Division of Fractions (1 of 3) 21st Century Lessons Division of Fractions (1 of 3) Primary Lesson Designer(s): William Baga (Boston Public)

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Lesson Overview (1 of 4) Lesson Objective Students Will Be Able To… interpret the rationality of quotients and apply this skill to “real-world” problems. Lesson Description This lesson is the first of 3 regarding the concept of division of fractions and standard CCSS.Math.Content.6.NS.A.1 . Lesson 1 supports students in making general sense of division and applying these generalities to fractional division. It seeks to take the comfort students have with whole numbers into fractions. The second lesson will focus more on the visual model aspect of the standard. Lesson 3 takes students from the visual to numerical and efficiency with fractional division calculations. Each of the 3 lessons culminates with word problems in order to ground the math in reality for students. The word “interpret” is specified in the standard, thus in objective.

Lesson Overview (2 of 4) Lesson Vocabulary Dividend, Divisor, Quotient Materials Classwork notes should be a double-sided copy. Homework sheet is single-sided. Summary “key to leave” copies needed. Common Core State Standard (To be broken into a 3-part lesson, lesson 1 focuses on interpreting.) CCSS.Math.Content.6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?

Lesson Overview (3 of 4) Scaffolding This is the into lesson and proceeds in a manner that is concrete and paced for all learners. The homework problems escalate in difficulty, so accessible to all. Enrichment This lesson focuses on rationality of solutions. Advanced learners will be tempted to solve. Tell these learners that the following 2 lessons will give them more of an opportunity to expand critically on what they know about fractional division. Solving is not important for this lesson’s objective. Online Resources for Absent Students As this lesson is narrow in focus, I caution teachers in directing students to resources that might go more into solving fractional division. A quick run through with a hard copy of the slides is a better option for students that miss class.

Lesson Overview (4 of 4) Before and After The standard is trisected. Lesson 1 focuses on interpretation and rationality. Lesson 2 deals with solving using visual models. Lesson 3 solves numerically and with equations. All lessons embed the objective into word problems. Topic Background Division of fractions has been a source of consternation for teachers and students alike. Most often, the “trick” of flipping the second fraction and multiplying has been taught and then memorized by students. This, however, is not an algorithm. In fact it does not teach the concept of division, but the idea of the reciprocal nature of multiplication and division…a separate skill. Review slide 8 for more about the thinking and research that went into the creation of this series of lessons on fractional division.

*Please Note* - Not the traditional approach to division of fractions Division of fractions is a difficult concept for many, students and teachers alike, to understand. The “flip and multiply” method that many of us were taught is functional but not necessarily logical for those trying to dig into mathematical meaning. The approach taken in this lesson is based on the fact that all fractional operations (addition, subtraction, multiplication, and division) can be computed using equivalent ratios (common denominators.) This approach was piloted in my 7th grade class. Instead of being asked to memorize four different ways to approach fractional operations, students were told to create equivalent ratios no matter the operation required. Two periods of instruction were devoted to “re-teaching” these skills that continued to hinder students even after three years of practice. Students were given mixed-operation pre and post quizzes that were identical except for different numbers used. For the post quiz, students were directed to use only the “equivalent ratios method.” The pre mean score was 57.5% while the post was 78.6%. The number of perfect scores increased 212.5%. (Time on this slide - 7 min) Time passed – 7 Solicit responses and key on answers that explain distance as a positive quantity. Agenda

Warm Up OBJECTIVE: Students Will Be Able To… interpret the rationality of quotients and apply this skill to “real-world” problems. We learned division long ago, so we know that 10 ÷ 2 = 5. Still, can you explain, in words, what dividing 10 by 2 means? Explain in as many different ways as you can think of. ex) 10 is cut into pieces that are 2 in length ex) The number of times 2 fits into 10 ex) How many quantities of 2 a quantity of 10 can hold (Time on this slide - 5 min) Time passed – 5 It is assumed students have a routine of doing “warm-up” activity in notebooks. Solicit responses prior to clicking to examples, students will likely provide similar examples. Agenda

Agenda and Objectives 1) Warm Up 2) Launch 1 3) Launch 2 4) Explore Learning Objective: Students Will Be Able To… interpret the rationality of quotients and apply this skill to “real-world” problems. Language Objective: SWBAT define and identify Dividend, Divisor, and Quotient as they relate to division of fractional quantities. 1) Warm Up independent 2) Launch 1 independent and partner 3) Launch 2 independent and partner 4) Explore independent and partner (Time on this slide - 1 min) Time passed – 6 Call on students to read objectives aloud 5) Summary “key to leave” check

Lesson Vocabulary and Language Objectives Word Definition Example/Symbol Dividend A number to be divided Divisor The number a dividend is divided by Quotient The number that results from division, solution of division = 5 (Time on this slide - 5 min) Time passed – 11 Depending on class norms, students can either fill vocab on CW notes handout or in designated vocab section of their notebooks. Preparation Notes I have chosen not to introduce “or equal to” inequalities in the notes. I feel this will complicate the initial understanding. It will be hinted at in the HW and must be addressed in a future lesson. Contributing to this decision is the idea that these inequalities are actually 2 symbols in 1. I want to stress the inequality aspect alone. Agenda

When the dividend is larger than the divisor, Launch 1 – Division: Larger Dividend Dividend Divisor Quotient Once Twice Concept 1 When the dividend is larger than the divisor, the quotient is always greater than 1. (Time on this slide - 5 min) Time passed – 16 Reassure students that you know this seems easy but that you want to really get to the heart of what division is. You may want to supply more examples off the top of your head to reinforce the idea that concept 1 always “works.” Agenda

