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1Running head: Greatest Common Factor

Scaling Down a Recipe: Finding the Greatest Common Factor to Represent Ratios in Simplest Form

Keishauna Banks

Towson University


Section I: Plan Overview

Introduction

With the implementation of the Common Core State Standards (CCSS), mathematics

students in grade 5 have added, subtracted, multiplied and divided fractions, but they do not

understand ratio concepts and how to use ratio reasoning to solve problems. Therefore the

focus of the instructional design is Ratio and Proportion Relationships domain of the CCSS.

The level of instruction begins with introducing students to the concept of a ratio to compare

parts with wholes with the scope of instruction addresses writing equivalent ratios. The

students will develop fluidity in using multiple forms of ratio language and ratio notation. The

instructional target is using the GCF in a real-world context of scaling down a recipe. The

instructor implemented this lesson over a two week time-frame in a sixth grade mathematics

classroom.

Front-End Analysis

Front end analysis (FEA) is a systematic approach which clearly defines the goals of

instruction, conducts a needs analysis and learner analysis. The purpose of FEA is to gather

information in order to identify problems and propose potential solutions. The instructor

selected the following models in the FEA:

1. Need Analysis- the Discrepancy-based needs assessment model from Smith and

Ragan(1999)

2. Learner Analysis – the Dick, Carey and Carey model (2001)

Needs analysis. Needs analysis assists instructors and Subject Matter Experts (SME) in

making data-driven decisions to address instruction. Needs analysis is the initial inquiry allowing

the instructional designer the opportunity to gather information and to determine if the instruction


will result in learning. The instructor selected the Smith and Ragan (1999) discrepancy-based

needs assessment model because this model concurrently identifies gaps in prerequisite skills and

advanced skills that the learner may possess. It is vital that the instructional designer identify skill

deficits in order to adapt instruction to meet the needs of learners. If the learner presently is at the

mastery stage of instruction and achieving the instructional target, then the learner does not need

instruction. Learners who are at the remediation level of instruction and failing to meet the

instructional target are in need of instruction. There are five distinct phases in the Smith and

Ragan (1999) Discrepancy Model of needs analysis.

The initial phase requires the instructional designer to list all the goals of the existing

instructional system (Smith & Ragan, 1999, as cited in Brown & Green, 2006, p. 97). The

instructional goal addressed through instruction a via the Pebble-in- the-Pond (PITP) model, is

that learners will be able to find the greatest common factor (GCF) of the numerator and

denominator in order to scale down a cooking recipe. The PITP model allows content to be

introduced sequentially progressively increasing in difficulty achieving the instructional target

step- by-step.

The second phase is to “determine how well the identified goals are already being

achieved” (Smith & Ragan, 1999, as cited in Brown & Green, 2006, p. 97). To determine the

learners present instructional level, the instructor created a pre-analysis administering the pre-

assessment during the analysis phase. The pre-assessment (see Appendix G) concurrently

assesses prerequisite skills and skills that will be obtained by the end of instruction in two

sections. The purpose of Section II is to determine the learner’s prerequisite skills while the

Section II assesses skills regarding the instructional target. The responses to the second section

identify how well the student already meets the planned goal of the instruction. Before


deciding which students should be given the pre-assessment, the instructor interviewed all

students to determine if any student had skills too advanced for the instruction.

Phase three is to determine the difference between what the students know and what

they should know (Smith & Ragan, 1999, as cited in Brown & Green, 2006, p. 97). The

results show areas where the students have skill deficiencies, which identifies what gaps exist

between what the student knows and what the student should know. All students entering

Middle School should know how to find the factors of multiples and find the GCF. All students

were identified as needing additional instruction.

The fourth phase is to “prioritize gaps according to agreed criteria” (Smith & Ragan,

1999, as cited in Brown & Green, 2006, p. 97). The gap skill identified was listing the factors

of numbers. In order to find the GCF, students must be able to first list the factors of the

numerator and the denominator, therefore a review of those skills should be included in the

instruction.

The fifth phase is to “determine which gaps are instructional needs and which are most

appropriate for design and development of instruction” (Smith & Ragan, 1999, as cited in

Brown& Green, 2006, p. 97). The deficit skill identified in phase 4, listing the factors of a

number must be addressed through instruction as this a prerequisite skill needed to solve the

end problem. with The PITP identifies problems is a particular sequence, so it was easy to

identify where in the sequence of problems each problem should be addressed.

In examining the results of the pre-assessment and taking into account that the students

did not receive Math instruction in grades K-5 using the CCSS, the instructor identified a skill

that needed to be assessed. The pre-assessment required students to list the factors of each

multiple but did not ask students to find the prime factorization of the multiples. Therefore, the


pre-assessment did not identify the students that did not know how to find the prime

factorization of a multiple. The Pre-Assessment should have asked the students to list the

prime factorization of the numbers because it would have identified students who did not

know how to identify prime numbers and ultimately prime factors of multiples.

