MATHEMATICS GRADE 7 HISD Curriculum: Unit...

MATHEMATICS GRADE 7 HISD Curriculum: Unit Framework

- English Language Proficiency Standards (ELPS) - Literacy Leads the Way Best Practices - Aligned Readiness - Process Standards

Ⓟ - HISD Power Objective Ⓡ - STAAR Readiness Standards Ⓢ - STAAR Supporting Standards Ⓣ - TAKS Tested Objective

© Houston ISD Curriculum

2011 – 2012

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Unit Framework User Information

Grading Cycle Instructional Days Recommended Time Allocation

2nd Six Weeks

25 Days 5 or

10

Oct 3- Nov 4, 2011 90-minute lessons 45-minute lessons

Unit 2.1 Overview

Integer Operations Students model and solve real-world problems involving integer operations and develop an understanding of the algorithms involved.

Outline of Unit(s) in the Six Weeks

Unit 2.1 – Integer Operations [ this unit ] link to Unit Planning Guide and supporting materials Unit 2.2 – Ratios and Rates Unit 2.3 – Proportional Reasoning

Essential Understandings

Numeric operations are used on integer quantities in real-world problems.

Key Concepts

integer operation real-world problem

Academic Vocabulary Content-Specific Vocabulary

patterns models generalization

additive identity property of zero community property additive inverse identity property

inverse operation multiplicative identity opposite integer zero pair

HISD Objectives / TEKS

Ⓢ MATH.7.2C

Use models, such as concrete objects, pictorial models, and number lines to add, subtract, multiply, and divide integers

and connect the actions to algorithms.

Ⓡ MATH.7.2F

Select and use appropriate operations to solve problems and justify the selections, problem solving process, and

reasonableness of answer, especially in terms of an estimate.

Ⓢ MATH.7.2G

Determine the reasonableness of a solution to a problem using a variety of strategies such as estimation using rounding

or compatible numbers.

MATH.7.13B

Use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and

evaluating the solution for reasonableness.

MATH.7.15A

Make conjectures from patterns or sets of examples and nonexamples.

MATH.7.15B

Validate conclusions using mathematical properties and relationships.

Performance Expectation(s)

Students will use a problem-solving strategy and use concrete objects, pictorial models, and number lines to perform numeric operations on integer quantities in real-world problems to develop an understanding of the algorithms involved.

Texas English Language Proficiency Assessment System (TELPAS): End of year assessment in Listening, Speaking, Reading, and Writing for all students coded as LEP (ELL) and students who are LEP but have Parental Denials for Language Support Programming (coded WH). For the Writing TELPAS, teachers provide 5 writing samples (1 narrative about a past event, 2 academic {Science, Social Studies, Mathematics}, and 2 other).

https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0BzaY2soe3hc_MThiNmFmZjgtZmQ0ZC00Yjk1LWE2MWYtZjZjNzJjNWY1YjQ0&hl=en_US&authkey=CJC51eML

http://www.tea.state.tx.us/student.assessment/ell/telpas/





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2nd Six Weeks

25 Days 2 or

4


Unit 2.2 Overview

Ratio and Rates Students begin their study of proportionality with ratios, rates, and unit rates.


Unit 2.1 – Integer Operations Unit 2.2 – Ratios and Rates [ this unit ] link to Unit Planning Guide and supporting materials Unit 2.3 – Proportional Reasoning


A ratio is a multiplicative comparison of two quantities. Every rational number can be expressed as a ratio or rate.

Key Concepts

ratio rate comparison rational number


relationship quantity

ratio rate

unit rate proportion/proportional


Ⓢ MATH.7.2D

Use division to find unit rates and ratios in proportional relationships such as speed, density, price, recipes, and student-

teacher ratios; and model those relationships concretely, pictorially, and in tabular form.

MATH.7.14A

Communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or

algebraic mathematical models.


Students will communicate mathematical ideas about ratios, rates, and unit rates in proportional relationships using graphical, numerical, physical, or algebraic mathematical models.

Texas English Language Proficiency Assessment System (TELPAS): End of year assessment in Listening, Speaking, Reading, and Writing for all students coded as LEP (ELL) and students who are LEP but have Parental Denials for Language Support Programming (coded WH). For the Writing TELPAS, teachers provide 5 writing samples (1 narrative about a past event, 2 academic {Science, Social Studies, Mathematics}, and 2 other).







2011 – 2012

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2nd Six Weeks

25 Days 3 or

6


Unit 2.3 Overview

Proportional Reasoning Students write and solve real-world problems involving proportional reasoning, such as measurement comparisons and cost ratios. Students use intuitive methods such as unit rates and factor of change strategies.


Unit 2.1 – Integer Operation Unit 2.2 – Ratios and Rates Unit 2.3 – Proportional Reasoning [ this unit ] link to Unit Planning Guide and supporting materials


Proportional reasoning involves real-world problems and relationships between ratios or rates. Comparison of attributes can be made within and between customary and metric measurement systems.

Key Concepts

comparison measurement

proportional reasoning rate

ratio real-world problem relationship


graphs measurement

tables

cost ratio conversion ratio equivalent ratios similarity

factor of change ratio table scale factor Unit Rate Method


Ⓡ MATH.7.3B

Estimate and find solutions to application problems involving proportional relationships such as similarity, scaling, unit

costs, and related measurement units using intuitive methods (such as unit-rate method, factor of-change approach, or a

graphical/visuals approach).

Ⓢ MATH.7.4A

Generate formulas involving unit conversions within the same system (customary and metric), perimeter, area,

circumference, volume, scaling, and sequences of numbers from a variety of representations including verbal

descriptions, tables of data, and diagrams.

MATH.7.13A

Identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and

with other mathematical topics.

MATH.7.13D

Select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math,

estimation and number sense to solve problems.


Students will write and solve real – world problems involving proportional reasoning such as similarity, scaling, unit costs, and related measurement units using various methods involving manipulatives, paper/pencil, and technology. Texas English Language Proficiency Assessment System (TELPAS): End of year assessment in Listening, Speaking, Reading, and Writing for all students coded as LEP (ELL) and students who are LEP but have Parental Denials for Language Support Programming (coded WH). For the Writing TELPAS, teachers provide 5 writing samples (1 narrative about a past event, 2 academic {Science, Social Studies, Mathematics}, and 2 other).



MATHEMATICS GRADE 7 HISD Curriculum: Unit 2.3 Planning Guide

English Language Proficiency Standards (ELPS) - Literacy Leads the Way Best Practices - Aligned Readiness - Process Standards



2011 – 2012

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Unit Planning Guide User Information

Unit 2.3 Proportional Reasoning

3 or

6

90-minute lessons 45-minute lessons


Ⓡ MATH.7.3B

Estimate and find solutions to application problems involving proportional relationships such as similarity, scaling, unit

costs, and related measurement units using intuitive methods (such as unit-rate method, factor of-change approach, or a

graphical/visuals approach).

Ⓢ MATH.7.4A

Generate formulas involving unit conversions within the same system (customary and metric), perimeter, area,

circumference, volume, scaling, and sequences of numbers from a variety of representations including verbal

descriptions, tables of data, and diagrams.

MATH.7.13A

Identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and

with other mathematical topics.

MATH.7.13D

Select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math,

estimation and number sense to solve problems.

English Language Proficiency Standards College and Career Readiness Standards

ELPS C.1g Demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one. commensurate with grade-level learning expectations

ELPS C.2i Demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling, or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs.

ELPS C.4i Demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with content area needs.

CCRS 1.B1 Perform computations with real (not complex) numbers.

CCRS 4.A1 Select or use the appropriate type of unit for the attribute being measured

CCRS 8.A3 Determine a solution. CCRS 8.B2 Use various types of reasoning. CCRS 9.A3 Use mathematics as a language for

reasoning, problem solving, making connections, and generalizing.

CCRS 9.C1 Communicate mathematical ideas reasoning and their implications using symbols, diagrams, graphs, and words.

Essential Understandings / Guiding Questions

Proportional reasoning involves real-world and relationships between ratios or rates. 1. How is a proportion constructed? 2. What kinds of questions can be answered using proportional reasoning? 3. How can proportional reasoning be used to make predictions?

Comparison of attributes can be made within and between customary and metric measurement systems. 1. How do you convert between units of measure within the same system? 2. How are the customary and metric measurement systems different and/or similar? 3. How is proportional reasoning used to solve measurement problems?

Instructional Considerations Instructional Strategies / Activities

Prerequisites and/or Background Knowledge for Students In sixth grade, students used multiplication and division to solve problems involving equivalent ratios and rates. Students used tables and symbols to represent and describe proportional and other relationships.

Nonlinguistic Representations Students should work in groups to measure different objects using standard and jumbo paper clips, to graph the data and to discuss the factor of change relationship (Activity: Paper Clip Chains – Factor of Change –see Resource column).

https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0BzaY2soe3hc_MTAzODNkODQtZmZiMi00ZjYxLWJhZTMtNmZhY2U5NjFmNjMx&hl=en_US&authkey=CP2_wxA

https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0BzaY2soe3hc_YTdkNTA5NWMtMWU5YS00N2RiLWE0MTItMjA3MTIyNTc2ZWU5&hl=en_US&authkey=CJqE4-IJ





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They converted measurements within the same system.

Background Knowledge for Teacher Critical Content Write proportional statements; Use factor of change to solve proportions; Use unit rate method to solve problems involving cost

ratios; and Solve problems involving measurement units.

Readiness Standard 7.3B asks students to write and build proportions, and to find the missing number in a proportion using intuitive methods such as unit rates and the factor of change method. The students should master these methods before they begin the use of the cross product algorithm. Students must have a conceptual understanding of the factor of change in order to transition to the algebraic concept of rate of change. While considering the “Instructional Strategies” column, note the verbs that align with the 5E Lesson Model: Engage, Explore, Explain, Elaborate, and Evaluate. These cues indicate appropriate strategies, level of rigor, and level of questioning to use during instruction. Vocabulary

Academic Content-Specific

measurement factor of change

tables scale factor

graphs unit rate

cost ratio

ratio table

Assessment Tips Students must be able to recognize abbreviations of measurement units of all types. Educational Technology Connection: Students use formulas in the design of a spreadsheet to assist in computing the measurement conversions. A step-by-step tutorial for this using formulas in Microsoft Excel is available online at About.com

Cues, Questions, and Advance Organizers

KWL Engage students via a brainstorming activity and a poster-sized KWL chart to build on their previous experience with proportional reasoning. For example, students many recall that “scaling up” a recipe is a process that is mathematically equivalent to solving a proportion. Students then create a table of equivalent ratios for a situation problem such as one involving a recipe. They pull proportional statements from the table and illustrate how to verify proportionality using a factor of change or scale factor. After they complete the activity, students should return to their class KWL chart and fill in the Learn column with what was learned in the activity (Activity: Applying the Factor of Change- see Resource column). Instructional Accommodations for Diverse Learners Cooperative Learning Students working in pairs use grocery store ads or circulars to assist in activities involving cost ratios and finding best buys (Activities: How Much, How Many?, How Much for One? C.1g, C.2i, C.4 Nonlinguistic Representations Various real-life problems that enable students to apply

the skills involved in proportional reasoning are available in MS TEXTEAMS, Proportionality Across the TEKS-(Activities: Slithering Solids, Unit Rates – see Resource column). Students examine measurement conversions in the context of proportionality by creating tables and graphs to illustrate various conversions in the activity Lost and Gained – see Resource column.

Middle School Mathematics Assessments contains problems that focus on students’ conceptual understanding as well as their procedural knowledge. The tasks require more than right or wrong answers; they focus on how students are thinking about a situation. Teachers should use evidence of student insight, student misconceptions, and student problem solving strategies to guide their instruction. Teachers may also use the questions included with the assessments to guide learning and to assess student understanding. The use of these assessments should help teachers enhance student learning and provide them with a source of evidence on which they may base their instructional decisions. Each problem

includes a mathematics task

includes “scaffolding” questions that the teacher may use to help the student to analyze the problem

provides one sample solution (which may not be the only solution possible)

http://spreadsheets.about.com/od/excelformulas/ss/formula_begin.htm


https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0BzaY2soe3hc_YzJiYzM3NjgtZWM2OS00YzMzLTg2NTgtNzg3NGFjYjhkY2Q4&hl=en_US&authkey=CPKiw68F






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includes extension questions to bring out additional mathematical concepts in a summative discussion of solutions to the problem.

Middle School Mathematics Assessment Solution Guide is a problem-solving checklist that teachers and students use to understand what is necessary for a complete problem solution. When assigning the problem, the teacher will give the students the solution guide and will indicate which of the criteria to consider in the problem analysis. In most problems, all of the criteria are important, but initially the teacher may want to focus on only two or three criteria (Formative Assessment 2.3 – Measurement Ratios)

Assessment Connections

Formative Assessment 2.3 – Measurement Ratios- student use a problem-solving checklist that teachers and students use to understand what is necessary for a complete problem solution. When assigning the problem, the teacher will give the students the solution guide and will indicate which of the criteria to consider in the problem analysis.

Resources

Clarifying Activities: The following activities involve multiple hands-on opportunities. Paper Clip Chains – Factor of Change Applying the Factor of Change How Much, How Many? How Much for One? Lost and Gained Slithering Solids Unit Rates

Textbook Resources: Glencoe, Texas Mathematics, Course 2. TWE/SE, pp. 342 – 354 Ch. 7 resource masters, pp. 21 – 32

Technology Resources: A step-by-step tutorial for using formulas in Microsoft

Excel may be found on About. com. Understanding Math 2008 from Neufeld Learning

Systems is a software program available on HISD Middle School servers.

Middle School Mathematics Assessment Solution Guide



http://www.neufeldmath.com/correlations/TEKS/TEKS-6-Aug08.pdf

99TEXTEAMS Rethinking Middle School Mathematics: Proportionality

Slithering Solids

Institute Notes

Concept: Investigate using ratios to represent rates of change;use fractions and decimals to describe these rates ofchange as unit rates, and compare the unit rates to solveproblems.

TEKS Focus: 6.3—The student solves problems involving proportionalrelationships.7.3—The student is expected to estimate and find solu-tions to application problems involving proportional re-lationships such as similarity, scaling, unit costs, andrelated measurement units.8.3—The student identifies proportional relationships inproblem situations and solves problems.

Overview: Participants will collect data based on dry ingredientsbeing emptied through a funnel into a cylindrical con-tainer. They will record the amount (in scoops) of dryingredient, the time (in seconds) for the funnel to empty,and the resulting height (in cm) of the column of dryingredient in the container. Three different sets of ratioswill be formed from the data to describe three differentrates (seconds per scoop, cm per scoop, and cm persecond). The three sets of ratios will be graphed, andthe results analyzed according to the properties of pro-portional relationships.

Materials: Clear cylinder1" graph paper (optional)Stop watchesCm ruler4 oz. or smaller scoopsPaper plate or paper towelsFunnel (store-bought or made by cutting top from plas-tic bottle)Markers or Peel-and-stick dotsGraphing calculators (optional)Dry ingredients (salt, rice)5 x 5" cardboard

Also:Grade 66B, 6BGrade 77B, 7DGrade 88B, 8D


Slithering Solids

Math Notes:There are several functionalrelationships that can beexplored in this experiment.Since the amount of the in-gredient poured into the fun-nel determines the numberof seconds, one can saythat the number of secondsis a function of the numberof scoops. Also, the heightof the ingredients in the cyl-inder is dependent upon thenumber of scoops or thetime which has elapsed. (Ifthere was a constant sup-ply of ingredient, it couldalso be said that the num-ber of scoops that flowedthrough the funnel is depen-dent upon the time.)

