ALEX | Alabama Learning Exchange

Scale City | Scaling Up Recipes and Circles in the Real World

Classroom Resource Information

Title:

Scale City | Scaling Up Recipes and Circles in the Real World

URL:

https://aptv.pbslearningmedia.org/resource/mket.math.rp.chickfest/scaling-up-recipes-and-circles-in-practice/

Content Source:

PBS

Type:

Audio/Video

Overview:

In this video, students visit a small-town festival that features the world’s largest stainless steel skillet. In addition to a question about scaling recipes, they also are asked how increasing or decreasing the radius of a circle affects its area. The accompanying classroom activity requires students to compare the areas of the world’s largest skillet and a standard 12-inch skillet through reasoning and computation and to explore the meaning of pi through a hands-on activity. This resource is part of the Math at the Core: Middle School Collection.

More About This Resource

Although the Scaling Up Recipes and Circles in Practice video ("Greetings from the World’s Chicken Festival") and the Scaling Up Recipes and Circles in the Real World interactive ("Sunnyside Up") can be used independently, they are deliberately designed to complement each other.

The video takes students to a small-town fall festival that features the world’s largest stainless steel skillet as well as food preparations for a crowd of 8,000 people. They are asked how they can use proportional reasoning to scale recipes and how increasing or decreasing the radius of a circle affects its area.

The interactive explores the questions asked in the video as students scale up recipes and food portions to feed a family reunion of 108 people and as they discover the mathematical relationship between the length of a circle’s radius and its area. To enhance classroom use, refer to the Interactive Guide handout and Questions worksheet that students can reference and complete as they work through the interactive.

Be sure to use the Scaling Up Recipes and Circles in the Real World Activity that can be found in the Support Materials for Teachers section for a great activity that teaches the standard(s).

Content Standard(s):

