Grade 7 – Module 4 Module Focus Session

Grade 7 – Module 4 Module Focus Session
February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 3 minutes 3 minutes MATERIALS NEEDED: G7M4 Module Overview and Topic Openers Student Example and Exercises Document Mid- and End of Module Assessments A Story of Ratios Grade 7 – Module 4 Session 1 NOTE THAT THIS SESSION IS DESIGNED TO BE A TOTAL OF 270 MINUTES IN LENGTH. Welcome Participants! This module focus session unfolds and examines Grade 7 – Module 4: Percent and Proportional Relationships. Presenter introductions. We will be doing a variety of things including some individual and group work. Most of what we do today should be done from the students’ perspectives. ** The following does not need to be in the facilitator guide as it is specific to our delivery ** Please be aware that when we (the presenters) raise our hands high, we are requesting that the group reconvene and come to order. We will take a 15-minute break [this morning at approximately 11:00 a.m.] and then break for lunch at 12:30 p.m. Our focus on module 4 will continue in the afternoon session beginning at 1:30 p.m. There will be another 15-minute break at 3:15 p.m. and then we will try something new. At 3:30 we will reconvene for discussions regarding your next steps and planning with an emphasis on addressing, diagnosing, and meeting student needs, calendar and pacing concerns, and parent communications. Please note that on your table are post-it notes. We have provided parking lots [indicate location(s)] upon which we invite you to leave comments, concerns, questions, and of course any typos that you might recognize in any of our materials. We will attempt to address these items as soon as we can.

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 2 minutes 5 minutes MATERIALS NEEDED: Agenda Introduction to the Module Concept Development Module Review Our session begins by exploring the curriculum overview and identifying how module 4 aligns with the other modules in grade 7. Next we look at the module overview to paint a big picture of module 4. We continue to focus in on Module 4 by examining each Topic Opener and how that topic is developed through its particular section of lessons. To do this we have compiled lesson examples, discussion topics, and student exercises that you can experience from the lessons themselves. As we study module 4 we ask that you be cognizant of the concept development that you see within each lesson, each topic, and the overall module, and how that development aligns with the grade 7 curriculum as a whole as well as the Story of Ratios grade band. Finally, we’ll review the key concepts from module 4 and reflecting on personal experiences from this module focus session.

Curriculum Overview of A Story of Ratios
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 3 minutes 8 minutes MATERIALS NEEDED: Curriculum Overview of A Story of Ratios The 4th module in Grade 7 is called Percent and Proportional Relationships. How does it fit in the G7 curriculum? Like module 1, the focus of this module is proportional relationships. It challenges students to build on understandings from modules 1 through 3 by: Deepening their understanding of the different representations of rational numbers (decimal, fraction, and percent); Extending their understanding of percents (from grade 6) to include those greater than 100%, and those less than 1%; Requiring students to solve multi-step problems that involve percents by creating equations to represent relationships between quantities and solving them; Using understanding of ratios and proportional relationships (from Module 1) to create equations, graphs, and tables for relationships that involve percents. Module 4 requires 25 instructional days for lessons, remediation, assessments and return.

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 8 minutes 16 minutes MATERIALS NEEDED: G7-M4 Module Overview Document Grade 7 Module 4 Overview Questions to address: How many topics are there and what are those topics? How many days are required for lessons? Assessments? Remediation and return? What are some of the important topics and concepts discussed in the narrative? Please turn to the Module Overview (page 3 of the module). Read through the overview document and look for major conceptual ideas that determine how the content will develop through each topic and lesson. How many days are allotted for lessons? (18) Assessments? (3) Assessment return and remediation? (4) How many topics are in this Module and what are they? (4 topics) A: Finding the Whole B: Percent Problems Including More than One Whole C: Scale Drawings D: Population, Mixture, and Counting Problems Involving Percents What are some of the concepts, topics, and representations discussed in the narrative? Finding the “whole” quantity A part of a quantity versus the whole quantity Quantity versus a distinct quantity Application of percent in contextual problems Continued use of expressions and equations Module Overview (p. 3)

Topic A Finding the Whole
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 21 minutes MATERIALS NEEDED: G7-M4 Topic A Opener Topic A Finding the Whole What concepts do you expect to see in Topic A? What do you expect to see in Topic A? [Pause for participant inquiry] Mental conversion between forms of numbers (percents, fractions, decimals) Solving percent problems involving part-to-whole relationships [part = percent x whole] Solving percent problems comparing distinct quantities [quantity = percent x whole] Emphasis on identifying the whole quantity (or whole quantities) in percent problems Percent increase and decrease Alternative strategies (other than equations) for solving percent problems (numeric representations, double number lines, mental math) Topic A Opener (p. 11)

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 2 minutes 23 minutes MATERIALS NEEDED: Lesson 1: Percent Students understand that 𝑃 percent is the number 𝑃 100 and that the symbol % means percent. Students convert between fraction, decimal, and percent; including percents that are less than 1% or greater than 100%. Students write a non-whole number percent as a complex fraction. Read student outcomes. There is a fluency sprint to open the lesson, and the module, that asks students to draw upon their knowledge of percents from grade 6. Lesson 1 / Student Outcomes

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 6 minutes 29 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Percent Color in a 10 x 10 grid to represent each fraction: 30 100 3 100 Please complete the Opening Exercise in lesson 1. Discussion Questions: How are these fractions and representations related to percents? What are some equivalent representations of 30/100? (30%, 15/50, 3/10, 0.3) Lesson 1 / Opening Exercise

