Teaching Resources for 6-7 Measurement and Geometry
- Difference of Squares - This activity uses a series of related arithmetic experiences to prompt students to generalize into more abstract ideas. In particular, students explore arithmetic statements leading to a result that is the factoring pattern for the difference of two squares. A geometric interpretation of the familiar formula is also included. This lesson plan was adapted from an article by David Slavit, which appeared in the February 2001 edition of Mathematics Teaching in the Middle School.
- Proportioner- This applet allows students to investigate ratios of various objects as well as compare ratios, and find area and perimeter of objects.
- Scale Factor- Explore the relationship between two shapes as the scale factor changes.
- Math 6 Spy Guys- Interactive lessons that covers multiple standards from the Mathematics Framework for California Schools. Includes a glossary, strategies, and operations.
- Web Math - Students could use this as a resource for review or to clarify topics discussed in class but not fully understood. The resource is easy to use and navigate to either specific concepts or broader topics. The site also offers specific sections on the conversion of units (applicable to the sciences).
- Discovery Math: Exploring Geometry- This lesson plan includes a project that allows students to deepen their understanding of simialar and congruent figures while also learning about surface area and geometric transformations.
- Scavenger Hunt for Lengths - This resource helps students to understand the difference between and importance of estimation and measuring.
Title Difference of Squares URL http://illuminations.nctm.org/LessonDetail.aspx?id=L276 Materials needed Browser
Scissors (optional)Learning Objectives Students will:
- analyze and represent patterns with symbolic rules
- represent and compare quantities with integers
- write about their experiences with these patterns
Grade Levels Grade 6, Grade 7 CA 97 Standards Grade 6: MR 1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns,
Grade 7: MR 1.1 Compare weights, capacities, geometric measures, times, and temperatures within and between measurement systems. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning6 MP: 1 Make sense of problems and persevere in solving them. 6.MP.7: Look for and make use of structure. 6.MP.8: Look for and express regularity in repeated reasoning. 7.MP.7: Look for and make use of structure.
7.MP.8: Look for and express regularity in repeated reasoning. How could you use this resource? This resource is mainly a lesson to help students discover patterns and relationships. Students use patterns and represent them using symbolic rules.
EL and Special Needs The teacher may need to adapt this lesson to make the flow of information more understanding.
Lesson Plans Teacher Comments Cost Free Copyright (c) 2000-2010
Title Proportioner URL http://seeingmath.concord.org/resources_files/Proportioner.html Materials needed Computer
JavaLearning Objectives Students will:
- Investigate and compare ratios, area, and perimeter
Grade Levels Grade 6, Grade 7, Geometry CA 97 Standards Grade 6: NS 1.2 Interpret and use ratios in different contexts to show the relative sizes of two quantities, using appropriate notations (a/b, a to b, a:b).
Grade 7: MG 2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangle, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. Geometry: 8.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning6.RP.1: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. 6.RP.2: Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. 7. G.4: Know the formulas for the area and circumference of a circle and use them to solve problems: give an informal derivation of the relationship between the circumference and area of a circle. 7.G.6: Solve real world and mathematical problems involving area, volume and surface area of two-and three- dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms. Geometry-GPE.7; Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. How could you use this resource? Students can use this resource as a method of studying ratios and how ratios vary based on the two objects being compared. Students can also compare ratios, find the area, and find the perimeter of a given object. Teachers can use this resource as an introduction tool to ratios and geometric figures. Since the applet does the computations, it should be used to demonstrate the effects of ratios, perimeter, and area instead of how to find them.
EL and Special Needs This is a very visual applet that can help students link the concept to real life problems.
