Students will calculate and represent ratios using different notations to solve real-world problems in a lesson that highlights the value of math in careers.
Grades
6–8
Students will:
Step 1: Tell the class the following story: You know of a student, Tommy, with a strange snacking habit that you would not recommend. Every day after school, he drinks an 8-ounce custom-made beverage of coffee and chocolate. He instructs the barista to use 6 ounces of chocolate syrup and 2 ounces of coffee.
Step 2: Tell the class that a ratio shows a comparison between two or more related quantities. Another way to explain it is to say that a ratio shows the relationship between a part and another part. Throughout history, ratios have been widely used in real-world fields like recipes, architecture, and art.
In the case of Tommy’s beverage, the ratio would compare 6 parts chocolate syrup to 2 parts coffee. Point out that simplifying the ratio makes the relationship clearer. In the case of the coffee-chocolate beverage, one could make a drink with the same taste using 3 parts chocolate to 1 part coffee.
a) Show the class that the ratio of chocolate to coffee can be expressed using three different notations: 6 to 2, 6:2, and 6/2. The 6/2 notation might be tricky for some students. Point out that when saying 6/2 out loud, it should be read as “6 to 2.” In this case it means there is 6/2 as much chocolate as coffee, or, when simplified, three times as much chocolate as coffee.
b) If desired, the same information could be used to show that the ratio of coffee to chocolate is 2 to 6, 2:6, or 2/6, which is equivalent to 1:3.
Step 3: Remind the class that a fraction shows a part-to-whole relationship, and demonstrate that the same information about the drink can be expressed as fractions, e.g., that the whole 8-ounce drink is 2/8 coffee (1/4 simplified) and 6/8 chocolate syrup (3/4 simplified). Remind the class that a fraction can also be expressed as a percentage and decimal. For example, since the drink is ¾ chocolate syrup, the chocolate syrup makes up 75% or 0.75 of the beverage.
Step 4: Tell the class that, one day, Tommy had a particularly challenging day at school and felt he needed an extra-large beverage. He told the barista he wanted his drink with its usual level of “chocolatey-ness,” but to make enough for a 24-ounce cup. Ask the class how we could figure out how much chocolate syrup and coffee the barista should use to make the drink. The following approaches should be discussed:
a) Make a ratio table. Show that the ingredients of the large drink can be determined by completing a table:
As you set up the table (adding by sixes in the top row and adding by twos in the bottom row), explain how it works both vertically and horizontally. Looking at it vertically, within each serving size is the recipe for that size cup. For the first size cup, 8 ounces, the ingredients in that cup must add up to 8 ounces. Since the ratio of chocolate syrup to coffee is 6:2, the amount of chocolate syrup and coffee in an 8-ounce cup would be 6 ounces and 2 ounces, respectively. Point out that another way to think of it, using fractions rather than ratios, is that the drink is 3/4 chocolate syrup and 1/4 coffee—3/4 times 8 ounces equals 6 ounces of chocolate syrup, and 1/4 times 8 ounces equals 2 ounces of coffee.
Looking at the table horizontally shows how the ingredients increase in proportion as the quantity of the cup, the total of all ingredients, increases. For example, to use the recipe for an 8-ounce cup to make a 16-ounce serving with the same flavor, each ingredient is multiplied by 2 (2 x 8-ounce cup = 16-ounce serving; 2 x 6 ounces of chocolate syrup in the original recipe = 12 ounces of chocolate syrup in the 16-ounce serving, and 2 x 2 ounces of coffee in the original recipe = 4 ounces of coffee in the 16-ounce serving). This makes 2 the scale factor. Show how the scale factor for using the 8-ounce cup recipe to make a 24-ounce serving is 3. Also show how the scale factor would be 1/3 if we wanted to use the 24-ounce recipe to make an 8-ounce serving.
b) Find multiplicative relationships. A 24-ounce cup is three times larger than an 8-ounce cup, so the quantity of each ingredient in the 8-ounce cup should be multiplied by 3. In this case, 6 ounces of chocolate syrup x 3 = 18 ounces and 2 ounces of coffee x 3 = 6 ounces.