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 3 minutes 32 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Percent What are equivalent representations of ? 30 100 It may help some students understand the equivalencies if you show them the other representations as they are shown on this slide. The shaded area remains but the number of equal sized parts changes such as 15/50, or 3/10. Discussion Questions (continued): If these representations are all equivalent to 30/100, then what percentage are they all equivalent to? (30%) Why do these representations all equal 30%? (Percent means “per one-hundred” and so 30 percent means 30/100 or any equivalent form of that part-to-whole relationship) There are a number of ways that this grid can be shaded to represent these fractions. Lesson 1 / Opening Exercise

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 2 minutes 34 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Percent What are equivalent representations of ? Discussion Questions (continued): Given these other representations of 30/100, what are equivalent representations of (1/3)/100? (1/300) There are a number of ways that this grid can be shaded to represent these fractions. Lesson 1 / Opening Exercise

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 6 minutes 40 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Percent Use the meaning of the word “percent” to write each percent as a fraction and then a decimal. Percent Fraction Decimal 37.5% 100% 110% 1% 1 2 % Complete Example 1 at your tables. In a few moments, we will cover some discussion questions from the lesson. What is the pattern or process that you recall or notice when converting percents to fractions? Percent means per hundred, so a percent has a fraction equivalent with the percent value in the numerator and 100 in the denominator. If I gave you a number as a fraction, could you tell me what percent the fraction represents? How would you do this? Find the equivalent fraction with a denominator of 100. What mathematical process is occurring for the percent to convert to a decimal? When converting a percent to a decimal, division is used to find the value of the fraction that is equivalent to the percent. If I gave you a number as a decimal, could you tell me what percent the decimal represents? How would you do this? Yes. 100% is equivalent to 1, so multiplying by 100% is the equivalent of multiplying by the fraction 100/100, which of course has a value of 1, and therefore does not change the value of the decimal, but instead changes the form of the number from a decimal to a fraction with a denominator of 100. What are we not “teaching” here that is typically taught in many classrooms and why? Move the decimal two places. Students lose track of what direction to move the decimal and they further do not understand what they are doing by moving the decimal place. This method is a great shortcut, but it should come naturally through fluency. [Show the conversion of the fraction 2/7 by finding an equivalent fraction with a denominator of 100. Lesson 1 / Example 1

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 45 minutes MATERIALS NEEDED: Consider this… The following is not from the module, but is to help make a point… I will name a sequence of colors and for each one, do one of the following: If I name a red color, raise your right hand high. If I name a blue color, clap your hands together once. Color sequence: Red…Blue…Red…Blue…Maroon…Brick…Navy…Denim…Azure…Folly…Carmine…Iris…Zaffre Discussion questions: Who can tell me the point of this exercise? (Likely no one will guess this) There are several points to this exercise: Everybody could do this right away; why? (The color names were easy to recognize) Why did you raise your right hand when I said red? (Because the “teacher” told me to) What happened when the colors were not so easily recognizable? (We didn’t know which option to do) Who raised their hand when I said Zaffre? Why did you raise your hand? (It was a guess) Now can anyone tell me why we did this exercise? (possible that someone will guess this time) Teaching percents using the strategy of “key words” (i.e. is and of ) typically does not teach understanding because students start looking for the is and the of in a word problem instead of thinking about the quantities and their relationship. For this reason, our focus will be to identify the quantity that represents the whole, or the 100%. If you can identify the whole, the part is easy, and the percent that corresponds with the part is obvious (in general). This forces students to think about the quantities and how they are being compared.

Lesson 2: Part of a Whole as a Percent
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 2 minutes 47 minutes MATERIALS NEEDED: Lesson 2: Part of a Whole as a Percent Students understand that the whole is 100%, and use the formula 𝑝𝑎𝑟𝑡=𝑝𝑒𝑟𝑐𝑒𝑛𝑡×𝑤ℎ𝑜𝑙𝑒 to problem-solve when given two terms out of three from the part, whole, and percent. Students solve word problems involving percent using expressions, equations, numeric, and visual models. Review the student outcomes for lesson 2. What do part-of-a-whole percent problems involve? (Part of a whole comparisons can be either comparisons of generic numbers, or the comparison of part of a quantity to the whole quantity) A big focus of the next few lessons is being able to identify the whole, or the 100%. How should students identify the whole in a given problem? [Discuss at your table] The whole will be the number or quantity that another number or quantity is being compared to. Lesson 2 / Student Outcomes

Part of a Whole as a Percent
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 6 minutes 53 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Part of a Whole as a Percent Example 1: In Ty’s math class, 20% of the students earned an A on a test. If there were 30 students in the class, how many got an A? Read Example 1 aloud. This is an excerpt from Example 1. Which quantity represents the whole? Explain how you know. (The 30 students in the class represents the whole because the number of students that got an A is being compared to the total number of students in the classroom.) Visual models are excellent resources for organizing the quantities given in the problem and identifying how and where to enter into solving the problem. Visual modeling will come in very handy in subsequent lessons. Use a double number line to model the problem. Because 30 students is the whole, 30 aligns with 100%. We know that there are 100 intervals of 1% in 100%. What number of students does each 1% correspond with? (0.3 of a student) If 0.3 students corresponds with 1%, then what number of students corresponds with 20%? (6 students) We refer to this method as finding the 1% first and is an essential skill for students. Some will recognize a more efficient route (divide by 5) but this should not be forced upon them. Continued practice of finding the 1% will develop better understanding of percents and students will eventually recognize more efficient methods on their own. Example 2 is a similar problem that students solve using this numeric method on the screen. They then discuss their steps in solving the problem and using properties are able to generalize the formula: part = percent x whole. Lesson 2 / Example 1 and Exercise 1

