# MiddleSchoolPortal/Math Focal Points: Grade 6

<keywords content="middle school mathematics focal points division multiplication fractions decimals rate ratio concepts skills interactive activities teaching curriculum algebra learning" /> <metadescription content="This free, standards-based, online publication developed for teachers features digital resources that support the teaching of the NCTM Curriculum Focal Points for grade 6: operations on fractions and decimals, ratio and rate, algebraic expressions." />

## Math Focal Points - Grade 6 - Introduction

In an effort to highlight the most important mathematical topics at each grade level, the National Council of Teachers of Mathematics (NCTM) has developed Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics. This second in a series of Middle School Portal publications supports the teaching of the topics selected for grade 6. The first in the series featured teaching of the focal points for grade 5.

NCTM emphasizes that a “focal point” is a “cluster of related knowledge, skills, and concepts,” rather than a discrete topic to be checked off a list. The focal points, three at each grade level, specify “the mathematical content that a student needs to understand deeply and thoroughly for future mathematics learning.”

In the sections of this publication we offer resources on the three areas of emphasis highlighted for sixth grade. (For a complete statement of the NCTM Curriculum Focal Points for grade 6, please see below.)

NCTM recommends that students at the sixth-grade level work on developing an understanding and mastery of multiplication and division of fractions and decimals.. In the section titled Fractions and Decimals, you will find visual and interactive explanations of the difficult concepts underpinning multiplication and division of fractions and decimals, plus opportunities for your students to practice their skills.

## Contents

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Students at this level should also be connecting ratio and rate to multiplication and division. The games, problems, and lesson ideas in Ratio and Rate are intended to help students extend their understanding of equivalent fractions to ratio and rate and their mastery of whole number multiplication and division to solve problems that use these concepts.

In grade 6, students should also be writing, interpreting, and using mathematical expressions and equations. The resources in Expressions and Equations focus on work with variables and simple equations: writing and evaluating expressions, solving one-step equations, or applying algebraic thinking to problems.

In Background Information for Teachers, you will find professional learning resources. Finally, in NCTM Standards we discuss the focal points as they are related to the Principles and Standards for School Mathematics.

We hope the resources in this and the other publications in this series open your classes to a wider view of mathematics!

NCTM Curriculum Focal Points for Grade 6

Number and Operations: Developing an understanding of and fluency with multiplication and division of fractions and decimals. Students use the meanings of fractions, multiplication and division, and the inverse relationship between multiplication and division to make sense of procedures for multiplying and dividing fractions and explain why they work. They use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain the procedures for multiplying and dividing decimals. Students use common procedures to multiply and divide fractions and decimals efficiently and accurately. They multiply and divide fractions and decimals to solve problems, including multistep problems and problems involving measurement.

Number and Operations: Connecting ratio and rate to multiplication and division. Students use simple reasoning about multiplication and division to solve ratio and rate problems (e.g., “If 5 items cost \$3.75 and all items are the same price, then I can find the cost of 12 items by first dividing \$3.75 by 5 to find out how much one item costs and then multiplying the cost of a single item by 12”). By viewing equivalent ratios and rates as deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings that indicate the relative sizes of quantities, students extend whole number multiplication and division to ratios and rates. Thus, they expand the repertoire of problems that they can solve by using multiplication and division, and they build on their understanding of fractions to understand ratios. Students solve a wide variety of problems involving ratios and rates.

Algebra: Writing, interpreting, and using mathematical expressions and equations. Students write mathematical expressions and equations that correspond to given situations, they evaluate expressions, and they use expressions and formulas to solve problems. They understand that variables represent numbers whose exact values are not yet specified, and they use variables appropriately. Students understand that expressions in different forms can be equivalent, and they can rewrite an expression to represent a quantity in a different way (e.g., to make it more compact or to feature different information). Students know that the solutions of an equation are the values of the variables that make the equation true. They solve simple one-step equations by using number sense, properties of operations, and the idea of maintaining equality on both sides of an equation. They construct and analyze tables (e.g., to show quantities that are in equivalent ratios), and they use equations to describe simple relationships (such as 3x = y) shown in a table.

Reprinted with permission from Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence, copyright 2006 by the National Council of Teachers of Mathematics. All rights reserved.

## Background Information for Teachers

If you are looking for teaching ideas on fractions, you may want to browse through the materials collected by the Math Forum in the first resource entry here. The following three resources in this section offer insights into the arithmetic and algebra underlying the Focal Points for sixth graders. Each is a free, online workshop session developed for K-8 teachers by Learning Math.

