# MiddleSchoolPortal/Math Focal Points: Grade 7

<keywords content="middle school mathematics curriculum focal points proportionality similarity ratio proportion percentages surface area volume circumference integers algebraic expressions equations problems lesson" /> <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 7: surface area and volume, proportions, integers and equations." />

## Math Focal Points - Grade 7 - Introduction

With the intention of streamlining the preK-8 mathematics curriculum, the National Council of Teachers of Mathematics (NCTM) has developed Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics. NCTM emphasizes that a curriculum 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 this publication, Math Focal Points: Grade 7, the third in our Middle School Portal series, we offer resources that support the teaching of the three areas of emphasis highlighted for seventh-grade learners. The three focal points and the related sections of resources are:

NCTM recommends that students at this level should develop an understanding of and apply proportionality, including scale factor, percentage, and unit rate problems. Resources in the section titled ratio and proportion deal with real-world situations, such as finding percentages and building scale models, as well as online scenarios that help students visualize the mathematical concepts involved.

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In grade 7, students should develop an understanding of and use formulas for surface area and volume of three-dimensional shapes, and apply them in problems involving prisms and cylinders. In this publication, we offer measurement resources that will engage students in work with not only surface area and volume but also circumference and areas of circles, topics that also figure in this area of emphasis.

Students in grade 7 should develop as well an understanding of operations on all rational numbers and solutions of linear equations. In the integers and algebraic expressions section, the activities range from operations on the concepts to building mathematical models in algebraic language to solving equations. You will find a variety of formats as well: tutorials, well-constructed lesson plans, interactive problem solving, even games!

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

The first publications in our Middle School Portal series featured teaching of the Math Focal Points: Grade 5 and Math Focal Points for Grade 6.

NCTM Curriculum Focal Points for Grade 7

Number and Operations and Algebra and Geometry: Developing an understanding of and applying proportionality, including similarity. Students extend their work with ratios to develop an understanding of proportionality that they apply to solve single and multistep problems in numerous contexts. They use ratio and proportionality to solve a wide variety of percent problems, including problems involving discounts, interest, taxes, tips, and percent increase or decrease. They also solve problems about similar objects (including figures) by using scale factors that relate corresponding lengths of the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and identify the unit rate as the slope of the related line. They distinguish proportional relationships (y/x = k, or y = kx) from other relationships, including inverse proportionality (xy = k, or y = k/x).

Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areas and volumes of three-dimensional shapes. By decomposing two- and three-dimensional shapes into smaller, component shapes, students find surface areas and develop and justify formulas for the surface areas and volumes of prisms and cylinders. As students decompose prisms and cylinders by slicing them, they develop and understand formulas for their volumes (Volume = Area of base × Height). They apply these formulas in problem solving to determine volumes of prisms and cylinders. Students see that the formula for the area of a circle is plausible by decomposing a circle into a number of wedges and rearranging them into a shape that approximates a parallelogram. They select appropriate two- and three dimensional shapes to model real-world situations and solve a variety of problems (including multistep problems) involving surface areas, areas and circumferences of circles, and volumes of prisms and cylinders.

Number and Operations and Algebra: Developing an understanding of operations on all rational numbers and solving linear equations. Students extend understandings of addition, subtraction, multiplication, and division, together with their properties, to all rational numbers, including negative integers. By applying properties of arithmetic and considering negative numbers in everyday contexts (e.g., situations of owing money or measuring elevations above and below sea level), students explain why the rules for adding, subtracting, multiplying, and dividing with negative numbers make sense. They use the arithmetic of rational numbers as they formulate and solve linear equations in one variable and use these equations to solve problems. Students make strategic choices of procedures to solve linear equations in one variable and implement them efficiently, understanding that when they use the properties of equality to express an equation in a new way, solutions that they obtain for the new equation also solve the original equation.

