Rational Numbers

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Solve problems with positive rational numbers.
Plot positive rational numbers on a number line.
Understand the equal sign.
Use the greater than and less than symbols with positive numbers (not variables) and understand their relative positions on a number line.
Recognize the first quadrant of the coordinate plane.

Lesson Flow

The first part of this unit builds on the prerequisite skills needed to develop the concept of negative numbers, the opposites of numbers, and absolute value. The unit starts with a real-world application that uses negative numbers so that students understand the need for them. The unit then introduces the idea of the opposite of a number and its absolute value and compares the difference in the definitions. The number line and positions of numbers on the number line is at the heart of the unit, including comparing positions with less than or greater than symbols.

The second part of the unit deals with the coordinate plane and extends student knowledge to all four quadrants. Students graph geometric figures on the coordinate plane and do initial calculations of distances that are a straight line. Students conclude the unit by investigating the reflections of figures across the x- and y-axes on the coordinate plane.

Gallery OverviewAllow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.Gallery DescriptionThe SurveyorStudents will read and follow a land surveyor's instructions in order to draw property boundaries on a coordinate plane.The Ink BlotOops! Part of a land surveyor’s report is covered by an ink blot. Students will figure out what the missing instructions are in order to draw property boundaries on a coordinate plane.A Mistake in the SurveyThe surveyor made a mistake! Students will correct the mistake in order to draw property boundaries on a coordinate plane.More ReflectionsStudents will reflect a figure located in two quadrants across the x-axis.Reflect Across the OriginStudents will reflect a figure across the origin and observe what happens to the coordinates of the figure.Reflect LettersStudents will reflect letters of the alphabet across the origin and observe what happens to the letters.Graphing a Coordinate Plane VideoStudents will create a video about graphing on the coordinate plane.HistoryStudents will create a report, poster, or video about the history of the coordinate plane.

Ratios

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Calculate with whole numbers up to 100 using all four operations.
Understand fraction notation and percents and translate among fractions, decimal numbers, and percents.
Interpret and use a number line.
Use tables to solve problems.
Use tape diagrams to solve problems.
Sketch and interpret graphs.
Write and interpret equations.

Lesson Flow

The first part of the unit begins with an exploration activity that focuses on a ratio as a way to compare the amount of egg and the amount of flour in a mixture. The context motivates a specific understanding of the use of, and need for, ratios as a way of making comparisons between quantities. Following this lesson, the usefulness of ratios in comparing quantities is developed in more detail, including a contrast to using subtraction to find differences. Students learn to interpret and express ratios as fractions, as decimal numbers, in a:b form, in words, and as data; they also learn to identify equivalent ratios.

The focus of the middle part of the unit is on the tools used to represent ratio relationships and on simplifying and comparing ratios. Students learn to use tape diagrams first, then double number lines, and finally ratio tables and graphs. As these tools are introduced, students use them in problem-solving contexts to solve ratio problems, including an investigation of glide ratios. Students are asked to make connections and distinctions among these forms of representation throughout these lessons. Students also choose a ratio project in this part of the unit (Lesson 8).

The third and last part of the unit covers understanding percents, including those greater than 100%.

Students have ample opportunities to check, deepen, and apply their understanding of ratios, including percents, with the selection of problems in the Gallery.

Students design and work on their projects in class. They review the project rubric and, as a class, add criteria relevant to their specific projects.Key ConceptsStudents apply their knowledge about ratios to solve a problem. They represent ratios using models such as tables, tape diagrams, double number lines, or graphs.Goals and Learning ObjectivesUse and interpret ratios to solve a problem.Model ratios—including tables, tape diagrams, double number lines, graphs—to represent a problem situation.Articulate strategies, thought processes, and approaches to solving a problem and defend why the solution is reasonable.

This lesson introduces the concept of a glide ratio and encourages students to use appropriate tools strategically (Mathematical Practice 5). Students use tape diagrams, double number lines, ratio tables, graphs, and equations to represent glide ratios.Key ConceptsA glide ratio for an object or an organism in flight is the ratio of forward distance to vertical distance (in the absence of power and wind). For a given object or organism that glides, this ratio has a constant value and is treated as a feature of the object or organism.Goals and Learning ObjectivesUnderstand the concept of a glide ratio.Make connections within and between different ways of representing ratios.