Lesson Plans Teacher Comments Cost Free Copyright (c) 2005 Concord Consortium
Title Scale Factor URL http://illuminations.nctm.org/ActivityDetail.aspx?ID=176 Materials needed Computer
JavaLearning Objectives Students will:
- Investigate and compare ratios, area, and perimeter
- Use an applet to discover the effects of scale factors
Grade Levels Grade 7 CA 97 Standards Grade 7 MG 2.0 Students compute the perimeter, area, and volume of common geometric objects and use the results to find measures of less common objects. They know how perimeter, area, and volume are affected by changes of scale. CA Common Core State Standards Standards for Mathematical Practice:
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
5. Use appropriate tools strategically
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning4 MD 3. Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. 5-MD-5.b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems 6-G-6. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. 7-G-6. Solve real-world and mathematical problems involving area, volume and surface area of two- and three dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. 7-G-7. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. 8.G 9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Geometry-GMD 1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Geometry-GMD-5. Determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids. How could you use this resource? This resource can be used by students as an exploratory tool on scale factors. It can also be used by students to visualize different scale factors and how they affect an object. Teachers can use this resource in a demonstration or as a tool for reviewing scale factors. EL and Special Needs This is a visual applet that will allow students to manipulate an object and visually see the affect of scale factors on an object. Lesson Plans Teacher Comments Cost Free Copyright (c) 2000-2010 NCTM
Title Math 6 Spy Guys URL http://www.learnalberta.ca/content/mesg/html/math6web/math6shell.html#
Materials needed Computer with high speed internet
Adobe Flash Player
Learning Objectives Learning objectives vary by lesson selection and concept.
Grade Levels Grade 6 CA 97 Standards Multiple Grade 6 standards from the Mathematics Framework for California Public School. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoningCovers multiple CaCCSS across grade levels and strands that involve: 1) Area and Perimeter, 2) Equations and Expressions, 3) Prime Factorization, 4) Percents, 5) Graphing, 6) Probability, 7) Integers, 8) Geometric Transformations, 9) Volume and 10) Angles. How could you use this resource? Students can use this resource to clarify concepts not understood in class, interact with virtual manipulatives, and see virtual representations of concepts. This resource could be used by a teacher to show to a class as a demonstration or an introduction. This is a fairly simple site to use EL and Special Needs Visual graphics display concepts and are interactive. Lesson Plans Teacher Comments Cost Free Copyright © 2003 Alberta Education
Title Web Math URL Materials needed Computer
Learning Objectives Learning objectives vary by concept.
Grade Levels K-8, Algebra 1, Geometry, Calculus, Trigonometry CA 97 Standards Multiple standards from the Mathematics Framework for California Public School. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoningCovers multiple CaCCSS across grade levels and strands. How could you use this resource? Students could use this as a resource for review or to clarify topics discussed in class but not fully understood. The resource is easy to use and navigate to either specific concepts or broader topics.
The site also offers specific sections on the conversion of units (applicable to the sciences).
EL and Special Needs Visual graphics display concepts and are interactive. Lesson Plans Teacher Comments Cost Free Copyright (c) 2009 WebMath.com
Title Discovering Math: Exploring Geometry URL http://www.discoveryeducation.com/teachers/free-lesson-plans/discovering-math-exploring-geometry.cfm
Materials needed Computer
Tape, scissors, paper, cardboard or boxes to build city, compass, measuring tools
Learning Objectives Students will:
1. Create three-dimensional figures.
2. Construct a three-dimensional model of a city using similar and congruent figures and geometric transformations.
3.Create a two-dimensional representation of their city.
4. Find the surface area of their three-dimensional figures.
Grade Levels Grade 6, Grade 7, Geometry CA 97 Standards Grade 6 NS 2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division. Grade 6 MG 1.0 Students deepen their understanding of the measurement of plane and solid shapes and use this understanding to solve problems:
Grade 6 MG 1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area of base ⋅ height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid.Grade 7 AF 1.0 Students express quantitative relationships by using algebraic terminology, expressions, equations, inequalities, and graphs:
Grade 7 MG 1.2 Construct and read drawings and models made to scale.
Grade 7 MG 2.0 Students compute the perimeter, area, and volume of common geometric objects and use the results to find measures of less common objects. They know how perimeter, area, and volume are affected by changes of scale:
Grade 7 MG 2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.