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

These professional resources come from Learning Math, a series of online workshops developed for elementary and middle school teachers by WGBH-Annenberg Media. Each course includes video, problem-solving activities, and case studies that show you how to apply what you have learned in your own classroom. You can use the material for personal review of a topic, but you may profit more from working on it with colleagues. You will find that group discussion is built into the workshop format.

Solving Equations In this online workshop session, K-8 teachers explore different methods for solving equations. You would investigate, in depth, the meaning of the equal sign, equality and balance, and various strategies for solving equations.

Proportional Reasoning In another algebra session, teachers work together to develop proportional reasoning skills. You would investigate ideas of scale, explore what is meant by a proportion, and interpret graphs showing direct variation. Created for K-8 teachers, the session includes online video, class problems, and case studies for discussion.

Similarity 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.

Volume Consider volume from its basic definition and, most important for middle school teaching, explore how volume formulas are derived and related to one another. Problems and their solutions, questions for group discussion, and even a video of the workshop session are available in this online workshop.

Surface Area and Volume Do prisms with the same volume have the same surface area? Here you can explore the relationship between surface area and volume through insightful problems and their solutions. A particularly interesting section deals with human measurements, such as the surface area of the body in relation to its volume.

## Ratio and Proportion

Students extend their work with ratios to develop an understanding of proportionality that they apply to solve single and multistep problems in numerous contexts. They use ratio and proportionality to solve a wide variety of percent problems, including problems involving discounts, interest, taxes, tips, and percent increase or decrease. They also solve problems about similar objects (including figures) by using scale factors that relate corresponding lengths of the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and identify the unit rate as the slope of the related line. They distinguish proportional relationships (y/x = k, or y = kx) from other relationships, including inverse proportionality (xy = k, or y = k/x).

The problems here deal with ratio, in the concrete as well as the abstract. Middle school learners will make actual scale models with paper or clay and find percentages in real-world situations. But they will also work hands-on with online images that make visual the abstractions of ratio and percentage.

Neighborhood Math Questions and hands-on work guide your class in these excellent investigative activities. In Math at the Mall, students calculate the ratio used in making a scale map of the mall, then figure areas and percentage of mall space for each category of shop. Math in the Park or City involves setting up proportions to find heights of buildings. In Gearing Up, students compare the ratio of the turn of the wheel to the turn of the pedal in various bikes.

Capture-recapture : how many fish in the pond? A real application of the ideas of proportion! To estimate the number of fish in a pond, scientists tag a number of them and return them to the pond. The next day, they catch fish from the pond and count the number of tagged fish recaptured. From this, they can set up a proportion to make their estimation. Hints on getting started are given, if needed, and the solution explains the setup of the proportion.

The full question is: "The arm of the Statue of Liberty is 42 feet. How long is her nose?" To answer the question, students first find the ratio of their own arm length to nose length and then apply their findings to the statue's proportions. The solution sets out different approaches to the problem, including the mathematics involved in determining proportion. Extension problems deal with shrinking a T-shirt and the length-to-width ratios of cereal boxes.

What's My Ratio? What would happen to a picture in the pocket of someone who is shrunk or enlarged? This question hooks students into a study of similar figures. As they compare the measurements of corresponding parts of pictures that have been either decreased or increased in size, they can investigate concepts of similarity, constant ratio, and proportionality.

Percentages (grades 9-12) In this interactive activity, students can enter any two of these three numbers: the whole, the part, and the percentage. The missing number is not only calculated but the relationship among the three is illustrated as a colored section of both a circle and a rectangle. The exercise is an excellent help to understanding the meaning of percentage.

Majority vote : what percentage does it take to win a vote? This problem challenges students' understanding of percentage. Two solutions are available, plus hints for getting started. Clicking on "Try these" leads to different but similar problems on percentage. Questions under "Did you know?" include Can you have a percentage over 100? and When can you add, subtract, multiply, or divide percentages? These questions can lead to interesting math conversations.

