Transcript Document

21st Century Lessons
Division of Fractions (1 of 3)
Primary Lesson Designer(s):
William Baga (Boston Public)
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This project is funded by the
American Federation of Teachers.
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21st Century Lessons – Teacher Preparation
Please do the following as you prepare to deliver this lesson:
•
Spend AT LEAST 30 minutes studying the
Lesson Overview, Teacher Notes on each
slide, and accompanying worksheets.
•
Set up your projector and test this PowerPoint file to make
sure all animations, media, etc. work properly.
•
Feel free to customize this file to match the language and
routines in your classroom.
*1st Time Users of 21st Century Lesson:
Click HERE for a detailed description of our project.
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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.
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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
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?
(To be broken into a
3-part lesson, lesson
1 focuses on
interpreting.)
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Lesson Overview (3 of 4)
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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.
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*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 “reteaching” 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%.
Agenda
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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
Agenda
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Agenda and Objectives
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.
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1) Warm Up
independent
2) Launch 1
independent and partner
3) Launch 2
independent and partner
4) Explore
independent and partner
5) Summary
“key to leave” check
Lesson Vocabulary and Language Objectives
Word
Definition
Dividend
A number to be divided
Divisor
The number a dividend is
divided by
Quotient
The number that results from
division, solution of division
Example/Symbol
10
10 ¸ 2 or
2
10
10 ¸ 2 or
2
=5
Agenda
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Launch 1 – Division: Larger Dividend
4¸2= 2
Dividend
Divisor
Quotient
Once
Twice
Concept 1
When the dividend is larger than the divisor,
the quotient is always greater than 1.
Agenda
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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 !
Click on the timer for internet
timer window to open!
Agenda
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Launch 2 – Division, smaller dividend
1
2¸4=
2
Dividend
Divisor
Quotient
Concept 2
When the dividend is smaller than the divisor,
the quotient is always less than 1.
Agenda
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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 !
Click on the timer for internet
timer window to open!
Agenda
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(SW #3) Don’t freak out if not whole numbers…
… the concepts still apply to fractions and decimals !!
1 1
¸
2 4
Dividend
Dividend
Greater than/Less than
Concept 1 or 2 (write out)
is __________
because _____________
Divisor
Divisor
3.01 ¸ 5.65
Greater than/Less than
Concept 1 or 2 (write out)
is __________
because _____________
Agenda
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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)
2)
5¸7
Less than
<
1
11.3 ¸ 8
Greater than
>
1
3)
1
3 ¸2
3
>
1
4)
1
2 ¸3
2
<
1
4.33 ¸ 4.32
>
1
6) 1.001¸1.0009
>
1
5)
1 4
¸
3 5
8)
1
1
2 ¸5
4
2
7 2
9)
¸
8 9
1 1
10)
¸
3 6
3 2
11)
¸
2 3
12) 9 10
¸
8 3
7)
<
1
<
1
>
1
>
1
>
1
<
1
Agenda
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Summary & Key to Leave
Name
1)
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.
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)?
An Italian sausage is 8 inches long. How many pieces of sausage can be cut from the 85inch piece
2 of sausage if each piece is to be
6
¸
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?
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Agenda
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21st Century Lessons
The goal…
The goal of 21st Century Lessons is simple: We want to assist teachers, particularly in
urban and turnaround schools, by bringing together teams of exemplary educators
to develop units of high-quality, model lessons. These lessons are intended to:
•Support an increase in student achievement;
•Engage teachers and students;
•Align to the National Common Core Standards and the Massachusetts curriculum
frameworks;
•Embed best teaching practices, such as differentiated instruction;
•Incorporate high-quality multi-media and design (e.g., PowerPoint);
•Be delivered by exemplary teachers for videotaping to be used for professional
development and other teacher training activities;
•Be available, along with videos and supporting materials, to teachers free of charge via the
Internet.
•Serve as the basis of high-quality, teacher-led professional development, including mentoring
between experienced and novice teachers.
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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
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