Module Focus: Grade 7 – Module 3 Sequence of Sessions

Overarching Objectives of this November 2013 Network Team Institute Module Focus sessions for K-5 will follow the sequence of the Concept Development component of the specified modules, using this narrative as a tool

for achieving deep understanding of mathematical concepts. Relevant examples of Fluency, Application, and Student Debrief will be highlighted in order to examine the ways in which these elements contribute to and enhance conceptual understanding.

High-Level Purpose of this Session Focus.  Participants will be able to identify the major work of each grade using the Curriculum Overview document as a resource in preparation for

teaching these modules. Coherence: P-5.  Participants will draw connections between the progression documents and the careful sequence of mathematical concepts that

develop within each module, thereby enabling participants to enact cross- grade coherence in their classrooms and support their colleagues to do the same .  (Specific progression document to be determined as appropriate for each grade level and module being presented.)

Standards alignment.  Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module addresses the major work of the grade in order to fully implement the curriculum.   

Implementation.  Participants will be prepared to implement the modules and to make appropriate instructional choices to meet the needs of their students while maintaining the balance of rigor that is built into the curriculum.    Instructional supports.  Participants will be prepared to utilize models appropriately in promoting conceptual understanding throughout A Story of Units.

Related Learning Experiences This session is part of a sequence of Module Focus sessions examining the Grade 7 curriculum, A Story of Ratios.

Key Points From Grade 6, students understand that the properties of operations also apply to letters (variables) which represent numbers; in Grade 7 they use the

properties to create equivalent expressions in order to problem solve. (p. 8 Progressions)

Rectangular arrays model the equivalent relationship between standard form and factored form of expressions; and the distributive property is key. Its application also shows, that like terms can be collected, as in: 2n + 5n = (2+5)n = 7n. (p. 6 - 8 Progressions)

Students transition from the use of tape diagrams to the use of algebraic expressions and equations to solve problems, as problems get more complex. (p. 9 Progressions)

Students use if-then statements, inverses, and identities to solve equations; and understand the goal is to find the value of the variable that make the equation true (resulting in a true number sentence). Mental math and estimation skills are used to check the reasonableness of a solution. (p. 8 Progressions)

Students recognize that multiplying or dividing an inequality by a negative number reverses the order of the comparison.

Students discover the value of pi to be the value of the ratio “circumference to diameter” for any circle, and understand that 3.14 and 22/7 are reasonable approximations of the value of pi.

Students will understand and apply area properties and use developed formulas to determine areas and surface areas of various 2-dimensional and 3-dimensional figures.

Students connect the relationship between 2-dimensional and 3-dimensional objects, focusing on the right prism, and use knowledge of surface area and volume of 3-dimensional objects in solving real-world contextual problems.

Session Outcomes

What do we want participants to be able to do as a result of this session?

How will we know that they are able to do this?

Focus.  Participants will be able to identify the major work of each grade using the Curriculum Overview document as a resource in preparation for teaching these modules.

Coherence: P-5.  Participants will draw connections between the progression documents and the careful sequence of mathematical concepts that develop within each module, thereby enabling participants to enact cross- grade coherence in their classrooms and support their colleagues to do the same .  (Specific progression document to be determined as appropriate for each grade level and module being presented.)

Standards alignment.  Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module addresses the major work of the grade in order to fully implement the curriculum.

Implementation.  Participants will be prepared to implement the modules and to make appropriate instructional choices to meet the needs of their students while maintaining the balance of rigor that is built into the curriculum.

Instructional supports.  Participants will be prepared to utilize models appropriately in promoting conceptual understanding throughout A Story of Units.

Participants will be able to articulate the key points listed above

Session Overview

Section Time Overview Prepared Resources Facilitator Preparation

Introduction to module and Topic A-

105 Minutes

Introduction to the instructional focue of Grade 7 Module 3 of

Grade 7 – Module 3 Grade 7 – Module 3 PPT

Review Grade 7 Module 3 Overview, Topic Openers, and

B Lessons 1-9

A Story of Ratios.

Examination of the development of mathematical understanding across the module using a focus on Concept Development within the lessons in Topic A and B.

Grade 7 – Module 3 Lesson Notes

Assessments.

Topic B, Mid-Module Assessment and Topic C Lessons 10-20

150 Minutes

Examination of the development of mathematical understanding across the module using a focus on Concept Development within the lessons in Topic C.

Grade 7– Module 3 Grade 7– Module 3 PPT Grade 7 – Module 3 Lesson

Notes

Review Grade 7 Module 3 Overview, Topic Openers, and Assessments.

Topic C Lessons 21-26 and End-of-Module Assessment

60 Minutes

Articulate the key points of this session.

Grade 7– Module 3 Grade 7– Module 3 PPT Grade 7 – Module 3 Lesson

Notes

Review Grade 7 Module 3 Overview, Topic Openers, and Assessments.

Session Roadmap

Section: Introduction to module and Topic A-B Lessons 1-9 Time: 150 Minutes

[105 minutes] In this section, you will…Introduce the mathematical models and instructional strategies to support implementation of A Story of Ratios.

Materials used include:

Time Slide #Slide #/ Pic of Slide Script/ Activity directions GROUP

1 1 NOTE THAT THIS SESSION IS DESIGNED TO BE A TOTAL OF 315 MINUTES IN LENGTH.Introduction to module and Topic A-B Lessons 1-9: 105 minutesTopic B, Mid-Module Assessment and Topic C Lessons 10-20: 150 minutesTopic C Lessons 21-26 and End-of-Module Assessment: 60 MinutesWelcome!Presenter introductions.In this module focus session, we will examine Grade 7 – Module 3, Expressions and Equations. The module will be divided amongst three sessions as there is a great deal of content to cover.

1 2 Our objectives for this session are to:• Examine the development of mathematical understanding

across the module using a focus on Concept Development within the lessons.

• Introduce mathematical models and instructional strategies to support implementation of A Story of Ratios.

1 3 We will begin by exploring the module overview to understand the purpose of this module.Then we will focus on each of the three topics in Module 3. We will start by looking at how the topics are introduced in the Module Overview, discussed in more detail in the Topic Openers, then look closely at how the topics are developed in the Examples and Exercises in the lessons themselves. Along the way we ask you to be cognizant of the lesson components and how they function in collaboration with the concept development.We will also take a look at how the concepts in Module 3 are tied to the Progressions for the Common Core State Standards in Mathematics (Referred to as “Progressions documents”).Finally, we’ll take a look back at the module and reflect on all of its parts as one cohesive whole.

Let’s get started with the module overview.

10 4 The Expressions and Equations Progressions document explains the cohesive flow of Algebra through the middle grades. First, we would like you to independently take 4 minutes to read about Grade 7 on pages 8-10; (and outside of this session we encourage you to read the entire document so that you understand the full 6-8 progression). Then, each tables’ members should discuss the reading, but with a focus on a specific page, for which you will share-out to the whole group, a summary of the material on that page. Pay close attention to vocabulary, representations, and connections to past, present, and future standards.[Assign a different page number to each table, using page numbers 8 through 10. Then allow 4 minutes for the reading.][ At 4 minutes, announce that table members should discuss their assigned page for the next 2 minutes.][At 6 minutes, call the whole group back in session and allow table members who were assigned page 8 to share-out key findings, followed by table members who were assigned page 9, followed by table members who were assigned page 10.][Click to show key points.]