Perplexing percentages : so how much does it cost? A problem straight from the mall! Here is a rack of clothing, originally on sale for 30 percent off the original price, but now discounted by an additional 50 percent. Is the new price actually 80 percent of the original price? Two complete solutions are set out, and several more problems in the shopping scenario are offered under "Try these."

Math-Kitecture Math-Kitecture is about using architecture to do math (and vice versa). The author provides activities that engage students in doing real-life architecture while learning estimation, measuring skills, proportion, and ratios. In Floor Plan Your Classroom, for example, students make a rough sketch of the classroom, followed by a more exact scale drawing. Scaling the pyramids Students compare the Great Pyramid to such modern structures as the Statue of Liberty and the Eiffel Tower. The site contains all the information needed, including a template, to construct a scale model of the Great Pyramid. In other activities, they must find the scale heights for the tallest building in their neighborhood and create models for two other pyramids, given only their dimensions. A beautifully illustrated site!

Figure and Ratio of Area A page shows two side-by-side grids, each with a blue rectangle inside. Students can change the height and width of these blue rectangles and then see how their ratios compare — not only of height and width but also, most important, of area. The exercise becomes most impressive visually when a tulip is placed inside the rectangles. As the rectangles' dimensions are changed, the tulips grow tall and widen or shrink and flatten. An excellent visual!

What's round, hard, and sold for 3 million dollars? This activity challenges students to determine which is worth more today: Babe Ruth's 1927 home-run record-breaking ball or Mark McGwire's 70th home-run record-breaking ball that sold in 1999 for three million dollars. The activity involves compound interest and rate of change.

Grid and Percent It Using a 10 x 10 grid, students first represent simple percents, then move to percents less than 1 and greater than 100. Problems involving percent increase and decrease are illustrated on grids, offering visuals that reinforce the instruction in this one-period lesson.

Scaling Away For this one-period lesson, students bring to class either a cylinder or a rectangular prism, and their knowledge of how to find surface area and volume. They apply a scale factor to these dimensions and investigate how the scaled-up model has changed from the original. Activity sheets and overheads are included, as well as a complete step-by-step procedure and questions for class discussion.

## Surface Area and Volume

By decomposing two- and three-dimensional shapes into smaller, component shapes, students find surface areas and develop and justify formulas for the surface areas and volumes of prisms and cylinders. As students decompose prisms and cylinders by slicing them, they develop and understand formulas for their volumes (Volume = Area of base × Height). They apply these formulas in problem solving to determine volumes of prisms and cylinders. Students see that the formula for the area of a circle is plausible by decomposing a circle into a number of wedges and rearranging them into a shape that approximates a parallelogram. They select appropriate two- and three dimensional shapes to model real-world situations and solve a variety of problems (including multistep problems) involving surface areas, areas and circumferences of circles, and volumes of prisms and cylinders.

These activities involve middle school learners in scenarios that challenge them to apply their understanding of measurement of solids. They will also work on circumference and circle area problems, included in this area of emphasis. Keeping cool : when should you buy block ice or crushed ice?

Keeping cool : when should you buy block ice or crushed ice? Which would melt faster: a large block of ice or the same block cut into three cubes? The prime consideration is surface area. A complete solution demonstrates how to calculate the surface area of the cubes as well as the large block of ice. Related problems involve finding surface area and volume for irregular shapes and examining the relationship between surface area and volume in various situations.

Cordwood This problem scenario, set in Alaska, asks students to find the volume of a shed, applying the standard formula, and then to determine the number of cords of wood needed to fill it. Finally, they must calculate the cost of the wood.

Popcorn : if you like popcorn, which one would you buy? Students are directed to use popcorn to compare the volumes of tall and short cylinders formed with 8-by 11-inch sheets of paper. A simple but visual and motivating way of comparing volume to height in cylinders! The solution offered explains clearly all the math underlying the problem.

Windshield wipers : it's raining! Who sees more? The driver of the car or the truck? In this activity, students compare the areas cleaned by different wiper designs. An animation shows the movement of the two windshield wipers, each cleaning off a different geometric shape on the window. Students are encouraged to draw the shape cleaned by each wiper and find its area.