Part of a Whole as a Percent
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 58 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Part of a Whole as a Percent Is the expression equivalent to from our steps in Example 1? What does represent? What does 30 represent? What does their product represent? These discussion questions are based on the solution process from Example 1. Is the expression ________ equivalent to _______ from the steps in Example 1? Why or why not? Yes, any order any grouping property of multiplication. What does 20/100 represent? 20/100 is 20%. What does 30 represent? 30 represents the whole quantity. What does their product represent? Their product (6) represents the part. Write a true multiplication sentence relating the part, the whole, and the percent. (20/100)(30) = 6 Translate your sentence into words. Is the sentence valid? 20% of 30 is 6. Yes the sentence is valid. Generalize the terms in your multiplication sentence by writing what each terms represents. Part = Percent x Whole Students can now begin using this percent formula and algebraic reasoning to solve percent problems. Lesson 2 / Discussion

Lesson 3: Comparing Quantities w/ Percent
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 2 minutes 60 minutes MATERIALS NEEDED: Lesson 3: Comparing Quantities w/ Percent Students use the context of a word problem to determine which of two quantities represents the whole. Students understand that the whole is 100% and think of one quantity as a percent of another using the formula 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦=𝑃𝑒𝑟𝑐𝑒𝑛𝑡×𝑊ℎ𝑜𝑙𝑒, to problem solve when given two terms out of three from a quantity, whole, and percent. When comparing two quantities, students compute percent more or percent less using algebraic, numeric, and visual models. Outcome 1: You can see here that identifying the whole is still a major focus. Outcome 2: The wording in this formula permits use in both types of percent problems that we’ve discussed. The reason that the “part” has been replaced is that is can present a misconception to students that all parts are smaller than the whole quantity. Outcome 3: The percent more and percent less prepares students for lesson 4 in which they work with percent increase and decrease. There is a fluency sprint at the end of the lesson that asks students to find the part, the whole or the percent of generic percent problems. Lesson 3 / Student Outcomes

Comparing Quantities with Percent
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 8 minutes 68 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Comparing Quantities with Percent Six club members decided to evenly split the total number of bracelets to be produced [300 bracelets]. Of the 54 bracelets produced over the weekend, Anna produced 32 bracelets. Compare the number of bracelets that Emily produced [22] as a percent of those that Anna produced. Compare the number of bracelets that Anna produced as a percent of the number that Emily produced. In lesson 3, students change the wording of the percent formula [“part”= percent x whole] to [“quantity”=percent x whole] so that it applies to a greater variety of problems. If two disjoint quantities are being compared as in this example, identifying which quantity represents the whole is a little bit trickier. Students continue to practice identifying the whole quantity first and justifying why that quantity represents the whole. What should a student’s response sound like when identifying the whole? (The number of bracelets that Anna produced is the whole since the number that Emily produced is being compared to it.) Lesson 3 / Example 1(a) and (b)

Comparing Quantities with Percent
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 3 minutes 71 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Comparing Quantities with Percent What percent more did Anna produce in bracelets that Emily? What percent fewer did Emily produce in bracelets than Anna? Following these examples the teacher prompts students to think about the percent more (or less) one quantity is of the other. By this point students associate whole quantity A with 100% and that a quantity B, greater than quantity A, must be greater than 100% of quantity A. This question helps prepare the students for lesson 4 which is Percent Increase and Decrease. [Problem set #12 is a 7-part question that compares several pairs of quantities on a graph and connects percents to proportional relationships] – Provide the problem separately and complete if time is available. There is a Sprint at the end of lesson 3 on calculating the part, the percent, or the whole. Lesson 3 / Example 1(a) and (b)

Lesson 4: Percent Increase and Decrease
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 1 minute 72 minutes MATERIALS NEEDED: Lesson 4: Percent Increase and Decrease Students solve percent problems when one quantity is a certain percent more or less than another. Students solve percent problems involving a percent increase or decrease. Read the student outcomes for the lesson. Lesson 4 / Student Outcomes

Percent Increase and Decrease
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 3 minutes 75 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Percent Increase and Decrease A sales representative is taking 10% off of your bill as an apology for any inconveniences. Following Example 1 students discuss a percent decrease situation and compare it to the previous percent increase problems. They use this discussion to write expressions that represent the situation, then show the equivalence of those expressions using familiar properties of operations. What does this statement imply? (Think like a student) How does this problem differ from the percent increase problems? Instead of the percent being added to 100%, it is less than 100% so must be subtracted from 100%. What expressions represent your implications? “I will only pay 90% of my bill” : 0.90(bill) “10% of my bill will be subtracted from the original total” : (original) – 0.10(original) These expressions are equivalent. Show or explain why. Lesson 4 / Discussion

Lesson 5: Find One Hundred Percent Given Another Percent
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 1 minute 76 minutes MATERIALS NEEDED: Lesson 5: Find One Hundred Percent Given Another Percent Students find 100% of a quantity (the whole) when given an quantity that is a percent of the whole by using a variety of methods including finding 1%, equations, mental math using factors of 100, and double number line models. Students solve word problems involving finding 100% of a given quantity with and without using equations. Read the student outcomes for the lesson. Lesson 5 / Student Outcomes

Find One Hundred Percent Given Another Percent
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 81 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Find One Hundred Percent Given Another Percent What are the whole number factors of 100? What are the multiples of those factors (up to 100)? How many multiples are there of each factor (up to 100)? The opening exercise in lesson 5 helps students identify more efficient means for working with percents. Take a few minutes and complete the opening exercise. How do you think we can use these whole number factors in calculating percents on a double number line? The factors represent all the ways that 100% can be broken into equal intervals. The multiples would be the percents represented by each cumulative interval. The number of multiples is equal to the number of intervals. Lesson 5 / Opening Exercise