Learner analysis. Learner attributes impact learning goals and the mechanism of

learning. The ability of the subject matter expert (SME) and instruction to consider and adapt

learning according to the characteristics of the learners can ultimately determine the

meaningfulness and overall effective of instruction. "They will help the designer develop a

motivational strategy for the instruction and will suggest various types of examples that can

be used to illustrate points, ways in which the instruction may (or may not) be delivered, and

ways to make the practice of skills relevant for learners" (Dick, Carey, & Carey, p. 98). The

subject matter expert is the instructor for this module of instruction.

A learner analysis is conducted to determine the manner that instruction will be delivered

for a group of learners with the goal of individualizing instruction. The instructional designer

considers traits such as general characteristics, specific entry competencies and learning style

when designing instruction. The instructor selected the Dick, Carey and Carey model (2001) to

conduct the learner analysis. The model identifies several types of useful information about the

target learners; entry behaviors, prior knowledge, attitudes toward content; academic

motivation, educational levels, general learning preferences, attitudes towards the instructor,

and group characteristics (Dick, Carey and Carey, 2001, as cited in Brown & Green, 2006, p.

129).

The Dick, Carey and Carey model (2001) was selected to conduct the learner analysis

because this model considers prior skills and knowledge, which is an essential factor regarding


the content and subsequent level of complexity to be included in the instruction. An additional

consideration is the student learning preferences because this determines learning modality,

delivery and range of instructional materials and media that should be used. For example, if

the majority of the students possess a visual learning preference, they need to see directions,

use visual diagrams, view films and take notes in order to successfully access content. In

order to obtain data regarding the learners, the instructor used the following instruments,

Student Perception survey, Learner Survey, Style to Content Learning Preferences Inventory,

and the Pre-assessment.

Student entry behaviors and competencies were determined by the instructor

administering a Student Perception survey and a Pre-assessment. The pre-assessment tool was

described in the needs analysis. The goal of the pre-assessment was to determine the specific

skills that students had prior to instruction. The goal of the Pre-Assessment was

objective(skill level) in nature whereas the goal of the perception survey was

subjective(feelings toward content). Through the Perception Survey, the instructor sought to

determine the attitudes towards learning about ratios and GCF and to gauge student efficacy-

how they felt regarding their current skill level.

Prior knowledge was determined using the Student Perception survey tool and the pre-

assessment as described in the previous section regarding entry behaviors. The perception

survey accessed learner efficacy and their attitudes and beliefs regarding their ability to find the

GCF of a ratio. The Pre-Assessment verified if the students truly possessed the skillset to find

the GCF of a ratio. The prerequisite skills for the instruction require performing operations

with whole numbers and fractions. Therefore, it was important to determine if the students

had those skills. It was also important to find out if the students already knew how to find the


GCF of ratios equations, because that is the end goal addressed through instruction. If the

students had already achieved the goal of the instruction, there would it be unnecessary for the

students to participate in this module of instruction.

To determine learner attitudes toward content, the students were asked in a

Perception Survey whether they liked math (see Appendix J). Five students indicated that

they did not like math, three students indicated that they liked math and the remaining four

students indicated their preference for math depended on the content be delivered. Nine

students indicated that they would attempt a concept even if they thought it was difficult.

Four students indicated that they felt they could find the GCF of multiples.

Academic motivation was determined through informal interview and through the Style

to Content Learning Preferences inventory. The instructor posed questions such as:

1. How relevant is scaling down a recipe to you?2. What aspects of scaling down a recipe interest you most?3. How confident are you that you could successfully use the GCF to scale down a recipe?4. How satisfying would it be to you to be able to use the GCF to scale down a recipe?

This information was obtained in order to gauge student interest and their motivation to engage in instruction.

Educational and ability levels were determined by the use of student testing data,

Maryland State Assessment (MSA) scores. The MSA Mathematics tool measures proficiency

in grade 5 mathematics. The levels of performance are Advanced, Proficient, and Basic.

Students scoring Advanced have performance above grade level, Proficient students are on

grade level with Basic are performing below grade level. One student scored Advanced, 6

students scored Proficient and 5 students scored at the Basic level. Two of the students who

scored Basic are 10 points from Proficient level. The level of instruction for the class is middle

school.


Determining the learning styles of students will help the instructional designer create a

lesson plan the students will understand as the instructional designer incorporates the styles

into the instruction. The instructor determined student learning preferences by administering

the online version of The Style to Content Learning Preferences Inventory (STC-LPI) is a

survey that enables students to indicate how they prefer to learn (see Appendix E). This vital

information in turn will assist instructors in modifying instruction adapting their teaching

strategies to address students’ varied approaches to learning, referred to as learning

preferences, a term often used synonymously with learning styles. Learning preferences

(styles) are essentially rooted in three domains – cognitive, affective, and psychological

(Irvine and York, 1995). Most students were field dependent, visual learners who learn

sequentially. The instructor designed instruction based upon these results.