Big tree : have you ever seen a tree big enough to drive a car through? Students must consider the girth and height of ten National Champion trees and determine which, if any, of the trees is large enough to drive a car through. The solution relies on knowledge of the formula for finding the circumference of a circle. MSP full record

Surface area and volume This applet enables students to form and rotate both rectangular and triangular prisms. They can set the dimensions (width, depth, and height), observing how each change in dimension affects the shape of the prism as well as its volume and surface area. This is a quick way to collect data for a discussion of the relationship between surface area and volume. Users can rotate the figure and call for its frontal, side, or back view — very interesting with a triangular prism!

Three dimensional box applet : working with volume With this applet, students create boxes online in order to explore the relationship between volume and surface area. The screen first shows a rectangular piece of graph paper. Students "cut" four squares of a size determined by the student from the corners of the rectangle. The cut surface then folds to form a box whose dimensions, surface area, and volume are displayed onscreen. Since various sizes of graph paper can be selected, data can quickly be collected and the relationship between volume and surface area explored.

Measuring the Circumference of the Earth Here is a real-world project that will engage your class in measuring the circumference of the Earth! You will find all the information you need to enable students to re-create the measurement as done by the Greek librarian Eratosthenes over 2,000 years ago. The procedure is based on measurements of shadows taken at high noon local time on a designated day in March; results from several schools are posted online and used to calculate the circumference. Included are detailed explanations and illustrations of the mathematics involved.

## Integers and Algebra

Students extend understandings of addition, subtraction, multiplication, and division, together with their properties, to all rational numbers, including negative integers. By applying properties of arithmetic and considering negative numbers in everyday contexts (e.g., situations of owing money or measuring elevations above and below sea level), students explain why the rules for adding, subtracting, multiplying, and dividing with negative numbers make sense. They use the arithmetic of rational numbers as they formulate and solve linear equations in one variable and use these equations to solve problems. Students make strategic choices of procedures to solve linear equations in one variable and implement them efficiently, understanding that when they use the properties of equality to express an equation in a new way, solutions that they obtain for the new equation also solve the original equation.

You will find that the first five activities require students to apply the properties of arithmetic to all rational numbers, including negative integers. Beginning with the activity titled “Building Bridges” below, the resources center on creating algebraic models of mathematical scenarios. These may appear as games or puzzles, but each moves from a concrete problem to an abstract representation. The last two activities deal directly with solving equations.

Integer Arithmetic These guided, interactive activities use number lines, color chips, and a variety of scenarios to help students understand what an integer is and how to do signed addition, subtraction, and multiplication.

eNLVM : Interactive online math lessons Check out this online tutorial offering a pre-test, practice exercises, and a post-test. It is especially handy as a quick review for students working independently.

Late delivery In this game, the student helps the mail carrier deliver five letters to houses with numbers such as 3(a + 2) and (2a + 5)/5. The value of ais held by the dog. This is a good exercise in substituting for variables. Three levels of difficulty are available; levels 2 and 3 are most appropriate for seventh-grade learners.

Number Line Bounce (grades 6-8) This number line game challenges the student to find a sequence of operations with four numbers that results in a given target number. The numbers are illustrated as bouncing balls on a number line. Each bounce can be in either a positive or negative direction. The student can use a guess-and-check approach to solving the problem or a more sophisticated strategy. In the final step, the student forms the number sentence that illustrates the sequence of operations used to arrive at the target number.

Algebraic Factoring An excellent set of lesson plans introduces factoring through finding areas of rectangles. Each step in the procedure is well explained and illustrated. Questions for the class are included. This unit is meant to be worked with algebra tiles, either the usual plastic ones or cut-out paper shapes.

