Students discover the formula for finding the volume of a pyramid and apply the formula to solve problems.Key ConceptsThe volume of a pyramid is one-third the volume of a prism with the same base and height. The shape of the base does not matter (including if it’s a circle), and students will see the same one-third comparison between a cylinder and cone.GoalsUnderstand the formula for the volume of a pyramid.Apply the volume formula to solve problems.

Lesson OverviewStudents will compare the formula for the area of a regular polygon to discover the formula for the area of a circle.Key ConceptsThe area of a regular polygon can be found by multiplying the apothem by half of the perimeter. If a circle is thought of as a regular polygon with many sides, the formula can be applied.For a circle, the apothem is the radius, and p is C.A=a(p2)→A=rC2→A=rπd2→A=rπ2r2→A=rπr=πr2 GoalsDerive the formula for the area of a circle.Apply the formula to find the area of circles.SWD: Consider the prerequisite skills for this lesson: understanding and applying the formula for the area of a regular polygon. Students with disabilities may need direct instruction and guided practice with this skill.Students should understand these domain-specific terms:apothemparallelogramderivationheightapproximate (estimate)scatter plotpiperimetercircumferenceIt may be helpful to preteach these terms to students with disabilities. 

Students will complete the first part of their project, deciding on two right prisms for their models of buildings with polygon bases. They will draw two polygon bases on grid paper and find the areas of the bases.Key ConceptsProjects engage students in the application 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 choose a partner or a group to work collaboratively on solving that problem. Students gain valuable skills in problem solving, reasoning, and communicating mathematical ideas with others.GoalsSelect a project shape.Identify a project idea.Identify a partner or group to work collaboratively with on a math project.SWD: Consider how to group students skills-wise for the project. You may decide to group students heterogeneously to promote peer modeling for struggling students. Or you can group students by similar skill levels to allow for additional support and/or guided practice with the teacher. Or you may decide to create intentional partnerships between strong students and struggling students to promote leadership and peer instruction within the classroom.ELL: In forming groups, be aware of your ELLs and ensure that they have a learning environment where they can be productive. Sometimes, this means pairing them up with English speakers, so they can learn from others’ language skills. Other times, it means pairing them up with students who are at the same level of language skill, so they can take a more active role and work things out together. Other times, it means pairing them up with students whose proficiency level is lower, so they play the role of the supporter. They can also be paired based on their math proficiency, not just their language proficiency.

Students will extend their knowledge of volume to find the volume of right prisms, seeing that the volume is the area of the base multiplied by the height.Key ConceptsVolume is measured in cubic units. The area of the base of a prism indicates how many cubic units are in the first unit “layer” of the prism. Multiplying by height gives the number of layers, and therefore the volume.GoalsFind the volume of right prisms.SWD: Some students with disabilities may have difficulty connecting newly introduced information with previously learned concepts. Consider ways to help students with disabilities to make connections between what they have learned in previous lessons about volume and right prisms and finding the volume of right prisms.Consider the prerequisite skills for this lesson. Students with disabilities may need direct instruction and guided practice with the skills, measurement, and concepts needed for this lesson.Students should understand these domain-specific terms:volumeright (domain-specific)prismcubicIt may be helpful to preteach these terms to students with disabilities.ELL: As new vocabulary is introduced, be sure to repeat it several times and allow students to repeat after you as needed. Write the new words as they are introduced, and allow enough time for ELLs to check their dictionaries or briefly consult with another student who shares the same primary language if they wish.

Students further explore scale, taking a scale drawing floor plan and redrawing it at a different scale.Key ConceptsStudents explore change from one scale to another, focusing on the ratios. Students will draw a scale model of a house.GoalsRedraw a scale drawing at a different scale.Find measurements using a scale drawing.

Students will explore the cross-sections that result when a plane cuts through a rectangular prism or pyramid. Students will also see examples of cross-section cuts in real-world situations.Key ConceptsStudents are very familiar with rectangular prisms, and to a lesser degree, they are familiar with rectangular pyramids. However, students haven’t been exposed to the myriad possibilities for solids that result from planar slices. The purpose of the lesson is for students to explore these possibilities.GoalsIdentify the plane figures that result from a plane cutting through a rectangular prism or pyramid.

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.Problem DescriptionsSprinklersExplore different sprinkler layouts, looking at circular areas (and partial circles) to decide which will be best to water a lawn.Leaning TowerChoose a scale and use a ruler and protractor to make a simple scale drawing.Pizza DoublerIf you could choose between doubling the fraction of the pizza that a slice is, or doubling the radius, which option would give you more pizza? In this problem you will investigate which choice gives a bigger slice.Area and ScaleWhen a figure is redrawn at a larger scale the side lengths increase by the factor of the scale (if the scale doubles the size, the side lengths double also). But, does the area increase the same way? Explore a dynamic sketch and see how area changes when the scale changes.Tree House 1Given plans for a tree house, redraw the plans at a different scale.