Mathematics MA2019 (2019) Grade: 6	3. Use ratio and rate reasoning to solve mathematical and real-world problems (including but not limited to percent, measurement conversion, and equivalent ratios) using a variety of models, including tables of equivalent ratios, tape diagrams, double number lines, and equations. Unpacked Content Evidence Of Student Attainment: Students: Given contextual or mathematical situations involving ratio and rate (including those involving unit pricing, constant speed, and measurement conversions), Represent the situations using a variety of strategies (tables of equivalent ratios, changing to unit rate, tape diagrams, double number line diagrams, equations, and plots on coordinate planes) in order to solve problems, find missing values on tables and interpret relationships and results. Change given rates to unit rates in order to find and justify solutions to problems. Given contextual or mathematical situations involving percents, Understand the relationship between ratios, fractions, decimals and percents. Interpret the percent as rate per 100. Solve problems and justify solutions when finding the whole, given a part and the percent. Solve problems and justify solutions when finding the part, given the whole and the percent. Solve problems and justify solutions when finding percent, given the whole and the part. Teacher Vocabulary: Rate Ratio Rate reasoning Ratio reasoning Transform units Quantities Ratio Tables Double Number Line Diagram Percents Coordinate Plane Ordered Pairs Quadrant I Tape Diagrams Unit Rate Constant Speed Knowledge: Students know: Strategies for representing contexts involving rates and ratios including. tables of equivalent ratios, changing to unit rate, tape diagrams, double number lines, equations, and plots on coordinate planes. Strategies for finding equivalent ratios, Strategies for using ratio reasoning to convert measurement units. Strategies to recognize that a conversion factor is a fraction equal to 1 since the quantity described in the numerator and denominator is the same. Strategies for converting between fractions, decimals and percents. Strategies for finding the whole when given the part and percent in a mathematical and contextual situation. Strategies for finding the part, given the whole and the percent in mathematical and contextual situation. Strategies for finding the percent, given the whole and the part in mathematical and contextual situation. Skills: Students are able to: Represent ratio and rate situations using a variety of strategies (e.g., tables of equivalent ratios, changing to unit rate, tape diagrams, double number line diagrams, equations, and plots on coordinate planes). Use ratio, rates, and multiplicative reasoning to explain connections among representations and justify solutions in various contexts, including measurement, prices and geometry. Understand the multiplicative relationship between ratio comparisons in a table by writing an equation. Plot ratios as ordered pairs. Solve and justify solutions for rate problems including unit pricing, constant speed, measurement conversions, and situations involving percents. Solve problems and justify solutions when finding the whole given a part and the percent. Model using an equivalent fraction and decimal to percents. Use ratio reasoning, multiplication, and division to transform and interpret measurements. Understanding: Students understand that: A unit rate is a ratio (a:b) of two measurements in which b is one. A symbolic representation of relevant features of a real-world problem can provide for resolution of the problem and interpretation of the situation. When computing with quantities the transformation and interpretation of the resulting unit is dependent on the particular operation performed. Diverse Learning Needs: Essential Skills: Learning Objectives: M.6.3.1: Define ratio, rate, proportion, percent, equivalent, input, output, ordered pairs, diagram, unit rate, and table. M.6.3.2: Create a ratio or proportion from a given word problem, diagram, table, or equation. M.6.3.3: Calculate unit rate or rate by using ratios or proportions with or without a calculator. M.6.3.4: Restate real-world problems or mathematical problems. M.6.3.5: Construct a graph from a set of ordered pairs given in the table of equivalent ratios. M.6.3.6: Calculate missing input and/or output values in a table with or without a calculator. M.6.3.7: Draw and label a table of equivalent ratios from given information. M.6.3.8: Identify the parts of a table of equivalent ratios (input, output, etc.). M.6.3.9: Compute the unit rate, unit price, and constant speed with or without a calculator. M.6.3.10: Create a proportion or ratio from a given word problem. M.6.3.11: Identify the two units being compared. M.6.3.12: Define percent. M.6.3.13: Calculate a proportion for missing information with or without a calculator. M.6.3.14: Identify a proportion from given information. M.6.3.15: Solve a proportion using part over whole equals percent over 100 with or without a calculator. M.6.3.16: Form a ratio. M.6.3.17: Convert like measurement units within a given system with or without a calculator. (Example: 120 min = 2 hrs). M.6.3.18: Know relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz; l, ml; and hr, min, sec. Prior Knowledge Skills: Recognize arithmetic patterns (including geometric patterns or patterns in the addition table or multiplication table). Examples: Continued Geometric Pattern by drawing the next three shapes. Complete the numerical pattern for the following chart when given the rule, "Input + 5 = Output". Recognize that comparisons are valid only when the two fractions refer to the same whole. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. Recognize key terms to solve word problems. Examples: times, every, at this rate, each, per, equal/equally, in all, total. Recall basic multiplication facts. Recognize equivalent forms of fractions and decimals. Recognize a fraction as a number on the number line. Label numerator, denominator, and fraction bar. Alabama Alternate Achievement Standards AAS Standard: M.AAS.6.1 Demonstrate a simple ratio relationship using ratio notation given a real-world problem.