Find One Hundred Percent Given Another Percent
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 86 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Find One Hundred Percent Given Another Percent Nick currently has 7,200 points in his fantasy baseball league which is 20% more points than Adam has. How many points does Adam have? Lesson 5 offers students alternatives to using equations in solving percent problems. The first is the modified double number line that was introduced in lesson 3, and then the use of the factors of 100% and their multiples. Give one example from the opening exercise…for example, 20. Which quantity represents the whole? (Adam’s points) [DOCUMENT CAMERA] Draw a percent number line and a bar representing the whole quantity (Adam’s points). The bar representing Nick’s points must extend to 120% because it is given that Nick has 20% more than Adam, or 100% + 20% = 120%. Students recognize that 100 and 120 have the common factor of 20 and that there is no need to find the 1% as was done in earlier lessons. This provides opportunities to use mental math in certain situations. Strategies for using mental computations using factors of 100 and their multiples are covered in greater depth in the second half of the lesson. Adam has 6000 points in the fantasy baseball league. Lesson 5 / Exercise 2

Mental Math using Factors of 100
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 91 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Mental Math using Factors of 100 If 39 is ___% of a number, what is that number? How did you find your answer? Students are further challenged to use their knowledge of percents to begin problem solving by mental calculation. Example 2 is a multi-part questions in which the part is constant, but the percent that it represents changes, and hence the value of the whole also changes. Take a moment to complete parts a-e of Example 2. (1%, 3900); (10%, 390); (5%, 780); (15%, 260); (25%, 156) <<<TAKE A 15-MINUTE BREAK at the conclusion of this slide>>> Lesson 5 / Example 2

Lesson 6: Fluency with Percents
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 1 minute 92 minutes MATERIALS NEEDED: Lesson 6: Fluency with Percents Students solve various types of percent problems by identifying the type of percent problem and applying appropriate strategies. Students extend mental math practices to mentally calculate the part, the percent, or the whole in percent word problems. Read the student outcomes for the lesson. There is a fluency sprint at the end of the lesson that asks students to answer percent questions involving percent more or percent less. Lesson 6 / Student Outcomes

Lesson 6: Fluency with Percents
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 97 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Lesson 6: Fluency with Percents Richard works from 11:00 a.m. to 3:00 a.m. His dinner break is 75% of the way through his work shift. What time is Richard’s dinner break? Lesson 6 is initially a day of extra practice with a variety of percent problems and an extension into problems with more steps. One such example is a percent problems involving unit conversions. Provide time for participants to complete the problem. Lesson 6 / Exercise 2

Topic B Percent Problems Including More than One Whole
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes minutes MATERIALS NEEDED: G7-M4 Topic B Opener Topic B Percent Problems Including More than One Whole What concepts do you expect to see in Topic B? What do you expect to see in Topic B? [Pause for participant inquiry] Markup and markdown problems Connection to proportional relationships Percent error Problems involving changing percents Simple Interest, tax, commission, other fees Modeling real world scenario using percents Topic B Opener (p. 101)

Lesson 7: Markup and Markdown Problems
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 2 minutes minutes MATERIALS NEEDED: Lesson 7: Markup and Markdown Problems Students understand the terms original price, selling price, markup, markdown, markup rate, and markdown rate. Students identify the original price as the whole and use their knowledge of percent and proportional relationships to solve multi-step markup and markdown problems. Students understand equations for markup and markdown problems and use them to solve markup and markdown problems. Read the student outcomes for the lesson. Lesson 7 / Student Outcomes

Markup and Markdown Problems
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Markup and Markdown Problems Black Friday: A mountain bike is discounted by 30% and then discounted an additional 10% for shoppers who arrive before 5:00 a.m. Find the sales price of the bicycle. In all, by how much has the price of the bicycle been discounted in dollars? Explain. After both discounts were taken, what was the total percent discount? Students discuss and understand what markups and markdowns are, and why we have them. They then develop the formulas used to calculate markups and markdowns, and are able to effectively use those formulas in solving multi-step problems. Complete Example 2. Lesson 7 / Example 2

Markup and Markdown Problems
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Markup and Markdown Problems Exercise 4: Write an equation to determine the selling price, 𝑝, on an item that is originally price 𝑠 dollars after a markup of 25%. Create a table (and label it) showing five possible solutions to your equation. Create a graph (and label it) of your equation. Interpret the points (0,0) and (1,𝑟). p = (1+0.25)(s) On document camera (0,0) represents an item with a price of $0 and its selling price of $0. The point (1,1.25) tells us that the unit rate is 1.25, that is for every $1 that the original price goes up, the selling price goes up by $1.25. Lesson 7 / Exercise (4)

Lesson 8: Percent Error Problems
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 2 minutes minutes MATERIALS NEEDED: Lesson 8: Percent Error Problems Given the exact value, 𝑥, of a quantity and an approximate value, 𝑎, of the quantity, students use the absolute error, 𝑎−𝑥 , to compute the percent error by using the formula 𝑎−𝑥 |𝑥| ×100%. Students understand the meaning of percent error: the percent the absolute error is of the exact value. Students understand that when an exact value is not known, an estimate of the percent error can still be computed when given a range determined by two inclusive values; (e.g., if there are known to be between 6,000 and 7,000 black bears in New York, but the exact number is not known, the percent error can be estimated at most (100%), which is %. Read the student outcomes for the lesson. Lesson 8 / Student Outcomes