Grade 7 MG 2.3 Compute the length of the perimeter, the surface area of the faces, and the volume of a three-dimensional object built from rectangular solids. Understand that when the lengths of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor.Grade 7 MG 3.0 Students know the Pythagorean theorem and deepen their understanding of plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures:
Grade 7 MG 3.3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement.
Grade 7 MG 3.4 Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures.
Grade 7 MG 3.6 Identify elements of three-dimensional geometric objects (e.g., diagonals of rectangular solids) and describe how two or more objects are related in space (e.g., skew lines, the possible ways three planes might intersect).Geometry 9.0 Students compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres; and students commit to memory the formulas for prisms, pyramids, and cylinders.
Geometry 15.0 Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles.Geometry 22.0 Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections.
CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning6.G: (Cluster Statement) Solve real-world and mathematical problems involving area, surface area, and volume.
6.G.2: Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V=lwh and V=bh to find the volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. 7.NS. (Cluster Statement) Apply and extend previous understandings of operations with fractions to add, subtract, multiply and divide rational numbers. 7.G.1: Solve problems involving scale drawings of geometric figures, including actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. 7.G.4: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. 7.G.6: Solve real world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms. 7.EE. (Cluster Statement) Solve real-life and mathematical problems using numerical and algebraic expressions and equations. 7.EE.4a: Solve word problems leading to equations of the form px+q=r and p(x+q)=r, where p, q and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution, identifying the sequence of the operations used in each approach. 8.G. Geometry Cluster Statement: Understand and apply the Pythagorean Theorem.
8.G.2: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
8-G.7: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two- and three-dimensions.How could you use this resource? This resource could be used as a project to solidify the related concepts.
EL and Special Needs This resource uses manipulatives that allow students to better understand the concepts of surface area and similar and congruent figures. Lesson Plans Teacher Comments Cost Free Copyright (c) 2011 Discovery Education
Title Scavenger Hunt for Lengths URL http://www.discoveryeducation.com/teachers/free-lesson-plans/scavenger-hunt-for-lengths.cfm
Materials needed Equipment for measuring length (rulers, string, tape measures, etc)
Reference materials like maps or atlases
Scissors, colored markers and/or computer (all optional).Copies of the Classroom Activity Sheet
Learning Objectives Students will:
1. uderstand how to measure and estimate lengths;
2. understand the difference between measuring and estimating; and
3. become more aware of linear measurements in the world and communicate better about the significance of these measurements.
Grade Levels Grade 6, Grade 7 CA 97 Standards Grade 6 AF 2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches).
Grade 7 NS 1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications.
Grade 7 MG 1.0 Students choose appropriate units of measure and use ratios to convert within and between measurement systems to solve problems. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
5. Use appropriate tools strategically
6. Attend to precision
8. Look for and express regularity in repeated reasoning6.RP.3: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
6.RP.3a: Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
6.RP.3b: Solve unit rate problems including those involving unit pricing and constant speed.
6.RP.3d: Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
7.NS.2: Apply and extend previous understandings of multiplication and division of fractions to multiply and divide rational numbers: present addition and subtraction on a horizontal or vertical number line.
7.NS.2a: Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property leading to products such as (-1)(-1)=1 and the rule for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
7.NS.2b: Understand that integers can be divided, provided the divisor is not zero, and every quotient of integers (with non-zero divisors) is a rational number. If p and q are integers, then –(p/q)=(-p)/q=p/(-q). Interpret quotients of rational numbers by describing real-world contexts.
7.NS.2c: Apply properties of operations as strategies to multiply and divide rational numbers.
7.NS.2d: Convert a rational number to a decimal using long division: know that the decimal form of a rational number terminates in 0s or eventually repeats.
7.EE.2: Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities are related. 7.EE.3: Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation or estimation strategies. How could you use this resource? Teachers can use this as an introduction to measurement and estimation and send the worksheet home as practice.
EL and Special Needs Allows differentiation for advanced learners. Lesson Plans Teacher Comments Cost Free Copyright (c) 2011 Discovery Education