Fractions : Middle School Lessons and Materials for Teachers The lessons and materials available here come from a variety of sites and organizations. Some include offers for video or software, but most are lesson plans, activities, and practice exercises. Topics include operations with fractions, fractions and decimals, fractions and algebra, and equivalent fractions.

Models for the Multiplication and Division of Fractions In this resource, part of a workshop session, you work with area models to demonstrate visually what happens when you multiply or divide fractions. Visual and insightful! After hands-on practice with these models, the lesson continues with a common denominator model to connect division of fractions with the actual procedure we all use.

Algebraic Thinking In this first session of a workshop for teachers, participants consider the role of algebra as a thinking tool. You will work with ways to describe and represent mathematical situations through pictures, charts, graphs, or words.

Patterns in Context This is the second session of the workshop described above. In these sessions, teachers explore the uses of variables in describing patterns and relationships. It’s more interesting than it sounds! You will find that algebra is one language used to describe and explain patterns that may look at first like only random facts. Using the tools of algebra can help us to reason and make sense of situations.

## Fractions and Decimals

Students use the meanings of fractions, multiplication and division, and the inverse relationship between multiplication and division to make sense of procedures for multiplying and dividing fractions and explain why they work. They use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain the procedures for multiplying and dividing decimals. Students use common procedures to multiply and divide fractions and decimals efficiently and accurately. They multiply and divide fractions and decimals to solve problems, including multistep problems and problems involving measurement (NCTM, 2006, p.18).

These resources offer support in explaining the difficult concepts underpinning multiplication and division of both fractions and decimals. Visual, interactive models are provided where possible. You will also find opportunities for your students to practice their skills in this area of arithmetic.

Multiplication of Fractions Visualize and practice multiplying fractions using an area representation. With the “Show Me” option selected, the virtual manipulative is used to graphically demonstrate, explore, and practice multiplying fractions. A rectangular grid, representing a whole, shows the areas of two fractions to be multiplied, one fraction in red on the left and another in blue at the bottom. The area of the overlapping region, shown in purple, represents the product of their multiplication. The “Test Me” option provides problems to be solved using the same graphical representation.

Multiplying Fractions This tutorial site offers instruction as well as practice in multiplication of fractions. The fractions are modeled with either circles or lines (rectangular areas). The visual display matched with the numerical makes an effective demonstration.

Divide Fractions This tutorial site offers instruction as well as practice in multiplication of fractions. The fractions are modeled with either circles or lines (rectangular areas). The visual display matched with the numerical makes an effective demonstration.

Divide and Conquer This lesson is based on the idea that middle school students can better understand the procedure for dividing fractions if they analyze division through a sequence of problems. Students start with division of whole numbers, followed by division of a whole number by a unit fraction, division of a whole number by a non-unit fraction, and finally division of a fraction by a fraction. Activity sheets and guiding questions are included.

Dividing Fractions Students divide fractions using area models. They can adjust the numerators and denominators of the divisor and dividend and see how the area model and calculation change. Full access to ExploreLearning is available through an annual subscription, but you can apply for a month’s free access in order to test out the applets.

Decimals This site has explanatory lessons and interactive practice on most aspects of decimals, including multiplying decimals and dividing them.

Learning about Multiplication Using Dynamic Sketches of an Area Model In this applet, a rectangle represents the familiar area model of multiplication. By changing the height of the rectangle, students can explore the effect of multiplying a fixed positive number, in this case 3, by decimal numbers greater than 1 and less than 1. The visual is powerful!

Too Big or Too Small? Of these three activities on developing number sense, go to Activity 3: Exploring the Effect of Operations on Decimals. Through playing the cleverly crafted game presented here, students explore the effect of addition, subtraction, multiplication, and division on decimal numbers. Students begin with the number 100 as they enter a maze. For each segment chosen on the maze, the student keys in the assigned operation and number; for example, “+ 1.2” or “? 0.8.” The goal is to choose a path through the maze that results in the largest value at the finish. A copy of the maze playing board is included.

## Ratio and Rate

Number and Operations: Connecting ratio and rate to multiplication and division.

Students use simple reasoning about multiplication and division to solve ratio and rate problems (e.g., “If 5 items cost \$3.75 and all items are the same price, then I can find the cost of 12 items by first dividing \$3.75 by 5 to find out how much one item costs and then multiplying the cost of a single item by 12”). By viewing equivalent ratios and rates as deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings that indicate the relative sizes of quantities, students extend whole number multiplication and division to ratios and rates. Thus, they expand the repertoire of problems that they can solve by using multiplication and division, and they build on their understanding of fractions to understand ratios. Students solve a wide variety of problems involving ratios and rates (NCTM, 2006, p.18).