Gallery 2Allow 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 to review the unit’s concepts or have fallen behind on work.Gallery OverviewOne World Trade CenterThis task gives students an opportunity to further explore figures that have been intersected by a plane. The task also allows students to revisit scale and think about the net of a sliced prism.Sketch ThreeThis task extends students’ knowledge of nets as they think about surfaces that are triangular and won’t line up parallel. Students may need to use a protractor to keep the angles of the sides consistent.Partial Cube NetThis task provides students with further experience in thinking about the revealed surface in a sliced prism, constructing a more complex net, and estimating area based on area formulas and measuring.Round PrismsThis task extends students’ knowledge of prism measurement to cylinders, which are really no different. Students will see that the only difference is that the base is circular, and they know how to find the circumference (perimeter) and area.Project Work TimeStudents may use a Gallery day to work on their projects and get help if needed.Cube Volume and NetsUsing the 2-D/3-D tool or the parallelogram cubes, students create a solid made of cubes. Using the 2-D views as a guide, they make a net for the figure and find its surface area. Students are challenged to make the net with one piece of paper.Same Surface Area, Different VolumeStudents create two solids with the same surface area but very different volumes. They that surface areas are the same by drawing the 2-D views.Tree House 2This task gives students further practice making a scale drawing and thinking about the net of a solid. Students should also realize that the plans for a building are the 2-D views of the building and are similar to a net.

Students are introduced to real-world applications of geometry and measurement by looking at architectural plans. Students also begin to get familiar with reading architectural plans and thinking about scale.Key ConceptsSince this lesson is exploratory, all of the mathematics discussed will be informal. However, most of the mathematics that students will see in the unit is introduced in this lesson. Students look at length, area, surface area, and volume and examine how these measurements pertain to architectural plans and determining building costs. Students will also consider scale and how scale is used in architectural plans and math drawings.GoalsThink about what measurements are needed to build a building.Think about what measurements determine the cost of a building.Think about how scale is used in math drawings.SWD: Check in to ensure that students understand the meaning of domain-specific vocabulary terms such as dimensions, scale, and area. You may also need to clarify the meaning of the word contract for some students.ELL: Consider having students compile a list or resource with key vocabulary terms for this unit.

Students will measure the circumference and diameter of round things in the classroom and discover the ratio pi. They will see that the ratio of a circle's circumference to its diameter can be used to solve for the circumference when the diameter is known.Key ConceptsStudents have seen circles before, but have not analyzed the relationships between parts of a circle. The ratio of the circumference to the diameter is pi, about 3.14 or about 227. Students see that all of the objects they measure have this ratio (or close, depending on accuracy) and that the ratio is true for all circles. Students also see that the ratio can be used to solve for the circumference of a circle if the diameter (or radius) is known.GoalsMeasure round things looking for similarities.Find the ratio of the circumference to the diameter of those round things.Find a formula to find the circumference of a circle.SWD: Make sure students understand these domain-specific terms:It may be helpful to preteach these terms to students with special needs. If possible, reinforce the definitions of these terms with visual supports (diagrams).ELL: As new vocabulary is introduced, be sure to repeat it several times and to allow students to repeat after you as needed. Write the new words as they are introduced and allow enough time for ELLs to check their dictionaries or briefly consult with another student who shares the same primary language if they wish.ratiocircumferencecirclediameterscatter plot

Students will continue to explore surface area, looking at more complex solids made up of cubes. Students will look at the 2-D views of these solids to see all of the surfaces and to find a shorter method to calculate the surface area.Key ConceptsThe 2-D views of 3-D figures (front top and side) show all of the surfaces of the figure (the area of the three views is doubled or the back, bottom, and other side) and so can be used to calculate surface area. The only exception is when surfaces are hidden or blocked and must be accounted for.GoalsExplore the relationship between 2-D views of figures and their surface area.Find the surface area of different solids.

Students find the area of regular polygons, recalling what they already know about the area of geometric shapes and generalizing a formula for any regular polygon.Key ConceptsStudents will recall what they already know about the area of geometric shapes and apply that to find the area of regular polygons. Any regular polygon can be divided into congruent isosceles triangles. If the length from the center to the midpoint of a side (the apothem) is known, the area of the triangles and the area of the polygon can be found. Students will see the similarities between the area of a polygon and derive the formula for the area of a circle.GoalsReview area of triangles, rectangles, and parallelograms.Find the area of regular polygons.Generalize an area formula for any regular polygon.ELL: This lesson offers students a rich opportunity to learn academic vocabulary. Display the new terms in writing somewhere visible in the classroom so that you can refer to them. Allow ELLs to use their bilingual dictionaries to help with understanding the terms. When possible, have ELLs discuss the terms in their language of choice with other ELLs who share the same primary language.