Mathematics MA2019 (2019) Grade: 7	2. Represent a relationship between two quantities and determine whether the two quantities are related proportionally. a. Use equivalent ratios displayed in a table or in a graph of the relationship in the coordinate plane to determine whether a relationship between two quantities is proportional. b. Identify the constant of proportionality (unit rate) and express the proportional relationship using multiple representations including tables, graphs, equations, diagrams, and verbal descriptions. c. Explain in context the meaning of a point (x,y) on the graph of a proportional relationship, with special attention to the points (0,0) and (1, r) where r is the unit rate. Unpacked Content Evidence Of Student Attainment: Students: Decide whether a relationship between two quantities is proportional. Recognize that not all relationships are proportional. Use equivalent ratios in a table or a coordinate graph to verify a proportional relationship. Identify the constant of proportionality when a proportional relationship exists between two quantities. Use a variety of models (tables, graphs, equations, diagrams and verbal descriptions) to demonstrate the constant of proportionality. Explain the meaning of a point (x, y) in the context of a real-world problem. Example, if a boy charges $6 per hour to mow lawns, this relationship can be graphed on the coordinate plane. The point (1,6) means that after 1 hour of working the boy makes $6, which shows the unit rate of $6 per hour. Teacher Vocabulary: Equivalent ratios proportional Coordinate plane Ratio table Unit rate Constant of proportionality Equation ordered pair Knowledge: Students know: (2a) how to explain whether a relationship is proportional. (2b) that the constant of proportionality is the same as a unit rate. Students know: where the constant of proportionality can be found in a table, graph, equation or diagram. (2c) that the constant of proportionality or unit rate can be found on a graph of a proportional relationship where the input value or x-coordinate is 1. Skills: Students are able to: (2a) determine if a proportional relationship exists when given a table of equivalent ratios or a graph of the relationship in the coordinate plane. (2b) identify the constant of proportionality and express the proportional relationship using a variety of representations including tables, graphs, equations, diagrams, and verbal descriptions. (2c) model a proportional relationship using coordinate graphing. Explain the meaning of the point (1, r), where r is the unit rate or constant of proportionality. Understanding: Students understand that: (2a) A proportional relationship requires equivalent ratios between quantities. Students understand how to decide whether two quantities are proportional. (2b) The constant of proportionality is the unit rate. Students are able to identify the constant of proportionality for a proportional relationship and explain its meaning in a real-world context. (2c) The context of a problem can help them interpret a point on a graph of a proportional relationship. Diverse Learning Needs: Essential Skills: Learning Objectives: M.7.2.1: Define proportions and proportional relationships. M.7.2.2: Demonstrate how to write ratios as a fraction. M.7.2.3: Define equivalent ratios and origin. M.7.2.4: Locate the origin on a coordinate plane. M.7.2.5 Show how to graph on Cartesian plane. M.7.2.6: Determine if the graph is a straight line through the origin. M.7.2.7: Use a table or graph to determine whether two quantities are proportional. M.7.2.8: Define a constant and equations. M.7.2.9: Create a table from a verbal description, diagram, or a graph. M.7.2.10: Identify numeric patterns and finding the rule for that pattern. M.7.2.11: Recall how to find unit rate. M.7.2.12: Recall how to write equations to represent a proportional relationship. M.7.2.13: Discuss the use of variables. M.7.2.14: Define ordered pairs. M.7.2.15: Show how to plot points on a Cartesian plane. M.7.2.16: Locate the origin on the coordinate plane. Prior Knowledge Skills: Recall basic addition, subtraction, multiplication, and division facts. Define ordered pair of numbers. Define x-axis, y-axis, and zero on a coordinate. Specify locations on the coordinate system. Define ordered pair of numbers, quadrant one, coordinate plane, and plot points. Label the horizontal axis (x). Label the vertical axis (y). Identify the x- and y- values in ordered pairs. Model writing ordered pairs. Define quantity, fraction, and ratio. Reinterpret a fraction as a ratio. Example: Read 2/3 as 2 out of 3. Write a ratio as a fraction. Create a ratio or proportion from a given word problem, diagram, table, or equation. Alabama Alternate Achievement Standards AAS Standard: M.AAS.7.2 Use a ratio to model or describe a real-world relationship.