Student attitude toward the instructor was determined by informal interviews and the

instructor’s rapport with the student as their classroom instructor. Students expressed a desire

to learn more about the topic despite perceived difficulty because of their positive

relationship with their teacher. The other student is the instructor’s peer in another course,

who also expressed a desire to help out with the plan. Student expressed a positive attitude

toward the instructor.

Learning context analysis. This module of instruction will be delivered in a Math 6

classroom on the sixth grade hallway. The room is arranged with desks in groups of fours. The

arrangement allows for collaborative groupings and increases the instructors’ access to students

by allowing more physical space in which to navigate.

The instructor will provide paper, pencils, a calculator, handouts, pre-assessments,

post- assessments, surveys, and manipulatives for each student. There is technology available


to the instructor and students. Technology available to students is the hp elitebook revolve

810 g2 on a one device to student ratio to access online content such as virtual manipulatives

and the pre-assessment. The hp elitebook revolve 810 g2 is a classroom set of devices. The

instructor has access to the laptop computer, lcd projector and document camera to show

students descriptions and diagrams of concepts and to assign students practice problems.

Time constraints within the classroom are limited to the duration of the class. The

duration of each class period is 50 minutes. The instructor must have all materials available

and ready for instruction to maximize time on task and learning opportunities.

Progression of Problems

After the needs analysis and learner analysis is completed, the next phase in the

instructional design process is instructional design. The progression of problems based around

the instructional problem; finding the greatest common factor (GCF) of the numerator and

denominator in order to scale down a beverage recipe to one serving. This instructional module

uses Merrill’s Pebble-in-the-Pond model (2002) which uses a progression of problems for the

design of the instruction. Merrill’s approach involves breaking instruction down into

progression of problems gradually increase in difficulty or skills that are required to solve an

instructional problem. Students solve the problems by obtaining the required skills in a step-

wise progression as defined by the instructional goal (2002, p. 42). The progression of

problems for this plan consists of five problems.

The first problem in the progression is the understanding of the ratio concept. This

problem introduces the basics of identifying a ratio and its parts. A ratio will be represented in

the form ab with a representing the numerator or part and b the denominator or the whole.

During this first problem, construct a fraction kit to model different part to whole using the


length module. Students will receive 8 strips of colored paper that are pre-divided. The

students will cut the strips into the divisions. Each kit will be composed of the following:

Students will model fractions initially with the teacher and then explore independently.

Once the students have mastered identifying the part and whole portions of the ratio,

they will be ready to move on to the next problem in the progression. At the end of the

instruction sequence, the instructor will administer Formative Assessment I (see Appendix H).

The first problem is defined in more detail in Section II.

After the students understand the concept of the ratio from the first problem, they

will be creating concrete representations of ratios using fraction strips in the second

problem. The students will apply what they learned in the first problem. Students will

manipulate the fraction strips to make the connection that a whole any color strip is equal to

one whole unit.

The third problem in the progression introduces modeling equivalent fractions. Students

will model different ratios of the same set and showing that they get the same number of

counters with different ratios. The instructor will not provide as much guidance as in the first

and second problems, and ask the students to work more independently. The students will then

be asked to complete a few practice problems and complete an assessment after completing the

practice problems. For this problem, the instructor will use only the National Library of Virtual

manipulatives math website for practice problems.

The fourth problem in the progression introduces finding the GCF then dividing in

order to write equivalent ratios. This step involves moving from the concrete using

manipulative to using the abstract to use mathematical reasoning and expression. The process

involves writing factors for the numerator and denominator and dividing by the largest factor


common to both. The students will then be asked to complete a few practice problems. The

instructor will provide less guidance and will assign more work on practice problems than in

the first three problems. Students will complete a formative assessment after completing the

practice problems. For this problem, the instructor will use only the math website

http://www.scoop.it/t/factor that she curated web resources for practice problems, and a

handout for the assessment (see Appendix K).

End problem. With the final problem in the progression, the students should

have learned all the prerequisite skills introduced in the first four problems. The students

should know how to find the GCF of a ratio and write an equivalent ratio. The end problem is

a word problem depicting a real-life scenario regarding about how to figure out the amount

ingredients required maintain the recipe taste (see Appendix F). The assessment criterion for

success is completing three out of the four steps required to solve the problem.

Standards

This plan integrates standards from the National Educational Technology Standards

(International Society for Technology in Education, 2008), the Maryland Teacher Technology

Standards (Maryland State Department of Education, 2003), the Common Core State

Standards for Mathematics (Common Core Standards Initiative, 2014) and the Standards for

Mathematical Practices (Common Core Standards Initiative, 2014).