1 5 The third module in Grade 7 is called Expressions and Equations. The module includes 26 lessons and is allotted 35 instructional days. It challenges students to build on understandings from previous models by:

1) Extending the use of properties of operations to create equivalent expressions involving signed numbers.

2) Applying the properties of operations to add, subtract, factor and expand linear expressions involving rational numbers.

3) Relating previous understandings of numerical solutions and visual models to algebraic steps in a clear and natural way.

4) Increasing the complexity of equations and inequalities; relating them to real-world situations.

5) Modeling geometric relationships related to circumference, area, surface area, and volume through the use of expressions and equations.

6) Using equations to represent angle relationships and solve multi-step problems related to angle measures.

5 6 Take two minutes to read the Table of Contents and Module Overview Narrative for Module. While doing this, be cognizant of major conceptual ideas that determine how the content will develop through the lessons. Make note of any questions you have, and we will address them aloud after everyone has had an opportunity to review the document.[After 2 minutes:]

• How many topics are in this Module? ( Three – A, B, and C.)

• After which lesson does the Mid-Module Assessment fall? (Lesson 15)

• How many days are allotted for administering this assessment, and its return, and remediation? (5 days) And the End-of-Module Assessment? (4 days)

• What are some of the concepts, topics, and representations discussed in the narrative? (Answers will vary. Equivalent Expressions, Area models, Multi-step Equations, Inequalities, Missing Angle Measures, Pi and its approximations, Area and Circumference, Surface Area, Volume

10 7 Now take two minutes to look over the Focus and Foundational standards that this Module is built upon.[After 2 minutes:]

• What are the Focus Standards? [7.EE.A.1, 7.EE.A.2, 7.EE.B.3, 7.EE.B.4. 7.G.B.4, 7.G.B.5, 7.G.B.6)

• What Foundational Standards are referenced? (1.OA.3, 3.OA.5, 4.MD.C.5, 4.MD.C.6, 4.MD.C., 6.EE.A.3, 6.EE.A.4, 6.EE.B.7, 6.EE.B.8, 6.G.A.1, 6.G.A.2, 6.G..4, 7.NS.A.1, 7.NS.A.2)

• Following these standards are the Mathematical Practices most closely tied to the Module. Take a moment to read the specific way these are embodied in this Module.

[After 1 minute:]• How do students Reason Abstractly and

Quantitatively? [Elicit response.]• Model with Mathematics? [Elicit response.]• Attend to Precision? [Elicit response.]• Look for and Make Use of Structure? [Elicit

response.]• Look for and Express Regularity in Repeated

Reasoning? [Elicit response.]The terminology, both definitions and descriptions, are developed carefully in accordance with grade-level standards, the progressions’ language, as well as future mathematical coursework in high school and beyond. Now take a moment to read through these.[After 1 minute:]

• What are some of the “New or Recently Introduced Terms”? [An Expression in Expanded Form, An Expression in Standard Form, An Expression in Factored Form, Coefficient of the Term, Circle, Diameter of a Circle, Circumference, Pi, Circular Region or Disk.]

In a few moments, we will focus on understanding the three forms of Expressions.

3 8 Locate the Topic A Opener, and read through it as it discusses the concepts in Topic A in more detail than the Module Overview Narrative.Look for key ideas that reveal how the concepts in this topic will be developed through the lessons.ALLOW 2 MINUTES TO REVIEW DOCUMENTSWhat do you expect to see in Topic A? [Pause for participant inquiry]

• Associative, commutative, distributive, inverse, and identity properties extended from arithmetic to algebra

• Making sense of different forms of expressions using visual models

• Application of algebraic expressions to contextual problems

• Combining like terms with positive and negative rational coefficients

3 9 Before we jump into equivalent expressions, it is important that we first understand how we are defining equivalent expressions, as well as other terminology related to the different forms of expressions.First, we “describe” equivalent expressions based on standard 6.EE.4 by saying that two expressions are called equivalent when both expressions evaluate to the same number for every substitution of numbers into all the letters in both expressions. This description is not real precise, however its meaning becomes more clear with practice and linking to the properties of operations.An expression in Expanded Form is an expression written as sums (and/or differences) of products whose factors are numbers, variables, or variables raised to whole number powers. This includes a single number, variable, or a single product of numbers and/or variables.An expression in Standard Form is an expression written in Expanded Form where all of its like terms have been collected. this is traditionally referred to as “simplified”An expression in Factored Form is an expression that is the product of

two or more expressions.

2 10 • Understanding the concept of combining like terms in expressions is obviously an important foundation for algebra. Lesson 1 begins by using the relationship between multiplication and addition to expand the like terms into the form of sums.

[Advance Slide]• This shows that the overall expanded forms of the like terms are

simply parts of a larger expanded sum.[Advance slide]• Similar reasoning is used for subtraction of like terms. Use this

discussion to help students recognize that the coefficient of the sum of like terms is the sum of the coefficients of the addends (likewise the coefficient of a difference of like terms is equal to the difference of the coefficients of the like terms).

• Once this expansion method has been used and is understood, there is not necessarily a need to continue expanding like terms for the purpose of combining. It is always a good backup to use for scaffolding with struggling students.

1 11 • We view combining like terms as an application of the distributive property. Factoring the common variable factor(s) from like terms results in a grouped sum (or difference) of the coefficients of those terms times the common variable factor. This shows mathematical reasoning for the pattern just mentioned, i.e. the coefficient of a sum is the sum of the coefficients of the addends.

[Advance slide]• If you refer to the Expressions and Equations Progression

document on page 6, “Collecting like terms, i.e. 5b+3b=(5+3)b, should be seen as an application of the distributive law, not as a separate method.”

• We’ll refer to this concept again in Lesson 6 where we look at combining like terms with rational coefficients.

2 12 • As we help students build understanding, we do not want to make any assumptions about what they do and do not understand about the properties of operations, so we next look at how the properties of operations come into play when like terms are not grouped together. Let’s write this sum in standard form using the properties of operations.

[Complete the problem as shown on the slide]• That was a lot of work to combine the 2x and the 5x. Let’s take a

closer look and see if we can ease the workload any.[Advance slide]• The three steps in the outlined region are all equivalent forms of

the expression. How do we know?• [The terms are all the same, just in different orders and

groupings][Advance slide]• Refer to the Expressions and Equations Progression document on

page 5. “The ‘any order any grouping’ property is a combination of the commutative and associative properties. It says that the sequence of additions and subtractions may be calculated in any order, and that the terms may be grouped together any way.” Students began to use “any order any grouping” to justify the regrouping of terms in Grade 6, and continue to do so.

• Using this reasoning, we can consolidate numerous steps in our process, removing monotonous copying. This does not however downplay the importance of the associative and commutative properties.

3 13 • The any order, any grouping property can be used with addition because addition is both commutative and associative. Can the any order, any grouping property be extended to other operations as well? Take a moment to talk with your neighbor about this question.