Understanding Percent Error
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Understanding Percent Error How Far Off? Using a 12-inch ruler, measure the diagonal of a "×11“ sheet of paper and record your value. How many teachers have seen students get stuck because they can’t seem to remember if the denominator of the formula is the measured value or the actual value? To gain the best understanding of what percent error is, it is important that students understand absolute error first. Then percent error is the percent that the absolute error is of the exact value. Once absolute error is understood, finding the percent error is a comparison of two very similar to the problems from previous lessons. In the beginning of lesson 8 students develop the concept of absolute error by considering the amount of error in three measurements of the diameter of a computer monitor. This can be simulated using almost anything of which you know the exact measurement. We will use a standard 8 ½ by 11 sheet of paper and follow the student/teacher dialog. Step 1: Measure like the typical 7th grader would measure Display 3-4 measurements that are above and below the known value and reveal that known value of inches. How could you determine the error of each measurement to the actual diameter? (x – a or a – x ) What is the difference of table ___’s measurement and the actual measurement? (2 possible answers) Which is correct? Why? (the positive answer because we are working with measurements and measurements must be positive) How can we make sure that the error is always a positive value? (absolute value) You have just developed what is called absolute error, which is |a – x| where a is a is the approximate value and x is the exact (or accepted) value. Lesson 8 / Example 1

Understanding Percent Error
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Understanding Percent Error How Far Off? Use the formula for absolute error to find the absolute errors of the given measurements. Provide time to calculate the absolute errors of the given measurements. Do you think that absolute error should be large or small? Why? (small…approximate should be as close to the exact as possible) If we found the percent that the absolute error is of the exact value, what would this tell us? (tells us by how much our approximation differs from the exact measurement) Additional question to consider: Why does percent error matter? Can’t we just always aim for an absolute error of less than 0.01 units? Consider using the given absolute errors on much larger exact measurements and how the percent error is affected. Lesson 8 / Example 1

Percent Error Problems
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Percent Error Problems The attendance at a musical event was counted several times. All counts were between 573 and If the actual attendance number is between 573 and 589, inclusive, what is the most and least the percent error could be? Explain your answer. Extending the concept of percent error, students understand that it is possible to estimate the percent error where the exact value is not known, but that value lies within a known interval (inclusive endpoints). The problem on the screen is an example of one such problem. Solution to problem: Since the counts and the actual number both lie on the interval 573 to 589, the absolute error cannot be greater than the difference of the upper and lower bounds, which is 16, and cannot be less than exact. If the count was exactly the same as the known attendance, then the percent error would be 0%. If the count was 589 and the actual attendance 573 (or vice versa), the absolute error would be 16, giving a percent error of 2.8%. So the percent error of the count of attendance is less than 2.8%. Lesson 8 / Example 3

Lesson 9: Problem Solving when the Percent Changes
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 1 minute minutes MATERIALS NEEDED: Lesson 9: Problem Solving when the Percent Changes Students solve percent problems where quantities and percents change. Students use a variety of methods to solve problems where quantities and percents change, including double number lines, visual models, and equations. Read the student outcomes for the lesson. Lesson 9 / Student Outcomes

Solving Problems when the Percent Changes
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Solving Problems when the Percent Changes Tom’s money is 75% of Sally’s money. After Sally spent $120 and Tom saved all of his money, Tom’s money is 50% more than Sally’s money. How much money did each have at the beginning? Try to solve this problem…but remember…equations with variables on either side of the equal sign are an 8th grade skill! This problem can be easily solved with a visual model like those used in earlier lessons. The key is to identify the whole(s) in the context of the problem. The first whole is Sally’s beginning money. The second whole is Sally’s ending money. Lesson 9 / Example 1

Problem Solving when the Percent Changes
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Problem Solving when the Percent Changes Kimberly and Mike have an equal amount of money. After Kimberly spent $50 and Mike spent $25, Mike’s money is 50% more than Kimberly’s. How much money did Mike and Kimberly have at first? Equation method first: Let x represent Kimberly’s final amount after spending $50. Mike only spent $25 so he now has $25 more than Kimberly. Since Mike’s money is 50% more than Kimberly’s, the $25 must correspond with the 50%, so the resulting equation is: 0.5x=25 Visual method next: Start at the end…Mike’s money is 50% more than Kimberly’s, or 150% of Kimberly’s. The 50% more must be the $25 difference, so each bar represents $25. Each persons started with $100. Lesson 9 / Example 3

Lesson 10: Simple Interest
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 1 minute minutes MATERIALS NEEDED: Lesson 10: Simple Interest Students solve simple interest problems using the formula: 𝐼=𝑃𝑟𝑡, where 𝐼=𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡, 𝑃=𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙, 𝑟=𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒, and 𝑡=𝑡𝑖𝑚𝑒. When using the formula 𝐼=𝑃𝑟𝑡, students recognize that units for both interest and time must be compatible; students convert the units when necessary. Read the student outcomes for the lesson. There is a fluency sprint at the end of the lesson that asks students to solve generic problems involving fractional percents. Lesson 10 / Student Outcomes

Understanding Simple Interest
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Understanding Simple Interest Larry invests $100 in a savings plan. The plan pays % interest each year on his $100 account balance. The following chart shows the balance on his account after each year for the next five years. He did not make any deposits or withdrawals during this time. Time (in years) Balance (in dollars) 1 104.50 2 109.00 3 113.50 4 118.00 5 122.50 Lesson 10 begins with a look at how simple interest works to give students a better understanding of what exactly simple interest is. Read the problem aloud to the group then follow up with discussion questions below. What pattern do you notice from the table? (Each year, the balance of the account increases $4.50) What does that $4.50 represent? (interest earned per year) How is that interest calculated? (it’s a percentage of the $100 invested) Can you create a formula to represent the pattern of change in the table? (I=100(0.045)t) where I represents the interest earned and t represents the number of years.) What kind of relationship is represented by your formula? (A proportional relationship) At this point you would reveal the simple interest formula I=Prt, define the variables, and model how the values would be substituted in to calculate a year ending balance. What is the importance of this first example? (it shows that simple interest earned represents a proportional relationship and also encourages students to understand what simple interest is and how it works rather than just a formula to use.) Lesson 10 / Example 1