c) Create a double number line (see image below):
Step 5: Tell the class that, one day, the barista made a mistake with Tommy’s beverage order and poured 9 ounces of chocolate syrup into a cup. How much coffee should the barista add to reach Tommy’s preferred level of chocolatey-ness? Show how cross multiplying can solve this problem, using x to represent the number of ounces of coffee to maintain the right taste with 9 ounces of chocolate syrup:
6/2 = 9/x
cross multiply by 2 and x
6x = 18
x = 3
Step 6: Demonstrate how equations can be used to depict ratio relationships. If the ratio of chocolate syrup (s) to coffee (c) is 3:1 when simplified, then we can say that s = 3c.
Step 7: Have students use the Ratio Design Challenge Digital Interactive Tool to practice what they have learned, or distribute the practice worksheet.
The Ratio Design Challenge is an online, interactive tool designed to engage students in grasping an understanding of ratio and proportional reasoning while highlighting the value of math in real-world situations and careers.
The interactive tool enables the student user to construct scale designs of an architectural venue by applying ratio/proportion calculations. Each word problem includes a guiding hint if a student answers incorrectly and, after several tries, an explanation of the correct answer will display. Critical-thinking reflection questions at the end of the user experience will challenge students to draw overarching conclusions about the math concepts and to reflect on the real-world implications.
The interactive tool has three stand-alone modules, each with a different venue theme:
Each theme of the tool can be used for whole-class instruction, independent practice, and/or homework. The tool includes a Teacher’s Guide and a student-facing “Using the Tool” page.
Launch use of the online Ratio Design Challenge by first inviting students’ thoughts on ways ratios and proportions might be used in the real world, besides recipes. If architecture or construction isn’t suggested, bring it up. Solicit students’ thoughts on why it might be important to use precise mathematical calculations in architecture. What could go wrong if precise calculations are not used? (Possible answers include: If walls were accidentally built at different heights, a roof might not attach properly and would fall; if doors were cut too long, they wouldn’t fit in doorways; if there weren’t enough support beams, higher floors might collapse; etc.) Ask students to share any initial ideas on how ratios and proportions specifically might be used in architecture; tell them the online tool will provide more examples.
Using an interactive whiteboard or computer/projector hookup, select one venue theme module in the Ratio Design Challenge to model for the class — choose from either "Baseball Stadium" or "Amusement Park," as these themes focus on ratio basics. Perform a “Think Aloud” as you answer the first 1–2 questions. Encourage students to take the lead on the remaining questions and to reach a consensus through discussion on what answer to input and why.
*The Digital Interactive Tool requires Internet access and either an interactive whiteboard or a computer/projector hookup.
Similar to the online tool above, the activity sheets for the Designing RATIOnally unit employ an architecture storyline, but can be used independently or in conjunction with the online tool as needed. In the activity sheets, the story line centers on the crucial real-world issue of constructing a water treatment facility in a country where clean water is scarce.
As a class, complete the first two problems of the Designing RATIOnally Worksheet: A Fresh, Clean Design printable. Have students explain their reasoning and methods for coming up with the answers.
If needed, remind students that the prime symbol ′ can be used to represent feet, and the double prime symbol ″ to represent inches.
Step 8: Assign students to complete additional practice using the Ratio Design Challenge Digital Interactive Tool, or have students complete the remainder of the Designing RATIOnally Worksheet: A First-Rate Design printable.
Assign small groups to a Ratio Design Challenge module to work on independently — either “Baseball Stadium” or “Amusement Park,” whichever was not used during Guided Practice.
*The Digital Interactive Tool requires Internet access and either an interactive whiteboard or a computer/projector hookup.
Have students complete the Designing RATIOnally Worksheet: A Fresh, Clean Design printable.
Note: This worksheet can also be assigned for homework if the online tool was used in class.
Step 9: Checking for Understanding: Have students report their findings or answers to the class and share their conclusions. Address any misconceptions yourself or via student volunteers.
CCSS and NCTM:
For more information, download the comprehensive Standards Chart: Designing RATIOnally printable.