[provide 1 minute for discussion then reconvene]• Can any order any grouping be used with subtraction? If so how?

• Answer: Essentially, yes it can because subtracting a number is equivalent to adding the number’s opposite

which changes the expression to a sum.• Can any order any grouping be used with multiplication? If so

how?• Answer: Yes it can because multiplication is both

associative and commutative.• Can any order any grouping be used with division? If so how?

• Answer: Essentially, yes it can because dividing by a non-zero number is equivalent to multiplying by that number’s reciprocal which changes the expression to a product.

4 14 • [Read problem aloud]• Students are aware by this time that in order for two expressions

to be considered equivalent, they must evaluate to the same value for every number that is substituted for the variable or variables in both expressions. This example challenges students to test the equivalence of two expressions where one is the result of a misunderstanding.

• Take a few moments to complete the problem.[Provide 2 minutes to complete the problem then solicit responses from participants]

• Answer: Alexander is not correct. The expressions do not result in equal values when x=-1 and y=-2. [Reveal sample response]

• What is the purpose of this problem? [Seek participant response]• Answer: Any order any grouping can be used in addition,

can be used in multiplication, but cannot be used to mix addition with multiplication.4

4 15 • Lesson 3 begins the transition from using arithmetic models, such as the tape diagram, to using algebraic reasoning with expressions. Example 1 begins by drawing upon student knowledge about tape diagrams and rectangular arrays that they have used in previous grades to multiply two numbers, and extends this model to multiplying a number and a sum.• Step (a) – [follow the slide]• Step (b) – A tape diagram utilizes the idea of a unit, and

even though this may not be apparent, each unit of tape

has a height of 1-unit.• Step (c) – Multiplying in tape diagrams is represented by

stacking a given number of exact copies on top of one another. Notice how useful that 1-unit height is!

• The number of units in the left rectangle (or square in this case) is 9 units, and the number of units in the right rectangle is 6 units. What do these numbers represent?• [Area of the rectangles in square units]

• Rewrite the given product as a sum using the rectangular array.• What important concept is this example leading to?

• [the distributive property]

3 16 • Now we extend the idea from arithmetic to algebra. How do we represent a unit of unknown size (e.g., x)?• Step (d) – [follow the slide]• Step (e) – Even though we do not know the width of x, we

do know that its height is 1 unit just as in the previous diagram.

• Step (f) – How many units are there in the 3 unit by x unit rectangle, and how many square units in the 3 x 2 rectangle?• Area=lengthx width A=3∙ x=3 xunit s2

A=3∙2=6unit s2

• What is the total number of square units in the rectangular array?• (3 x+6 ) unit s2

• What is the product of 3 and the sum x + 2 equivalent to?• 3 ( x+2 )=3 x+6• You can see how students are able to make sense of the

distributive property based on the idea of area and square units.

3 17 • This example takes a different visual approach to showing the equivalence of various expressions that represent the same area.

[Read the problem aloud]• Take a few moments to write different expressions that represent

the number of tiles needed to surround the fountain and show using a diagram where your expression came from.

[Allow 2-minutes to complete the task then move on to the next slide.]

3 18 • By drawing the diagram in different configurations, we can see where the different forms of the expression come from. The beauty of this example is that the equivalence of the expressions is verified because they all represent the same number.

• To confirm the equivalence of the expressions, expand each product to a sum and use any order, any grouping to combine like terms. The expressions in standard form are each 4s+4.

3 19 • After using the distributive property to write several products as sums in Example 1 of Lesson 4, students are challenged to think backwards and write several other simple sums as products essentially by factoring the greatest common factor of the terms. Example 2 takes a look at this reverse thinking using the rectangular array.

• Students are given that x and y represent positive integers and that 2x, 12y, and 8 stand for the number of unit squares in a rectangular array (the array is not given). Students are left with the task of figuring out how many rows there must be in each rectangular array for them all to have the same height (our goal of course is to find the greatest common factor).

[Advance slide]• What does the large rectangle that contains the three smaller

rectangles represent?• [A rectangular array that contains 2x + 12y + 8 square

units]• How many rows of square units does the rectangular array have?

• [2 because each can be given array can be split into 2 rows]

• What are the missing values and how do you know?• [x, 6y, and 4. Since the products represent the areas of the

rectangular regions, dividing each by 2 yields the missing values.]

• Finally we write the equivalent expression as a product of the height and the sum of the remaining values.

• The lesson transitions from factoring using the rectangular array to factoring algebraically using the distributive property. This transition is necessary to deal with negative factors, since it doesn’t make mathematical sense to represent a negative value using an area model.

2 20 • Lesson 5 focuses first on additive and multiplicative identities.[Advance]• We want students to understand why in an expression, if there is

a 0 term in a sum or difference, that 0 term does not affect the value of the expression because of the additive identity property of 0, [a + 0 = a for all numbers a] and can therefore be removed from the expression.

• Likewise, we want students to understand that if a term in an expression contains a factor of 1, that factor of 1 does not affect the value of the term because of the multiplicative identity property of 1, [1a = a for all numbers a] and can therefore be removed from the term.

[Advance]• Students then come to understand that every number has an

additive inverse (opposite) such that their sum is 0, and that every non-zero number has a multiplicative inverse (reciprocal) such that their product is 1.

[Advance]• Why are these properties considered so important? [Pause for

participant answers.]We focus on these properties in this lesson because this knowledge is another key foundation of algebraic reasoning. We will use these

properties a great deal when we work to solve equations beginning in Topic B. These properties should be emphasized with students because they often provide a means by which to condense expressions by eliminating unnecessary symbols (summands of 0 and factors of 1).

6 21 In Lesson 6 we take a look at some different approaches to combining rational number like terms. Take a few moments and complete Example 4 in the way that you would show your students. [Provide 3 minutes to complete the problem][Ask if any participants would like to share their work on the document camera and describe their process.]• Now let’s take a look at a couple of approaches. The first is a more

traditional approach where we find a common denominator for all terms then combine terms as appropriate. This gets us the expression 5/4 x+ 2/5.

• If we think back to earlier in the Expressions and Equations Progression document however, combining like terms should be looked at as what?• […application of the distributive property. So the second

approach does just that.]• In the second approach we first use the any order, any grouping

property to group the like terms together. Then we can write the sum of like terms as a product by factoring out the common variable factor from each term. Once this is done, we simply combine the grouped sum of fractions using known rules for adding rational numbers.

• Aside from being in the Progressions document, why do you think this might be a valuable approach to use with 7th grade students?• [Use of the distributive property is consistent for integer

coefficients and rational number coefficients]• Given a value for x, which form of the expression (given or

standard form) is easier to evaluate? (obviously standard form). The lesson provides a note at this point for teachers to make your students aware that if a standardized test asks them to “simplify” and expression, it is directing them to write the expression in standard form. Our curriculum does not use the term “simplify”.

Expressions in various forms shed light on the ways quantities are related (7.EE.A.1), and so we don’t want to provide the misconception that all expressions need to be “simplified”.

6 22 • Please find the Topic B Opener and read through it in preparation for the next set of lessons.

• Look for key ideas that reveal how the concepts in the topic will be developed through the lessons.