Standards Applies toISTE Standard 1:Facilitate and Inspire Student Learning and Creativity

Problems 1,3, and 4:Students are using computers to access math websites.

ISTE Standard 2:Design and Develop Digital-Age Learning Experiences and Assessments

Problems 1,3, and 4:Students are using math websites to view math-related diagrams and to solve practiceproblems.

MTTS Standard 5:Integrating Technology into the Curriculum and Instruction

Problems 1,3, and 4:Students are using computer technology to learn content and solve practice problems.

http://www.scoop.it/t/factor


Common Core Curriculum6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities

6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use ratelanguage in the context of a ratio relationship

6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems.

All problems:Students are completing practice examples and experiential learning experiences :

1. Identifying the parts of a ratio2. Identifying that the numerator is

the part of the denominator whole3. Representing equivalent ratios4. Finding the factors of multiples5. Applying instruction to a real-

world scenario

Standards for Mathematical Practices

1. Make sense of problems & persevere in solving them.

All problems

Students understand the problem, find a way to attack it, and work until it is done.

4. Model with mathematics. All problems: Concrete models using fraction

kits, grid paper, virtual manipulatives and beans.

Draw a picture or diagram Draw a graphic organizer or chart

to organize work5. Use appropriate tools strategically. Problems 4 and 5:

Selecting division to scale down Selecting multiplication to list

factors of multiples

6. Attend to precision. Problem 5: Checking and verifying work to

make sure work is correct


Section II: Detailed Lesson Plan

Task Analysis

This lesson plan will address the problem of students scaling down ingredients in a

beverage recipe so that it will taste the same for a single serving. The task analysis for solving

Problem 1 is based on the component analysis described by Merrill in his Pebble-in-the-Pond

model (2002). Each step that is required to solve the instructional problem must be addressed in

detail (Merrill, 2002, p.41). Further, solving the instructional problem involves identifying all

knowledge and skills needed to solve the problem (Merrill, 2002, p.42). The knowledge and

skills required to solve each problem is identified in the PITP graphic organizer (see Appendix

A). The detailed steps to solve Problem 1 are shown in the subsequent instructional sequence.

For problem 1, the component knowledge and skills needed are the concept of a ratio

and the ratio parts. During the instruction, the students will be construct ratio kits. As a result

of the instruction of Problem 1, the students will learn that numerator is the part of the whole

of the denominator and that there can be different parts of different wholes and be able to

identify each fractional part and each whole of ratios.

Instructional Strategy (Tell, Ask, Show, Do)

This plan is presented using Merrill’s (2007) instructional strategy form of Tell

(present), Ask( recall), Show(demonstrate), and Do(apply). In order to maintain the logical

and sequential manner of problem progression while accessing prior knowledge and building

upon skill mastery, the elements are repeated throughout the instructional events. During the

Tell step, the instructor presents information to the students; during the Ask step, the instructor

asks students to recall prior knowledge; during the Show step, the instructor demonstrates

each step to solve a problem in detail; during the Do step, the students apply the knowledge


and skills they have learned to solve a problem (Merrill, 2007, p13).

Problem 1 Instructional Sequence. Prior to beginning the instruction for Problem 1, the

instructor will give an overview of ratios using a real-world context of ordering pizza.

Instruction will be expanded from this initial discussion.

TELL The instructor presents the following scenario.“We are going to discuss a real-life situation that you may encounter.”Alyssa is ordering a pizza for her family but everyone wants a different topping. Alyssa wants pepperoni, her mom and dad want mushroom, and her brother wants green pepper. Instead ordering four pizzas, they will just order one.

ASK The instructor will ask, “How do you think we can give everyone what they want with only one pizza?”

SHOW The instructor will place the pizza (see Appendix C) on the board stating that “This is pizza that has arrived at Alyssa’s house. The pizza will have 1 peperoni slice, 2 mushroom slices and one green pepper slice.”

The instructor will project a chart on the board (see Appendix B) and place each pizza slice in the appropriate place. For example, the peperoni slice will be placed in the column labeled ‘Alyssa.’

TELL The instructor will tell students that to describe this pizza they need to use ratios because it is not a pizza which only has one topping and to think about all the slices making that make up the whole pizza.

ASK The instructor will ask students “ How many slices will make the pizza whole.” “If we give Alyssa her slice, how many slices does she receive out of four slices?” The instructor will pose this question for each member of the family. “What is the part?” “ What is the whole”

SHOW The instructor will label each column with the appropriate ratio , ¼ (see Appendix B) and then will place each pizza slice in a blank pizza to illustrate the concept of part of a whole (see Appendix D). The part is the numerator and the whole is the denominator.