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Simple Interest Complete Lesson 10, Problem Set #3 It is important for students to understand that the units on the interest rate and the time in a simple interest problem must be compatible. For example, if an interest rate is given as 4% per year, and the time given is 6 months, then either the 6 months must be converted to ½ year, or the 4% per year must be converted to 2% per 6-months. Problems in this lesson not only vary in time units, but also vary in units of time for the interest rate. Take a few moments to complete Lesson 10, Problem Set #3. We’ll reconvene to discuss the solution process and to make some reflections in a few minutes. Lesson 10 / Problem Set #3

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 2 minutes minutes MATERIALS NEEDED: Lesson 11: Tax, Commissions, Fees, and other Real-World Percent Problems Students solve real-world percent problems involving tax, gratuities, commissions, and fees. Students solve word problems involving percent using equations , tables, and graphs. Students identify the constant of proportionality (the tax rate, commission rate, etc.) in graphs, equations, tables, and in the context of the situation. Read the student outcomes for the lesson. Lesson 11 / Student Outcomes

Tax, Commissions, Fees, and other Real-World Percent Problems
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Tax, Commissions, Fees, and other Real-World Percent Problems Complete modeling Exercise 5 (parts a, b, and c) from lesson 11. Write up your solutions on poster paper to present to the group. Exercises 1 and 2 are part of a modeling lesson in which students apply their knowledge of percents to realistic situations. Lesson 11 / Exercise 5

Mid-Module Assessment
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes minutes MATERIALS NEEDED: G7-M4 Mid-Module Assessment Mid-Module Assessment Complete Problem #1 from the Mid-Module Assessment If time allows, complete problem #1 from the mid-module assessment without looking at the rubric or sample student responses. When finished, compare participant answers to the rubric and student sample response. <<<THIS IS IDEAL TIME FOR LUNCH>>> Mid-Module Assessment / Problem #1

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME XX minute minutes MATERIALS NEEDED: A Story of Ratios Grade 7 – Module 4 Session 1 This concludes the Grade 7 the first half of the Module 4 focus session.

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 5 minutes MATERIALS NEEDED: G7-M4 Topic C Opener Topic C Scale Drawings What concepts do you expect to see in Topic C? What do you expect to see in Topic C? [Pause for participant inquiry] Scale factors in scale drawings expressed as percents Horizontal and vertical scale factors Scale factors of scale drawings A and B have reciprocal relationships A/B and B/A Comparing scale factors between three (or more) scale drawings Using scale drawings and scale factors to determine actual measurements Area problems in scale drawings Topic C Opener (p. 170)

Lesson 12: The Scale Factor as a Percent for a Scale Drawing
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 2 minutes 7 minutes MATERIALS NEEDED: Lesson 12: The Scale Factor as a Percent for a Scale Drawing Given a scale factor in percent, students make a scale drawing of a picture or geometric figure using that scale, recognizing that the enlarged or reduced distances in a scale drawing are proportional to the corresponding distances in the original picture. Students understand scale factor to be the constant of proportionality. Student make scale drawings in which the horizontal and vertical scales are different. Read the student outcomes for the lesson. Lesson 12 / Student Outcomes

The Scale Factor as a Percent for a Scale Drawing
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minute 12 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises The Scale Factor as a Percent for a Scale Drawing Create a scale drawing of the following drawing using a horizontal scale factor of %, and a vertical scale factor of 25%. Complete Exercise 1 with participants on document camera. Students will need to imagine or draw horizontal and vertical auxiliary lines. Students have worked with several scale drawings to this point where they drew in any diagonal line segments by joining the corresponding vertices from the original drawing. For this exercise they might imagine (or draw in) horizontal and vertical auxiliary lines to obtain the vertices of the scale drawing. Sometimes it is helpful to make a scale drawing where the horizontal and vertical scale factors are different, such as when creating diagrams in the field of engineering. Having different scale factors may distort some drawings. For example, when you are working with a very large horizontal scale, you sometimes must exaggerate the vertical scale in order to make it readable. This can be accomplished by creating a drawing with two scales. Unlike the scale drawings with just one scale factor, these types of drawings may look distorted. [We can compare this to the use of different scales on coordinate grids. ] When a scale factor is given as a percent, why is it best to convert the percent to a fraction? (as opposed to a decimal) Sometimes a fractional percent results in a repeating decimal which may lead to an approximate answer. Percents can always be converted to a fraction by dividing the percent value by 100 and reducing the fraction. Consider 166 2/3%. Describe the scale drawing this creates (enlargement) and the scale factor in fraction form (5/3 or 1 2/3) Lesson 12 / Exercise 1

The Scale Factor as a Percent for a Scale Drawing
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minute minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises The Scale Factor as a Percent for a Scale Drawing Create a scale drawing of the original drawing given below with a horizontal scale factor of 80% and a vertical scale factor of 175%. Write numerical equations to find the horizontal and vertical distances. Read the problem aloud. Provide 2-3 minutes for participants to complete the problem. Share solution on the document camera. Lesson 12 / Problem Set #2

