Scrambled Eggs II – Human Ratios and Constructing Ratio Tables

This resource was created by Big Ideas in Beta, a Big Ideas Fest project, with acknowledgement to Pat C. Browne

LEARNING OUTCOMES:

• Students will recognize that a ratio is a comparison of two numbers or quantities 
• Students will represent ratios arithmetically (4 to 1, 4:1, and  4/11) and algebraically (a to b, a:b, and  a/b)  
• Students will demonstrate how to simplify a ratio represented as a fraction in simplest form. Students will use proportional reasoning to solve problems.

COMMON CORE STANDARDS ADDRESSED:

Analyze proportional relationships and use them to solve real-world and mathematical problems.

7.RP.1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

7.RP.2. Recognize and represent proportional relationships between quantities.

7.RP.2a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

7.RP.2b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

TIME REQUIRED FOR LESSON:

45 minutes - 50 minutes

TIME REQUIRED FOR TEACHER PREPARATION:

20 minutes - 30 minutes

MATERIALS FOR LESSON:

• Blackboard, whiteboard, or chart paper 
• Markers 
• A class set of 60 large egg signs printed on colored cardstock (30 red and 30 blue); one color of each for all students to wear during activity. Attach egg with yarn around the neck of each student.  (A free egg template is available at http://www.firstpalette.com/tool_box/printables/eastereggs.html.) Paper and pencils (for each student)
• Pens or pencils for each student
  

OVERVIEW OF LESSON:

1.  Review with students that ratios can be expressed arithmetically (4 to 1,  4:1, and  4/11) and algebraically (a to b, a:b,  and  a/b)   Explain that today we will use proportional thinking to solve proportions between quantities.

2. Write the equation:  2/3 = 4/6 on the board. Divide the class into four groups. Distribute one color of egg signs to each the students in the four groups – there should be a red group, yellow group, blue group, and green group..  Ask the students in each group to stand together with each student in the group wearing the colored egg signs designated for their group. Ask the students in the red and yellow groups to form the ratio 2/3 and the blue and green group to form 4/6. Ask the students: “How do we know that these two ratios are equivalent?” Wait for students to volunteer to respond and discuss.

3. Bring all the students together again, and draw three red eggs and four yellow eggs on the board.  Ask students, “How would you write this ratio of red to yellow eggs?”   Write their response on the board. Ask, “How would you write this ratio of red to yellow eggs?”   Write their response on the board. Ask, “How many blue eggs and green eggs would be needed to create an equivalent ratio?” Explain that their task now is to create another ratio that is equivalent to 3/4.  Ask the  students in the red and yellow eggs to form the ratio on the board and the remaining blue and green eggs to  assemble themselves into groups that represent the equivalent ratio.  Ask any remaining students to serve as “checkers” to determine if the new groups (blue:green) are equivalent to the first (red:yellow) ratio.

4.  Explain:  “A proportion is an equation which states that two ratios are equal.  When the terms of a proportion are cross- multiplied, the cross products are equal.   Cross multiplication is the multiplication of the numerator of the first ratio by the denominator of the second ratio and the multiplication of the denominator of the first ratio by the numerator of the second ratio.” 

Demonstrate the proportionality  by writing on the board:

    21      3       

    --  =  --      

    70     10

Circle the “cross” in the equation so that students see that you are multiplying

(3 X 70 and 21 X 10).

State: “This is called the “Property of Proportions.”  Write:

             The cross products of a proportion are equal.

             If a/b = c/d, then ad = bc.

Explain: “So in our previous equation, 3 X 70 = 210 and 21 X 10=210.  Because both products are equal, we know that the two ratios (written as fractions) are equivalent or are proportional.”

5.  Ask students to again organize themselves in groups to demonstrate a ratio that is equivalent to the following ratio:  (write on the board or display on chart paper): 12:60.  Discuss that since there aren’t enough students to increase the proportion to an equivalent ratio beyond 60 their groups will need to be smaller.

6.  Ask the partnered groups to switch  (red and blue as a team and yellow and green as a team).    Ask the students to demonstrate and solve the following proportion by using cross products, then demonstrating the new ratio (X / 21) in their groups:

            2                      X

         _____     =        _____                           

            7                      21

Ask for a volunteer to write the numeric version of solution on the board: 2 X 21  = 42,  6 X  7 = 42 )  x = 6

(We know that X = 6 because of the inverse operation: 42  /  7 = 6

7.    Repeat the process by asking the groups to solve one more problem with one more problem:

            3                      4

         _____     =        _____                           

            X                      16

Ask for a volunteer to write the numeric version of solution on the board: 3 X 16  = 48,  4 X  12 = 48)   We know that X = 16  because of the inverse operation: 48  /  4 = 16

8.   Assess students’ understanding by asking them to solve the following (write on board):

Which of the following ratio proportions are equivalent?  How do you know?

(1)        12/18 and 9/12  (no)

(2)        150/15 and 3/1   (yes)

(3)        5/36  and  8/72   (no)

(For enrichment, write the ratios in alternative forms, such as fractions or decimals)

ADDITIONAL INFORMATION:

Sousa, D. A. How the Brain Learns Mathematics, Corwin Press, 2008.

Van de Walle, j. A., K. S. Karp, and J. M. Bay-Williams, Elementary and Middle School Mathematics – Teaching Developmentally, Allyn & Bacon, 2010.