Each set of points plotted onthe three graphs will lie ap-proximately on a line. Ineach set of ratios, the ratesof change represented maynot be exactly constant inthe experiment, but they arevery close. It is important tomake a connection betweenthe set of points, the lineused to represent the set ofpoints, and a rate of changethat can be used to makepredictions based on thatset of data. By drawing aline through each point andthe origin to represent whatthe data would look like witheach constant rate ofchange, you can lead theparticipants to identify theone line that they think bestrepresents the data (or leadthem to the use of thegraphing calculator’s re-gression capabilities to findthe line of best fit). The con-nection can then be madebetween the unit rate ofchange that the points onthat line represent (e.g., theconstant number of sec-onds per each scoop) and

Procedure: 1. Divide participants into groups of four.2. Distribute experiment instructions and the materials.

Pose the following discussion questions before be-ginning the experiment:How will ratios play a part in describing the results ofthis experiment?What do you expect the graph of the set of orderedpairs (number of scoops, number of seconds) to looklike? Why?If two different groups have the same ingredient butdifferent funnels, how do you expect the graphs tobe different? Why?

3. Ask participants to follow the directions, work on theexperiment, construct the three graphs, and answerthe questions.

4. If large grid paper and peel-and-stick dots are avail-able, have each group graph their results on the largepaper and post the graphs. If different groups havedifferent ingredients or different funnels, ask them toguess which group made which graph.

Debriefing: Conclude the activity with a whole group discussion ofthe ratios that can be used to describe the rates repre-sented in the data collected. Use the Reason and Com-municate questions to have participants analyze thethree graphs for properties of proportional relationships.

Describe how to change this experiment to affect therates of change (answers vary.)

Describe what you would change in this experiment sothat it represents a non-proportional relationship.Start with some of the solid material already in the con-tainer.

The following sample data is used to present a discus-sion of the results:

# of scoops 1 2 3 4 5 6 7 8

Time in seconds 0.41 0.84 1.63 1.91 2.07 2.63 3.07 3.57

Height in cm 0.5 1.1 1.5 2.1 2.6 3.1 3.7 3.42


Slithering Solids

the slope of that line (theconstant change in y foreach unit change in 2). Thissingle line through the ori-gin, then, represents a pro-portional situation (e.g., thenumber of seconds is pro-portional to the number ofscoops) in which the con-stant of proportionality, theunit rate, can be used in theform y = kx to make predic-tions.

How can we use ratios to describe the results of theexperiment? The ratios 0.41 sec/1 scoop, 0.84 sec/2scoops, 1.63 sec/3 scoops, and so forth describe therelationship of number of seconds to number of scoops(for Graph 1). Similarly, ratios can be made from thedata to describe the relationships in Graph 2 (number ofcentimeters to number of scoops) and Graph 3 (num-ber of centimeters to number of seconds).

How can we compare the ratios, e.g., do 3 scoops flowthrough the funnel at the same rate (number of secondsper 1 scoop) as 2 scoops?Using division to find decimal equivalents for each frac-tion, each of these ratios can be described with anequivalent unit rate: 0.41 sec/1 scoop. 0.84 sec/2 scoops= 0.41 sec/scoop, 1.63 sec/3 scoops is about 0.54 sec/scoop, and so forth.

What should Graph 1 look like if all the rates (of sec-onds per scoop) represented by our data were the same?If all the ratios were equivalent, that would mean thatthe relationship between the time to drain and the num-ber of scoops is a proportional relationship, and thepoints would form a straight line throught the origin. Forexample, we can find all the points that have thecharateristics = 0.4 sec/scoop if we draw a line con-necting (0,4) and (0,0).

What does Graph 1 actually look like?…points scattered around, but near the line that repre-sents y/x = 0.4 seconds per scoop

If you had to pick one point from our data to draw a linethrough the origin to represent the whole set of data,which point would you pick? Why? What would this linerepresent?It would represent the set of points that all have the sameunit rate as the data point you picked.

How can you find the line that best represents the set ofpoints on Graph 1?You can use the regression function on a graphing cal-culator. For example, for this data, Graph 1 is best rep-resented by the line y = 0.423x + 0.104.

yx


Is the relationship of time to the number of scoops aproportional relationship? Why or why not?Not exactly, because the data do not lie exactly on astraight line through the origin. However, the regressionline (the line of best fit) goes pretty close to the origin,meaning that the data are fairly well represented by aproportional relationship described by the unit rate of0.423 seconds per scoop.

How can you use the unit rate you decide upon to de-scribe the relationship to make predictions?For example, if you decide upon the unit rate of 0.4 sec-onds per scoop to describe a proportional relationshipthat is close to the data in Graph 1, then you can usethe y = kx representation of proportionality to determinethat the number of seconds, y, is equal to the unit ratetimes the number of scoops, or y = (0.4 seconds perscoop) (15 scoops) = 6 seconds. The same questionsshould be asked about Graph 2 (cm per scoop) andGraph 3 (cm per second).

Extensions: Try the experiment again using odd-shaped bottles in-stead of cylinders and predict which graphs you thinkwill be different from the cylinder graphs and how youthink they will be different.Develop your own experiment that models equivalentratios describing a constant rate of change.

Assessment: Journal Entry—Describe other situations in which a ra-tio describing a rate of change would be an importantcomponent in an activity.

Notes:

Slithering Solids


Reason and Communicate:How can we use ratios to describe theresults of this experiment?

What do you expect each graph of thesets of ordered pairs (e.g., number ofcups, time in seconds) to look like?Why?

If two different groups have the sameingredient but different funnels, howdo you expect the graphs to be dif-ferent? Why?

If two groups have the same funnels,but different ingredients, how do youexpect the graphs to be different?Why?

How can we compare the ratios?

What should the graph look like if allthe ratios (rates) represented by ourdata were the same?

What does the graph actually looklike?

If you had to pick one point from ourdata to draw a line through the originto represent the whole set of data,which one would you pick? Why?What would this line represent?

How can you find the line that bestrepresents the set of points on thegraph?

Is the relationship a proportional one?Why or why not?

How can you use the unit rate youdecide upon to describe the relation-ship to make predictions?

Slithering Solids

TEXTEAMS Rethinking Middle School Mathematics: Proportionality Activity-104

Slithering Solids

Activity 1

Set-up: A group member will need to be assigned to each of thefollowing jobs: Measurer, Funnel Holder, Time Keeper,Recorder

Procedure: 1. Place the funnel over the cylinder with your finger,cardboard, or a ruler under the spout.

2. Measure 1 scoop of the dry ingredient and pour itinto the funnel.

3. Slip your finger or cardboard off the spout. Time thenumber of seconds from the time you release your finger until the funnel isemptied.

4. Shake the cylinder to level the dry ingredient in the cylinder. Measure theheight of the cylinder from the table to the top of the column of dry ingredient.

5. Record the data in the table on the data page.6. Empty the cylinder by pouring the dry ingredient into another container. You

may use this scoop of dry ingredient again as part of the next measurementusing 2 cups.

7. Repeat the steps 1-6 for the other measurements shown in the table.8. Graph the sets of ordered pairs described on Graph 1, Graph 2, and Graph 3

using the graph grid or a graphing calculator.

Data

Number of Time to Nearest Hundredth Height toScoops of a Second Nearest Tenth

of a Centimeter

1

2

3

4

5

6

7

8

9


Slithering Solids

Reason and Communicate:How can we compare the ratios?







If you had poured the ingredient intoa container which was in the shape ofa cone, would this graph be any dif-ferent? How? Why?

Answers:a. Answers will vary. But, all should compare y = time in

xseconds : number of scoops or x = number of scoops : time

yin seconds.

b. This ratio is a unit rate and is written number of scoops . 1 second

c. (unit rate) time in 1 second • 15 scoops = time in seconds for 15 number of scoops

scoops to be emptied from the funnel

d. If the spout of the funnel were smaller in diameter, it would take moretime for the funnel to empty. This would make the graph steeper.


Slithering Solids

Activity 2

Make a graph of the ordered pairs (number of scoops, time in seconds) using 1"graph paper or the grid below.

Graph 1

Number of scoops

a. Use ratios to describe the relationship between the number of scoops and thetime in seconds to empty the funnel.

b. Use a ratio to describe the rate in scoops per second at which the dry ingredi-ent is being emptied from the funnel.

c. Use a unit rate to predict how long (in seconds) it would take for 15 scoops tobe emptied from the funnel.

d. Suppose the spout of the funnel were smaller in diameter. What effect wouldyou expect this to have on the graph? Explain.

Tim

e in

sec

onds


Slithering Solids








If you had poured the ingredient intoa container which was in the shape ofa cone, would this graph be any dif-ferent? How? Why?

If you had used a funnel with a largerdiameter, would this graph be differ-ent? How? Why?

Answers:a. Answers will vary. All ratios, however, should compare y or

xheight in cm : number of scoops.

b. This ratio is a unit rate and is written as number of cm . 1 scoop

c. number of cm • 15 scoops = number of cm in the height of the 1 scoop column of ingredient

d. The ingredient would empty faster from the funnel but the height perscoop of ingredient would not be affected. The graph would not be changed.


Hei

ght i

n cm

Slithering SolidsActivity 3

Make a graph of the ordered pairs (number of scoops, height in centimeters) using1" graph paper or the grid below.

Graph 2

Number of scoops

a. Use ratios to describe the relationship between the number of scoops and theheight in centimeters.

b. Use a ratio to describe the rate in cm per scoop at which the height is changing.c. Use the unit rate you chose in part “b” to predict how high the column of

ingredients would be for 15 scoops.d. Suppose the opening of the funnel were made twice as large as the original

funnel. What effect would you expect this to have on the graph? Explain.









Slithering Solids

Answers:a. Ratios will vary and will be of the form y or x .

x y

b. This is a unit rate and is written number of cm . 1 second

c. number of cm • 15 scoops = number of cm in the height of the 1 secondcolumn of ingredient

d. The ingredient would flow faster from the funnel, e.g., it would take moretime for the height to rise. The graph would be steeper than the original.


Slithering Solids

Activity 4

Make a graph of the ordered pairs (time in seconds, height in centimeters) using 1"graph paper or the grid below.

Graph 3

Time in seconds

a. Use ratios to describe the relationship between the time in seconds and theheight in centimeters.

b. Use a ratio to describe the rate in cm per second at which the height is changing.c. Use the unit rate you chose in part “b” to predict how high the column of ingre-

dients would be after 15 seconds (assuming a constant supply of ingredients).d. Suppose the opening of the funnel was twice as large as the original funnel.

What effect would this have on the graph? Explain.

Hei

ght i

n cm


Institute Notes

Concept: Use ratios and unit rates as conversion factors betweenmeasurement units and rates.

Overview: Manipulatives, tables, and graphs are used to demonstratethe proportional relationship between equivalent measures.The scientific method of dimension conversion (dimen-sional analysis) is related to equivalent ratios.

TEKS Focus: 6.3—The student solves problems involving proportionalrelationships.7.3—The student solves problems involving proportionalrelationships.8.3—The student identifies proportional relationshipsin problem situations and solves problems.

Materials: 1 yard strips of adding machine tape or butcher paper

Procedure: 1. Give each participant a strip of adding machine tapeor butcher paper strip, that is one yard long. Ask themto use the tape to show you a length of one foot.

2. Ask each participant to use the strip of paper to showhow 5 feet could be described in yards.

3. Organize the participants into groups of two.4. Show your transparency of Activity 1. Give the par-

ticipants a copy, directing them to complete the tableand to draw the graph. Partners should discuss thequestions.

5. Debrief with the whole group using the Reason andCommunicate questions.

6. Next, have participants use their one-foot strip ofpaper to show a strip of paper one inch long.

7. Ask participants to use strips of paper to show how27 inches could be described in feet.

8. Ask the participants to complete the tables, graphs,and questions in Activities 2, 3 and 4. Debrief oneach activity.

9. In Activity 5, participants connect the proportionalrelationships to Dimensional Analysis, a method usedin science for unit conversions. Model the first twoproblems with the group, and then let them work inpairs on the remaining problems.

Also:Grade 6:1B, 2C, 2D, 4A, 4B, 5, 8B,8D, 11A, 11B, 11C, 11D,12A, 12B, 13A, 13BGrade 7:2A, 2D, 2F, 2G, 4A, 4B, 9,13A, 13B, 13C, 13D, 14A,14B, 15A, 15BGrade 8:1B, 2A, 2B, 2C, 2D, 4, 5A,14A, 14B, 14C, 14D, 15A,15B, 16A, 16B

Math Notes:It is assumed that the par-ticipants know that one yardis equal to three feet. Toshow one foot they must di-vide the tape into threeequal parts. Emphasize thatthey are finding the follow-ing unit rate:1 yard or 1 yard per foot3 feet 3

To use symbols to deter-mine the equivalent value infeet, one may multiply 5 feetby the unit rate of 1 yard perfoot. 3

Note that you want to losethe unit of yards and gainthe unit of feet. Thus, theactivity is called Lost andGained.1 yard ( 5 feet) = 5 yard3 feet 3

Lost and Gained


Extensions: Activity 6 is an activity in which the participant uses di-mensional analysis to convert the units within rates.

Assessment: Use the following prompt for a journal entry:Equivalent measures can be shown to be proportionalby ________________________.

Notes:

Lost and Gained


Reason and Communicate:Write a sentence to describe the rela-tionship between feet and yards.The number of yards is 1 thenumber of feet. 3

What is the equation that describes theproportional relationship between thenumber of feet and the number ofyards?

xy3

1=

How could you use the graph to findthe number of feet that are equivalentto18 yards?Draw a horizontal line through y = 18and find the x coordinate for the pointwhere y = 18 intersects the graph.

If the graph were the graph of the or-dered pairs (yards, feet), how wouldthe graph be the same? How wouldthe graph be different?The graph would still be a straight linethrough the origin, but the line wouldhave a different slope. The equationof the line would be y = 3x.

Answers:b. The graph is a set of points which lies on a straight line through theorigin.c. This relationship appears to be a proportional one because the graph isa straight line passing through the origin. The equation of the relationship is

xy3

1=

where y is the number of yards, x is the number of feet, and 1 is the 3

constant of proportionality y (and also the slope of the line). x

d. There are two unit rates that can describe the proportional relationship: 1 yard per foot or 3 feet per yard.

3e. Yes, because the coordinates x = 7.5 and y = 2.5 satisfy the equation:

2.5 = 1 (7.5)3

Lost and Gained

1/35/32

364.524

345

0.9(1/3) x


Lost and Gained

Activity 1—Converting Feet to Yards

a. Using the ratio1 yard : 3 feet,complete thistable and graphthe ordered pairs.

feet (x) yards (y)3 1156

121.58

915

2.7x

b. Describe the graph.

c. Explain how you know that there is a proportional relationship between thenumber of feet and the number of yards.

d. What two unit rates appear in this relationship?

e. Does the ordered pair (7.5, 2.5) belong in your table? Explain.