Lesson 13: Changing Scales
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 1 minute 18 minutes MATERIALS NEEDED: Lesson 13: Changing Scales Given Drawing 1, and Drawing 2 (a scale model of Drawing 1 with scale factor), students understand that Drawing 1 is also a scale model of Drawing 2, and compute the scale factor. Given three drawings that are scale drawings of each other, and two scale factors, students compute the other related scale factor. Read the student outcomes for the lesson. Lesson 13 / Student Outcomes

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 23 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Changing Scales A regular octagon is an eight-sided polygon with side lengths that are all equal. All three octagons are scale drawings of each other. Use the chart and the side lengths to compute each scale factor as a percent. How can we check our answers? Students find the scale factors that exist between three drawings of an octagon and write equations to illustrate the relationships between measurements in those drawings. Read problem aloud. Complete first row with participants, then let them complete at least next two rows (depending on time) What problem arises between drawings 1 and 2, and 2 and 3? How can this problem be avoided? (convert the percent to a fraction instead of a percent.) Lesson 13 / Example 2

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minute 28 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Changing Scales The scale factor from Drawing 1 to Drawing 2 is 112%, and the scale factor from Drawing 1 to Drawing 3 is 84%. Drawing 2 is also a scale drawing of Drawing 3. Is Drawing 2 a reduction or an enlargement of Drawing 3? Justify your answer using the scale factor. The drawing is not necessarily drawn to scale. In this example, students use the scale factors between 2 of 3 scale drawings to determine the remaining scale factor(s). Complete Example 3 and discuss your strategy with a neighbor. Share strategy when you’re finished. Also, consider the question that follows Example 3: Explain how you could use the scale factors from drawing 1 to drawing 2 (112%) and from drawing 2 to drawing 3 (75%) to show that the scale factor from drawing 1 to drawing 3 is 84%. Lesson 13 / Example 3

Lesson 14: Computing Actual Lengths from a Scale Drawing
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 1 minute 29 minutes MATERIALS NEEDED: Lesson 14: Computing Actual Lengths from a Scale Drawing Given a scale drawing, students compute the lengths in the actual picture using the scale factor. Read the student outcomes for the lesson. Lesson 14 / Student Outcomes

Computing Actual Lengths from a Scale Drawing
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 34 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Computing Actual Lengths from a Scale Drawing The distance around the entire small boat is 28.4 units. The larger figure is a scale drawing of the smaller sketch of the boat. State the scale factor as a percent, and then use the scale factor to find the distance around the scale drawing. Good example of MP1 – students may overthink this problem and give up when they are unsure what to do. However, all that is needed is to determine the scale factor using two corresponding vertical measurements and two corresponding horizontal measurements. Lesson 14 / Example 1

Lesson 15: Solving Area Problems Using Scale Drawings
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 1 minute 35 minutes MATERIALS NEEDED: Lesson 15: Solving Area Problems Using Scale Drawings Students solve area problems related to scale drawings and percent by using the fact that an area, 𝐴′, of a scale drawing is 𝑘 2 times the corresponding area, 𝐴, in the original picture, where 𝑘 is the scale factor. Read the student outcomes for the lesson. Lesson 15 / Student Outcomes

Solving Area Problems Using Scale Drawings
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 40 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Solving Area Problems Using Scale Drawings What percent of the area of the large disk lies outside the smaller disk? Students were introduced to the relationships between areas of scale drawings in module 1, and lesson 15 in this module is much the same, but the scale factors are instead in percent form. Lesson 15 / Example 2

Solving Area Problems Using Scale Drawings
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 45 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Solving Area Problems Using Scale Drawings Use Figure 1 below and the enlarged scale drawing to justify why the area of the scale drawing is 𝑘 2 times the area of the original figure. Students were introduced to the relationships between areas of scale drawings in module 1, and lesson 15 in this module is much the same, but the scale factors are instead in percent form. Area original = lw Area scale drw = (kl) x (kw) = k^2 (lw) The area of the scale drawing is equal to the area of the original drawing times the square of the scale factor k. Lesson 15 / Example 4

Topic D Population, Mixture, and Counting Problems Involving Percents
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 50 minutes MATERIALS NEEDED: G7-M4 Topic D Opener Topic D Population, Mixture, and Counting Problems Involving Percents What concepts do you expect to see in Topic D? What do you expect to see in Topic D? [Pause for participant inquiry] More word problems related to percents Population and mixture problems related to percents (extending use of percent to other areas of math and science) Counting problems involving percent to prepare for future work with probability. Topic D Opener (p. 227)

Lesson 16: Population Problems
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 1 minute 51 minutes MATERIALS NEEDED: Lesson 16: Population Problems Students write and use algebraic expressions and equations to solve percentage word problems related to populations of people and compilations. Read the student outcomes for the lesson. Lesson 16 / Student Outcomes

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minute 56 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Population Problems A school has 60% girls and 40% boys. If 20% of the girls wear glasses and 40% of the boys wear glasses, what percent of all students wears glasses? Students are faced with multi-step percent problems in which there are multiple whole quantities to consider, some of which are sub-quantities, and multiple percents as well. They use visual modeling to organize and make sense of the information given and algebraically solve problems. Have participants complete Example 1 then go over the solution. Lesson 16 / Example 1

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 61 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Population Problems The weight of the ﬁrst of three containers is 12% more than the second, and the third container is 20% lighter than the second. The first container is heavier than the third container by what percent? This problem is a little trickier. Do this problem with participants if time allows. Lesson 16 / Example 2

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minute 66 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Population Problems In one year’s time, 20% of Ms. McElroy’s investments increased by 5%, 30% of her investments decreased by 5%, and 50% of investments increased by 3%. By what percent did the total of her investments increase? This is a different type of “population” problem that can be solved using the same methods. Have participants complete Example 3 then go over the solution. Lesson 16 / Example 3