[ALLOW 4 MINUTES TO REVIEW DOCUMENT]What do you expect to see in Topic B? [Pause for participant inquiry]

• Students move from the pictorial to the abstract with the use of algebraic equations versus the tape diagram used in the past.

• Equations (and inequalities) are in context so students can exercise judgment of the reasonableness of their answers.

• Students use skills from Topic A in finding solutions to equations and inequalities given in real-world context.

• Students are introduced to angle relationships and use expressions and equations to solve problems involving angles.

• Students explore and understand the directionality of inequalities.

• Students use inequalities to solve problems given in real world context.

3 23 The emphasis of Lesson 7 is for students to build an algebraic expression and set it equal to a number to form an equation that can be used to solve a word problem. As part of the activity, students are asked to check whether a number (or set of numbers) is a solution to the equation. Solving an equation algebraically is left for future lessons.Before we get into equations, let’s take a look at the Expressions and Equations Progressions document on page 4. Toward the bottom is a problem that starts with “Daniel…” Read to the end of page 4, including the bulleted note in the right margin. [Provide 1 minute to read.]How do you think we will approach equations based on what you read?

[Participant responses]• Lots and lots of contextual word problems

Given a problem situation, where does an equation come from?• When a quantity can be expressed in distinct expressions, those

expressions are equal to each other because they represent the same quantity.

Our goal is to find the unknown value (the variable) that makes that equation fit the problem situation.

4 24 In Lesson 7 we take first an arithmetic approach to solving a word problem then build a bridge to an algebraic solution.[Read problem aloud]Take a few moments to solve the problem in Example 1, using a tape diagram. [Provide 2-minutes]What are the ages of the sisters? [14, 15, and 16]How did you set up your tape diagram? [ Participant share on document camera if time allows.]By taking away the 3 additional years, our 3 remaining units are equivalent to 42 years. Dividing this by 3, we find that one unit in our tape diagram is equivalent to 14 years.

2 25 Now lets build a bridge to an algebraic solution:[Read the problem aloud]How old is the youngest sister? [ x ]How old is the middle sister? [x+1]How old is the eldest sister? [x+2]An expression representing the sum of their ages?And we know from the problem that the sum of their ages is?So here we have two different expressions that represent the same quantity, so their values are equal.

2 26 Remember that a solution to an equation is a value for the variable that makes the equation true. Check the value that you found using the tape diagram to see if it is in fact a solution to the equation. [Provide 1 minute]Is x=14 a solution to the equation? [YES][Advance]How does the unknown unit in a tape diagram represent the unknown integer, represented by 𝑥?

[Consecutive integers begin with the unknown unit then every consecutive integer thereafter increases by 1 unit.]

2 27 • If-then moves were introduced in the second half of Module 2 using the integer game, and they refer to the properties of inequality. On page 80 of the CCSS, Table 4 lists the properties of equality by name and defines them using If-Then statements. Specifically we refer to the addition, subtraction, multiplication and divisions properties when we refer to if-then moves.

• In Module 2, equations were solved using a series of if-then statements that were justified by properties. The properties of operations are not considered if-then moves; they are defined using identities and used for transforming a single expression. The properties of equality however are used in equation solving to transform both expressions in an equation simultaneously. If you compare Tables 3 and 4 from page 80 of the CCSS, you will see the difference in their definitions.

[advance slide]• If-then moves are used to promote and encourage algebraic

reasoning skills. If students look at the process of solving an equation as a sequence of reasoning, they will develop a much better understanding of the process rather than “undo”-ing an equation as is referred to in many textbooks.

3 28 In Lesson 8, we finally explore the algebraic reasoning used in solving equations. We again construct a bridge from the arithmetic approach to the algebraic approach.[Read Example 1 aloud.]Take 2 minutes to solve Example 1 using a tape diagram. [Provide 2 minutes; then seek participants’ solutions.][Advance slide] reveal solutionWhat operations did we use to get our answer?

We added 42.50 and 27.50 to get 70. Then we subtracted 70 from 125.95. Then we divided 55.95 by 3 to get 18.65.

1 29 • If we let the letter j represent the amount of money in dollars that Julia collected, write an expression to represent the amount of money in dollars that Keller collected.

• Using the expressions for Julia and Keller’s amounts collected, write an expression to represent the amount of money in dollars that Isreal collected.

• Using these expressions, write an equation in terms of j that can be used to find the amount each person collected.

4 30 • How does the original tape diagram translate into the equation? [the unknown unit represents how much money Julia collected: j dollars.]

• Challenge students (participants) to try to solve this equation on their own. Think of how a student would approach solving this equation with what they’ve learned through the last several lessons.

• What do you think they would do first? [probably write the left expression in standard form first]

[reveal the first step]• At this point there are no like terms to combine, and no 0’s or 1’s.

How can we make a zero or one?• [Can make a zero by subtracting 70 from both sides OR can

make a 1 by multiplying both sides by 1/3. [Of course making a 1 at this point would create extra calculations]

• Let’s subtract the 70 from both sides. What justification do we use for this “move”? [If-then: If a=b, then a – c = b – c ; the subtraction property of equality.]

• By subtracting 70 from both sides, we get a new set of equivalent expressions.

• After exhausting the use of properties of operations again, we’re left with 3j=55.95. How can we make a zero or one?• [We can multiply both sides of the equation by the reciprocal

of 3, 1/3, to obtain a factor of 1.]• By multiplying both sides by 1/3, we get a new set of equivalent

expressions again. Using properties of operations, we are quickly able to find the solution to the equation.

• What if we instead used the amount of money that Keller collected: k dollars? How would the others’ amounts of money collected then be defined?

[Advance to next slide]

2 31 The expressions defining each person’s amount differ depending on who we choose to represent the other two people. Complete the chart to show how the expressions vary when the value of the variable changes.If time allows in the classroom, set up an solve the equation in terms of k .Will the equation be the same as on the last slide? [No]Will the solution be the same as on the last slide? [No]Will anything be the same as on the last slide? [Yes; Keller will still have collected $61.15, Isreal $46.15, and Julia $18.65]

3 32 Word problems can (and do) get really complex sometimes and Lesson 9, in addition to providing practice solving equations, provides some strategies to help organize the information enclosed in a word problem.[Read Example 2 aloud.]There is a lot to keep track of in this problem, so let’s build a table to help organize the information. [Advance once to show table.]• How can we represent Bonnie’s present age? [Click to Advance :

We are trying to find it, so let’s represent it with a variable, such as, x.]

• How can we represent Shelby’s present age? [Click to Advance :

Since she is seven times as old as Bonnie, 7x.]• How can we represent Bonnie’s future age? [Click to Advance : In

5 years, her age is she is, x + 5.]• How can we represent Shelby’s future age? [Click to Advance : In

5 years, her age is she is, 7x + 5.]• What did your equation and solution look like? [Click to Advance :

Are we in agreement? Comments?]

2 33 Following Lesson 9, we move into angle relationships in Lesson 10. We will begin tomorrow’s session with Lesson 10. Let’s take a moment to reflect upon the concepts and representations highlighted today in Lessons 1 through 9. What are your thoughts on this Algebra Module thus far?[Elicit participant responses and remind participants that we will resume where we left off, in tomorrow’s session.]