Feet

Yar

ds


( )

Reason and Communicate:Explain how you know that the rela-tionship between equivalent measuresof feet and inches is proportional.The ordered pairs represent equiva-lent ratios and the graph is a linethrough the origin.

Answers:b. The graph is a straight line through the origin.c. Possible response: the number of inches in a length is 12 times thenumber of feet.d. y = 12x where y is the number of inches and x is the number of feet.e. The unit rates are 12 inches per foot and 1 foot = 1 foot per inch.

12 inches 12If you choose a point on the graph and go up vertically 12 units and acrosshorizontally one unit, you are using y = 12 inches (the slope) to find

x 1 footanother point on the graph. If you choose a point on the graph and movehorizontally one unit and vertically 12 units, you are using x = 1 footto find another point on the graph. y 12 inchesf. 56 = 12x, so x = 56 = 4 2

12 3g. When this problem is discussed, emphasize the use of the units and thefact that you start with inches, but want to end up with a measurement infeet. You wish to lose the inch units and gain the feet units; so the rate youneed is 1 foot . 12 inches 156 inches 1 foot = 13 feet

12 inches

Lost and Gained

241/22/35/422.5

366012(x)


Activity 2—Converting Feet to Inches

a. Using the ratio12 inches : 1 foot,complete this tableand graph theordered pairs.


c. Write a sentence to describe the relationship.

d. Write an equation of the graph.

e. What two unit rates appear in this relationship?

f. Find the value for “feet” in the ordered pair (feet, 56) so that the point lies on the line.

g. If you need to change 156 inches to feet, which rate would you use?

Lost and Gained

Feet

Inch

es

feet (x) inches(y)1 122

68152430

35x


Reason and Communicate:Describe at least three things aboutthis proportional relationship.The points lie on a straight line throughthe origin. The constant of proportion-ality, y, is 1, x 10meaning that the number of centime-ters is 1 the number of millimeters. 10The equation is y = 1 x where y

10represents the number of centimetersand x, the number of millimeters.

Answers:b. The graph is a straight line through the origin.c. Possible repsonse: The number of centimeters is 1 the number ofmillimeters. 10d.

where y is the number of centimeters and x is the number of millimeters.e.

f.

g. 10 mm per 1 cm because you wish to lose the cm units and gain themm units.

Lost and Gained

32.56

8025120

0.9150

14(1/10)x

(20, 2)

(50, 5)

(70, 7)

xy10

1=

centimeter 1for smillimeter 10or centimeter 1

smillimeter 10 and

millimeter onefor centimeter 10

1or

millimeter 10

centimeter 1

cm 1.6)mm 16(mm 10

cm 1 ==y

s


mm (x) cm (y)10 1302560

82.512

915

140x

Lost and Gained

Activity 3—Converting Millimeters to Centimeters

a. Using the ratio1 cm: 10 mm,complete this tableand graph theordered pairs.


c. Write a sentence to describe the relationship.

d. What is the equation of the graph?

e. What two unit rates appear in this relationship?

f. Find the value of y so that the ordered pair (16, y) is on the graph.

g. Suppose you wanted to change 23 centimeters to millimeters. Which unit ratewould you use?

10 20 30 40 50 60 70 80 90 100 110 120

Millimeters

Cen

timet

ers

12

34

56

78

910

1112

13


Lost and Gained

Reason and Communicate:What is the constant of proportionalityfor this relationship?The constant y is 4, meaning that

xthe number of quarts in a measure-ment is four times the number of gal-lons in the same measurement.

How is that constant evident on thegraph?The slope of the line is 4. For every 4units on the vertical axis, one moves1 unit on the horizontal to reach an-other point on the graph.

Answers:a. The table values will vary. This is a possible table.Gallons (x) Quarts (x)1 42 83 124 165 206 247 288 329 3610 40x 4xb. The graph is a set of points which lie on a straight line that passes throughthe origin.c. Possible response: the number of quarts is 4 times the number of gallons.d. 4 quarts per gallon or 1 gallon per quart

417 gallons 4 quarts = 68 quarts

1 gallon

e. The ordered pair (6, 26) would notlie on the graph because26 quarts (1 gallon) =

4 quarts

26 gallons = 6.5 gallons 4

The point on the graph is (6.5, 26).f. Use the unit rate 4 quarts for 1 gal-lon: 17 gallons ( 4 quarts ) = 68 quarts

1 gallon

(8.3, y)

(5, 20)

(2, 8)

( )


Lost and Gained

Activity 4—Converting Gallons to Quarts

a. Using the ratio4 quarts : 1 gallon,choose values tocomplete a table andgraph the orderedpairs.

Gallons (x) Quarts (y)

x


c. Write a sentence to describe this relationship.

d. What two unit rates appear in this relationship?

e. Would the ordered pair (6, 26) lie on the graph? Why or why not?

f. Suppose you needed to change 17 gallons to quarts. Which unit rate wouldyou use? Explain.

Gallons

Qua

rts


Reason and Communicate:How are proportional relationshipsbeing used in this method of unit con-version?There is a proportional relationshipbetween the units, e.g. number ofyards = 1 times the number of feet. 3This relationship gives a unit rate, e.g.1 yard per foot, which can be used3as the multiplier.

Lost and Gained

Answers:a. 2.4 yards • 3 feet = 2.4 • 3 feet = 7.2 feet 1 1 yard

b. 14 cups • 1 pint • 1 quart • 1 gallon = 14 gallon = 7 gallon 1 2 cups 2 pints 4 quarts 16 8

c. 24 mm • 1 cm • 1 meter = 24 = 0.024 meter 1 10 mm 100 cm 1000

d. 2 gallons • 60 seconds • 4 quarts = 12 quarts per minute 40 seconds 1 minute 1 gallon

e. 25 miles • 1cm • 48mm = 60 miles 2 cm 10 mm

Math Notes:a. The beginning unit is divided by one.You wish to lose the yard units andgain the feet units. You must multiplyby a unit rate of feet per yard.b. You must use the connecting ratios(which may be found in a conversiontable, if necessary) that represent theconstants of proportionality betweenunits. For example, to change cups topints, the ratio 1 pint : 2 cups repre-sents the unit rate 1/2 pint per cup andthe proportional relationship “numberof pints = 1/2 the number of cups.” Mul-tiply by these ratios until the desiredunit is found.d. The answer is expressed in quartsper minute. In 2 minutes, the amountwould be 12 quarts/minute or 24 quartsin 2 minutes.


Activity 5—Dimensional Analysis

a. Convert 2.4 yards to feet.

2.4 yards • ________ 1

b. How many gallons is 14 cups?

14 cups • ________ • ________ • ________ 1

c. Convert 24 millimeters to meters.

d. Water is flowing into a tank at a rate of 2 gallons every 40 seconds. At this rate,how many quarts of water will flow into the tank in 2 minutes?

e. A map shows a scale of 2 centimeters representing 25 miles. If the distancebetween two towns on the map is 48 mm, how many miles are the townsactually apart?

Lost and Gained


Reason and Communicate:What unit rates are being used to solvethis problem?Miles per minute, minutes per second

Answers:700 miles • 1 minute = 70 miles 1 minute 60 seconds 6 seconds

70 miles per second is the same as 6 second per mile, 6 70or 6 seconds per 70 miles.

6 seconds • 4 miles = 24 second70 miles 70

The cannon should have been fired 24/70 second before 8:00 a.m. Ap-proximately one-third of a second before 8:00 a.m., the signal should havebeen given.

Lost and Gained


Lost and Gained

Activity 6—Cannon SIgnal

The Rebel army was to signal thebeginning of a battle at exactly 8:00a.m., when an opposing army was

beginning an attack. Rebel CaptainHines and his men were 4 miles from

the opposing army. They shot the cannonat exactly 8:00 a.m., but the Rebel forces did not hear it intime. Sound travels at a rate of 700 miles per minute.

When should the cannon have been shot to have been heardon time?


Unit VII: Reflect and Apply

Answers:1. Determine the ratios that describe the constants of proportionality:1000 grams or 1 kilogram1 kilogram 1000 grams

Use the appropriate ratio (unit rate) as a multiplier (scale factor):148 grams • 0.001 kg = 0.148 kg

1 gram

2. 100 yards • 3 feet • 60 seconds • 2 minutes = 1800 feet 20 seconds 1 yard 1 minute

3. 24 mL • 1 L • 60 seconds • 3 minutes = 4.32 liters 1 second 1000 mL 1 minute


Unit VII: Reflect and Apply

1. Explain how you can use proportionality to change a measurement from oneunit to another. For example, how would you use proportionality to determinehow to express 148 grams in kilograms?

2. Keitha can run 100 yards in 20 seconds. At this rate, how many feet could sherun in 2 minutes?

3. A box is being filled with sand at a rate of 24 ml per second. At this rate, howmany liters of sand would be in the box at the end of 3 minutes?

Assessment Solution Guide

Student Name _________________________________________ Date ___________ Problem Name _________________________________________________________ This solution guide is a problem-solving checklist to help you understand what is necessary for a complete problem solution. Your teacher will indicate which of the criteria on this list you should consider in your problem analysis. After you have completed your solution, check if you feel you have sufficiently satisfied the criteria your teacher indicated. Be prepared to justify your check marks with examples in your solution.

Criteria to be considered in the solution of

problem (check all that

apply)

Criteria

Check if solution satisfies

this criteria

Describes mathematical relationships.

Recognizes and applies proportional relationships.

Develops and carries out a plan for solving a problem that includes understand the problem, select a strategy, solve the problem, and check.

Solves problems involving proportional relationships using solution method(s) including equivalent ratios, scale factors, and equations.

Evaluates the reasonableness or significance of the solution in the context of the problem.

Demonstrates an understanding of mathematical concepts, processes, and skills.

Uses multiple representations (such as concrete models, tables, graphs, symbols, and verbal descriptions) and makes connections among them.

Communicates clear, detailed, and organized solution strategy.

Uses appropriate terminology, notation, and tools.

States a clear and accurate solution using correct units.

Adapted from: MS Mathematics Assessments, The Charles A. Dana Center at the University of Texas at Austin, 2004, Austin, TX

Notes to the Teacher Materials 1-foot strips of adding machine

tape or butcher paper Transparency of Blackline Master

B23 TAKS Grade 8 Mathematics Chart

B25b (optional) It is hoped that the students will recall that 3 feet are equal to one yard. If necessary, refer students to the TAKS Chart.

A23 Measurement Benchmark Rulers Give each student 6 strips of adding machine tape that are each 1 foot long. Instruct students to use their tapes to show a length of one yard. (To do this they must put together 3 strips.)

Show a transparency of the Discussion Questions (B23) and discuss the questions in the first section (Feet to Yards) with the students.

Have students consider their “yardstick” now in terms of feet and discuss the questions in the second section (Yards to Feet) of the transparency.

Sample Responses: Feet to Yards

feetyard 3 feet

1 yard or 3 feet per yard 2 feet 2 • 3

Yards to Feet: yard

feet 13 yard

131

feetyard or 1

3 yard per foot

23 yard

2 • 13

number of feet or yards • unit rate

Source: Middle School TEXTEAMS: Proportionality Across the TEKS Materials 1 copy of Blackline Master B24a

for each pair of students Transparency of Blackline Master

B24a

A24 Linear Measurement Rates and Conversions Group the students into pairs and hand out a copy of the Blackline Master of Feet & Yards (B24a) to each group. Ask them to complete the tables and graph the resulting ordered pairs.

Use discussion questions listed below to debrief with the whole class. Sample Responses: 1. Describe the graphs. (They are both sets of points that lie

on a straight line through the origin.) 2. Explain how you know that there is a proportional relationship

between the two quantities in each on the graphs. (Because the graph is a straight line passing through the origin.)

3. What are the unit rates for each graph? (Yards to Feet: unit rate of 3 feet per yard; Feet to Yards: unit rate of 1

3 yard per foot)

4. What is the equation for each graph? (Yards to Feet: y = 3x; Feet to Yards: y = 1

3 x) 5. How could you use the graphs to find the number of feet

that are equivalent to 18 yards? (Yards to Feet: Draw a vertical line through x = 18 and find the y coordinate for the point where x = 18 intersects the graph. Feet to Yards: Draw a horizontal line through y = 18 and find the x coordinate for the point where y = 18 intersects the graph.)

Notes to the Teacher 6. How are the two graphs the same? How are they different? (Both are

straight lines going through the origin. The first line is steeper than the second.)

7. Does the ordered pair (7.5, 2.5) belong in one of the tables? Which one? Explain. (Yes. Explanations will vary.)

YArds (x) Feet (y)12350.5

3.5

x

369

1.5

1510.5

3x Yards

Feet

•1

2

3

45

6

78

9

10

1 2 3 4 5 6 7 8 9 10

•

•

•

•

11

12

13

14

15

•

Feet (x) Yards (y)36912

4.5

x

5

123

1.5

13

13 x

23

53

Feet

Yard

s

1

2

3

45

6

78

9

10

1 2 3 4 5 6 7 8 9 10

•••

•

11

12

13

14

15

•11 12 13 14 15

A25 Inching Along Ask students to use one of their foot-long strips from the introductory activity to fold into a 12-inch ruler. (They must fold the strip into 12 sections.) Relate this action to the actions taken in the first activity with feet and yards. Distribute a copy of the Blackline Master (B25a) to each pair of students and ask them to fill out the table, graph the ordered pairs, and answer the questions at the bottom of the sheet.

Materials 1-foot strips of adding machine

tape or butcher paper One copy per student of

Blackline Master (B25a) TAKS Grade 8 Mathematics

Chart (optional)(B25b)

Notes to the Teacher

Students might use 121

inchesfoot as

a unit rate. This is a conversion rate but it is not a true unit rate.

Sample Responses: 1. The graph is a straight line through the origin. 2. The number of inches in a length is 12 times the number

of feet. 3. y = 12x

4. 1

12foot

inches = 12 inches per foot; 1

121

inchfoot =

121 foot per inch

5.Yes, the ordered pairs represent equivalent ratios and the graph is a line through the origin.

6. 56 = 12x, so x = 1256 = 4

32

7. 121 foot per inch or the conversion rate of

121

inchesfoot ;

156 inches (121

inchesfoot ) = 13 feet

Source: Middle School TEXTEAMS: Proportionality Across the TEKS

Remind students of the importance of the “word ratio” in helping them set up a proportion of this type.

A26 Using Cross Products to Solve Measurement Problems After the students complete the above activity sheet, write the following excerpt from their table on the board or overhead: Feet 2 3 Inches 24 36 Illustrate to the students that the values from this chart may be written as proportions:

inchesfeet =

242 =

363

Introduce the cross-product rule for proportions. That is, 2 X 36 = 24 X 3 72 = 72 Therefore, if one of the values in a conversion is missing the above

method can be used as a short cut. In the problem, 54 inches are equal to how many feet?, a proportional statement can be set up using the conversion ratio:

inchesfeet =

121 =

54?

Solution: cross product statement: 1 X 54 = 12 X ?

1254 = ?