Here you will find games, problems, and lesson ideas that can help middle school students extend their understanding of equivalent fractions and their mastery of whole-number multiplication and division to the concepts of ratio and rate. Communicating about Mathematics Using Games: Playing Fraction Track

Center for Digital Curriculum Research In this math game, students think about how fractions are related to a unit whole, compare fractional parts of a whole, and find equivalent fractions. Building on this experience, a teacher can help students see the connection between equivalent fractions and ratios.

These two lessons are based on an online simulation of two runners along a track. Users can change the step size of each runner, control starting points, and examine a time-versus-distance graph created as the runners move. As students analyze each situation, they learn about constant rates of change and their description as formulas relating distance, speed, and time. Questions for guiding the lesson and questions for assessment are included.

Which tastes juicier? Students are challenged to decide which of four cans of grape juice concentrate requiring different amounts of water would have the strongest grape juice taste. A hint suggests forming ratios that are fractions to compare quantities. Two solutions are given, each fully illustrated with tables. Students are then offered further mixture-related questions.

Drip drops : how much water do you waste? A leaky faucet is dripping at the rate of one drop every two seconds. Students are asked to decide if the water lost in one week would fill a drinking glass, a sink, or a bathtub. The only hint is that a teaspoon holds about 20 drops. The solution demonstrates how to convert the drops to gallons using an equation or a table. Students then consider: How much water is lost in one year by a single leaky faucet? By two million leaky faucets?

How far can you go on a tank of gas? Which car will go the farthest on a single tank of gas? Students are given the mileage and gasoline tank capacity of three models of automobiles. They are encouraged to begin the problem by calculating how far each car could go in the city and on the highway. In follow-up problems, students compare the fuel efficiency of two sports cars and calculate how often a commuter would need to refuel.

Understanding Rational Numbers and Proportions To work well with ratios, learners need a solid basis in the idea of rational numbers. This complete lesson includes three well-developed activities that investigate fractions, proportion, and unit rates — all through real-world problems students encounter at a bakery.

Constant dimensions This lesson plan requires students to measure the length and width of a rectangle using both standard and nonstandard units of measure, such as pennies and beads. As students graph the ordered pairs, they discover that the ratio of length to width of a rectangle is constant, in spite of the units. This leads to the definition of a linear function and to the rule that relates the dimensions of the rectangle. An activity sheet and questions for class discussion are included.

## Expressions and Equations

Students write mathematical expressions and equations that correspond to given situations, they evaluate expressions, and they use expressions and formulas to solve problems. They understand that variables represent numbers whose exact values are not yet specified, and they use variables appropriately. Students understand that expressions in different forms can be equivalent, and they can rewrite an expression to represent a quantity in a different way (e.g., to make it more compact or to feature different information). Students know that the solutions of an equation are the values of the variables that make the equation true. They solve simple one-step equations by using number sense, properties of operations, and the idea of maintaining equality on both sides of an equation. They construct and analyze tables (e.g., to show quantities that are in equivalent ratios), and they use equations to describe simple relationships (such as 3x = y) shown in a table (NCTM, 2006, p. 18).

Working with variables and simple equations is the focus in these resources. Each engages middle school students in an interactive experience in which they write and evaluate expressions, work with one-step equations, or apply algebraic thinking to problems.

Late delivery This interactive math game provides exercises in substituting for variables. Players must help the mail carrier deliver letters to houses with addresses like 3(a + 2). The value of a is held by Dougal, the dog guarding the house. The algebraic expressions become more complex according to the level of difficulty selected. Tips for students are available as well as an explanation of the key ideas underlying the game.

Number tricks : bet I can guess your color! This activity opens with a cartoon in which two characters present students with directions for arriving at a special number. Students then correlate their special number with a letter of the alphabet and choose a color that begins with that letter. The characters bet that the color is yellow, and students are challenged to discover why the bet would be correct. The activity introduces algebraic procedures by demonstrating an algebraic representation in which the students' special number is added and subtracted in such a way that the operations do not affect the answer. Related questions present additional number tricks for students to solve.

Pan Balance - Expressions This interactive pan balance allows students to enter numeric or algebraic expressions. The students can "weigh" the expressions by entering them on either side of the balance and seeing if they are equivalent. The Exploration section offers teachers an interesting class activity.