DO Students will go to the virtual manipulative sites to relate parts of a whole unit to a written description and ratio. http://nlvm.usu.edu/en/nav/frames_asid_102_g_2_t_1.html?from=search.html?qt=ratios

DO Students will complete the Problem 1 formative assessment (see Appendix H)

TELL The instructor will tell students“Clear your desks complete of all materials.”


“Pick up a blue strip”

ASK The instructor will ask students,“What is this?” “How many strips of paper do you have?” “Are there any parts?”

SHOW The instructor will hold up the blue strip and write 1 whole unit on the blue strip

DO The instructor will label with the class the blue strip as 1 unit and the class will be asked “What does this represent?” and they will respond “One unit.”The instructor then hands out a brown strip.

TELL “I need you to tell me something new about this brown strip. When we compare this brown strip to the blue strip they are the same. They both are equal to one unit.”

ASK What would happen if we were to cut the brown strip down the middle?” “ How many pieces would we have?” “ What is our new whole”

SHOW The instructor will fold her strip down the middle and place her strip on top of the blue strip to show that 1 brown strip is half of the blue strip.

DO The instructor and class will cut the brown strip down the middle making two strips. The students will place the two brown strips on the blue strips and label the strips ½ units.

DO Students will go to the virtual manipulative site relate parts to wholes of ratios by manipulating ratio partshttp://nlvm.usu.edu/en/nav/frames_asid_274_g_3_t_1.html?open=activities&from=search.html?qt=ratios

This instruction sequence concurrently teaches how students should strategically

approach problem solving and how to identify the parts of a ratio using concrete and abstract

representations. Although this problem could be solved in fewer steps, the instructor seeks to

establish instructional relevancy making real-world connections and arriving at solutions

through problem solving.

http://nlvm.usu.edu/en/nav/frames_asid_274_g_3_t_1.html?open=activities&from=search.html?qt=ratios

http://nlvm.usu.edu/en/nav/frames_asid_274_g_3_t_1.html?open=activities&from=search.html?qt=ratios


Materials and Media

For Problem 1, the instructional materials used are ratio kits, virtual manipulatives

website accessed via the hp elitebook revolve 810 g2 , Pizza Discussion Pizza, Pizza

Discussion Chart, the Blank Pizza and Formative Assessment 1. The Pizza Discussion

Pizza, Pizza Discussion Chart and Blank Pizza, and Formative Assessment 1 will be created

using Microsoft Word and Microsoft PowerPoint. Students will use tablet technology to

access the virtual manipulatives websites. The ratio kits will be assembled by the students

with the use of pre-labeled and cut construction paper strips. The instructor will bookmark

the websites with the names, Fraction Parts and Fraction Exploration. Paper, pencils, and

provided by the instructor. Each student will be given a folder containing all materials

needed for class.

Assessment and Evaluation. Brown and Green (2006) state that, “ (l)Learner

evaluation helps determine the level of performance or achievement that an individual has

attained as a result of instruction” (p. 208). Although the term evaluation is used in the

statement above, this plan uses the term assessment when referring to learner evaluation.

Ongoing assessment during the learning process is performed by the instructor (formative)

and at the completion of instruction (summative) (Brown & Green, 2006, p 230).

The formative assessment, assess how the students are progressing during the

instructional strand, the instructor will use a formal formative assessment during problem

1 and problem 4, observations and quick write activities. “A quick write

involves having a learner write a response to either a question the instructor has asked or to

the major ideas he or she has learned to that point in the instruction” (Brown & Green, 2006,

p. 230).


Summative assessment. For the summative assessment, the instructor will use a word

problem based upon a real-world scenario. The students will have to find the GCF of each

ratio in a beverage recipe (see Appendix F).

Student performance on the Post Assessment will be considered successful if the students

are able to correctly scale down 3 out 4 components. All students were able to successfully list

the factors of each numerator and denominator of the ratios. Some students struggled with

remembering to divide both the numerator and denominator by the GCF when scaling down a

ratio.

Formative evaluation. According to Brown and Green (2006), “ (f)Formative

evaluation is used throughout the instructional design process to gather data that can be used

to provide feedback on the process” (p. 238). The instructor selected Phase 1 of Dick, Carey,

and Carey’s approach (2001) to formative evaluation by obtaining feedback from individual

students on the instructional materials. This plan did not use Phase 2 because the purpose of

this phase is to determine the effectiveness of the changes made in Phase 1 (Brown & Green,

2006). Due to the limited timeframe for the class instruction, there was insufficient time to

implement Phase 2.

The instructor received test results and comments that the instructions were clear and

easy to understand, which helped the instructor feel more confident about the instructional

design decision to use a pre-assessment.

Summative evaluation. The instructor used Smith and Ragan (1999) for the

summative evaluation. Smith and Ragan describe a summative evaluation process consisting

of eight steps (as cited in Brown & Green, 2006, p. 251).

