math-Assessment-Ratios-Proportions-with-Learning-Goals-and-Scales-1386553?aref=6369kafd

Thank you to Jasmine F. for your feedback!

“One of the best purchases. It has every standard and scaffolds all the standards as far as 4th grade. I use them as quizzes, test, and even pretest or a game. Totally worth it.” – September 2019

Tricia W. said;

“This resource has taught my students more about themselves as learners.” – JANUARY 2016

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Differentiate Sixth Grade Ratios with Proficiency Scales

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Proficiency Scales are all about helping students show growth in small steps instead of being overwhelmed by a year-end goal that seems way over their head. This video will walk you through one way…