ALEX Lesson Plan

     

How Big Should it Be?

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  This lesson provided by:  
Author:Tim McKenzie
Organization:UAB/UABTeach
The event this resource created for:CCRS
  General Lesson Information  
Lesson Plan ID: 33091

Title:

How Big Should it Be?

Overview/Annotation:

This lesson will allow students to become familiar with the concept of equivalent ratios and similar objects. Through an open investigation, students will develop methods to find equivalent ratios. This is a lesson to be used as part of a unit with Painter Problems and How Far Can You Leap found in ALEX.

This is a College- and Career-Ready Standards showcase lesson plan.

 Associated Standards and Objectives 
Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 6
1. Use appropriate notations [a/b, a to b, a:b] to represent a proportional relationship between quantities and use ratio language to describe the relationship between quantities.
Unpacked Content
Evidence Of Student Attainment:
Students: Given contextual or mathematical situations involving multiplicative comparisons.
  • Communicate the relationship of two or more quantities using ratio language.
Teacher Vocabulary:
  • Ratio
  • Ratio Language
  • Part-to-Part
  • Part-to-Whole
  • Attributes
  • Quantity
  • Measures
  • Fraction
Knowledge:
Students know:
  • Characteristics of additive situations.
  • Characteristics of multiplicative situations
Skills:
Students are able to:
  • Compare and contrast additive vs. multiplicative contextual situations.
  • Identify all ratios and describe them using "For every…, there are…"
  • Identify a ratio as a part-to-part or a part-to whole comparison.
  • Represent multiplicative comparisons in ratio notation and language (e.g., using words such as "out of" or "to" before using the symbolic notation of the colon and then the fraction bar. for example, 3 out of 7, 3 to 5, 6:7 and then 4/5).
Understanding:
Students understand that:
  • In a multiplicative comparison situation one quantity changes at a constant rate with respect to a second related quantity. -Each ratio when expressed in forms: ie 10/5, 10:5 and/or 10 to 5 can be simplified to equivalent ratios, -Explain the relationships and differences between fractions and ratios.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.6.1.1: Define quantity, fraction, and ratio.
M.6.1.2: Identify the units or quantities being compared.
Example: Read 2/3 as 2 out of 3.
M.6.1.3: Write a ratio in appropriate notation;[a/b, a to b, a:b].
M.6.1.4: Draw a model of a given ratio or fraction.
M.6.1.5: Identify the numerator and denominator of a fraction.

Prior Knowledge Skills:
  • Compare two fractions with the same numerator or the same denominator by reasoning about their size.
  • Addition and subtraction of fractions as joining and separating parts referring to the same whole.
  • Label numerator, denominator, and fraction bar.
  • Recognize fraction 1 as the quantity formed by 1 part when a whole is partitioned into b equal parts.

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: 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
17. Solve problems involving scale drawings of geometric figures, including computation of actual lengths and areas from a scale drawing and reproduction of a scale drawing at a different scale.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Solve problems involving scale drawings.
  • Use a scale factor to reproduce a scale drawing at a different scale.
  • Determine the scale factor for a scale drawing.
Teacher Vocabulary:
  • Scale drawing
  • Reproduction
  • Scale factor
Knowledge:
Students know:
  • how to calculate actual measures such as area and perimeter from a scale drawing.
  • Scale factor impacts the length of line segments, but it does not change the angle measurements.
  • There is a proportional relationship between the corresponding sides of similar figures.
  • A proportion can be set up using the appropriate corresponding side lengths of two similar figures.
  • If a side length is unknown, a proportion can be solved to determine the measure of it.
Skills:
Students are able to:
  • find missing lengths on a scale drawing.
  • Use scale factors to compute actual lengths, perimeters, and areas in scale drawings.
  • Use a scale factor to reproduce a scale drawing at a different scale.
Understanding:
Students understand that:
  • scale factor can enlarge or reduce the size of a figure.
  • Scale drawings are proportional relationships.
  • Applying a scale factor less than one will shrink a figure.
  • Applying a scale factors greater than one will enlarge a figure.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.7.17.1: Define scale, scale drawings, length, area, and geometric figures.
M.7.17.2: Locate/use scale on a map.
M.7.17.3: Identify proportional relationships.
M.7.17.4: Recognize numeric patterns.
M.7.17.5: Recall how to solve proportions.