1. Determine the goals of the evaluation. The goals in this plan were instructional and


determined by the instructor/SME. The desired goal of the instructional design plan is

to provide instruction that is coherent and sequential building upon a mastery of a

progression of problems leading to the final problem. The evaluation should ask

appropriate questions to determine if the instructional designer met the desired goals.

2. Select indicators of success. Success will indicated by students achieving a 75% on the

Post- Assessment.

3. Select the orientation of the evaluation. There will be an objective evaluation using the

Learner Survey and Post-Assessment, and a subjective evaluation based on observations

during the instructional module.

4. Select the design of the evaluation. The Post-Assessment will be created around a real-

world context to apply the skills obtained through instruction. Students will use have a

graphic organizer to organize the information needed to successfully demonstrate

scaling down a ratio. The Learner Survey will contain questions asking whether students

liked the instructional approach, instructional materials, and methods.

5. Design or select the evaluation measures. The instructor selected results from the Pre-

Assessment, Post- Assessment t, and Learner Survey as evaluation measures.

6. Collect the data. The instructor will collect, compile and analyze data in an excel

spreadsheet document during and post instruction.

7. Analyze the data. The instructor will review the feedback from the Perception and Learner

Surveys and compare the results. The Perception Survey will be administered pre-

instruction and the Learner Survey post-instruction.

8. Report the results. According to Smith and Ragan (1999) (as cited in Brown & Green,

2006), the report should include a “summary, background information, description of the

evaluation study, results, and conclusion and recommendations” (p. 251). The instructor

will report the results using this method.


Evaluation report. The purpose of the evaluation is to measure the success of the

instructional design approach implemented. The evaluation was designed using a Post-

Assessment (see Appendix F) and Learner Survey (see Appendix L). The success of the

instruction will be measured if students achieve a 75% or greater on the Post-Test, and

through feedback on the Learner Survey.

This instructional design plan was implemented with twelve students in a Math 6

classroom. Eight out of 12 achieved a 75% or greater after instruction, one student achieved a

100% (4/4), two students achieved a 50% (2/4) and two students achieved a 25% (1/4). The

instructor did not award partial credit but did provide substantive comments on the Post-

Assessment and the opportunity for students to redo with explanations the examples that they

did not get correct.

Based on the results, the instructor makes the following recommendations. Problem 2

could be eliminated and instead Problem 4 should be expanded into two problems. This will

enhance instructional coherency as well progression of problems in a sequential process. It

seemed like instruction was hurried in Problem 4 in order to get to Problem 5. Students needed

more practice with factor trees and dividing to find the equivalent fractions.

Conclusion

I would like to implement this plan with my other math classes because they could

benefit from this level of instruction on a challenging topic. With the CCSS, there is a need for a

progression of problems during instruction as the expectation is that the students are taught to

mastery in each content. This level of instruction will help instructors to design instruction to

teach to mastery. There is an system-wide opportunity to present during professional study day

in my content that I am considering giving an overview of success using the PITP approach to


instructional design in instruction. I feel that other instructors could benefit from this information

and enhance the quality of the instruction.


References

Brown, A., & Green, T. D. (2006). The essentials of instructional design: Connecting

fundamental principles with process and practice. Upper Saddle River, NJ: Pearson

Education.

Common Core State Standards Initiative. (2014). Retrieved May 1, 2014, from

http://www.corestandards.org

Dezmond, B. (2005). Style to Content™ Learning Preferences Inventory. Style to Content™

Learning Preferences Inventory Users Guide. : DEZMON™ Educational Strategies,

LLC.

International Society for Technology in Education. (2008). NETS for teachers 2008.

Retrieved April 20, 2014, from

http:// www .ist e .o r g /Cont e nt/ Na vig a tionM e nu/ N E T S/ F o rTe a c h er s/2008St a nd ar ds/ NET S_

f o r _ Te a c h e r s_2008.htm

Irvine, J. J. & York, D. E. (1995). Learning styles and culturally diverse students. In J. Banks

C. Banks (Eds.) Handbook of research on multicultural education. NY: Macmillan

Publishing.

Maryland State Department of Education. (2003). Maryland teacher technology

standards. Retrieved April 20, 2014 from http:// w ww .mttsonlin e .o r g /st a n d ar ds/ind e x .php

Merrill, M. D. (2002). A pebble-in-the-pond model for instructional design. Performance

Improvement, 41(7), 39-44.

http://www.corestandards.org/

Progression of Problems

Guidance Provided


Appendix A

Pebbles Plan Graphic Organizer

Instructional Problem: Students in my sixth grade math students are experiencing difficulties when finding the GCF to write ratios in simplest form.Instructional Goal: Students will be able to find the greatest common factor (GCF) of the numerator and denominator in order to scale down a beverage recipe.