Students will explore scale and use it to find measurements in scale drawings.Key ConceptsScale drawings are drawn proportionally so that there is a ratio between a given length on the drawing and the actual length. This ratio is used to set up a proportion to find other measurements.GoalsUnderstand that scale drawings are proportional.Use scale to find actual measurements.ELL: Define these terms in the context of the discussion:scalescale drawingscaled to fitproportionalAllow ELLs to use the dictionary if they wish.

Students critique their work from the Self Check and redo the task after receiving feedback. Students then take a quiz to review the goals of the unit.Key ConceptsStudents understand how to find the area of figures such as rectangles and triangles. They have applied that knowledge to finding the area of composite figures and regular polygons. The area of regular polygons was extended to understand the area of a circle. Students also applied ratio and proportion to interpret scale drawings and redraw them at a different scale.GoalsCritique and revise student work.Apply skills learned in the unit.Understand two-dimensional measurements:Area of composite figures, including regular polygons.Area and circumference of circles.Interpret scale drawings and redraw them at a different scale.SWD: Make sure all students have the prerequisite skills for the activities in this lesson.Students should understand these domain-specific terms:composite figuresregular polygonsareacircumferencescale drawingstwo dimensionalIt may be helpful to preteach these terms to students with disabilities.ELL: As academic vocabulary is reviewed, be sure to repeat it and allow students to repeat after you as needed. Consider writing the words as they are being reviewed. Allow enough time for ELLs to check their dictionaries if they wish.

Students will critique their work from the Self Check in the previous lesson and redo the task after receiving feedback. Students will then take a quiz to review the goals of the unit.Key ConceptsStudents understand how to find the surface area (using nets) and volume of rectangular prisms. They have extended that knowledge to all right prisms and were able to generalize rules for both measurements. Students also found the surface area (and volume) of figures made up of cubes by looking at the 2-D views.GoalsCritique and revise student work.Apply skills learned in the unit.Understand 3-D measurements:Surface area and volume of right prismsArea and circumference of circlesSurface area and volume of figures composed of cubesSWD: Consider the prerequisite skills for this Putting it Together lesson. Students with disabilities may need direct instruction and/or guided practice with the skills needed to complete the tasks in this lesson. It may be helpful to pull individual students or a small group for direct instruction or guided practice with the skills they have learned thus far in this unit. While students have had multiple exposures to the domain-specific terms, students with disabilities will benefit from repetition and review of these terms. As students move through the lesson, check to ensure they understand the meaning of included domain-specific vocabulary. Use every opportunity to review and reinforce the meaning of domain-specific terms to promote comprehension and recall.

Students will extend their knowledge of surface area and nets of rectangular prisms to generalize a formula for the surface area of any prism.Key ConceptsStudents know how to find the surface area of a rectangular prism using a net and adding the areas for pairs of congruent faces. Students have not seen that the lateral surface forms one long rectangle whose length is the perimeter of the base and whose width is the height of the prism.Using this idea, the surface area of any right prism can be found using the formula:SA = 2B + (perimeter of the base)hGoalsFind a general formula for surface area of prisms.Find the surface area of different prisms.SWD: Generalization of skills can be particularly challenging for some students with disabilities. Students may need direct instruction on the connection between what they already understand and a general formula.Some students with disabilities may have difficulty recalling formulas when it comes time to apply them. Once students discover the formula SA = 2B + (perimeter of the base)h, consider posting the formula in the classroom and encouraging students to add the formula(s) to the resources they have available when completing classwork and homework.

Lesson OverviewStudents will work on the final portion of their project which includes creating the nets for the sides, making a slice in one of their buildings, and putting their buildings together. Once their two model buildings are complete, they will find the surface area and volume for their models and the full-size buildings their models represent.Key ConceptsThe second part of the project is essentially a review of the second half of the unit, while still using scale drawings. Students will find the surface area of a prism as well as the surface area of a truncated prism. The second prism will require estimating and problem solving to figure out the net and find the surface area. Students will also be drawing the figure using scale to find actual surface area.GoalsRedraw a scale drawing at a different scale.Find measurements using a scale drawing.Find the surface area of a prism.SWD: Students with disabilities may have a more challenging time identifying areas of improvement to target in their projects. It may be helpful to model explicitly for students (using an example project or student sample) how to review a project using the rubric to assess and plan for revisions based on that assessment.Students with fine motor difficulties may require grid paper with a larger scale. Whenever motor tasks are required, consider adaptive tools or supplementary materials that may benefit students with disabilities.Students with disabilities may struggle to recall prerequisite skills as they move through the project. It may be necessary to check in with students to review and reinforce estimation skills.

This goal of this task is to give students familiarity using the formula for the area of a circle while also addressing measurement error.

CK-12 Foundation's Middle School Math € Grade 6 Flexbook covers the fundamentals of fractions, decimals, and geometry. Also explored are units of measurement, graphing concepts, and strategies for utilizing the book's content in practical situations.

Provides teaching tips, information on common errors, differentiated instruction, enrichment, and problem solving for teachers to use with the CK-12 Middle School Math - Grade 6, Student Edition.