Section: Topic B, Mid-Module Assessment and Topic C Lessons 10-20

Time: 150 Minutes

[minutes] In this section, you will…Examine the conceptual understandings that are built in Grade 7 Module 3, Topic C

Materials used include:PPT and Presenter Notes

Time Slide #Slide #/ Pic of Slide Script/ Activity directions GROUP

1 34 NOTE THAT THIS SESSION IS DESIGNED TO BE A TOTAL OF 315 MINUTES IN LENGTH.Introduction to module and Topic A-B Lessons 1-9: 105 minutesTopic B, Mid-Module Assessment and Topic C Lessons 10-20: 150 minutesTopic C Lessons 21-26 and End-of-Module Assessment: 60 MinutesWelcome!Presenter introductions.In this module focus session, we will continue to examine Grade 7 –

Module 3, paying close attention to lessons 10-20 and the mid module assessment. This section of lessons extends work with expressions and equations to inequalities, and to geometry applications.

3 35 Lesson 10 opens with a discussion about angle notation in diagrams, then reviews types of angle relationships including adjacent angles, vertical angles, angles on a line, and angles at a point. The opening exercise asks students to use a protractor to measure a given set of angles, and name the angles in a diagram that have the previously mentioned relationships.[Read Example 4 aloud.]Label the diagram with expressions that describe the relationship.[Advance]Write an equation that models the angle relationship and solve for x. Find the measurements of the obtuse and acute angles.[Provide 2 minutes for participants to complete the problem.]

4 36 Lesson 11 continues where lesson 10 left off, and delves into more difficult angle relationship problems. At the heart of these angle relationship problems is the need to model the angle relationships in an equation and then solve to find an unknown value and/or the unknown angle measurement. All drawings in the module are drawn to scale. Why do you think this is done?[students can verify the reasonableness of their answers by using their protractors to measure angles relevant to their solution.][Read Exercise 2 aloud.] [Provide 3 minutes for participants to complete the task.]What is the value of x? x=27 °

15 37 Standard 7.EE.B.4.a states: “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”.Since similar skills will be used in solving inequalities, lesson 12 starts with a fluency sprint on equations.Round 1: Directions on the slide1-minuteCorrect answers: Clap once, incorrect answer: Circle the problem – read through the answers starting with number 1. When the claps stop, stop giving answers. Record the number correct.Ask who answered 1 or more problems correctly to raise their hands, then continue raising the number by 1. Congratulate those who have their hands raised the longest and achieved high numbers of correct answers.Direct participants to complete the remaining problems in round 1. [1-2 minutes]Go over every answer. Students say “yes” or some variation when they have a correct answer, and say nothing when they have an incorrect answer.Please stand and give yourself some room. We are going to jog in place and skip count by 7’s up to 49, then back down again. We will say another multiple after I say “2-3-4-___”. Ready, begin.Now we’ll do arm circles and skip count by 3’s up to 24 then back down. Ready begin.Round 2: Same process as the first round.This time ask students to raise their hand if they got one MORE right, 2 more, 3 more, etc. until there are only 1 or 2 hands left in the air. Congratulate these students for their growth!

10 38 In lesson 12, students reason and justify properties of operations via a four station exercise that examines how inequalities react to the use of operations with different types of numbers.• You will find some number cubes that are labeled with various

integers ranging from -5 to 5. At your tables, complete at least two of the stations found in your examples and exercises packet. To do this exercise, roll the number cubes and record the numbers in the first and third columns (no particular order). Then fill in the remaining cells following the directions at the top of each column.

• When you’re finished, we will discuss at your tables what the students will recognize from each of the stations, then report those findings out to the group. [Provide 5 minutes to complete 2 stations]

• Discuss with your tables what each of the stations reveals about inequalities. [ 2 minutes]

[Report out to the group] [2 minutes]Participant responses should include:Station 1: When a number is added or subtracted to both numbers in an inequality, the symbol stays the same so the inequality symbol is preserved.Station 2: When both numbers are multiplied by −𝟏, the symbol changes so the inequality symbol is reversed.Station 3: When a positive number is multiplied or divided to both numbers being compared the symbol stays the same so the inequality symbol is preserved.Station 4: When a negative number is multiplied or divided to both numbers being compared the symbol changes and the inequality symbol is reversed.• What are these 4 observations ultimately? [The properties of

inequality] [If-then moves of inequality] See page 80 table 5 of CCSSM.

5 39 Lesson 13 reviews conceptual understandings of inequalities and introduces the “why” and “how” of moving from numerical expressions to algebraic inequalities. The Opening Exercise is a great way to model this transition.[Read Opening Exercise aloud then have participants complete the problem for 2-8 weeks.] [Provide 2 minutes to complete task.]• When will Tarik have enough money to buy the tablet?

[From 6 weeks and onward.]• Why is it possible to have more than one solution?

[The minimum amount of money that Tarik needs is $265.49. He can save more money than that but he cannot have less than that amount. As more time passes, he will save more money. Therefore any amount for time from 6 weeks onward will endure that Tarik has enough money to purchase the tablet.]

• So the minimum amount of money that Tarik needs is $265.49 and he could have more, but certainly not less. What inequality would demonstrate this? [The amount he saves must be greater than or equal to the amount he needs]

Examine each of the numerical expressions and write an inequality showing a comparison of the actual amount of money saved to what is needed, then determine if each inequality is true or false. [Go over results – advance as applies]How can this problem be generalized? [Instead of asking for the amount of money saved after a specific amount of time (guess and check), the question can be asked as to how long it will take for Tarik to save the necessary amount of money to buy the tablet.]Write an inequality that would generalize this problem for money being saved for w weeks.[Advance]Interpret the meaning of the 38 in the inequality. [The 38 represents the amount of money saved each week. As the weeks increase the amount of money saved increases by this amount for each week. The 40 represents the original amount of money saved, not money saved each week.]

4 40 Here we’ll take another look at an example that includes a class discussion as the solution unfolds. [Participants act as the students][Read Example 1 aloud]• First choose a letter (variable) to represent the unknown

quantity. [ let c represent the number of campers that can go to the carnival] - MP.6 (notice that we are not saying let c = campers)

• Write an inequality to represent the given situation.• Why is < used?

[The camp can spend less than the budgeted amount or the entire amount, but cannot spend over that amount]• Solve the inequality. [Provide 2 minutes to complete the

problem]• How many campers can go to the carnival? [77 campers]• Why did we round down instead of rounding up?

[In the context of this problem, the number of campers has to be less than 77.99 campers. Rounding up to 78 would be greater than 77.99 and would go over budget. Thus we round down.]• Describe the If-Then moves used in solving the inequality.

[First, 600 was subtracted from either side (If a<b, then a-600< b-600) to make a 0. Next the number 1/17.95 was multiplied times both sides of the inequality (if a<b, then ac<bc for c>0) to make a 1]

4 41 In Lesson 15, we go one step further with inequalities by representing our solutions graphically taking care to interpret the solutions in the context of the problem.[Read Example 1 aloud]• Write an inequality that can be used to find the number of w full

weeks. Since w is the number of full (or complete) weeks, when w=1, means at the end of 1 week.[Advance]

• Solve and graph the inequality. [Provide 1 minute for participants to complete the task]

• Interpret the solution in the context of the problem.[If the dealership sells 50 cars per week for more than 9 weeks, they

will then have less than 75 cars left on their lot.