Skills/Knowledge – P1

S/K – P2 S/K – P3 S/K – P4 S/K – End P

-Students will understand that the numerator is the part and the denominator is the whole.

-Students will know how to compare parts of whole.

-Students will know the number above the ratio bar is the numerator and the number below the fraction bar is the denominator.

-Students will understand how to compare parts with one whole with parts of another.

-Students know how to make concrete representations of fractions.

-Students understand that equivalent fractions have the same value but are just a different form.

Students will know how to make concrete representations of equivalent ratios to make legal trades.

i.e. 2 reds= 1 yellow

4 light blues=1 orange

-Students will know how to model and write equivalent fractions.

-Students will know how to list products of two quantities to find the GCF.

-Students will know how to factorize two numbers.

-Students will know how to factorize a numerator and denominator.

-Students will understand that you must divide the numerator and denominator by the same quantity to find to simplify the ratio.

-Students will find the GCF of the numerator and denominator by listing the factors of the numerator and denominator.

-Students will divide the numerator and the denominator by the GCF.

-Students will know how to write equivalent ratios by writing the ratio in simplest form

-Students will know how to scale down ingredients in a recipe to feed less people.

Understand the concept of ratios

Create concrete representation of fractions

Model equivalent ratios

Find the Greatest Common Factor

Scale a beverage recipe down

Learners: Sixth grade math students in an Urban fringe classroom in Suburban Baltimore County c who scored below 70%


Appendix B Pizza Order Chart

PIZZA ORDER CHART- What did each person order?

Alyssa Dad Mom Brother

Appendix CPizza Discussion Tool


Appendix DBlank Pizza


Appendix E


Style to Content Learning Preferences Inventory

STYLE TO CONTENT LEARNING PREFERENCES INVENTORY

This survey asks about what you like and don’t like about learning and your schoolwork. Please read each statement carefully and mark your answer sheet to show the one answer that tells how much you agree or disagree with the statement. SD Strongly Disagree D Disagree N Neutral A Agree SA Strongly Agree For example, if you strongly agree with a statement, you should circle “SA” on your answer sheet. If you make a mistake or change your mind, mark an “X” through your first answer. Then circle the answer you want. There are no right or wrong answers to these questions. Mark your answer sheet to show how you like to learn. If you have questions at any time, raise your hand. © Copyright 2006 Barbara Dezmon, PhD. All rights reserved.

1. I like to study with a partner or in a group.


2. I work best in a group instead of alone. 3. I like to study with my friends. 4. When I have a problem, I like to talk to others about how to solve it. 5. I like friends to help me do well in school. 6. I’d rather complete a group project than do one alone. 7. I like class discussions better than working alone. 8. I like class best when I’m working with other students. 9. I like group work more than independent work. 10. I like class best when we are working with the teacher. 11. I learn something better if I can think it through before doing it. 12. I study best when I am by myself. 13. I like best to work by myself on projects. 14. When I am learning something new, I would rather work on it alone. 15. I like it when the teacher gives me directions and then leaves me alone to work. 16. I like independent work more than group work. 17. I like to set my own goals for learning. 18. I like the teachers to give me directions and then let me do the work on my own. 19. I like to work on my own. 20. I like to solve problems by myself. 21. I like to use flash cards or pictures to help me remember. 22. I learn best when books have pictures, maps, or diagrams. 23. I do better when the teacher gives written directions rather than speaks them. 24. I get mixed up when there is a lot of movement or clutter in the classroom. 25. I like to watch someone show me how to do something. 26. It is easier to learn if I can see a picture. 27. Graphs and charts help to make information clear for me. 28. I can solve problems when they are written better than when someone speaks them to me. 29. I can remember what I learned best when I write it down. 30. Usually I notice when people change something about their appearance, like getting a haircut or getting new glasses. 31. I like to listen and take notes. 32. I remember what someone tells me better than when they write it. 33. I remember best when I hear the directions. 34. I like to listen and take notes as the teacher talks. 35. I learn best if someone tells me how to do something. 36. I know the words to most of the songs I listen to. 37. Repeating something aloud helps me remember it. 38. I can remember what I learned best when I say it to myself. 39. I sing the words to songs and commercials I’ve heard. 40. I do best when I hear a math problem.