Materials Transparency of Blackline Master

B27 TAKS Conversion Chart B25b

(optional)

A27 Units Lost and Gained Through Dimensional Analysis In science class, students are often taught a method of converting measurements that is called dimensional analysis. This is a technique used to “chain” conversions using the unit rates. For example, if a problem involves converting yards to inches, a conversion equation is set up that looks like this: Convert 2.4 yards to inches: 2.4 yards • 3 feet • 12 inches = 2.4 • 3 • 12 = 72 inches 1 1 yard 1 foot

Notes to the Teacher In this conversion, you wish to “lose” the yard units and “gain” the inch units. To do this, the connecting ratios (which may be found in the TAKS conversion chart) that represent the conversion ratios of the “in-between” units must be used. Notice that as the multiplications are done, that the yards and feet units are “lost” (cancelled out) and the inches units are “gained”. Show a transparency of B27 and work through the problems with the students. The first two problems show conversions from yards to inches and from inches to yards. Explain to the students that the conversion ratios used are dictated by which units are to be “cancelled out.” The last problem on the page involves 2 sets of conversion ratios, the

problem asks the students to convert gallonsseconds

to quartsminute

. That means

that gallons must be converted to quarts and seconds must be converted to minutes. Once the unit rate is computed that must then be multiplied by 2 to get the amount of water for 2 minutes. Have students make up conversion problems and trade them with other students to work. Source: Middle School TEXTEAMS: Proportionality Across the TEKS

It is important to emphasize the inclusion of the units in this method. As you work through the problems, line through the “lost” units so the students can see what is being cancelled out.

B23 Benchmark Ruler: Discussion Questions

Feet to Yards What “conversion ratio” is being using? What is the unit rate? What is the measure (in yards) of 6 strips/feet? What computation was done to get the above answer? Yards to Feet What conversion ratio must we use to talk about feet? What is 1 foot (in terms of a yard)? The unit rate for this conversion is _____? What is the measure (in yards) of two feet? What computation was done to get the above answer? How do we generalize the process for computing these conversions using the unit rate?

B24a Feet and Yards Complete the tables below and graph the ordered pairs.

Yards

Feet

Yards (x) Feet (y)1235

0.53.5

x

Feet

Yar

ds

Feet (x) Yards (y)36912

4.5

x

5

B25a Inching Along

Feet (x) Inches (y)12

x5

68

152430

3

Feet

Inch

es

1. Describe the graph. 2. Write a sentence to describe the relationship. 3. Write an equation of the graph. 4. What two unit rates appear in this relationship? 5. Is this relationship proportional? Explain. 6. Find the value for “feet” in the ordered pair (feet, 56) so that the point lies on

the line. (hint: use the equation above) 7. If you need to change 156 inches to feet, which rate would you use?

B25b TAKS Mathematics Chart

LENGTH

Metric Customary

CAPACITY AND VOLUME

MASS AND WEIGHT

TIME

Metric Customary

Metric Customary

1 kilometer = 1000 meters

1 meter = 100 centimeters 1 centimeter = 10 millimeters

1 mile = 1760 yards

1 mile = 5280 feet 1 yard = 3 feet

1 foot = 12 inches

1 liter = 1000 milliliters 1 gallon = 4 quarts

1 gallon = 128 ounces 1 quart = 2 pints

1 pint = 2 cups 1 cup = 8 ounces

1 kilogram = 1000 grams

1 gram = 1000 milligrams

1 ton = 2000 pounds

1 pound = 16 ounces

1 year = 365 days

1 year = 12 months 1 year = 52 weeks

1 week = 7 days 1 day = 24 hours

1 hour = 60 minutes 1 minute = 60 seconds

B27 Dimensional Analysis

Convert 2.4 yards to inches. 2.4 yards • 3 feet • 12 inches = 2.4 • 3 • 12 = 72 inches 1 1 yard 1 foot Convert 125 inches to yards. 125 inches • 1 foot • 3 yards = 375 = 31.25 inches 1 12 inches 1 foot 12

How many cups is 3 gallons? 3 gallons • 4 quarts • 2 pints • 2 cups = 48 gallons 1 1 gallon 1 quart 1 pint How many gallons is 14 cups? 14 cups • 1 pint • 1 quart • 1 gallon = 14 gallon = 7 gallon 1 2 cups 2 pints 4 quarts 16 8

Water is flowing into a tank at a rate of 2 gallons every 40 seconds. At this rate, how many quarts of water will flow into the tank in 2 minutes? 2 gallons • 60 seconds • 4 quarts = 12 quarts per minute 40 seconds 1 minute 1 gallon In two minutes, 24 quarts will flow into the tank.

Notes to the Teacher Materials Transparency of Blackline Master

B6a to display on overhead projector

One copy of Blackline Master B5b for each group.

Jumbo paper clips Standard paper clips

A6 Paper Clip Chains Measurement – Factor of Change

Students will work in small groups of no more than four students per group. Distribute 15 jumbo paper clips and 24 standard paper clips to each group of students. Students will measure different items that are either pre-selected by the teacher, or student choice. Either way, the items measured need to be of different lengths (not extended beyond the number of jumbo clips). The following procedure needs to be followed by the students for the activity: • Clip both sizes of the paper clips together to make a chain of

each size. • Create a chart for each item measured as follows (B6a):

ITEM JUMBO CLIPS LONG

STANDARD CLIPS LONG

STANDARD CLIPS FOR

EACH JUMBO CLIP RATIO

1. 2. 3. 4.

• Graph the data with the jumbo Paper Clips on the x-axis and the standard Paper Clips on the y-axis.

• Answer questions at the bottom of the page. Possible answers: 1) For every two jumbo clips there are three standard clips.

2) 32

= 1.5 is the factor of change.

3) Divide 3 by 2. 4) The pattern shows up on the graph in a straight line which is

proportional going through (0,0). 5) To determine the length of something in jumbo clips from a given

standard clip measurement, multiply the number of standard clips by the scale factor of 1.5.

Discuss in whole group that the 1.5 is a conversion rate from standard clips to jumbo clips. This conversion rate can be informally demonstrated by showing on the graph that for every point plotted each point therefore is up 3 units and over 2 units. (Do not go any further at this time with slope.) However, the graphing of the measured objects is proportional. Ask students to validate the statement of proportionality using the factor of change and what they have learned about proportional relationships.

Student answers will vary, but focus on the starting ratio of 3 to 2 and that if there are 6 standard clips there are 4 jumbo clips (factor of change of 2) and if there are 9 standard clips there are 6 jumbo clips (factor of change of 3). Also, the line is a straight line passing through the origin (0,0).

B6a Paper Clip Chains Measurement

ITEM JUMBO CLIPS LONG STANDARD CLIPS LONG

STANDARD CLIPS FOR EACH JUMBO

CLIP RATIO 1.

2.

3.

4.

5.

6.

ANSWER THE FOLLOWING QUESTIONS:

1) What patterns do you see in the chart?

2) Looking at the ratio column how much bigger is the standard clip number than the jumbo clip

number?

3) How did you arrive at the answer in number 2? What does this number mean? AFTER THE DATA HAS BEEN GRAPHED, ANSWER THE FOLLOWING QUESTIONS:

1) How do the patterns in the chart show up on the graph?

2) How could you use the patterns to determine the length of something in standard clips if you know its length in jumbo clips?

B6b Graph Paper

Notes to the Teacher Materials Blank transparency for writing the problems onto overhead. (Problems can be placed on board.)

One copy of Blackline Master B8 for each student.

Graph paper (or Graphing Calculaotrs)

Colored pencils Remind students of the importance of attaching the labels to the ratios.

A8 Applying Factor of Change Display the following problem on the board or overhead transparency: A recipe calls for 5 tbsp. of applesauce for every 8 muffins. How much applesauce do you need for 48 muffins?

• Ask students to set up the initial ratio? • Then ask students: How would they set up the ratio to show 48

muffins? • How could this be written as a proportion using these two ratios?

Generate thought about equivalent ratios. Once this is established review that proportions are two equal ratios.

• What operation is performed when equivalent ratios are

generated? Multiplication. Show the ratio in a table and multiply by scale factors of 2, 3, 4, 5, and 6 to show that the ratios are equal. This may also be demonstrated with rainbow or color tiles to show they are still equivalent.

If the students need a concrete experience to assist them, show the steps to create a table of equivalent ratios with tiles. Have the students lay down 1 group of tiles (5 of one color and 8 of a second color). They then lay down an additional group of the same ratio and “read” the equivalent ratio for that group, e.g. two groups would be 10 of the first color and 16 of the second. Continue this until they have 48 of the second color of tiles showing and have them “read” the number of the first color of tiles that “matches”. The students should record their actions on a table as they create them with the tiles as shown below.

Number of groups

1

2

3

4

5

6

Tbsp of applesauce

5

10

15

20

25

30

Number of muffins

8

16

24

32

40

48

Have the students pull the equivalent fractions from the table and write them as a series of equivalent fractions. For example,

Next, draw in arrows to reinforce that the number of groups of tiles corresponds to the use of a factor of change or scale factor. For example:

Notes to the Teacher Remind students that in the original problem the situation was:

, which used a factor of change (scale factor) of 6, yielding the equivalent ratio which gives the solution “30 tbsp. of applesauce are necessary to make 48 muffins”. Display the next problem: An electric bill is $360 for 60 days. What is the average cost for 5 days? Ask the same questions as above, setting up the proportion in a similar manner. In this situation, ask the students to set up a table of equivalent ratios and discover the factor of change that is used to create the necessary ratio. Make sure that the students understand that the factor of change is a division operation in this case (factor of change = 12). Once these situations have been discussed, distribute one copy of Blackline Master B8 to each student. Students work independently to create a table of values (equivalent ratios) for each situation to reach the scale factor necessary for finding a solution. Graph one of the tables of values on paper and one on the graphing calculator. If calculators not available then students should choose at least two situations to graph by hand. Have them look at the graphs they created and make a list of similarities that they see between the graphs. (Similarities might include: both are straight lines, both go upwards from left to right, both (if extended) go through the origin.) Answers to Blackline Master B8: 1. No; the scale factor (factor of change) for this situation is 15. (4 X 15

= 60) which means that 90 circles are needed (6 X 15 = 90), therefore, 80 circles are not enough.

2. 340 inches would have fallen. The scale factor is 5 since we want to know how much in the 5th year.

3. No, each person has approximately $2.60 as a share of the $13.00. If two people pooled the money they would have $5.20. Or one ticket cost $3.25 then 5 tickets would cost $15.75.

4. The scale factor is 3 since we want to know how far in 3 hours. The plane would have flown a total of 2460 miles.

5. No. The rate is fifty miles per hour and if you use a factor of change of 8 (the number of hours to drive) the family would only have driven 400 miles.

B8 Applying Factor of Change

Solve the following problems. Show the original ratio, the second ratio, write the proportional equation, and show the scale factor used to find the answer. 1. A design on a banner uses 4 stars for every 6 circles. If the design is extended to

contain 60 stars, will 80 circles be enough? 2. A newspaper reported that 68 in. of snow fell last year. If the same amount falls

every year, how much snow will have fallen by the end of year 5? 3. You and four friends are in line for the roller coaster ride. Together you have $13.

The sign at the door reads “Tickets 2 for $6.50.” Do you have enough money for all of you?

4. If an airplane travels at an average speed of 820 mile/hr, how far will it have flown in

3 hours? 5. A family drove about 800 miles from New York City to Chicago in 16 hours. At that

rate, can they make a 530 mile drive in 8 hours? 6. Choose two of the above situations and graph the table of values. Make a list of the

similarities you see in the two graphs.

Notes to the Teacher Materials Copies or transparency of Blackline Masters B11a and B11b

Calculators

A11 How Much for One? In the previous activity “How Much, How Many?”, students might have noticed the possibility of using a different strategy for solving the proportion problems. In fact, if the factor of change is not an even number, students often prefer to use the unit rate strategy and find the price for one item and then determine the total amount for the number of items desired. The unit cost strategy is used in this activity. Begin by displaying a transparency of Blackline Master (B11) and asking the students to speculate how they could decide which store offers the “best deal” on the items shown. If the suggestion to figure out the price each store charges for one bag of carrots or one apple is not made, remind students that they cannot generally compare two quantities unless they have a “common” unit. In this case, the cost ratios should have a common denominator. The simplest cost ratios are the “for one” ratios. This is called the unit cost or unit price. As the class works through the examples provided on the transparency, be sure to include discussion concerning the “price per” statements. In the apple examples, the price is “per apple”. This entails using the number of apples as the denominator in the unit cost ratio. In the carrot examples, the unit price is “per pound.” Answers to first part of transparency B10a:

Store Proportion Unit Cost

Rite-Price 1600.1$ x

appleprice

== $1.00 ÷ 6 =

$0.17 or 17¢ per apple

Space City 110

59.1$ xappleprice

== $1.59 ÷ 10 =


Tom’s 14

75.0$ xappleprice

== $0.75 ÷ 4 =


Fresh … 12

59.0$ xpoundprice

== $0.59 ÷ 2 =

$0.30 or 30¢ per pound

Bargain … 13

95.0$ xpoundprice

== $0.95 ÷ 3 =

$0.32 or 32¢ per pound

Best buy on apples: Space City Foods Best buy on carrots: Fresh & Fast Other information needed might include: type of apples, size of

apples, Are carrots packaged in bags? Does Kelly need a 3 lb. bag of carrots?

Grocery stores try to help shoppers do comparison shopping by including the unit prices for items on their shelf labels. An example of two labels is shown at the bottom of the Blackline master (B11a). As the students look at these ask them to think about the discussion questions above the picture. It is of particular importance that the students notice the different units used in the unit prices of the two different items.


Sample Answers to bottom of transparency B11a: A better-buy decision cannot be made unless the unit price ratio has a common denominator or unit. Ask students to convert the ratios to a common denominator. To make the units common, they must again use a conversion ratio (1 pound = 16 ounces). The second ratio

becomes 16

59.1$=

ounceprice which must be made into a unit price ratio for

ounces ($1.59 ÷ 16 = $0.099 or 9.9¢ per ounce). Distribute copies of Blackline Master B11b for a homework assignment. Answers to B11b: Iced Tea: $9.52 per gallon Cranberry Juice: $10.00 per gallon Sport Drink: $10.17 per gallon Diet Drink: $10.32 per gallon Correction Fluid: $25.42 per gallon Brake Fluid: $33.60 per gallon Mouthwash: $84.48 per gallon Liquid Antacid: $123.20 per gallon Cold Medicine: $178.13 per gallon Designer Water: $21.19 per gallon Designer Perfume: $4,800.00 per gallon

Students may need help with writing the number $0.099 in terms of cents. Students may need to be reminded that 128 ounces = 1 gallon

A12 Supermarket Best Buys Distribute copies of the Blackline Master “Supermarket Best Buys” or display a transparency on the overhead projector. Have the student complete the practice problems using the unit cost method illustrated in the previous activity. Answers to B12: The best buys are indicated in bold italic type.

1. $1.75 ≈ $0.583

; $1.19 $0.602

≈

2. $0.59 $0.106

≈ ; $0.75 ≈ $0.093

3. $3.40 ≈ $0.854

; $1.80 $0.902

≈

4. $0.77 $0.155

≈ ; $1.56 ≈ $0.1312

Materials Copies or transparency of Blackline Master B12

Calculators

Notes to the Teacher On some of the answers, students may need to use exact answers to make the choice for the best buy.

5. $3.56 = $0.894

; $

6.

4.59 $0.925

≈

$0.99 = $0.198 ≈5

$0.20 ; $=

7.

0.60 $0.203

$0.56 ≈ $0.096

; $

8.

1.69 $0.1116

≈ (16 ou = 1 lb)

$0.79 $0.0250

≈ ; $1.05 $0.01475

≈ ; $10

9.