Student groups continue to make their presentations, provide feedback for other students’ presentations, and get evaluated on their listening skills.Key ConceptsIn this culminating event, students continue presenting their project plan and solution to the class. The presentation allows students to explain their problem-solving plan, to communicate their reasoning, and to construct a viable argument about a mathematical problem. Students also listen to other project presentations and provide feedback to the presenters. Listeners have the opportunity to critique the mathematical reasoning of others.Goals and Learning ObjectivesPresent project to the class.Give feedback on other project presentations.Exhibit good listening skills.Reflect on the problem-solving process.

Student groups make their presentations, provide feedback for other students’ presentations, and get evaluated on their listening skills.Key ConceptsIn this culminating event, students must present their project plan and solution to the class. The presentation allows students to explain their problem-solving plan, to communicate their reasoning, and to construct a viable argument about a mathematical problem. Students also listen to other project presentations and provide feedback to the presenters. Listeners have the opportunity to critique the mathematical reasoning of others.Goals and Learning ObjectivesPresent project to the class.Give feedback on other project presentations.Exhibit good listening skills.

Students choose a project idea and a partner or group. They write a proposal for the project.Key ConceptsProjects engage students in the applications of mathematics. It is important for students to apply mathematical ways of thinking to solve rich problems. Students are more motivated to understand mathematical concepts if they are engaged in solving a problem of their own choosing. In this lesson, students are challenged to identify an interesting mathematical problem and to choose a partner or a group to work with collaboratively in order to solve that problem. Students gain valuable skills in problem solving, reasoning, and communicating mathematical ideas with others.Goals and Learning ObjectivesIdentify a project ideaIdentify a partner or group to work with collaboratively on a math project

Surface Area and Volume

Type of Unit: Conceptual

Prior Knowledge

Students should be able to:

Identify rectangles, parallelograms, trapezoids, and triangles and their bases and heights.
Identify cubes, rectangular prisms, and pyramids and their faces, edges, and vertices.
Understand that area of a 2-D figure is a measure of the figure's surface and that it is measured in square units.
Understand volume of a 3-D figure is a measure of the space the figure occupies and is measured in cubic units.

Lesson Flow

The unit begins with an exploratory lesson about the volumes of containers. Then in Lessons 2–5, students investigate areas of 2-D figures. To find the area of a parallelogram, students consider how it can be rearranged to form a rectangle. To find the area of a trapezoid, students think about how two copies of the trapezoid can be put together to form a parallelogram. To find the area of a triangle, students consider how two copies of the triangle can be put together to form a parallelogram. By sketching and analyzing several parallelograms, trapezoids, and triangles, students develop area formulas for these figures. Students then find areas of composite figures by decomposing them into familiar figures. In the last lesson on area, students estimate the area of an irregular figure by overlaying it with a grid. In Lesson 6, the focus shifts to 3-D figures. Students build rectangular prisms from unit cubes and develop a formula for finding the volume of any rectangular prism. In Lesson 7, students analyze and create nets for prisms. In Lesson 8, students compare a cube to a square pyramid with the same base and height as the cube. They consider the number of faces, edges, and vertices, as well as the surface area and volume. In Lesson 9, students use their knowledge of volume, area, and linear measurements to solve a packing problem.

Lesson OverviewStudents revise their packing plans based on teacher feedback and then take a quiz.Students will use their knowledge of volume, area, and linear measurements to solve problems. They will draw diagrams to help them solve a problem and track and review their choice of problem-solving strategies.Key ConceptsConcepts from previous lessons are integrated into this assessment task: finding the volume of rectangular prisms. Students apply their knowledge, review their work, and make revisions based on feedback from the teacher and their peers. This process creates a deeper understanding of the concepts.Goals and Learning ObjectivesApply your knowledge of the volume of rectangular prisms.Track and review your choice of strategy when problem-solving.