• Verify the solutions. Is 9 weeks a solution? NO Is 10 weeks a solution? YES

• In the original inequality, why was 50w subtracted from 525, and why was the inequality < used?[Subtraction was used because the cars are being sold and

therefore the inventory of cars is being reduced. Less than was used because the question asked for the number of cars on the lot to be less than 75 cars.]• In one of the steps, the inequality was reversed. Why?

[Both sides of the inequality were multiplied by a negative value, and if a<b ,∧c<0 ,then ac>bc.

6 42 Please spend the next 5 minutes answering Question #6 from the Mid-Module Assessment. A copy of the problem has been included in your student examples and exercises packet.[After 5 minutes:]Now locate the Mid-Module Assessment, Rubric, and Sample Student Responses in your binder.

10 43 Let’s take a look at some sample solutions and rate them using the rubric. Then we’ll look at your work and do the same.Let’s see how you did? Exchange your paper with a neighbor.• Part (a): Student clearly defines the variables and writes an

expression such as 2d+4 r with appropriate work shown. The definition of the variables must indicate the cost of each admission.

• Part (b): An expression is written, such as d=2r or r=12

d, to

demonstrate the cost of 3D admission is double or two times the cost regular.

• Part (c): Student writes a correct equation such as 2d+4 r+18.50=94.50, solves it correctly by substituting 2 r for dresulting that d=9.5 and writes the correct answer of the cost of

regular admission, $9.50.Note the following standards are addressed in this task:7.EE.4 – Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

a. 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 to an arithmetic solution, identifying the sequence of the operations used in each approach.MP.1: Solving complex word problems requires perseverance.MP.2: Make sense of the quantities given in the problem, represent those quantities in expressions and equations, find a solution and contextualize that solution back into the problem.MP.6: Must clearly and completely define variables, build expressions to represent the given quantities and represent them in equation form, find a solution and represent that solution with correct units of measure.

4 44 Locate the Topic C Opener and read through it to gain a better insight into this last section of lessons.Look for key ideas that reveal how the concepts in the topic will be developed through the lessons. [Provide 3 minutes to read.]What do you expect to see in Topic C? [Pause for participant inquiry.]

• Discovery of the value of the ratio pi• Make sense of the area of a circle by decomposing into a

rectangular-like shape• Composite areas applied in real-world contexts• Surface areas and volumes of solids in real-world and

mathematical contexts

5 45 Topic C starts out with students exploring the circle and coming to a clear understanding of its definition. They then explore distances related to circles to begin understanding the most famous ratio of all, pi.[Participants draw a circle using a geometer’s compass.]• Mark a point on the circle and another point where the spike of

the compass was. The distance between C and B is called the radius of the circle.

• Thinking like a student, write your own definition of the term “circle”. [Provide 1 minute to try then ask for participant definitions.][Be careful because ovals and curves are round as well.]

• Think about the tool used to draw the circle. How does that tool construct the circle?[The distance between the spike and the pencil is fixed.]

• Try defining the term circle again using the theory behind the compass.

• The definition of circle is…• What does the distance between the spike and pencil on the

compass represent in the definition? [radius r]• What does the spike of the compass represent in the definition?

[Center C]• What does the image drawn by the pencil represent in the

definition above? [The “set of all points”]• The exercise continues, looking at the relationship between the

radius and diameter, then concludes by asking students to draw a circle that has a radius of 6cm.

10 46 Given a bicycle wheel, a length of string, masking tape and chalk, how do you think students will try to measure the circumference of the wheel?[Advance]Dramatically walk from one mark to the other declaring that the distance between the marks is the circumference of the wheel and is the distance around the wheel.[Have two participants mark the distance using a piece of string and hold it up for all to see.]The circumference of a circle is always the same multiple of the circle’s

diameter. Mathematicians call this number pi. It is one of the few numbers that is so special that it has its own name. Let’s see if we can estimate the value of that number.[Measure out three diameter lengths along the string using the wheel itself as the unit of measure.]About how many diameter lengths is the circumference of the circle? [ A little more than 3 diameters.]How much is “a little more than”? [Guide participants to a little more than a tenth of a diameter.]The circumference of a circle is a little more than 3 times its diameter. The number pi is a little more than 3. We use the symbol π to represent this special number.

2 47 This information was previously brought to our attention and we know that this is probably a question on everyone’s mind. We respect the reasoning behind this but we also respect the ability to approximate, especially when a calculator is not available. As a result, this curriculum in addition to seeking exact answers in terms of pi, will utilize three approximations of the value of pi in problems.There are times where one approximation seems more appropriate than the others. Many times a problem will direct the student toward one particular approximation as an exercise of operations with fractions or decimals, and other times the student may be asked to choose which seems most appropriate given the information in the problem. Teaching students to work with the pi symbol in an equation or expression just as they would a variable however will leave any approximations for the last step in solving a problem.

10 48 This demonstration is performed by the teacher and its parts discussed with the students as it progresses. We will take a few minutes to make these today and you can take them with you!• Mark a circle on card stock with your compass. Use a pair of

scissors to carefully cut out the circle.• Cut the circle into 16 equal sized circle sectors as is shown on the

screen. This can be done in several ways:• Fold in half repeatedly (folds interfere with precision)• Marking 22.5 degree increments using a protractor• Compass and straight edge construction (perpendicular

bisector of diameter and angle bisectors) – I found this to be the most precise method.

You can extend this activity by performing actual measurements for comparison but to obtain accurate results requires a great deal of precision in this step. For in class demonstrations, understanding of the argument in the lesson is convincing enough.

• At your tables, try one of the above methods to construct this circle, its sectors, and cut the sectors from the circle. [Provide 10 minutes work time]

• Comments or concerns about this task? [Allow participant feedback.]

[Advance]• Arrange the sectors by alternating their direction as is shown on

the slide. This forms a parallelogram “ish” figure. If you cut one sector in half and move it to the opposite side you create more of a rectangle “ish” figure.

[Advance]

3 49 • What do the longer sides of the resulting rectangle correspond to on the circle?[the circumference or 2πr]

• So the width of this rectangle then is approximately what?[Half the diameter or πr]

• What does the height of the rectangle correspond to on the circle? How do you know?[The radius of the circle; the point at the thin end of each sector is

the center of the circle and the distance from that point to any point on the opposite curved end is the radius of the circle]• What is the area?

[Advance]• Are the area of the circle and the area of the rectangle the same?

Explain.[Yes they are the same because they are composed of the same

figures and therefore cover the same amount of area.]• If their areas are the same, what is the area of the circle?

[A=π r2]

5 50 Here is another approach to finding the area of a circle.[Read Example 1 aloud]What is the radius of the circle? [10 cm]What would be a quicker method for determining the area of the circle other than counting all the squares in the entire circle?

[Just count the squares needed to cover ¼ of the circle then multiply that number by 4 to cover the entire circle.How many squares did Michael use to cover one-fourth of the circle? Do you think this is reasonable?