1.20 ≈ $0.0120

$=

1.95 $0.653

; $ ; $0.64 X 6 = $3.84

10. Biz Brite

1.28 ≈ $0.642

: $5.446

= $0.0854

White Plus: $4.32

on s:

$0.0948

=

Biz Brite: $0.085 X 32 = $2.72

B u $0.954

$0.042

≈ ; $

$0.10 – $0.04 = $0.06 more

3.32 $0.1032

≈

B11a How Much for One?

Kelly is on a strict budget and wants to get the best buy possible on several items she needs to buy. She has gotten prices from several stores for the items. How can she decide which is the best deal?

Store Item Price Rite-Price Grocery Apples 6 for $1.00 Space City Foods Apples 10 for $1.59 Tom’s Market Apples 4 for 75¢ Fresh & Fast Carrots 2 pounds for 59¢ Bargain Town Carrots 3 pounds for 95¢

Unit Cost = Total Price

Number of Items Store Proportion Unit Cost

Rite-Price Grocery (apples) 16

00.1$ xappleprice

== $1.00 ÷ 6 = _____ per apple

Space City Foods (apples) 110

.......... xappleprice

== _____ per apple

Tom’s Market (apples) 1.........

......... xappleprice

== _____ per apple

Fresh & Fast (carrots) 1........

59.0$ xpoundprice

== _____ per pound

Bargain Town (carrots) 13

......... xpoundprice

== _____ per pound

Where should she buy apples? Carrots? What other information might she need to know about the items to make sure she is getting the best buy? Look at the following unit price labels that might be found on the grocery store shelves. Kelly needs to buy peanut butter. Can she use these labels as they are shown? What does she have to do to make a better-buy comparison for these two products? What other factors might enter into her decision?

B11b Another Way to Look at Unit Costs

•

How much would these items cost per gallon?

Iced Tea 16 oz. $1.19

Cranberry Juice 16 oz. $1.25

Sport Drink 20 oz. $1.59

Diet Drink 16 oz. $1.29

Correction Fluid 7 oz. $1.39

Brake Fluid 12 oz. $3.15

Mouthwash 1.5 oz. $0.99

Liquid Antacid 4 oz. $3.85

Nighttime Cold Medicine 6 oz. $8.35

Designer Water 9 oz. for $1.49

Designer Perfume 1.6 oz. for $60

B11b Another Way to Look at Unit Costs

Find the unit price of the items below. Round to the nearest cent whenever necessary. Then tell which is the better buy. 1. 3 bottles of tomato juice for $1.75 or 2 jars for $1.19. 2. A package of 6 rolls for 59¢ or a package of 8 rolls for 75¢. 3. 4 light bulbs for $3.40 or 2 light bulbs for $1.80. 4. 5 pencils for $0.77 or 12 pencils for $1.56. 5. 4 notebooks for $3.56 or 5 notebooks for $4.59. 6. 5 bars of soap for 99¢ or 3 bars of soap for 60¢. 7. A 6-ounce bag of chips for $0.56 or a 1-lb. bag of chips for $1.69. 8. 50 envelopes for 79¢, 75 envelopes for $1.05, or 100 envelopes for

$1.20. 9. Three cans of Corny corn sell for $1.95. Two cans of Econo corn sell

for $1.28. You buy 6 cans at the best price. How much do you spend?

10. BizBrite Detergent is selling at the rate of 64 ounces for $5.44.

WhitePlus Detergent is selling at the rate of 48 ounces for $4.32. You buy 32 ounces at the best price. Which brand do you buy and how much do you spend?

Bonus A can of frozen orange juice concentrate is on sale for $0.95. The can makes 24 ounces of juice. 32 ounces of fresh orange juice sells for $3.32. How much more do you spend per ounce for fresh juice than for frozen?

Grade Seven Mathematics: Unit 3 Lesson 7 CLEAR Model Lessons © 2003 Houston Independent School District

Notes to the Teacher Materials Transparency of Blackline Master B9a

Copies or transparency of Blackline Master B9b

Calculators (optional) Definition: A proportion is a statement that two ratios are equal. In this case, the initial cost ratios have many equivalent ratios but the ones desired have 12 items purchased and an unknown cost. Emphasize predictions in answering these questions to build an intuitive sense of proportionality. For example, ask questions such as: Will it cost more or less? Will the number of items be smaller or larger?

A9 How Much, How Many? Display a transparency of Blackline Master B9a on the overhead projector. Ask the students to write these relationships as cost ratios, making sure to write the word ratio as well as the numeric ratio. Emphasize the setup of the ratios in the form Some students might write the ratios in a different form incorporating

the units into the word ratio such ascents

pencils204 . Discuss the fact that

either is correct but the units must be noted somewhere. Tell the students that they are to use the above cost ratios to find out how much a given number of items will cost. Use 12 pencils, 12 oranges, and 12 cans of juice as the number of items. Remind the students that these questions can be set up into proportions and the factor of change method may be used to find the unknown value (the cost). Guide the students through the following steps in answering the question: 1) Write the proportion. 2) Determine a factor of change. 3) Solve for the missing number.

Answers (top of B9a):

1. proportion: pencilscost

= x

1220.0$

4=

factor of change: 12 ÷ 4 = 3 missing number: x = $0.20 • 3 = $0.60 solution: 12 pencils will cost $0.60

2. proportion: orangescost

= x

1250$.6

=

factor of change: 12 ÷ 6 = 2 missing number: x = $0.50 • 2 = $1.00 solution: 12 oranges will cost $1.00

3. proportion: ca s of juicecost

= nx

121$

3=

factor of change: 12 ÷ 4 = 3 missing number: x = $1.00 • 3 = $3.00 solution: 12 cans of juice will cost $3.00.

Using the same cost ratios and process as above, ask the students to determine how many pencils, oranges, or cans of juice can be bought for $5. Answers (bottom of B9a):

1. proportion: pencilscost

= 5$20.0$

4 x=

factor of change: $5 ÷ $.20 = 25 missing number: x = 4 • 25 = 100 solution: 100 pencils can be bought for $5.00

CLEAR Model Lessons Grade Seven Mathematics: Unit 3 Lesson 7 © 2003 Houston Independent School District


2. proportion: orangescost

= 6

$.50 $5x

=

factor of change: $5 ÷ $.50 = 10 missing number: x = 6 • 10 = 60 solution: 60 oranges will cost $5.00.

3. proportion: cans of juicecost

= 5$1$

3 x=

factor of change: $5 ÷ $1 = 5 missing number: x = 3 • 5 = 15 solution: 15 cans of juice will cost $5.00.

Distribute copies or display a transparency of Blackline Master B9b for the students to work on independently. Answers: How much?

Spicey Dicey cans 2 6= =cost $0.49 x

; 6 ÷ 2 = 3; $0.49 • 3 = $1.47

Mild Mannered cans 3 12= =cost $0.79 x

; 12 ÷ 3 = 4; $0.79 • 4 = $3.16

Italiano Deluxe cans 4 8= =cost $0.89 x

; 8 ÷ 4 = 2; $0.89 • 2 = $1.78

Mama’s Best jars 6 3= =cost $0.95 x

; 3 ÷ 6 = 0.5; $0.95 • 0.5 = $0.48

Mario’s Max jars 4 2= =cost $1.60 x

; 2 ÷ 4 = 0.5; $1.60 • 0.5 = $0.80

Answers: How many?

Spicey Dicey cans 2 x= =cost $0.49 $2.45

; 2.45 ÷ 0.49 = 5; 2 • 5 = 10

Mild Mannered cans 3 x= =cost $0.79 $1.58

; 1.58 ÷ 0.79 = 2; 3 • 2 = 6

Italiano Deluxe cans 4 x= =cost $0.89 $2.67

; 2.67 ÷ 0.89 = 3; 4 • 3 = 12

Mama’s Best jars 6 x= =cost $0.95 $2.85

; 2.85 ÷ 0.95 = 3; 6 • 3 = 18

Mario’s Max jars 4 x= =cost $1.60 0.40

;0.40 ÷ 1.60 = 0.25; 4 • 0.25 = 1

Make sure that the students see that the number of jars of the last two brands to be purchased is less than the number of jars in the initial cost ratio. The factor of change is therefore less than one and the amount spent will be less that the cost in the initial ratio. Here the students should see that the amount of money spent on the last brand is less than cost in the initial cost ratio. The factor of change is therefore less than one and the number of jars purchased will be less that the number in the initial ratio.

A10 Supermarket Ratios Divide the class into small groups and distribute several grocery ads or circulars to each group. Have the students look through the ads and locate items that can be expressed in cost ratios. They should cut out the ads that they find and glue them to the chart paper. Next to the ads they should write How many? or How much? questions similar to those they used in the last activity. Each group creates a poster with several different grocery items and cost ratio problems to go with them. They write the solutions to their problems on the reverse side of the chart paper in pencil. After the posters are complete, display them on the wall to be used for assessment purposes. All of the students then do a gallery tour of the other posters and solve the problems for an assessment grade. Each group visits each poster to record the problems displayed, which the students then solve individually for a homework grade.

Materials Copies of grocery ads or

circulars Chart paper Scissors Glue Markers Masking tape Calculators (optional) A time limit at each poster may need to be enforced so there is time for each group to get to all the posters.

Grade Seven Mathematics: Unit 3 Lesson 7 CLEAR Model Lessons © 2003 Houston Independent School District

B9a Factor of Change Method

4 pencils cost 20¢ 6 oranges cost 50¢

3 cans of juice cost $1

Write a cost ratio for each of the above quantities.

How much would 12 of each of the above items cost?

cositems

t

Follow these steps to answer the question: 1) Write a

proportion for the relationship.

2) Determine a factor of change.

3) Solve for the missing number in the proportion.

How many of the above items can be bought for $5?

CLEAR Model Lessons Grade Seven Mathematics: Unit 3 Lesson 7 © 2003 Houston Independent School District

B9b How Much, How Many? Kelly went to the grocery store to buy ingredients for homemade pizza. She first went to find tomato paste and found many different brands and prices. Set up the proportions for the cost problems below and show the factor of change solution strategy for each. How much will she pay for the indicated purchase of each of the following brands?

Brand Price Purchase Spicey Dicey 2 cans for 49¢ 6 cans Mild Mannered 3 cans for 79¢ 12 cans Italiano Deluxe 4 cans for 89¢ 8 cans Mama’s Best 6 jars for 95¢ 3 jars Mario’s Max 4 jars for $1.60 2 jars Kelly spent the indicated amount of money for each purchase (no tax included). How many did she get of each brand?

Brand Price Purchase Spicey Dicey 2 cans for 49¢ $2.45 Mild Mannered 3 cans for 79¢ $1.58 Italiano Deluxe 4 cans for 89¢ $2.67 Mama’s Best 6 jars for 95¢ $2.85 Mario’s Max 4 jars for $1.60 $0.40


Unit Rates

Institute Notes

Concepts: Use unit rates to solve problems involving scaling upand scaling down.

Overview: Participants will use measurements, tables, and graphsto develop unit rates. Participants will then use the unitrates to solve problems involving scaling up and scalingdown.

TEKS Focus: 6.3—The student solves problems involving proportionalrelationships.6.4—The student uses letters as variables in mathemati-cal expressions to describe how one quantity changeswhen a related quantity changes.7.3—The student solves problems involving proportionalrelationships.7.4—The student represents a relationship in numeri-cal, geometric, verbal, and symbolic form.8.3—The studnet identifies proportional relationships inproblem situations and solves problems.8.4—The student makes connections among variousrepresentations of a numerical relationship.

Materials: RulersGraphing calculators (optional)

Procedure: 1. Distribute the activity sheet for Activity 1, Kendrea’s DollHouse, and show a transparency of the activity sheet.Lead the group in a discussion of the problem.

2. Have the whole group work together to answer Ques-tions a ,b, and c.

3. Then, have participants work in pairs to answer theremaining questions in Activity 1.

4. Debrief on this activity, using the Reason and Com-municate questions.

5. Allow participants to work in pairs on Activity 2, ABushel and a Peck, and Activity 3, Clean Sweep.

Debriefing: Summarize Activities 2 and 3 using the Reason andCommunicate questions.

Also:Grade 6:2C, 3A, 3B, 3C, 4A, 8A,8B, 11A, 11B, 11C, 11D,12AGrade 7:2D, 2F, 2G, 3B, 4A, 4B,7A, 13A, 13DGrade 8:1B, 2A, 2B, 2C, 3A, 3B,5A, 14A


Unit Rates

Answers and Math Notes:a. You may multiply by a scale factor of one-fifth.

Number of feet 2 2 (1) = 0.4 5

Number of cm 5 5 (1) = 1 5

One cm represents 2 or 0.4 of a foot. 5

b. 2 foot : 1 cm 5This ratio is a unit rate because the second term is 1. This rate may alsobe read 0.4 foot per cm or 2 foot per cm.

5c. To determine how many feet are represented by 25 cm, use the scalefactor of 25.Number of feet 2 2 2 (25) 2 (25) = 10 ft or

5 5 5Number of cm 5 1 1 (25) 0.4 (25) = 10 ft

d. To determine the number of cmneeded to represent one foot, use ascale factor of 1 .

2

Number of feet 2 (1) 1 2

Number of cm 5 (1) 2.5 2

e. This rate can be read as 1 foot per2.5 cm. It may also be read as the unitrate of 2.5 cm per foot.f. To complete the table, the participantmay use either unit rate (feet per cmor cm per foot) depending upon whichvalue is given.g. Refer to graph above.h. y = 0.4x where y is the number offeet and x is the number of cmi. 5 cm per foot or 3 feet per cm 3 5

4.8 8

2.5 7.5 10 17.5


Unit Rates

Activity 1—Kendrea’s Doll House

Kendrea is making a doll house from a drawing. A side which measures five centimeters long inthe drawing will measure 2 feet long in the doll house.

The scale may be expressed as one of the following ratios: 5 cm : 2 feet or 2 feet : 5 cm

a. If a side measures 1 centimeter long in the drawing, how many feet long will it measure inthe doll house?

Number of feet 2 b. What is the unit rate of feet per cm? Number of cm 5 1

c. How many feet is represented by 25 centimeters in the drawing?

d. If a side on the doll house is to be 1 foot long, how many centimeters will be used to repre-sent it in the drawing?

Number of feet 2 1 e. What unit rate is represented in this Number of cm 5 situation?

f. Complete this table to show other possible equivalent ratios:

Number of feet 2 2 1 3 4 75

Number of cm 5 1 12 20

g. Graph the data and write a descriptionof your graph.

h. Write an equation which describesthe proportional relationship betweenthe measurements on the drawingand the measurements of thedoll house.

i. If Kendrea had decided to use thescale, 5 cm represent 3 feet, whattwo unit rates would have beenevident?

Num

ber

of fe

et

Number of cm


Unit Rates

Reason and Communicate:

What rates are you assuming as youcomplete the table?6 apples for $1.50 and 1 apple for$0.25 and 4 apples for $1.00

Explain how you would determine thecost of 1 apple.

Show the unit rates on the graph.

Is the graph of your data a straight line?Emphasize that it is not appropriate tosay that the data forms a straight line,because the data is discrete. You maynot purchase one-half of an apple.However, a line can be used to de-scribe the “shape” of the data.