Gallery OverviewStudents who are caught up can spend this time working on gallery problems while you work with other students in study groups. Students have a choice of problems, and they can work on however many problems time permits.Gallery DescriptionsFinding the Missing BaseStudents will find the length of one of the bases of a trapezoid given the length of the other base, the height, and the area.Utah UnitsStudents will estimate the area of Utah and then estimate the area of the United States in “Utah” units.Growing RectanglesStudents know how to find the lengths of the sides of polygons on the coordinate plane. They will use this knowledge to find the area of each rectangle in a series of growing rectangles.The Volumes of SolidsStudents will find the volume of solids that are built out of cubes. They will also build their own solid out of cubes and have their partner find its volume.From 3-D to 2-D and BackStudents will investigate the net of a milk carton.Geometry of GardeningStudents will use their knowledge of perimeter and area to design a garden on grid paper.Net of a Number CubeStudents will draw a net of a number cube.Dividing ParallelogramsStudents will prove or disprove Emma's statement about dividing parallelograms into four triangles, all with the same area.Area of TrianglesStudents will find the areas of two different triangles on the coordinate plane.Placing a RugStudents will place a square rug exactly in the middle of a floor and find the number of square feet not covered by the rug.

Four full-year digital course, built from the ground up and fully-aligned to the Common Core State Standards, for 7th grade Mathematics. Created using research-based approaches to teaching and learning, the Open Access Common Core Course for Mathematics is designed with student-centered learning in mind, including activities for students to develop valuable 21st century skills and academic mindset.

Algebraic Reasoning

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Add, subtract, multiply, and divide rational numbers.
Evaluate expressions for a value of a variable.
Use the distributive property to generate equivalent expressions including combining like terms.
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?
Write and solve equations of the form x+p=q and px=q for cases in which p, q, and x are non-negative rational numbers.
Understand and graph solutions to inequalities x<c or x>c.
Use equations, tables, and graphs to represent the relationship between two variables.
Relate fractions, decimals, and percents.
Solve percent problems included those involving percent of increase or percent of decrease.

Lesson Flow

This unit covers all of the Common Core State Standards for Expressions and Equations in Grade 7. Students extend what they learned in Grade 6 about evaluating expressions and using properties to write equivalent expressions. They write, evaluate, and simplify expressions that now contain both positive and negative rational numbers. They write algebraic expressions for problem situations and discuss how different equivalent expressions can be used to represent different ways of solving the same problem. They make connections between various forms of rational numbers. Students apply what they learned in Grade 6 about solving equations such as x+2=6 or 3x=12 to solving equations such as 3x+6=12 and 3(x−2)=12. Students solve these equations using formal algebraic methods. The numbers in these equations can now be rational numbers. They use estimation and mental math to estimate solutions. They learn how solving linear inequalities differs from solving linear equations and then they solve and graph linear inequalities such as −3x+4<12. Students use inequalities to solve real-world problems, solving the problem first by arithmetic and then by writing and solving an inequality. They see that the solution of the algebraic inequality may differ from the solution to the problem.

Students write and solve inequalities in order to solve two problems. One of the problems is a real-world problem that involves selling a house and paying the real estate agent a commission. The second problem involves the relationship of the lengths of the sides of a triangle.Key ConceptsIn this lesson, students again use algebraic inequalities to solve word problems, including real-world situations. Students represent a quantity with a variable, write an inequality to solve the problem, use the properties of inequality to solve the inequality, express the solution in words, and make sure that the solution makes sense.Students explore the relationships of the lengths of the sides of a triangle. They apply the knowledge that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side to solve for the lengths of sides of a triangle using inequalities. They solve the inequality for the length of the third side.Goals and Learning ObjectivesUse an algebraic inequality to solve problems, including real-world problems.Use the properties of inequalities to solve an inequality.

Students use inequalities to solve real-world problems. They see that the solution of the algebraic inequality may differ from the solution to the problem it represents. For example, a fractional number or a negative number may not be an appropriate solution for a word problem.Students complete a Self Check. They are given an algebraic inequality that they need to solve. They then write and solve a word problem that the inequality could represent.Key ConceptsIn this lesson, students write and solve an algebraic inequality that matches a situation given in a word problem. They then interpret that algebraic solution in the context of the problem. For example, students write and solve an algebraic inequality to represent the number of T-shirts that can be bought given a certain amount of money and another purchase. The inequality produces the solution t < 2.5. Since a fractional part of a T-shirt does not make sense, students reason that 2 is the greatest number of T-shirts that can be purchased.Goals and Learning ObjectivesInterpret the solution to an algebraic inequality within the context of a word problem.