[The area of ¼ of the circle is approximately 79c m2]Why approximately? [The squares are not an exact fit]What is the area of the entire circle? [Seek answer from participant][Advance][316 c m2]

This is a great opportunity to look back to the area formula from

earlier…does the area formula seem consistent using this example?[Yes…the radius of the circle is 10cm, and if A=π r2, then

A=π (10cm )2. 102=100, so the area of the circle would be exactly 100π cm2. Using any of the approximations of pi, we get an area of approximately 314 c m2 which is pretty close to Michael’s answer.You may want to encourage students to brainstorm ways to approximate the area closer. [Such as ½ square unit triangles]

8 51 This problem is the last problem of the homework problem set and not only applies new knowledge, but also refreshes some prior knowledge as well. Take a few minutes at your table and determine an answer and justification that you would want to see from your students. Record your solution on chart paper. Also, let’s see who can figure out what other module and topic this question connects to. [Module 1- scale drawings][Allow 4 minutes to work, then share solutions]Does half of a radius yield half of a circle?

8 52 • Brainstorm some methods for finding half the area of the square and half the area of the circle.[Folding in half and counting the grid squares, cutting each in half

and counting the squares, etc. ]• Find the area of half the square and half the circle and describe to

a neighbor how you arrived at the area.[The area of half of the square is 72 cm2. The area of half of the

circle is 18π cm2. Some students may count the squares others may realize that half of the square is a rectangle with side lengths of 12 cm and 6 cm and use A=l ∙wto determine the area. Some students may fold the square vertically and some may fold it horizontally. Some students will try to count the grid squares in the semicircle and find that it is easiest to take half of the area of the circle.]• What is the ratio of the new area to the original area for the

square and for the circle?The ratio of the areas of the rectangle (half of the square) to the

square is 72cm2

144 cm2 or 12 . The ratio for the areas of the circles is

18πc m2

36π cm2

or 12 .

• Find the area of one-fourth of the square and the circle first by folding and then by another method. What is the ratio of the new area to the original area for the square and for the circle?Folding the square in half and then in half again will result in one-

fourth of the original square. The resulting shape is a square of side length 6 cm with an area of 36cm2. Repeating the same process for the circle will result in an area of 9 π cm2. The ratio for the areas of the

squares is 36cm2

72cm2 or 14 . The ratio for the areas of the circles is

9π cm2

36π cm2

❑

or

14 .

• Write an algebraic expression that will express the area of a semicircle and the area of a quarter circle.

Semicircle: A=12

π r2; Quarter circle: A=14

π r2

5 53 In Lesson 18, students begin to reason about the properties of area as they determine the area relationships of semi-circles, quarter circles, and use this reasoning to determine the areas of circular regions and other regions made out of rectangles, semicircles, quarter circles, and circles. They also begin solving problems with regard to circles where there are unknown dimensions.[Read aloud Example 3.]• What information is needed to solve this problem?

[To find the area, we need the radius. To get the radius, we need to use the given circumference.]Take 2 minutes to find the exact area of the circle.Reflect: How does this example meet the desired outcome of the expressions and equations module?

[It requires knowledge of manipulating expressions and equations to find unknown values.]

8 54 Lesson 19 takes the concepts of decomposition from Lesson 18 a step further into the coordinate plane. They find the areas of regions in the coordinate plane that can be decomposed into quadrilaterals and triangular regions. To get started, the lesson begins with a proof of the area formula for a parallelogram.[Allow 3 minutes for participants to complete the example in the student examples and exercises] Go over answers to through part e.• How did you find the base and height of the figures?

[Using the scales on the coordinate plane]• How did you find the area of the parallelogram?

[Subtracted the areas of the triangles from the area of the composed rectangle]• The second coordinate plane shows figure R which is a rectangle

that has the same base as figure P and also the same height.• Find the area of rectangle R.

[30 square units]• What do figures R and P have in common?

[They have the same area, the same base, and the same height.]

3 55 Here is an interesting type of problem from Lesson 19’s problem set. The trapezoid is given in the coordinate plane, but the y-axis is missing its scale. Students are challenged to find the height of the trapezoid.What skills are necessary to solve this problem? [Ability to build algebraic expressions and set up equations from expressions.]Take a few minutes and solve this problem. [Provide 2 minutes to complete the task.]What is the height of the trapezoid? [10units]

2 56 Since the larger rectangles have the same size, their areas are equal, so the following equation must be true: [Advance]Based on this equation, what must be true about the area of P ? [Advance]So how can we find the area of a parallelogram? [Advance]By the way, how did we get rid of the Areaof S and Areaof T? [If-then moves]

7 57 Lesson 20 extends these concepts even further to finding the areas of regions in the plane with polygonal boundaries and other composite areas by decomposing into familiar figures including triangles, quadrilaterals, circles, semicircles, and quarter circles. Some of these composite areas are regions that include polygonal and circular holes.Example 3 shown on the slide asks students to devise a plan for finding the area of the shaded region and discuss the optional paths as a class.• What two recognizable shapes make up the figure?• What is the square comprised of? [Be specific]• Redraw the figure separating the triangles [advance]• Do we know any of the lengths of the non-right triangle? [NO]• How can we use the information that we have to find the area of

the shaded region?[The area of the shaded triangle is equal to the difference of the

area of the square and the sum of the areas of the three right triangles][Provide 3 minutes for participants to complete the problem then ask for volunteer to share out their method]

5 58 Here is a student problem set for independent practice. Take a few minutes to solve this problem. Remember that we are aiming to use an algebraic approach to these problems. [Provide 4 minutes to complete the problem.]Share out participant solutions.[Note that the answer is given in terms of pi which teachers can use to explore the various approximation choices.]

3 59 Following Lesson 20, we move into 3-dimensional geometry applications of expressions and equations, which also sets the stage for later work in geometry in module 6. We will begin the last session with Lesson 21. Let’s take a moment to reflect upon the concepts and representations highlighted today in Lessons 10 through 20. What are your thoughts on this Algebra Module thus far?[Elicit participant responses and remind participants that we will resume where we left off, in tomorrow’s session.]

Section: Topic C Lessons 21-26 and End-of-Module Assessment

Time: 60 Minutes

[minutes] In this section, you will…Examine the conceptual understandings that are built in Grade 7 Module 3, Topic c and End of module assessment

Materials used include:PPT and assessment

Time Slide #Slide #/ Pic of Slide Script/ Activity directions GROUP

0 60 NOTE THAT THIS SESSION IS DESIGNED TO BE A TOTAL OF 315 MINUTES IN LENGTH.Introduction to module and Topic A-B Lessons 1-9: 105 minutesTopic B, Mid-Module Assessment and Topic C Lessons 10-20: 150 minutesTopic C Lessons 21-26 and End-of-Module Assessment: 60 MinutesWelcome!Presenter introductions.In this module focus session, we will examine Grade 7 – Module 3, Expressions and Equations.

1 61 Our next focus is on Lessons 21-26 in Topic C. In these lessons, students apply their work with expressions and equations in Topic A to model geometric relationships and efficiently solve surface area and volume problems.