What two equations could be used todescribe this relationship?C = 0.25A where C is the amount indollars and A is the number of apples.or A = 4 C

Answers and Math Notes:To fill in the third column many participants will divide both values in thesecond column of numbers by 2. This is a valid approach and should notbe discouraged. To complete the remaining cells in the table the participantmay determine the two unit rates of 1 apple for $0.25 and 4 apples for$1.00.

16 10 15

0.75 1.25 2.00 5.75 4.75


Unit Rates

Activity 2—A Bushel and a Peck

a. Complete this table.

Number 6 3 5 8 27 19 of apples

Cost in 1.50 4.00 2.50 3.75 dollars

b. Identify two different unit rates that describe this proportional relationship.

c. Draw a graph of the data.

6 apples for$1.50

Cos

t in

dolla

rs

Number of apples


Reason and Communicate:

Would it be appropriate to draw astraight line through the points on thegraph?Yes, because the number of squarefeet does not have to be a whole num-ber, the ray from the origin through thepoints and in the first quadrant wouldrepresent the problem.

Explain how you determined the unitrates.100 square feet divided by $40 gives2.5 square feet per dollar$40 divided by 100 square feet gives$0.40 per square foot.

Describe the constant of proportion-ality.$0.40

What is the equation that describes theproportional relationship? y = 0.40 x where y is the cost in dol-lars and x is the number of square feet

Unit Rates

Answers and Math Notes:a. The points lie on a straight line through the origin.b. The two unit rates related to this problem are: $0.40 per square foot(based on the ratio $40 for 100 square feet); and 2.5 square feet per dollar(based on the ratio 100 square feet for $40).c. 1200 square feet x $0.40 per square foot = $480d. If $40 is the charge for 100 square feet, the amount which could becleaned for $200 (5 x $40) would be 5 times 100 or 500 square feet. Usingthe unit rate of 2.5 square feet per dollar, $200 x 2.5 square feet per dollar= 500 square feet.

(100,40)

(300, 120)

(200, 60)

(400, 160)

(600, 240)


Activity 3—Clean Sweep

The graph shows the relationship between the amount charged to clean carpetand the number of square feet cleaned.

100 200 300 400 500 600 700 800 900 1000Number of square feet

a. Explain why this graph represents a proportional relationship.

b. Identify two different unit rates that describe this proportional relationship.Explain.

c. Since the rate of charge is constant, how much would you expect to pay for1200 square feet of carpet? How did you determine your answer? How canyou use a unit rate to determine your answer?

d. How many square feet of carpet can you have cleaned for $200? How didyou determine your answer? How can you use a unit rate to determine youranswer?

Cos

t in

dolla

rs40

8012

016

020

024

028

032

036

0

400

Unit Rates

(100,40)

(300, 120)

(200, 80)

(400, 160)

(600, 240)


Unit IV: Reflect and Apply

Possible Responses:A unit rate is a ratio between different measurements.

A unit rate is a ratio in which the second number is 1.

A unit rate is a constant of proportionality.

In any proportional situation there are two unit rates which are reciprocalsof one another.

THe unit rate is the slope of the equation of the line that describes theproportional relationship.

TEXTEAMS Rethinking Middle School Mathematics: Proportionality Transparency-119

Unit IV: Reflect and Apply

Write at least four sentences to describe unit ratesand how they can be used to solve problems in-volving proportional situations.

•

•

•

•





2011 – 2012

Page 1 of 11


Unit 2.1 Integer Operation Lesson Set: Addition of Integers

1 or

2



Ⓢ MATH.7.2C



MATH.7.15A


MATH.7.15B



ELPS C.3b Expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication.

ELPS C.5b Write using newly acquired basic vocabulary and content-based grade level vocabulary.


CCRS 8.A1 Analyze given information. CCRS 8.A3 Determine a solution. CCRS 9.A3 Use mathematics as a language for



Numeric operations are used on integer quantities in real-world problems. 1. How do operations with integers compare to operations with whole numbers? 2. How can addition of integers be modeled? 3. What rules can be developed from modeling integer operations?


Prerequisites and/or Background Knowledge for Students In sixth grade, students used integers to represent real-life situations; They modeled integers using manipulatives, and connected vocabulary words to integers. Background Knowledge for Teacher Critical Content Add integers

Instructional Accommodations for Diverse Learners Since this is the students’ first experience with integer operations, it is important to develop these concepts with varied concrete as well as pictorial experiences when implementing Supporting Standard 7.2C. Do not rush to the algorithm or expect students to memorize “rules” without having a solid conceptual understanding. C.3b Present problems in a real-world context (such as money, yards gained/lost, or floors in an elevator). Ask students to validate that properties of whole numbers such as the additive property of zero (additive identity) apply to integer operations (e.g. – 3 + 0 = – 3). They should give reasons why this is true (or prove that it is not by giving a

Nonlinguistic Representations Two-Column Notes

Students model addition of integers with manipulatives (such as two-color counters, pictorial models, or number lines). As the students explore with manipulatives, they record their steps in a two-column format using pictures, numbers, and symbols in order to facilitate the transfer of their experiences from concrete models to pictorial representations finally to the abstract level. See examples on next page.









2011 – 2012

Page 2 of 11


counterexample). Instructional Accommodations for Diverse Learners Use Intermath, an interactive virtual math dictionary, to help students in their vocabulary acquisition. C.5b Vocabulary


patterns Integer

generalizations Property of Zero

models Additive Identity

While considering the “Instructional Strategies” column, note the verbs that align with the 5E Lesson Model: Engage, Explore, Explain, Elaborate, and Evaluate. These cues indicate appropriate strategies, level of rigor, and level of questioning to use during instruction.

Example:

Pictorial Model

Symbolic Representation

3 + 2 = 5

- 3 + (- 2) = - 5

- 3 + 2 = - 1

3 + (- 2) = 1

Generalization of the Rule:

The following involve the use of hands on manipulatives. (Activities: Addition and Subtraction of Integers on the Number Line, Integer Addition and Subtraction – Hot Air Balloon; Virtual activities in Understanding Math 2008 ,and Math Continuum: Integers: Add & Subtract-see Resource column).


Performance Assessment - Pair students off and have them use playing cards to model addition of integers. Let the red cards represent negative numbers and black cards represent positive numbers. Players divide the cards evenly between themselves. Players turn over two cards each and add them. See example below:

Player One: red 5 + red 2 = -7 Player Two: black 3 + red 4 = -1. The player with the greatest sum wins that round of cards. Play continues until one player has collected all of the cards.

Narrative –after playing the card game, have students write a personal reflection.

Resources

Clarifying Activities: The UT Dana Center Mathematics Toolkit includes

hands-on experiences with student activity sheets and extensive teacher notes.

The following activities involve the use of hands on manipulatives. Addition and Subtraction of Integers on the Number Line Integer Addition and Subtraction – Hot Air Balloon

.

Textbook Resources: Glencoe, Texas Mathematics, Course 2 TWE/SE, pp. 95 - 99; Algebra Lab, p. 93; Ch. 2 resource masters, pp. 29 – 34;

Teaching Math with Manipulatives, p. 42; MathScape, Course 2, pp. 96 – 103. Technology Resources: Understanding Math 2008 from Neufeld Learning


= 1 = -1


http://intermath.coe.uga.edu/dictnary/howtouse.asp


http://www.learnalberta.ca/content/mec/flash/index.html

http://www.utdanacenter.org/mathtoolkit/instruction/activities/6.php



- - English Language Proficiency Standards (ELPS) - Literacy Leads the Way Best Practices - Aligned Readiness - Process Standards



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Resources

Math Continuum: Integers: Add & Subtract is an Interactive virtual lesson at LearnAlberta.ca.

Intermath, interactive virtual math dictionary.


http://www.learnalberta.ca/Main.aspx






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Unit 2.1 Integer Operation Lesson Set: Subtraction of Integers

1 or

2



Ⓢ MATH.7.2C



MATH.7.15A


MATH.7.15B




ELPS C.4b Recognize directionality of English reading such as left to right and top to bottom.





Numeric operations are used on integer quantities in real-world problems. 1. How do operations with integers compare to operations with whole numbers? 2. How can subtraction of integers be modeled? 3. What rules can be developed from modeling integer operations? 4. How does the commutative property apply to integer operation?


Prerequisites and/or Background Knowledge for Students In sixth grade, students used integers to represent real-life situations; They modeled integers using manipulatives, and connected vocabulary words to integers. This is the students’ first experience in operating with integers. Background Knowledge for Teacher Critical Content Subtract integers

Continue the implementation of Supporting Standard 7.2C with subtraction. As the students are working with the manipulatives, they should be recording their steps using pictures, numbers, and symbols in order to facilitate the transfer of their experiences to the abstract level. In the case of subtraction, it is particularly important students understand the number of zero pairs necessary is determined by the quantity that is being subtracted. A greater number of zero pairs may be used than is necessary; however, the students should come to realize that it is unnecessary and inefficient.

Nonlinguistic Representations When modeling subtraction of integers, address the concept of “zero pairs” prior to beginning the work with the operation of subtraction. The idea that the quantities 1 and (–1) are additive inverses and their sum is zero (the additive identify) is an important foundation concept. Example: 3 – (–2) = 5

In this case, there are no negatives to subtract, so add two zero pairs to the initial quantity of 3. Once the zero pairs have been added in, a quantity of (–2), 2 negative chips, may be removed or subtracted. The resulting quantity (difference) is +5 which is the original 3 positive chips and the 2 positive chips that remain from the zero pairs. Example: –3 – 2 = –5

In this case, there are no positives to subtract, so two zero pairs must be added to the original quantity of (–3).







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Instructional Accommodations for Diverse Learners Since this is students’ first experience with integer operations, it is important to develop these concepts with varied concrete as well as pictorial experiences when implementing Supporting Standard 7.2C. Do not rush to the algorithm or expect students to memorize “rules” without having a solid conceptual understanding. C.3b Vocabulary


patterns zero pairs

generalizations Identity property

models Inverse Operation

Additive inverse

Commutative Property

Use Intermath, an interactive virtual math dictionary, to help students in their vocabulary acquisition. While considering the “Instructional Strategies” column, note the verbs that align with the 5E Lesson Model: Engage, Explore, Explain, Elaborate, and Evaluate. These cues indicate appropriate strategies, level of rigor, and level of questioning to use during instruction.

Once the zero pairs are added in, two positive chips may be subtracted (“taken away”). The result (–5), is the original 3 negative chips and the 2 negative chips that remain from the zero pairs. (Activities: Addition and Subtraction of Integers on the Number Line, Integer Addition, and Subtraction – Hot Air Balloon.) Activity 7.2C in Mathematics Toolkit. Virtual activities in Understanding Math 2008, LearnAlberta.ca, and the National Library of Virtual Manipulatives; Math Continuum: Integers: Add & Subtract -see Resource column). Explain to students that they should look for patterns and use them to generalize an algorithm for subtraction. Examination of the use of the zero pairs should lead to a discussion of the fact that the final “operation” is really addition and the realization of the “add the opposite” rule. Have students write narrative statements detailing the process. Instructional Accommodations for Diverse Learners Ask students to explore the directionality of adding & subtracting integers and validate that the commutative property of addition is applicable to integer operations. Ask questions such as,

“Are –3 + 2 = 2 + (–3) and –3 – 2 ≠ 2 – (–3) true statements?”

“Is this true for all integer addition and subtraction problems?” C.4b


Formative Assessment – students will write a narrative comparing and contrasting addition and subtraction of integers. Reciprocal Teaching- working through the four stages: Questioning, Clarification, Summarization, and Prediction, in

cooperative groups, students will reinforce their learning.

Resources

Clarifying Activities: The following activities involve the use of hands on manipulatives. Addition and Subtraction of Integers on the Number Line Integer Addition and Subtraction – Hot Air Balloon Activity 7.2C, Mathematics Toolkit UT Dana Center.

.

Textbook Resources: Glencoe, Texas Mathematics, Course 2 TWE/SE, pp. 103 – 06 Algebra Lab, p. 101 – 02 Ch. 2 resource masters, pp. 35 – 41 Teaching Math with Manipulatives, p. 43 MathScape, Course 2, pp. 96 – 103

Virtual activities: Understanding Math 2008 from Neufeld Learning


LearnAlberta.ca.; Math Continuum: Integers: Add & Subtract

National Library of Virtual Manipulatives: Color Chips - Subtraction



http://utdanacenter.org/mathtoolkit/instruction/activities/6.php





http://nlvm.usu.edu/en/nav/vlibrary.html












http://nlvm.usu.edu/en/nav/frames_asid_162_g_3_t_1.html

http://nlvm.usu.edu/en/nav/frames_asid_162_g_3_t_1.html





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Unit 2.1 Integer Operation Lesson Set: Multiplication of Integers

1 or

2



Ⓢ MATH.7.2C



MATH.7.15A


MATH.7.15B



ELPS C.1a Use prior knowledge and experiences to understand meanings in English.

ELPS C.2h Understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations.






Numeric operations are used on integer quantities in real-world problems. 1. How do operations with integers compare to operations with whole numbers? 2. How can multiplication of integers be modeled? 3. What rules can be developed from modeling integer operations?


Prerequisites and/or Background Knowledge for Students In sixth grade, students used integers to represent real-life situations; They modeled integers using manipulatives, and connected vocabulary words to integers. Background Knowledge for Teacher Critical Content Multiply integers

Continue the implementation of Supporting Standard 7.2C with multiplication. As the students are working with the manipulatives, they should be recording their steps using pictures, numbers, and symbols in order to facilitate the transfer of their experiences to the abstract level. Instructional Accommodations for Diverse Learners In the case of multiplication, it is helpful for the students to observe the patterns that develop in the signs of the products and relate those patterns to the commutative property of multiplication. They should speculate whether or

Nonlinguistic Representations When modeling multiplication of integers, students recall the meaning of multiplication as “grouping.” C.1a

Example: 2 X (–3) = –6 Read this problem as two groups of negative 3. Students should be encouraged to say these orally and repeat for each example used. C.3b

Also, illustrate this problem on a number line. Students will read this problem as two moves of negative 3.

Example: –2 X (–3) = 6

When students illustrate a problem such as the one above, the negative sign takes on the meaning “the opposite of.”










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not the same property used in whole number multiplication is true for integer multiplication. Ask students to validate their conclusions by giving reasons or proving their conclusions invalid by providing counterexamples. C.2h Vocabulary


inference Commutative Property

patterns additive inverse

opposite

zero pairs

Relate this to the previous discussion of zero pairs and additive inverses. Model “the opposite of” a number by flipping over the two-color counters. Students read this problem (orally) as the opposite of two groups of negative 3.

Alternatively, represent the expression –2 X (–3) as two moves the opposite direction of negative three on the number line.

(Activities: Activity 7.2C in Mathematics Toolkit, and Virtual activities in Understanding Math 2008 and Math, Continuum: Integers: Multiply & Divide- see Resource column). Instructional Accommodations for Diverse Learners Since this is the students’ first experience with integer operations, it is important to develop these concepts with varied concrete as well as pictorial experiences when implementing Supporting Standard 7.2C. Do not rush to the algorithm or expect students to memorize “rules” without having a solid conceptual understanding. C.3b


Integer Multiplication War- place students in groups of fours. Players divide a deck of cards evenly between themselves. Players turn over two cards each and multiply them. The player with the highest product collects the pile. Play continues until one player has all of the cards. (In the event of a tie, each player displays two more cards and multiply; highest product collects the pile).