Building Bridges Designed expressly for middle school classes, this lesson is built on the premise that "teachers need help in building a bridge between their current instructional goals and new goals that emphasize an earlier introduction to algebraic thinking." As students work through tasks, they organize values into tables and graphs as they move toward symbolic representations of the functions involved. The problem situations, carefully explained, employ linear, quadratic, and exponential models.

Rectangle Pattern Challenges Students analyze a colorful rectangular pattern, composed of red, green, and blue squares, and find the number of squares of each color as the rectangle grows. Again, the goal is to express the general patterns algebraically in terms of n.

Function machine (grades 6-8) Applying a machine metaphor for the critical concept of function, this virtual manipulative allows the learner to examine the relationship between input (domain) and output (range). The learner inputs numbers from 1 to 4 and the virtual machine generates output information in a table. At this point, the student must find the output for numbers 5 to 7; in other words, the function rule. Using a new function button, different types of functions are randomly offered for investigation.

Hop to It! This excellent lesson emphasizes establishing patterns and developing general rules. A pre-assessment problem asks: How many small triangles are contained in a sequence of increasingly large similar triangles? The core problem of the lesson asks: How can 10 frogs lined up on the left swap places with 10 frogs lined up on the right? For each problem, students work in small groups to devise a model and recording system, list their findings, and use the pattern they find to write a general rule that solves the problem. Well-illustrated handouts and solutions are included.

Algebra balance scales : negatives (grades 6-8) This online manipulative features a virtual balance scale. The activity offers students an experimental way to learn about solving linear equations involving negative numbers. The applet presents an equation for students to illustrate by balancing the scale, using blue blocks for positive units and variables and red balloons for negative units and variables. Students then work with the arithmetic operations to solve the equation. A record of the steps taken by the student is shown on the screen and on the scale. The applet reinforces the idea that what is done to one side of an equation must be done to the other side to maintain balance.

Equation Match Students must solve equations, from the most simple to the more complex, and in this way find pairs of equations that "match"; that is, both equations in the pair have the same value of x. When a match is found, part of a picture is revealed. Levels 2 and 3 require multistep solutions. At each return to the game, a new set of equations is given.

## 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.

## Careers

The FunWorks Visit the FunWorks STEM career website to learn more about a variety of math-related careers (click on the Math link at the bottom of the home page).

## NCTM Standards

You may be wondering, What do the curriculum focal points have to do with the Principles and Standards for School Mathematics (PSSM)? The National Council of Teachers of Mathematics (NCTM) says that identifying areas of emphasis at each grade level is the next step in implementing those principles and standards. According to NCTM, Curriculum Focal Points for Prekindergarten Through Grade 8: A Quest for Coherence "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" (p.12).

The curriculum focal points draw on the content standards described in PSSM, at times clustering several topics in one focal point. Also, the process standards are pivotal to well-grounded instruction, for "it is essential that these focal points be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations" Curriculum Focal Points for Grade 7

This Middle School Portal publication offers resources intended to support you in teaching the key mathematical areas identified for grade 7: ratio and proportion, measurement of three-dimensional shapes, operations on integers, and solving linear equations. In particular, we aimed to provide a variety of formats (lesson plans, games, interactive problem solving, hands-on projects) for your use in teaching these critical topics. We hope the resources will engage your seventh graders in moving from concrete situations to more abstract formulas and algebraic modeling as they work through the range of activities.

The full description of the Curriculum Focal Points for Grade 7 is available at http://www.nctm.org/standards/focalpoints.aspx?id=338&ekmensel=c580fa7b_10_52_338_9

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

## Author and Copyright

Terry Herrera taught math several years at middle and high school levels, then earned a Ph.D. in mathematics education. She is a resource specialist for the Middle School Portal 2: Math & Science Pathways project.

Please email any comments to msp@msteacher.org.

Connect with colleagues at our social network for middle school math and science teachers at http://msteacher2.org.

Copyright May 2008 - The Ohio State University. This material is based upon work supported by the National Science Foundation under Grant No. 0424671 and since September 1, 2009 Grant No. 0840824. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.