1 62 In Lessons 21 and 22, students revisit surface area, and build on their understandings from Grade 6. They continue to use a net to decompose a right prism, and relate its surface area to the sum of the areas of its bases and lateral faces.

6 63 Locate Lesson 21, Example 1 in your additional materials. Take 5 minutes to answer all parts; notice variables serve as placeholders for the lengths of the edges. Students apply their understandings of expressions in Topic A to Geometry and Measurement in Topic C.(After 5 minutes, elicit verbal answers from the audience).(Click twice to display answers; and address any questions which may arise.)

1 64 In Lesson 22, students continue to recognize surface area as the sum of the area of a polyhedron’s bases and lateral faces. In this lesson, there is a focus on pyramids as well as right prisms.

6 65 Work with your elbow partner to complete the Exit Ticket for Lesson 22; located in your supplemental materials. There are two questions: One person should complete Question #1, and the other person should complete Question #2. When you are both finished, take time to share and explain your thought process and solution with your partner.(Allow 5 minutes for this to occur; fielding questions 1-on-1 as in circulating among the tables.)(Click to display answers; and address any final questions which were not addressed 1-on-1.)

1 66 In Lesson 23, students build on their understanding (from Grade 6) of finding the volume of right rectangular prisms, and use it to calculate the volume of other right prisms.

3 67 Locate this question from Lesson 23 in your supplemental materials packet: Student Examples and Exercises. Discuss the question at your table; and then illustrate the answer by sketching the new prism and stating its volume. (Advance to the next slide for the illustration.)

1 68 Since 5 (0.5)(36) = 5(18) = 90; the volume of the right triangular prism would be 90 cubic units.

5 69 Work with your elbow partner to complete the Exit Ticket for Lesson 23; located in your supplemental materials. Not only is the base more complex, but fractional lengths are incorporated into this problem. (Allow 4 minutes for this to occur; fielding questions 1-on-1 as in circulating among the tables.)(Click to display answer; and address any final questions which were not addressed 1-on-1.)

1 70 In Lesson 24, students use their conceptual understanding of the volume of a right prism equaling the area of the base x height, to determine the depth of liquid in a container when given the amount of liquid poured into the container, and the dimensions of the container. They use their proportional reasoning developed in Grade 6 and Grade 7- Module 1, to convert a liquid’s unit of measure to cubic units (that measure the prism’s volume).

3 71 To get a better idea of the types of questioning in this lesson which involves volume and conversions, take 3 minutes to independently complete Example 3 from Lesson 24.(Allow 3 minutes, then click to display answer.)How did we do? Any questions?

1 72 Students in Lesson 25, students real-world problems related to surface area and volume, as they have spent the past few lessons differentiating between the two concepts.

10 73 Turn to Lesson 25 - Exercise 1. You will have 6 minutes to work with your elbow partner to answer parts a) – d). Take note of the real-world application. What are some of the challenges students may face? I will ask you for your thoughts after you have completed this task.(After 6 minutes:)Think about Bloom’s Taxonomy; and the level of questioning you just encountered. What level(s) are these questions --- they reach the higher levels, right? Following this Exercise, students reach the highest level (Create) by creating a fish tank with certain parameters that holds a certain amount of water. Do you see the rigor?Let’s share-out our answers to parts a) – d). (Spend 4 minutes eliciting answers from various participants at each table, and discuss the opportunities for various answers to part b.)

1 74 Students continue to apply their understanding of surface and volume to problem-solve in Lesson 26; working with cubic units and units of liquid measure. To determine the volume of a shell that is formed by two nested prisms, students the prisms’ volumes.

10 75 Locate Lesson 26 - Exercise 1 in your materials. Take 8 minutes to work as a group with your table members to answer parts f) – j).(After 8 minutes, elicit answers for each part from members at various tables.)• What are the internal dimensions of the planter? ¿ ft. by 8 ft. by 2 ft.)

•  If you are going to fill the planter ¾ full of soil, how much soil will you need to purchase and what will be the height of the soil?

(132 ft3 with a height of 112 feet)

• What is the volume of bricks that will be used to construct the planter? (94 ft3¿162,432 in3 )

• If 1 brick has a volume of 6 in ×312 in ×2 in, how many

bricks are needed for the planter? Round to the nearest hundred. (3900 bricks)

• If soil is sold in 5 cubic-foot bags that cost $14/bag and bricks are sold in stacks of 100for $140/stack, how much will it cost to construct the planter? (

132 ft3× $145 ft3

+3900bricks× $140100bricks

=$369.60+$5460=$5829.60

)

3 76 Take the next three to complete the last question in the End-of-Module Assessment, Question #3. A copy of the question is located in your supplemental materials: Student Examples and Exercises.(After 3 minute advance to the next slide.)

2 77 (After 3 minutes when participants have answered question #3.)Which standard did this task address?• 7.G.B.4 – (Cluster: Solve real-life and mathematical problems

involving angle measure, area, surface area, and volume) - Standard: Know the formulas for area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between circumference and area.

• Which Mathematical Practices are embodied in the task? MP.1- Make Sense of Problems and Persevere in Solving Them, MP.2 – Reason Abstractly and Quantitatively, MP.4 – Model with Mathematics, MP.6 – Attend to Precision

2 78 Take two minutes to turn and talk with others at your table. During this session, what information was particularly helpful and/or insightful? What new questions do you have?Allow 2 minutes for participants to turn and talk. Bring the group to order and advance to the next slide.

2 79 Let’s review some key points of this session.• From Grade 6, students understand that the properties of

operations also apply to letters (variables) which represent numbers; in Grade 7 they use the properties to create equivalent expressions in order to problem solve. (p. 8 Progressions)

• Rectangular arrays model the equivalent relationship between standard form and factored form of expressions; and the distributive property is key. Its application also shows, that like terms can be collected, as in: 2n + 5n = (2+5)n = 7n. (p. 6 - 8 Progressions)

• Students transition from the use of tape diagrams to the use of algebraic expressions and equations to solve problems, as problems get more complex. (p. 9 Progressions)

• Students use if-then statements, inverses, and identities to solve equations; and understand the goal is to find the value of the variable that make the equation true (resulting in a true number sentence). Mental math and estimation skills are used to check the reasonableness of a solution. (p. 8 Progressions)

• Students recognize that multiplying or dividing an inequality by a negative number reverses the order of the comparison.

• Students discover the value of pi to be the value of the ratio “circumference to diameter” for any circle, and understand that 3.14 and 22/7 are reasonable approximations of the value of pi.

• Students will understand and apply area properties and use developed formulas to determine areas and surface areas of various 2-dimensional and 3-dimensional figures.

• Students connect the relationship between 2-dimensional and 3-dimensional objects, focusing on the right prism, and use knowledge of surface area and volume of 3-dimensional objects in solving real-world contextual problems.

0 80

Use the following icons in the script to indicate different learning modes.

Video Reflect on a prompt Active learning Turn and talk

Turnkey Materials Provided

● Grade 7 Module 3 PPT

● Grade 7 Module 3 Lesson Notes

Additional Suggested Resources

● A Story of Ratios Curriculum Overview

● CCSS Progressions Document: Expressions and Equations (6-8)