Narrative –after playing the card game, have students write a personal reflection.

Resources

Clarifying Activities: Activity 7.2C from the Mathematics Toolkit (UT Dana

Center) involves hands-on experiences with student activity sheets and extensive teacher notes.

Textbook Resources: Glencoe, Texas Mathematics, Course 2 TWE/SE, pp. 109-113 Ch. 2 resource masters, pp. 46-51 Teaching Math with Manipulatives, p. 44-45 MathScape, Course 2, pp. 96 – 103.

Technology Resources: Understanding Math 2008 from Neufeld Learning


Math Continuum: Integers: Multiply & Divide is an Interactive virtual lesson at LearnAlberta.ca.














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Unit 2.1 Integer Operation Lesson Set: Division of Integers

1 or

2



Ⓢ MATH.7.2C

Use models, such as concrete objects, pictorial models, and number lines to add, subtract, multiply, and divide integers and connect the actions to algorithms.

MATH.7.15A Make conjectures from patterns or sets of examples and nonexamples.

MATH.7.15B Validate conclusions using mathematical properties and relationships.







Numeric operations are used on integer quantities in real-world problems. 1. How do operations with integers compare to operations with whole numbers? 2. How can subtraction of integers be modeled? 3. What rules can be developed from modeling integer operations? 4. How does the commutative property apply to integer operation?


Prerequisites and/or Background Knowledge for Students In sixth grade, students used integers to represent real-life situations; they modeled integers using manipulatives, and connected vocabulary words to integers. This is the students’ first experience in operating with integers. Background Knowledge for Teacher Critical Content Divide integers

Continue the implementation of Supporting Standard 7.2C with division. In the case of division, it is generally more beneficial to relate division back to multiplication as the modeling is often difficult to use with signed numbers. The observation of patterns is crucial to the development of an algorithm. Vocabulary


patterns Multiplicative Identity

inverse operation

Nonlinguistic Representations Students must understand the operation of division as the inverse operation of multiplication. The question students should ask is “how many groups of ___ are there in ___?”

Example: (–6) ÷ (–3) = 2

How many groups of –3 are there in –6? (There are 2 groups of –3 in –6)

Instructional Accommodations for Diverse Learners Students should be encouraged to say these orally and repeat for each example used.3(B)

Modeling examples of a negative number divided by a negative number is often difficult for students to visualize. It is easier to relate the division problems to fact families. Example: (–6) ÷ (3) = ? ? • 3 = – 6 (–2) • 3 = – 6 Example:








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While considering the “Instructional Strategies” column, note the verbs that align with the 5E Lesson Model: Engage, Explore, Explain, Elaborate, and Evaluate. These cues indicate appropriate strategies, level of rigor, and level of questioning to use during instruction.

(–6) ÷ (–2) = ? ? • (–2) = 6 – 3 • (–2) = 6 (Activities: Activity 7.2C in Mathematics Toolkit, and Virtual activities in Understanding Math 2008. In addition, activities in Math Continuum: Integers: Multiply & Divide –see Resource column). Ask students to explore the existence of applicable rational number properties at this point. They should validate their conclusions by giving their reasons for their answers and providing examples or non-examples to support their reasoning. They should relate the commutative property to

multiplication and division of integers. Is it universally true that a • b = b • a and a ÷ b ≠ b ÷ a when a or b might be negative?

Does the identity property apply to multiplication and division of integers as well? If so, what is the identity element? That is, is it true that a • 1 = a if a is negative?

Instructional Accommodations for Diverse Learners Students should be encouraged to say the above properties orally and repeat for each example used. C.3b


Provide students with application problems involving integer operations. Students will use two-color counters to model each problem. They may use (+) and (–) to represent the two-color counters or draw counters using colored pencils. They will write a verbal description and number sentence for each problem.

Resources

Clarifying Activities: Activity 7.2C, Mathematics Toolkit (UT Dana Center)

involves the use of hands-on manipulatives and includes extensive teacher notes and scaffolding questions.

Textbook Resources: Glencoe, Texas Mathematics, Course 2 TWE/SE, pp. 114-117 Ch. 2 resource masters, pp. 46-51 Teaching Math with Manipulatives, p. 43 MathScape, Course 2, pp. 96 – 103



Math Continuum: Integers: Multiply & Divide is an interactive virtual lesson available at LearnAlberta.ca













2011 – 2012

Page 10 of 11


Unit 2.1 Integer Operations Lesson Set: Problem-Solving using Integer Operations

1 or

2



Ⓡ MATH.7.2F

Select and use appropriate operations to solve problems and justify the selections, problem solving process, and

reasonableness of answer, especially in terms of an estimate.

Ⓢ MATH.7.2G

Determine the reasonableness of a solution to a problem using a variety of strategies such as estimation using rounding

or compatible numbers.

MATH.7.13B

Use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and

evaluating the solution for reasonableness.


ELPS C.4j Demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs.

ELPS C.5g Narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.





Numeric operations are used on integer quantities in real-world problems. 1. What rules can be developed from modeling integer operations? 2. What kinds of real-life problems require the use of integers?


Prerequisites and/or Background Knowledge for Students In sixth grade, students used integers to represent real-life situations; They modeled integers using manipulatives, and connected vocabulary words to integers. Background Knowledge for Teacher Critical Content Identify which operation is necessary to solve a real-life problem involving integers and solve real-life problems involving integer operations. In implementing objective 7.13B, students move to the application level in solving application problems involving integer operations. The use of a problem-solving model is an important part of this process. It is also important to emphasize the part of the process involving evaluating the solution. Students must be able to explain how they solved a problem as well as how they know that the solution they obtained “makes sense.” Using a Problem- Solving Mat graphic organizer will guide students through the various steps of the process.

Cues, Questions, & Advance Organizers Graphic Organizers

Use a graphic organizer to help students deconstruct application problems involving integer operations (Textbook Word Problem Practice, Problem Solving Mat – see Resource column). Students should recognize the cues that help them to identify the operation necessary to solve each problem. Instructional Accommodations for Diverse Learners As student work through problems, have manipulatives available for students to use to model the operations within the problems if they feel it necessary. (Activities: Ch. 2 resource masters, Word Problem Practice, pp.33, 40, 50, and 56). Summarization & Note Taking

Two-Column Notes Divide students into pairs to create a “Pair- Share” journal. Start by vertically folding a sheet of paper in half. Partner A writes on the left half, and partner B writes on the right half. One student writes an integer application problem at the top of the paper. Next, everyone becomes silent for the










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Monitor student understanding of integer operations closely as students apply the skills in application problems.

duration of the problem. Partner A begins by writing one idea s/he has about the problem. Partner B then writes a response to Partner A. This response could be an observation, a question, an elaboration (such as drawing a picture), or a next step. This silent “Pair-Share” volleys back and forth until each partner agrees, in writing, that the problem is solved, and tells how they know it is solved. Their writings should include the strategies that they used to evaluate the reasonableness of their solution (e.g. estimation, using a different strategy, working the problem in reverse using the solution obtained, etc.) The final statement agreed upon by the pair is then recorded into the summary section of the note-taking table.


Students create a comparison matrix that summarizes each operation, its related algorithm, a student-generated application problem, and a pictorial representation of its solution. C.4j, C.5g

Resources

A Problem Solving Mat is a graphic organizer that is available as a template in the file Mathematics Graphic Organizer Templates.

Textbook Resources: Glencoe, Texas Mathematics, Course 2 Ch. 2 resource masters, Word Problem Practice Add Integers, p. 33 Subtract Integers, p. 40 Multiply Integers, p. 50 Divide Integers, p. 56.


Notes to the Teacher Materials Transparency of Blackline Master

B39 or One copy each for students needing additional assistance.

A39 Addition and Subtraction of Integers on the Number Line Some students have difficulty understanding addition and subtraction using the two-color counter models. The use of the number line is an alternative model that may be used to present these operations. A copy of the blackline master (B39) may be used to assist those students. Using the number line model, students should be shown how to break down the parts of a problem and equate them to actions on the number line. In an addition problem, the meanings of the “parts” of the problem are verbalized as:

The first number indicates a starting point, the addition symbol indicates a move, the sign on the second number indicates the direction of the move, and the value of the second value is the distance moved. For example,

+5 + –2 means “starting at +5, move 2 places to the left” –6 + +8 means “starting at –6, move 8 places to the right”

The operation of subtraction “reverses the direction” (is the inverse) so now the parts of the problem are changed so that the subtraction symbol means “move in the reverse direction indicated”. Examples of this are:

–4 – –7 means “starting at –4, move 7 places to the right (reversed from left)”

3 – 5 means “starting at +3, move 5 places to the left (reversed from the right)”

After introducing the model and several examples, ask the students to draw a number line on their own paper and work additional problems to practice this strategy. The students may use the problems on the activity sheets B35 and B36.

B39 Addition and Subtraction of Integers on the Number Line

In an addition problem the meanings of the “parts” of the problem are shown below.

+5 + -2 means “starting at +5, move 2 places to the left”

-6 + +8 means “starting at –6, move 8 places to the right”

The operation of subtraction “reverses the direction” (is the inverse) of the problem

-4 - -7 means “starting at –4, move 7 places to the right (reversed from left)”

3 – 5 means “starting at 3, move 5 places to the left (reversed from the right)”


Materials One copy of Blackline Master B5 to each student. Scissors In scientific terms: Gas bags add “lift”. In scientific terms: Sand bags add “drag”.

A5 Integer Addition and Subtraction – Hot Air Balloon The hot air balloon uses a vertical number line and gas or sand bags to demonstrate the addition and subtraction of integers. The balloon uses gas or sand bags to move up or down. Gas Bags: Each bag has enough lift to raise the balloon exactly one unit. Sand Bags: Each bag weighs enough to lower the balloon exactly one unit. There are basically two things you can do with one of these bags. You can either put it on or take it off. Since gas bags cause the balloon to move up when they are put on, they represent positive integers. On the other hand since sand bags cause the balloon to move down when they are put on, they represent negative integers. Distribute Blackline Master B5 and scissors to each student. Students cut the Hot Air Balloon from the page, and use the number line to model addition and subtraction. Orally present the following situations to the students. They model the situation using the hot air balloon cutout by moving it on the number line. Start the balloon at 0 and add 6 gas bags. Ask students: “Where does the balloon end up? Explain why the balloon stops at this point.”+ 6, (because gas bags make the balloon rise.) Start the balloon at 0 and add 6 sand bags. Ask students: “Where does the balloon end up? Explain why the balloon stops at this point.” - 6, (because sand bags make the balloon lower.) Do a few more examples of just reaching an integer from before moving to the next step of showing addition and subtraction with the hot air balloon. Students can write their own problems for integer representation at this point and share with the class, but only a few minutes should be spent on this exercise. Addition and subtraction of integers using the hot air balloon. Model the following situation.

• The balloon starts at +2 and ends up at +8. If the balloon ends at

+8 it rose by +6. Ask students what kind of bags were placed on the balloon to cause the change? Gas bags.

Do the following problems and ask the same questions as above. Ensure that student responses reflect that either gas bags are put on or that sand bags were put on. If bags are placed onto the balloon it becomes an addition problem. • The balloon starts at +8 and ends up at +2. Student responses

should reflect that 6 sand bags were placed on the balloon. • The balloon starts at -1 and ends up at +2. Student responses

should reflect that 3 gas bags were placed on the balloon. • The balloon starts at +4 and ends up at -3. Student responses

should reflect that 7 sand bags were placed on the balloon.


• The balloon starts at -3 and ends up at -1. Student responses should reflect that 2 gas bag were placed on the balloon.

• The balloon starts at +1 and ends up at +3. Student responses should reflect that 2 gas bags were placed on the balloon.

After students have completed these five problems have students write the addition equation that reflects each of these situations. The equations can be done as the action or missing addend. Answers: Action: Start Bags End Up Missing Addend • 8 + -6 = 2 OR 8 + _______ = 2 • -1 + 3 = 2 OR -1 + _______ = 2 • 4 + -7 = -3 OR 4 +_______ = -3 • -3 + 2 = -1 OR -3 +_______ = -1 • 1 + 2 = 3 OR 1 +_______ = 3 Now ask students how subtraction can be shown with the Hot Air Balloon. Expect student responses to be that bags will need to be taken off the balloon. If gas bags are taken off what can we expect to happen to the balloon? The balloon should go down. If sand bags are taken off what can we expect to happen to the balloon? The balloon should go up. Model with the students: • The balloon starts at -4 and ends up at +2. Ask students what was

taken off the balloon? Student responses should reflect that 6 sand bags were taken off. Taking the sand bags off makes it rise.

Do the following problems and ask the same question as above. Ensure that student responses reflect that either gas bags are taken off or that sand bags are taken off. If bags are taken off the Hot Air Balloon then it is a subtraction problem. • The balloon starts at +2 and ends up at -4. Student responses

should reflect that 6 gas bags are taken off. Taking the gas bags off would make the hot air balloon should go down.

• The balloon starts at +5 and ends at +7. Student responses should reflect that 2 sand bags are taken off. Taking the sand bags off would make the balloon rise.

• The balloon starts at -3 and ends at 0. Student responses should reflect that 3 sand bags are taken off. Taking the sand bags off would make the balloon rise.

• The balloon starts at +3 and ends at 0. Student • responses should reflect that 3 gas bags are taken off. Taking the

gas bags off would make the balloon go down. • The balloon starts at +4 and ends at +1. Student responses should

reflect that 3 gas bags are taken off. Taking the gas bags off makes the balloon go down.

After students have completed these five problems have students write the subtraction equation that reflects each of these situations. • 2 – 6 = -4 • 5 – -2 = 7 • -3 – -3 = 0 • 3 – 3 = 0 • 4 – 3 = 1 Adapted from: Bill Kring, Vancouver, WA: “Ride a Hot-Air Balloon”, NCTM Presentation. Used with permission from the author.

B5 Hot Air Balloon

20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9-10-11-12-13-14-15-16-17-18-19-20

Math Power

Name _________________________________ Date ______________

Problem Solving Mat

Problem

I know that… I need to find out… Strategy o Draw a Picture or Diagram o Look for a Pattern o Guess and Check o Act It Out o Make a Table/List o Simplify the Problem o Work Backwards o Eliminate Possibilities o Find the Subproblem o Use Algebra

Create a Representation (For example , use a picture/diagram/table/rule)

Show your work.

Final Answer (In a complete sentence restate what you found out, including your answer.) I found out that …

(Complete the following sentence) My answer is reasonable because …

Mathematics Graphic

Organizers

These instructional tools help students

construct meaning and organize their

knowledge before, during, and/or after

instruction. They can be completed

linguistically or non-linguistically, using the

computer or paper-based.

When using these tools, a teacher should:

• monitor students as they complete their

own graphic organizer with new content. • model how to use a specific organizer

with familiar content. • allow students to help complete a class

graphic organizer. • show several completed examples.

Frayer Model

This graphic organizer is used to help students deepen their understanding of a key concept or content-specific vocabulary term by analyzing its essential and non-essential characteristics, drawing an illustration, or giving examples and non-examples. (Frayer, Frederick, and Klausmeier, 1969) It is suggested that students:

• create the chart as shown on the next page.

• write the concept term inside the oval.

• label each quadrant.

• model the use of the Frayer Model with an easy term.