Mathematics MA2019 (2019) Grade: 7	20. Explain the relationships among circumference, diameter, area, and radius of a circle to demonstrate understanding of formulas for the area and circumference of a circle. a. Informally derive the formula for area of a circle. b. Solve area and circumference problems in real-world and mathematical situations involving circles. Unpacked Content Evidence Of Student Attainment: Students: Solve problems with the circumference and area of a circle. Teacher Vocabulary: Diameter Radius Circle Area Circumference π Knowledge: Students know: that the ratio of the circumference of a circle and its diameter is always π. The formulas for area and circumference of a circle. Skills: Students are able to: use the formula for area of a circle to solve problems. Use the formula(s) for circumference of a circle to solve problems. Give an informal derivation of the relationship between the circumference and area of a circle. Understanding: Students understand that: area is the number of square units needed to cover a two-dimensional figure. Circumference is the number of linear units needed to surround a circle. The circumference of a circle is related to its diameter (and also its radius). Diverse Learning Needs: Essential Skills: Learning Objectives: M.7.20.1: Define diameter, radius, circumference, area of a circle, and formula. M.7.20.2: Identify and label parts of a circle. M.7.20.3: Recognize the attributes of a circle. M.7.20.4: Apply the formula of area and circumference to real-world mathematical situations. Prior Knowledge Skills: Define center, radius, and diameter of a circle. Identify real-world examples of radius and diameter. Examples: bicycle wheel, pizza, pie. Alabama Alternate Achievement Standards AAS Standard: M.AAS.7.20 Identify the radius, diameter, and circumference of a circle.
Mathematics MA2019 (2019) Grade: 7 Accelerated	2. Represent a relationship between two quantities and determine whether the two quantities are related proportionally. a. Use equivalent ratios displayed in a table or in a graph of the relationship in the coordinate plane to determine whether a relationship between two quantities is proportional. b. Identify the constant of proportionality (unit rate) and express the proportional relationship using multiple representations including tables, graphs, equations, diagrams, and verbal descriptions. c. Explain in context the meaning of a point (x,y) on the graph of a proportional relationship, with special attention to the points (0,0) and (1, r) where r is the unit rate. [Grade 7, 2] Unpacked Content Evidence Of Student Attainment: Students: Decide whether a relationship between two quantities is proportional. Recognize that not all relationships are proportional. Use equivalent ratios in a table or a coordinate graph to demonstrate a proportional relationship. Identify the constant of proportionality when a proportional relationship exists between two quantities. Interpret a variety of models (tables, graphs, equations, diagrams and verbal descriptions) to identify the constant of proportionality. Explain the meaning of a point (x, y) in the context of a real-world problem. Example: If a boy charges $6 per hour to mow lawns, this relationship can be graphed on the coordinate plane. The point (1, 6) contains the unit rate or constant of proportionality, 6. Teacher Vocabulary: Equivalent ratios proportional Coordinate plane Ratio table Unit rate Constant of proportionality Equation Ordered pair Knowledge: Students know: (2a) how to explain whether a relationship is proportional. (2b) that the constant of proportionality is the same as a unit rate. (2b) where the constant of proportionality can be found in a table, graph, equation or diagram. (2c) that the constant of proportionality or unit rate can be found on a graph of a proportional relationship where the input value or x-coordinate is 1. Skills: Students are able to: (2a) model a proportional relationship using a table of equivalent ratios. Use a coordinate graph to decide whether a relationship is proportional by plotting ordered pairs and observing whether the graph is a straight line through the origin. (2b) translate a written description of a proportional relationship into a table, graph, equation or diagram. Read and interpret these to find the constant of proportionality. (2c) model a proportional relationship using coordinate graphing. Explain the meaning of the point (1, r), where r is the unit rate or constant of proportionality. Understanding: Students understand that: (2a) a proportional relationship requires equivalent ratios between quantities. Students understand how to decide whether two quantities are proportional. (2b) the constant of proportionality is the unit rate. Students are able to identify the constant of proportionality for a proportional relationship and explain its meaning in a real-world context. (2c) the context of a problem can help them interpret a point on a graph of a proportional relationship. Diverse Learning Needs:
Mathematics MA2019 (2019) Grade: 7 Accelerated	36. Explain the relationships among circumference, diameter, area, and radius of a circle to demonstrate understanding of formulas for the area and circumference of a circle. a. Informally derive the formula for area of a circle. b. Solve area and circumference problems in real-world and mathematical situations involving circles. [Grade 7, 20] Unpacked Content Evidence Of Student Attainment: Students: Solve problems with the circumference and area of a circle. Teacher Vocabulary: Diameter Radius Circle Area Circumference π Knowledge: Students know: the ratio of the circumference of a circle and its diameter is always π. The formulas for area and circumference of a circle. Skills: Students are able to: use the formula for area of a circle to solve problems. Use the formula(s) for circumference of a circle to solve problems. Give an informal derivation of the relationship between the circumference and area of a circle. Understanding: Students understand that: area is the number of square units needed to cover a two-dimensional figure. Circumference is the number of linear units needed to surround a circle. The circumference of a circle is related to its diameter (and also its radius). Diverse Learning Needs: