Proportional Relationships: Modeling Mathematics

(Standard 4)

Carolyn Balmages and Guadalupe SaldivarSouth Junior High School

Anaheim Union High School District

balmages_c@auhsd.us saldivar_g@auhsd.us

Goals for our time today

1. Understand and recognize the concept of proportionality 4 ways.

2. Use three different modeling strategies to estimate and/or solve proportion problems.

3. Watch students explain their reasoning of proportionality.

4. Apply proportional reasoning to real life mathematics.

5. Leave with 5 potential lessons for your students.

In June 2011, my mom, my sister, and I traveled to Denmark.

The Danish currency is Kroner. The exchange rate is 1 Krone = 0.18 US Dollar

We bought tickets into Tivoli Garden.

Was it expensive?

Entrance = 75 DKK

(Exchange rate: 1 Krone = 0.18 US Dollar)

We took a train to Sweden for a day.We spent 107 Danish Kroner (DKK) each.

How much did a ticket cost in USD?(Exchange rate: 1 Krone = 0.18 US Dollar)

Swedish money is called Kronor.

The exchange rate is 1 Danish Krone = 1.18 Krona

While in Sweden we had pastries at Mormors Bageri in Lund (40 Swedish Kronor/SEK)

and enjoyed a boat tour in Malmo (120 SEK)

How many Danish Kroner (DKK) did we need to change into Swedish Kronor (SEK)? (1 DKK = 1.18 SEK)

Modeling Proportional RelationshipsStandard 4: Model with mathematicsThe Standard:Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Modeling Proportional Relationships

Standard 4: Model with mathematicsThe Standard:

● In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community.

● They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas.

Developing the Concept of and Modeling Proportionality

● Double Number Lines / Ratio Tables / Charts

● Graphing data points

● Equivalent Fractions

● Recognizing Proportional Scenarios

Data Collection Activities and Observations of Proportionality Characteristics

ProportionalityCharacteristics

Graphs:

Double Number lines:

Fractions:

Scenario:

Quiz Quiz Trade

● Hand up to find a partner● Person A quizzes Person B● Person B quizzes Person A● Trade cards● Hand up to find a new partner

Question___________________________________________________________________________________________________________________________________

=

This is / is not proportional because1.____________________________________________________________2. ____________________________________________________________3. ____________________________________________________________4. ____________________________________________________________

Double Number Line Graph

Equivalent Fractions

We took a train to Sweden for a day. We spent 107 Danish Kroner (DKK) each. How much did a ticket cost in USD? (Exchange rate: 1 Krone = 0.18 US Dollar)

=

This is / is not proportional because1. The comparison of Dollars to Kroner is consistent2. The double number line starts at (0,0), follows a pattern, has a factor between rows.3. The graph is a straight line starting at (0,0)4. Comparison of Dollars and Kroner can be written as two equivalent fractions and can be solved as a proportion.

USD 0 0.18 0.36 0.54 0.72 0.90 1.08 1.26 1.80 18.00 D

DKK 0 1 2 3 4 5 6 7 … 10 … 100 107

0.18 USD

1 DKK

? USD

107 DKK

DKK 0 20 40 60 80 100 120 140

USD20

16

12

8

4

0

$19.76

$19.76

Which Wich?

Are the Small and the Large wiches proportional?

How do you know?

What should the medium wichcost?

Which wich should I buy? Why?

How much would a 35 inch wich cost?

How long should a wich be that costs 23 dollars?

Student Videos/Voices: Which Wich?

Students prove proportionality with equivalent fractions.

Students find the cost of the medium by finding the middle (or average) between the small and large costs.

Student uses a graph to find the approximate cost of the medium.

Student uses a proportion and cross products to find the cost of a medium.

Student uses partial sums or line segment addition to find the cost of a 35 inch sandwich.

Student uses a factor to multiply up to find the cost of a 35 inch sandwich.

Student uses doubling to find the length of a $23 sandwich.

Shopping Coupon LessonEquivalent Fractions

● How would you use this coupon to get the best deal?● How would you spend $75 to get the most merchandise?● How much money do you need to buy $180 worth of merchandise?● Is there only one answer? Why or why not?

$15 off of $30$30 off of $75$60 off of $120

Pencil Unit Rate

Which pencils are the best deal?

$5 for 24 pencils

$11 for 72 pencils

$4 for 18 pencils

Menu Project with Tax and Tip (Percentages)

Write Your Own Word Problem Project

Step 1: Write about a time when you and 3 friends went out to eat. Where did you eat? What did you each order? Include the menu. Be creative. Make sure you write the question for someone else to answer.

Step 2: Make the answer key:What was the subtotal? Find the tax. Find the 15% tip. What was the grand total? How much did each friend have to pay at the end?

Show your work and explain your thinking!

Additional Resources

Proportionality Lessons (Provided by University of California Irvine, Irvine Math Project)

http://auhsdmath.pbworks.com/ http://auhsdmath.pbworks.com/w/browse/#view=ViewFolder&param=Proportional%20Relationships%20Ahttp://auhsdmath.pbworks.com/w/browse/#view=ViewFolder&param=Proportional%20Relationships%20B%20Percent

Double Numberline Resources(Provided by Tad Watanabe from Kennesaw State University)

http://science.kennesaw.edu/~twatanab/DeKalb%20Title%20I%20Summit%202012.pdf

Additional Real Life Applicationshttp://blog.mrmeyer.com/?p=8713