Function machine (grades 6-8) Operating under a secret rule, the function machine uses numbers input by the students to generate output. Students compare the input (domain) to the output (range) to find the function rule. The analogy of a function machine is a basic, strong visual that holds up even in advanced study of functions. MSP full record

The Handshake Problem This two-lesson unit allows students to discover patterns in a fictional but real-world scenario: How many handshakes occur when the nine Supreme Court justices shake hands with each other? Students explore through a table, a graph, and finally an algebraic formula—the number of handshakes in any size group. A second pattern is explored, that of triangular numbers; again, students generalize the pattern with variables. The lessons are well illustrated and include background information for the teacher. (From Illuminations, National Council of Teachers of Mathematics Vision for School Mathematics — MSP full record)

Body mass? : what's your index? In this activity, students learn about the body mass index (BMI) formula and how it can be used to determine health risk. Essentially, this is an exercise in substituting values into an algebraic expression. The page contains links to a solution hint, the solution, and other math questions, such as how the BMI formula can be written to show weight in kilograms and height in centimeters. Students and their families are challenged to use the BMI formula to determine their health risks.

Smiles : which is worth more, a smile or a frown? This activity offers a logic problem in which students are shown an array of smiling, frowning, and neutral faces. Each row and each column add up to a different dollar amount, and students are challenged to determine how much a smile is worth. As noted on the page, "Reasoning about unknowns is essential to studying equations."

Equation Match Students find the matching pairs of equations by working out the value of x in each equation. Level 1 offers a set of one-step equations, which changes on each return to the game. This is an excellent interactive game.

Variables and Unknowns This is a PowerPoint lesson on variables, designed with the middle schooler in mind. The lesson uses color, sound, and animation to illustrate necessary concepts and procedures. Students are shown how to substitute the value of a variable in place of the variable's symbol in an algebraic expression. The concept of solving an equation for an unknown variable value is also presented.

Traffic Jam Activity The activity: there are seven stepping stones and six people, three on the left-hand stones, facing the center, and three on the right-hand stones. Everyone must move so that the people originally standing on the right exchange places with those on the left. How many moves does it take? What if there are eight people? 10? Students explore the situation through large movement experience, manipulatives, and an interactive Java applet. They look for patterns and, most importantly, write their conclusion algebraically. You’ll find the lesson plan well explained and complete.

Reaching New Heights Measuring, collecting and interpreting data, using variables—this complete lesson has it all! Students explore the relationship between two variables—height and arm span—then create a scatterplot of the data, and discuss the graph. This lesson is an excellent way to build the foundation for the study of functions. Teacher support and information about related research are included.

## SMARTR: Virtual Learning Experiences for Students

Visit our student site SMARTR to find related math-focused virtual learning experiences for your students! The SMARTR learning experiences were designed both for and by middle school aged students. Students from around the country participated in every stage of SMARTR’s development and each of the learning experiences includes multimedia content including videos, simulations, games and virtual activities.

## NCTM Standards

As mentioned in the first of this series, a question that naturally arises is: What do the curriculum focal points have to do with the Principles and Standards for School Mathematics? The National Council of Teachers of Mathematics answers that identifying areas of emphasis at each grade level is the next step in implementing the principles and standards. Curriculum Focal Points for Prekindergarten Through Grade 8 “provides one possible response to the question of how to organize curriculum standards within a coherent, focused curriculum, by showing how to build on important mathematical content and connections identified for each grade level, pre-K–8” (NCTM, 2006, p. 12).

The Curriculum Focal Points draw on the content standards described in the Principles and Standards, at times clustering several topics in one focal point. At each grade level, the Council reiterates that the process standards are pivotal to well-grounded instruction, for, according to the Curriculum Focal Points for Grade 6, “it is essential that these focal points be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations."

This Middle School Portal publication offers resources intended to support you in teaching the key mathematical areas identified for grade 6. In particular, we aimed to provide visual and interactive models for the multiplication and division of fractions, lesson ideas and problems for introducing ratio and rate through whole number operations, and innovative approaches to early work with algebra. Our focus throughout has been on students’ understanding of the concepts underpinning standard procedures in arithmetic and algebra as well as providing problems that would engage a sixth grader.

A description of the Curriculum Focal Points for grade 6 is found at http://www.nctm.org/standards/focalpoints.aspx?id=336&ekmensel=c580fa7b_10_52_336_8

Curriculum Focal Points for Prekindergarten Through Grade 8: A Quest for Coherence may be viewed in its entirety at http://www.nctm.org/standards/content.aspx?id=270