Click on the timer for internet timer window to open! Student Work (SW #1) *On the lines below, write what you think Concept 1 means in your own words. If needed, you can use the number example, 4 ÷ 2, to help in your explanation. Please use either the words "fits into" or "holds" in your explanation. Once finished, share your explanation with a table partner. (Everyone must share) 5 Minutes of Work Time ! (Time on this slide - 7 min) Time passed – 23 Teacher needs to direct how students are partnered, easiest is shoulder/table partners. Supply about 3 minutes for students to write their own explanations and about 2 to share with a table partner. 2 additional minutes to solicit answers from students for the whole class to hear aloud. Rationale for activity: I find it effective when students are able to internalize a math “rule” into words they can digest and then explain to others. This doesn’t diminish the need to know the vocabulary…just takes what may be abstract for a student to the concrete. The exchange with a partner may also provide clarity that simply writing the “rule” might not do. Click on the timer for internet timer window to open! Agenda

Concept 2 Launch 2 – Division, smaller dividend Won’t Fit Dividend Divisor Quotient Concept 2 When the dividend is smaller than the divisor, the quotient is always less than 1. (Time on this slide - 5 min) Time passed – 28 You may want to run through the animations several times and stop at different junctures to explain. Talk about how the “4” red blocks move as “1 unit” that can’t fit into the 2. Agenda

Click on the timer for internet timer window to open! Student Work (SW #2) *On the lines below, write what you think Concept 1 means in your own words. If needed, you can use the number example, 2 ÷ 4, to help in your explanation. Please use either the words "fits into" or "holds" in your explanation. Once finished, share your explanation with a table partner. (Everyone must share) 5 Minutes of Work Time ! (Time on this slide - 7 min) Time passed – 35 Supply about 3 minutes for students to write their own explanations and about 2 to share with a table partner. 2 additional minutes to solicit answers from students for the whole class to hear aloud. Looking for “half of the 4 fits into the 2” or “the 2 holds half of a 4” Rationale for activity: I find it effective when students are able to internalize a math “rule” into words they can digest and then explain to others. This doesn’t diminish the need to know the vocabulary…just takes what may be abstract for a student to the concrete. The exchange with a partner may also provide clarity that simply writing the “rule” might not do. Click on the timer for internet timer window to open! Agenda

(SW #3) Don’t freak out if not whole numbers… … the concepts still apply to fractions and decimals !! Greater than/Less than Concept 1 or 2 (write out) Dividend Divisor (Time on this slide - 5 min) Time passed – 40 Students complete individually, review answers. Review after “1/2 holds more than one 1/4” “5.65 does not fit once into 3.01” Some students, upon seeing fractions instead of whole numbers, become frustrated. This slide is meant to ease this by having them focus on the 2 concepts they now have at their disposal. As long as they can identify the larger quantity between the dividend and divisor, they now have the tools to answer these. The prep work with whole numbers that was done, heretofore, leads to this moment with fractions. Dividend Divisor Greater than/Less than Concept 1 or 2 (write out) Agenda

Explore – Rational Quotients Circle the larger number (Dividend or Divisor). Then, write greater than or less than, you may use inequality symbols ( < or > ). #1 is answered for you. 1) Less than < 1 2) 3) 4) 5) 6) 7) 1 8) 9) 10) 11) 12) < Greater than > < > > < > > > (Time on this slide - 12 min) Time passed – 52 Students complete individually for 5 minutes then review with “shoulder partner” for 3 minutes. Class review for 4 minutes This activity assesses the fundamental objective of the lesson…can students recognize quotients that make sense given dividends and divisors in all forms. The development of this “number sense” skill will go far towards a student’s intrinsic understanding of what division means. > < Agenda

Summary & Key to Leave Name An Italian sausage is 8 inches long. How many pieces of sausage can be cut from the 8-inch piece of sausage if each piece is to be Problems 2 & 3 involve division to solve: Place a box around the dividend and circle the divisor. Write a numerical expression or equation. Then write whether the quotients will be greater than or less than 1. Name Explain whether the quotient to the problem below is greater than or less than 1. Use the words “Dividend” and “Divisor” in your explanation. 2) How many halves, (1/2), are there in an eighth, (1/8)? 3) A hotdog is 7 ½ inches long. How many pieces can be cut from the hotdog if each piece is to be two-thirds of an inch? (Time on this slide - 8 min) Time passed – 60 5 minutes for students individually, then have students exchange with partner. Students should place slashes through the number of the problem that is incorrect as the whole class reviews. Teacher should then collect summary slips from class to assess later. Lesson 2 will begin with a Do Now similar to problems #2 and #3 for review. These 2 problems were included to assess students’ ability to transfer concrete skills to real world problems. Some will struggle with this but this summary will provide insight as to which students will need a little more support in this transfer process. Agenda

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21st Century Lessons The people… Directors: Kathy Aldred - Co-Chair of the Boston Teachers Union Professional Issues Committee Ted Chambers - Co-director of 21st Century Lessons Tracy Young - Staffing Director of 21st Century Lessons Leslie Ryan Miller - Director of the Boston Public Schools Office of Teacher Development and Advancement Emily Berman- Curriculum Director (Social Studies) of 21st Century Lessons Carla Zils – Curriculum Director (Math) of 21st Century Lessons Brian Connor – Technology Coordinator