Lesson 17: Mixture Problems
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 1 minute 67 minutes MATERIALS NEEDED: Lesson 17: Mixture Problems Students write and use algebraic expressions and equations to solve percent word problems related to mixtures. Read the student outcomes for the lesson. Lesson 17 / Student Outcomes

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 72 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Mixture Problems A 6-pint 25% oil mixture is added to a 3-pint 40% oil mixture. What percentage of the resulting mixture is oil? Have participants attempt exercise 1(a) before moving forward. Let’s take a look at a strategy used for mixture problems. Organize the information in a mixture problem using a table to compare and combine the amounts of liquids (or matter) that are combined and the amount of the ingredients within them. How much of liquid 1? (6-pints) How much of liquid 2? (3-pints) How much total liquid then when they are combined? (9-pints) How much oil is in liquid 1? (0.25(6)=1.5-pints) How much oil is in liquid 2? (0.4(3)=1.2-pints) How much oil is in the combined liquids? (2.7-pints) What percent of the total combined mixture is oil? (2.7/9)x100%=30% oil Let’s take the information from the table and put it into equation form: (0.25 x 6) + (0.4 x 3) = (p x 9) where p represents the percent oil in the final mixture. At the end of lesson 17, students reverse this process by dissecting an equation that represents a mixture problem. [Do if time allows, otherwise move to lesson 18] Lesson 17 / Exercise 1a

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 77 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Mixture Problems 0.2 𝑥 −𝑥 =(0.4)(6) is used to model a mixture problem. How many units are in the total mixture? What percentages relate to the two solutions that are combined to make the final mixture? The two solutions combine to make six units of what percentage solution? When the amount of a resulting solution is given (for instance 4 gallons) but the amounts of the mixing solutions are unknown, how are the amounts of the mixing solutions represented? Propose exercise 1(a) before moving forward. Let’s take a look at a strategy used for mixture problems. Organize the information in a mixture problem using a table to compare and combine the amounts of liquids (or matter) that are combined and the amount of the ingredients within them. How much of liquid 1? (6-pints) How much of liquid 2? (3-pints) How much total liquid then when they are combined? (9-pints) How much oil is in liquid 1? (0.25(6)=1.5-pints) How much oil is in liquid 2? (0.4(3)=1.2-pints) How much oil is in the combined liquids? (2.7-pints) What percent of the total combined mixture is oil? (2.7/9)x100%=30% oil Let’s take the information from the table and put it into equation form: (0.25 x 6) + (0.4 x 3) = (p x 9) where p represents the percent oil in the final mixture. At the end of lesson 17, students reverse this process by dissecting an equation that represents a mixture problem. [Do if time allows, otherwise move to lesson 18] Lesson 17 / Exercise 2

Lesson 18: Counting Problems
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 1 minute 78 minutes MATERIALS NEEDED: Lesson 18: Counting Problems Students solve counting problems related to computing percentages. Read the student outcomes for the lesson. Lesson 18 / Student Outcomes

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 83 minutes MATERIALS NEEDED: G7-M4 Student Examples and Exercises Counting Problems How many 4-letter passwords can be formed using the letters “A” and “B”? What percentage of the 4-letter passwords contain: No “A’s”? Exactly one “A”? Exactly 2 “A’s”? Exactly 3 “A’s”? 4 “A’s”? In Lesson 18, counting problems are presented that require understanding and use of percents. This lesson serves as an introduction to probability (module 5) without formally computing combinations or permutations. Discussion questions Lesson 18 / Exercises 1-2

End-of-Module Assessment
Grade 7 – Module 4 Module Focus Session February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 7 minutes 90 minutes MATERIALS NEEDED: G7-M4 End-of-Module Assessment End-of-Module Assessment The End-of-Module Assessment is given over two days. The first of two days is completed with the use of calculators whereas the second day does not. We’ve left this time for you to complete the End-of-Module Assessment using the concepts and strategies that we discussed in today’s session. End-of-Module Assessment

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minutes 95 minutes MATERIALS NEEDED: Your Biggest Takeaway Take a moment to reflect on today’s module focus session and prepare 1 or 2 key takeaways that you think are important to share with the group. Allow 1-minute for participants to reflect on the module focus session and identify a key takeaway(s) to share with the group.

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minute minutes MATERIALS NEEDED: Key Points Percent is a part to whole relationship. Focus on identifying the whole quantity (or quantities) in percent problems. Greater fluency with percent improves problem solving abilities. Percent problems can be represented in a variety of models including equations, visual models, and numeric models. Percent can compare a part of a quantity to the whole quantity, or can compare two separate quantities. A percent 𝑝 of a set of quantities represents a proportional relationship. Percent error is not just a formula to memorize … it has meaning. Many, if not all of these key points were brought up but we’ll review them now in case anything was missed.

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 5 minute minutes MATERIALS NEEDED: Key Points The scale factor is the unit rate (in percent form) Scale drawings may have more than one scale factor (horizontal and vertical scales). Given a drawing 𝐴, and scale drawing 𝐵 of drawing 𝐴 with a scale factor 𝑏 𝑎 , drawing 𝐴 is a scale drawing of drawing 𝐵 with scale factor 𝑎 𝑏 . Work with percents in module 4 ushers in the topics of probability and statistics in module 5.

February 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME XX minute minutes MATERIALS NEEDED: A Story of Ratios Grade 7 – Module 4 Session 1 This concludes the Grade 7 Module 4 focus session.

Grade 7 – Module 4 Module Focus Session

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Grade 7 – Module 4 Module Focus Session

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