Students match equations such as 3x − 50 = 90 and 3(x − 50) = 90 to real-world and mathematical situations. They identify the steps needed to solve these equations.Key ConceptsStudents solve equations such as 3x − 50 = 90 by using first the addition property and then the multiplication property of equality.Students also solve equations such as 3(x − 50) = 90. Equations with parentheses were introduced in the Challenge Problem of Lesson 6. Now, in this lesson, students use two methods to solve the equation. First method: use the multiplication property of equality and then the addition property of equality; second method: use the distributive property to eliminate the parentheses, then use the addition property of equality, and then the multiplication property of equality.Goals and Learning ObjectivesMatch equations to problems.Solve two-step equations.

Students discover how the addition and multiplication properties of inequality differ from the addition and multiplication properties of equality.Students use the addition and multiplication properties of inequality to solve inequalities. They graph their solutions on the number line.Key ConceptsIn this lesson, students extend their knowledge of inequalities from Grade 6. In Grade 6, students learned that solving an inequality meant finding which values made the inequality true. Students used substitution to determine whether a given value made an inequality true. They also used a number line to graph the solutions of inequalities. By graphing these solutions on a number line, they saw that an inequality has an infinite number of solutions.Now, in Grade 7, students work with inequalities that also contain negative numbers and learn to solve and graph solutions for inequalities such as −2x − 4 < 5. This involves first understanding how the properties of inequality differ from the properties of equality. When multiplying (or dividing) both sides of an inequality by the same negative number, the relationship between the two sides of the inequality changes, so it is necessary to reverse the direction of the inequality sign in order for the inequality to remain true. Once students understand this, they can apply the same steps they used to solve equations to solve inequalities, but remembering to reverse the direction of the inequality sign when multiplying or dividing both sides of the inequality by a negative number.Goals and Learning ObjectivesAccess prior knowledge of how to solve an inequality.Observe that when multiplying or dividing both sides of an inequality by the same negative number, the inequality sign must change direction.Solve and graph inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers.

Students solve a problem about a salesperson's compensation. They solve the problem first by arithmetic and then by writing and solving an inequality.Key ConceptsIn Lesson 11, students learned how to solve inequalities using the addition and multiplication properties of inequality. In this lesson, they solve word problems by writing and solving inequalities.To help students make connections and see how problems can be solved in different ways, students first solve the same problem using arithmetic.Goals and Learning ObjectivesWrite and solve an algebraic inequality to solve a word problem.

Students extend what they learned about solving equations in Grade 6. They learn to solve equations that require them to use both the addition and the multiplication properties of equality. They use what they know about solving equations such as 2x = 6 and x + 3 = 7 to solve equations such as 2x + 3 = 8. They connect solving problems using arithmetic to solving problems using equations. They solve equations containing both positive and negative rational numbers.Key ConceptsAddition property of equality: If a = b, then a + c = b + c.Multiplication property of equality: If a = b, then ac = bc.For any equation, add or subtract the same value from both sides of the equation and the equation will still be true.For any equation, multiply or divide both sides of the equation by the same value and the equation will still be true.In this lesson, students use both properties to solve equations. They then solve equations that contain both positive and negative rational numbers.Goals and Learning ObjectivesSolve equations using both the addition and multiplication properties of equality.Relate solving problems using arithmetic to solving problems using equations.Solve equations containing both positive and negative rational numbers.

Getting Started

Type of Unit: Introduction

Prior Knowledge

Students should be able to:

Understand ratio concepts and use ratios.
Use ratio and rate reasoning to solve real-world problems.
Identify and use the multiplication property of equality.

Lesson Flow

This unit introduces students to the routines that build a successful classroom math community, and it introduces the basic features of the digital course that students will use throughout the year.

An introductory card sort activity matches students with their partner for the week. Then over the course of the week, students learn about the routines of Opening, Work Time, Ways of Thinking, Apply the Learning (some lessons), Summary of the Math, Reflection, and Exercises. Students learn how to present their work to the class, the importance of students’ taking responsibility for their own learning, and how to effectively participate in the classroom math community.

Students then work on Gallery problems, to further explore the resources and tools and to learn how to organize their work.

The mathematical work of the unit focuses on ratios and rates, including card sort activities in which students identify equivalent ratios and match different representations of an equivalent ratio. Students use the multiplication property of equality to justify solutions to real-world ratio problems.