41. I like to put things together. 42. I learn best when the teacher shows us how to do something. 43. I learn best when I can touch things. 44. I like hands-on work like projects and labs. 45. I like to take things apart and put them together. 46. I like to use my hands and make things. 47. I am good at putting pieces together to solve a puzzle. 48. I understand math best when I can touch objects. 49. I like to show what I know by building a model or making a project. 50. I like drawing and designing things. 51. I like to demonstrate or show others how to do things. 52. I like to be physically active. 53. I like field trips that teach about things I am learning. 54. I do better on tests if I can take short breaks. 55. I like to work with things I can touch, build, or move. 56. I like to try something new right away so I can learn it faster. 57. I like to use my hands when I talk about something. 58. I can show what I have learned by acting it out. 59. I learn best when I get a chance to do physical things. 60. I like to study with music in the background. 61. I like learning games and competitions. 62. I like group work more than working alone. 63. I like to make connections about something I am learning. 64. I do my best when I am working in a group in class. 65. I like it when the teacher uses the whiteboard, overhead projector, or posters to teach new information. 66. I remember something better if I talk about it. 67. I understand something better if I can draw it. 68. I want to see how something works before I talk about it. 69. It is hard for me to sit still for very long. 70. I get bored when the teacher takes a very long time to explain something. 71. I like to express my ideas in a journal. 72. I like to solve problems by thinking carefully about the task. 73. I like to work in a quiet place. 74. I like to have time to think about something before starting to work on it. 75. Writing my thoughts helps me to understand things. 76. I like to think carefully about a problem before talking about solving it. 77. I learn best when I write. 78. I like to think for a while before I tell people how I feel about something. 79. I do my best work when I’m working alone in class.


80. I want time to think about a problem before I start to solve it. 81. I like to get the big picture first then work on details. 82. I like to make connections. 83. I like to work with ideas and theories. 84. I like it when my teacher corrects my work. 85. I like to know the main idea before I read something. 86. I read a short summary of the story first so I can understand the story better. 87. I want to know about the whole project I am supposed to complete before I break it down into steps. 88. I tend to make decisions quickly based on my first impressions. 89. I like to get big chunks of information rather than small details.

90. I learn best when I can look at the whole picture rather than the parts. 91. I like to follow directions step-by-step. 92. I like to work with formulas or procedures in order. 93. I like solving problems by thinking about the steps needed to find the answer. 94. I like to use diagrams, outlines, or flowcharts to organize new information. 95. I like to use my finger or a marker when I read new or difficult information. 96. I can do something well when I have a set of directions to follow. 97. I like to do things in the right order. 98. I like teachers to break directions down one step at a time. 99. I like to go step-by-step through a problem when I solve it. 100. I like to think about parts of the picture before I see the whole.


Appendix F Post Assessment Post ASSESSMENT

A caterer makes a special punch using fresh juices. Here is how the ingredients are mixed for its Super Punch.

810

of the mixture is apple juice

520

of the mixture is pineapple juice

315

of the mixture is pomegranate juice

721

of the mixture is grape juice

The caterer wants to make just one serving of the special punch. He needs to know the ratio of each fruit juice to make the special punch.

Part WholeScaling

down factor(GCF)

New Ratio

Apple Juice

Pineapple Juice

Pomegranate Juice

Grape Juice

Appendix G


Pre-AssessmentThe Pre-Assessment will be administered online via the link below: http://www.thatquiz.org/tq/previewtest?O/I/S/L/92341251696249

Below is a screenshot of the Pre-Assessment questions.

http://www.thatquiz.org/tq/previewtest?O/I/S/L/92341251696249


Appendix HProblem 1 Formative Assessment

Formative Assessment IA. Identify the part (shaded) and the whole (shapes) of each picture.B. Write the ratio that the shaded portion of the ratio represents

Part: Part:Whole: Whole:Ratio

❑❑

Ratio❑❑


Appendix IStudent Levels of Performance on the MSA

Advanced Proficient Basic0

1

2

3

4

5

6

7

Math 6 Period 1 - MSA Levels

Performance Level

num

ber o

f stu

dent

Scaling Down a Recipe: Finding the GCF to Represent Ratios in Simplest Form

Appendix JStudent Perception Survey

How I Feel About Math

A. Read each statementB. Place a check in the box that best describes how you feel.

Statement Strongly

Agree

Agree Neutral Disagree Strongly Disagree

I think mathematics is important in lifeI feel confident with my ability to work with ratios.In the past, I have not enjoyed math class.I am sure I can learn math.

My instructor will assist me if I do not understand.I am willing to try my best even if I do not know the answer.I know how to find the GFC of multiples.

Appendix KFormative Assessment- Problem 4


A. Select the best response

B. Use a factor tree to list the factors of each set of multiples

9 and 12

C. Find the GCF of the ratio

912

Appendix LLearner Survey

Learner Survey

Please tell me more about yourselves so that I can design activities that you enjoy and allow you to learn best.

Which choice is not afactor of 52?A. 4B. 26C. 3D. 13


Formative Evaluation Questions

1. Did you like working with the ratio kits? Why or Why not?

2. What do you like or what don’t you like about the websites?

3. How can I improve instruction?

Summative Evaluation Survey

1. Did you understand what the scenario asked you to find? Why or why not?

2. Did you feel that the instruction prepared you for the task?

3. Do you understand how you can use ratios in the real-world?

4. Is there something you would like to see changed in the way the instruction was done?

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