• refine their original answers as the unit of study progresses to deepen their understanding of the term.

Word/Concept to be defined

Non-Examples Examples

Illustration My Definition

Place Value Mat

Number Lines

Blank 100 Grids

Fraction:Decimal: Percent:






Obj:

Name: _______________________________________________ Date ____________________

Problem:

I know that ….

Answer

I need to find out …

Strategy Show your work …

Name _________________________________ Date ______________

Problem Solving Mat

Problem

I know that… I need to find out… Strategy o Draw a Picture or Diagram o Look for a Pattern o Guess and Check o Act It Out o Make a Table/List o Simplify the Problem o Work Backwards o Eliminate Possibilities o Find the Subproblem o Use Algebra

Create a Representation (For example , use a picture/diagram/table/rule)

Show your work.

Final Answer (In a complete sentence restate what you found out, including your answer.) I found out that …

(Complete the following sentence) My answer is reasonable because …

Common Factors

Common Factors of _____ and _____

Common Factors of _____ and _____

Factors of _____ Factors of _____

Factors of _____ Factors of _____

Coordinate Planes Coordinate Planes

Dot Paper

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Fraction Strips





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Unit 2.2 Ratio and Rates

2 or

4



Ⓢ MATH.7.2D

Use division to find unit rates and ratios in proportional relationships such as speed, density, price, recipes, and student-

teacher ratios; and model those relationships concretely, pictorially, and in tabular form.

MATH.7.14A

Communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or

algebraic mathematical models.


ELPS C.1c Use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary.

ELPS C.4c Develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials.

ELPS C.4e Read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned.

ELPS C.5b Write using newly acquired basic vocabulary and content-based grade-level vocabulary.

ELPS C.5e Employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as: (i) using correct verbs, tenses, and pronouns/antecedents; (ii) using possessive case (apostrophe s) correctly; and (iii) using negatives and contractions correctly.


CCRS 8.A1 Analyze given information. CCRS 8.A3 Determine a solution. CCRS 9.C1 Communicate mathematical ideas reasoning

and their implications using symbols, diagrams, graphs, and words.

CCRS 9.C2 Create and use representations to organize, record, and communicate mathematical ideas.

CCRS 10.1B Use multiple representations to demonstrate links between mathematical and real world situations.


A ratio is a multiplicative comparison of two quantities. 1. How are comparisons used in proportional reasoning? 2. What is the relationship between ratios and proportions?

Every rational number can be expressed as a ratio or rate. 1. How are ratios and rates related to fractions? 2. How are ratios and rates different? 3. What is special about a unit rate?


Prerequisites and/or Background Knowledge for Students In sixth grade, students used ratios to describe proportional relationships; they represented ratios with concrete models, fractions, and decimals. Background Knowledge for Teacher Critical Content Investigate ratios and rates; Find a unit rate; and Use unit rates to solve problems.

Nonlinguistic Representations

KWL Cues, Questions, and Advance Organizers Use a brainstorming activity and KWL chart to engage a general discussion about ratios and rates. Compare and contrast ratios and rates and discuss the special case of unit rate. Discuss different types of real-world situations that are modeled by ratios and rates.








2011 – 2012

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Objective 7.14A introduces the concept of proportionality, beginning with the relationships between ratios and rates. Refresh students’ knowledge about the notation and terminology used in proportional reasoning. Emphasize the importance of labels in expressing both rates and ratios and the differences between part-to-part ratios and part-to-whole ratios. Cues, Questions, and Advance Organizers

Frayer Model Continue to emphasize vocabulary development using a word wall, a graphic organizer such as a Frayer Model or a student-created glossary. Instructional Accommodations for Diverse Learners Students may use eGlossary, a multilingual virtual math dictionary, to help them create their glossaries or vocabulary foldables or Intermath, an interactive math dictionary. C.4e While considering the “Instructional Strategies” column, note the verbs that align with the 5E Lesson Model: Engage, Explore, Explain, Elaborate, and Evaluate. These cues indicate appropriate strategies, level of rigor, and level of questioning to use during instruction.

Cues, Questions, and Advance Organizers Use an Anticipation Guide to refresh students’ knowledge about the notation and terminology used in proportional reasoning. Anticipation Guide

Statement Agree Disagree

A unit rate has a denominator of one.

Proportions can be written as an equation.

If the denominator of two ratios is equivalent, write both ratios in simplest form.

Scale factors can be used to solve a proportion.

Instructional Accommodations for Diverse Learners

Summary Frames Use a series of sentence stems such as those below to highlight important elements to produce a summary. Provide opportunities for students to read in their textbooks or other printed material to find the information necessary. • A ratio is (a comparison of two quantities). • A rate is (a ratio that compares two different kinds of units). • If two (ratios) (written as (fractions) are equivalent, they can be (simplified) to the same (fraction). • If two (ratios) (written as (fractions) are equivalent, (multiplying) the (numerator) and (denominator) of one ratio by the same number will result in the second ratio. • That number is called a (factor) of (change). • A unit rate is (simplified) so that it has a denominator of (one). 5 Nonlinguistic Representations Use a variety of manipulatives such as color tiles, beans, rainbow cubes, or similar models to review rates and ratios concretely and pictorially and to create equivalent ratios and solve proportions (Activities: Ratios vs. Rates, Activity 7.2D in Mathematics Toolkit. –see Resource column). Use Cuisenaire Rods or paper manipulatives to illustrate proportional relationships between numbers and to make predictions. Cooperative Learning After presentations about their research, each group writes one or more representative problems that a practitioner of the job they researched would need to solve. The problems could involve ratios such as average speed, gas mileage, unit price, cost ratios, or hourly wage.



http://www.glencoe.com/apps/eGlossary612/grade.php










2011 – 2012

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At the end of the activity, students exchange their problems with another group and solve each other’s problems C.5e Homework and Practice Use interactive virtual activities and graphing calculator activities to assist students in their understanding of ratios and proportionality (Activities: Math Spy Guys, Understanding Math Plus, and Making Cookies – see Resources Column).


In Job-Related Ratios, students are separated into groups of two or three and each group picks an occupational field such as cooking, business, sport, or science. They then research how ratios are used in that field. Use a computer with Internet access, newspapers, or magazines. Have each group write a list of the ratios they discover, explaining how they are used. If time allows, each group will then present their findings to the class as a whole. C.1C, C.4, C.5b

Formative Assessment 2.2 Use multiple-choice questions as formative assessments by having students analyze not only the correct answers but also the incorrect answers.

Resources

Clarifying Activities: The UT Dana Center Mathematics Toolkit includes

hands-on experiences with student activity sheets and extensive teacher notes.

Resources: Online glossaries/ dictionaries are available for use at:: eGlossary, Glencoe Online Intermath, Univ. of Georgia at Athens, Mathematics

Education Dept.



http://www.learnalberta.ca/content/mesg/html/math6web/lessonLauncher.html?lesson=m6lessonshell03.swf&launch=true

http://www.neufeldmath.com/correlations/TEKS_GR7_070507.pdf

http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=4979



http://www.glencoe.com/apps/eGlossary612/grade.php



Notes to the Teacher Materials Colored cubes Overhead color tiles This is an important point and must be emphasized strongly. Ratios should always have the quantities labeled no matter what format is used to express them. It is important to discuss the differences in the two types of ratios and again mention that the labels are crucial in being able to differentiate them.

A1 Ratios vs. Rates These activities are an extension from the work begun in sixth grade with ratios and proportions. Sixth graders used ratios to describe and make predictions in proportional relationships as well as solving problems involving equivalent ratios and rates. To refresh the students’ knowledge about the notation and terminology used in proportional reasoning, write the following items on the overhead or chalkboard and ask the students to say out loud what these statements mean:

35

3 or 3/5 3:5 3 to 5 → 5

Most students will read the fraction notation as three-fifths while the remaining items will probably be read as 3 to 5 or 3 out of 5. These responses should lead to a discussion of the definition of ratio and the two different types of ratios (part/part and part/whole).

A ratio is defined as a comparison of any two quantities.

Ratio is the general concept as demonstrated in the graphic at the end of this section. The book defines ratios as two quantities that have the same unit of measure. This definition is misleading. A more accurate definition omits the reference to units of measure. A ratio is a comparison of two quantities (like or unlike). These quantities or numbers being compared are called the terms of the ratio. Remind the students that ratios are hard to define without a context. For example, the above ratios might have a meaning 3 girls to 5 boys or 3 blue cubes to 5 red cubes. If ratios are expressed as fractions, they should always be written with labels (or

“word ratios”) attached such as: 35

girlsoysb

or 35bluered

.

These different meanings should be illustrated using manipulatives such as colored cubes or color tiles. Distribute them to the students to be used during the following discussion.

Place 2 blue tiles and 3 red tiles on the overhead projector and ask the students to write the ratio for the situation they see.

Most students will probably write something like 23bluered

. Ask them if

they see any other possible ratios that could be read from the same

situation. Other possibilities include 32

redblue

, 25blueotalt

, 35

redotalt

. The

first two ratios are examples of part/part ratios and the second two are examples of part/whole ratios.

Ratios may also be used to compare measures of different types. Such a ratio is usually called a rate. Implicit in the use of the word “rate” is the additional assumption that the comparison describes a quality that is common to many situations. For example, a common rate such as $3 per dozen is a rate that describes the relationship between cost in dollars and number of dozen in all of the following instances: $6 for 2 dozen, $18 for 6 dozen, $24 for 8 dozen, and so on.

B B R R R

Notes to the Teacher Make sure that the students notice that the terms of the ratio can be set up in three different ways: using a fraction, a colon, or with the word to. Each way is read as a part to part or a part to whole relationship depending on the situation. Rate is a special ratio in which the two terms of the ratio have different units of measure. This is demonstrated as a subset of ratio in the diagram below. The textbook defines rate as having two different units of measure. This is correct, but incomplete. It does not state that it is a ratio with different units of measure. The different measures being compared are the terms of the ratio. Here again, a rate can be set up in three different ways: using a fraction, a colon, or with the word to. The following diagram may be used to summarize the differences between the different types of ratios:

This conceptual understanding is important to the later use of factors of change and scale factors.

Different measures

RATES

Same measures

P/P P/W

RATIOS

Assessment for Multiple Choice Questions Enhance multiple-choice questions to formative assessments by having students analyze not only the correct answers but also the incorrect answers. Using the graphic organizer attached, students first answer the question as indicated as if it is an open- ended question. Then students will look at all the possible answer choices. For the correct answer, the student may want to add addition justifications for the answer that they mark correct. For the incorrect answers, students must give reasons for why the answer is incorrect: word clues, process mistakes, or concept mistakes. Students may prove the mistake mathematically or in a written statement. If the formative assessment is being given after a multiple choice test is given, students may use this process to check and correct their work. Even if the student has the question correct in the original testing, he/she must also prove why the incorrect answers are not possible. Region IV Rubrics may be used for evaluation: 6-8 Assessment Rubric (student) – see attachment HS Assessment Rubric (student) – see attachment Assessment Rubric (teacher) – see attachment

2003 Region IV Education Service Center. All rights reserved.

Criteria 4 3 2 1 Part b)

Conceptual Knowledge

Attribute(s) of concept(s) Correctly identifies attributes of the problem, which leads to correct inferences. Inferences Combines the critical attributes of the problem in order to correctly describe the mathematical relationship(s) inherent in the problem.

Attribute(s) of concept(s) Correctly identifies attributes of the problem, which leads to correct inferences. Inferences Combines the critical attributes of the problem, which leads to a partial identification of the mathematical relationship(s) inherent in the problem.

Attribute(s) of concept(s) Identifies some of the attributes of the problem, which leads to partially correct inferences. Inferences Combines the identified attributes of the problem, which leads to a partial identification of the mathematical relationship(s) inherent in the problem.

Attribute(s) of concept(s) Lacks identification of any of the critical attributes of the problem. Inferences Combines few of the attributes of the problem which leads to an incomplete identification of the mathematical relationship(s) inherent in the problem.

Part c)

Procedural Knowledge

Appropriate strategy Selects and implements an appropriate strategy. Representational form Uses appropriate representation to connect the procedure to the concept of the problem. Algorithmic competency Correctly implements procedure to arrive at a correct solution.

Appropriate strategy Selects and implements an appropriate strategy. Representational form Uses appropriate representation to connect the procedure to the concept of the problem. Algorithmic competency Implements selected procedure but arrives at an incorrect solution.

Appropriate strategy Selects and implements an appropriate strategy. Representational form Uses inconsistent or insufficient representation for the selected solution strategy. Algorithmic competency Implements selected procedure but arrives at an incorrect or correct solution. (See Part a above)

Appropriate strategy Selects and implements an inappropriate strategy. Representational form Uses incorrect representations. Algorithmic competency Makes significant errors.

Part d)

Communication

Justification Fully answers the question of “why” for the strategy selection; explains procedure; and/or evaluates reasonableness of solution. Terminology Uses appropriate terminology and notation.

Justification Fully answers the question of “why” for the strategy selection; explains procedure; and/or evaluates reasonableness of solution. Terminology Uses some appropriate terminology or notation.

Justification Incompletely answers the question of “why” for the strategy selection; explains procedure; and/or evaluates reasonableness of solution. Terminology Uses some appropriate terminology or notation.

Justification Provides very little or no explanation of what was done and why. Terminology Uses limited or inappropriate terminology or notation.

Mathematics Performance Assessment Rubric Part a) Correct Solution YES NO

Assessment for Multiple Choice Questions Part 1: Problem: Solve the problem in the space provided below. Be sure to show all your work. Select the answer from the multiple-choice possibilities. In the multiple choice square for the correct answer, mark in the space “correct” and show how you checked your work in Part 2. In the other three squares, justify why the answer choice is incorrect by explaining either in words or numerically.

Part 2: Student work: Answer Choice (A)

Answer Choice (B)

Answer Choice (C)

Answer Choice (D)

2003 Region IV Education Service Center. All rights reserved.

Check List 4 3 2 1 Part b) Concept Understand the problem.

I understood how all of the parts of the problem fit together, so I could make sense of the problem.

I understood all of the parts of the problem, and I made partial sense of the problem.

I understood some of the parts of the problem.

I showed little to no understanding of the important facts of the problem that would help me find the answer.

Part c) Procedure Work the problem.

I used an appropriate strategy. I connected how I needed to do the problem with what I understood about the problem and my selected strategy. I did all of my math steps correctly.

I used an appropriate strategy. I connected how I needed to do the problem with what I understood about the problem and my selected strategy. I did some of my math steps correctly. I did not arrive at a correct solution.

I used an appropriate strategy. I showed little connection between how I needed to do the problem and my selected strategy. I did some of my math steps correctly, but reached an incorrect or correct solution. (See Part a.)

I used an inappropriate strategy. My work had lots of mistakes.

Part d) Communicate what you understand. Communicate how you worked the problem.

I explained why I did what I did and supported my explanation with information from the problem. I used correct math vocabulary and notation.

I explained why I did what I did and supported my explanation with information from the problem. I used some correct math vocabulary and notation.

I gave little explanation of why I did what I did. I only explained what I did. I used some correct math vocabulary and notation.

I gave very little or no explanation of what I did. I used little or incorrect math vocabulary and/or notation.

STUDENT RUBRIC, GRADES 6 - 8

Part a) I arrived at a correct solution YES NO

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