Prior Knowledge Skills:
  • Construct repeating and growing patterns with a variety of representations.
  • Continue an existing pattern.
  • Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.
  • Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories.
  • Define unit rate, proportion, and rate.
  • Create a ratio or proportion from a given word problem.

Local/National Standards:

Math Practice Standards:

  • Make sense of problems and persevere in solving them.
  • Reason abstractly and quantitatively.
  • Construct viable arguments and critique the reasoning of others.
  • Model with mathematics.
  • Use appropriate tools strategically.
  • Attend to precision.
  • Look for and make use of structure.
  • Look for and express regularity in repeated reasoning.

Primary Learning Objective(s):

I CAN identify and develop ratios in real world situations.

I CAN identify equivalent ratios.

I CAN compare ratios in real world situations.

Additional Learning Objective(s):

 
 Preparation Information 

Total Duration:

61 to 90 Minutes

Materials and Resources:

  • How Big Should It Be Activity Guide (found in attachments)
  • Investigative Activity Rubric (found in attachments)
  • Ratio 2 Exit Slip (found in attachments)
  • Chart paper
  • Yard stick
  • Math Toolbox which includes the following: pencil, paper, graph paper, markers, scissors, glue, calculator, and sticky notes

Technology Resources Needed:

Interactive Whiteboard (Optional) with required software, Document camera, projector, laptop or computer capable of showing videos (TV and DVD player can be used if the DVD is available). 

Background/Preparation:

  • The teacher must make the appropriate number of copies of the How Big Should It Be activity guide (found in attachments). Copies should be made so that students can work collaboratively. 
  • The teacher must make the appropriate number of copies of the Ratio 2 Exit Slip (found in attachments). Each student should have one.
  • Teacher must prepare the appropriate number of Math Toolboxes.
  • Teacher must download the video at the link below.  Zoolander Center for Ants (If YouTube is not available through your school system, you may download the video using www.keepvid.com)
  • The students must have knowledge of ratios.

 

  Procedures/Activities: 
  1. Teacher will show the students the video clip of Zoolander (video shows an unintelligent supermodel angered at the reveal of a model of a building, he claims the building is for ants and has to be at least three times as big). The teacher will ask the students, "Why did Zoolander get so upset at Mugato?" Ideal response, "He thought the model of the building was the actual building." The teacher will ask, "What do you think about his statement 'The building has to be at least three times that big'?" Ideal response, "three times as big would still not be big enough for anyone to fit in." 
  2. The teacher will introduce the idea of a scale factor as "a ratio used to enlarge or shrink any shape or object." 
  3. The teacher will present the How Big Should It Be activity (found in attachments). The teacher may have to demonstrate how to measure.
  4. Students will begin the investigative activity. They will be producing a poster with a shape 5 times the original size and 10 times the original size. The teacher will act as a facilitator and coach throughout the investigation. The teacher should address misconceptions and drive inquiry related to ratios. 
  5. Once adequate time (30-45 minutes) is given, the students will share their findings on the document camera. (If a document camera is not available, students may present their work in the front of the class, this is where the students would need chart paper). As the students are sharing, the teacher is acting as the facilitator and coach asking questions that drive ratio understanding. "How did you know to multiply by _____?" "How did you know to do _______?" "Did someone do this differently?" 
  6. Students will complete the Ratio 2 Exit Slip.


Attachments:
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  Assessment  

Assessment Strategies

Formal formative assessment: Ratio 2 Exit Slip

Formal Assessment: Using the Investigative Activity Rubric (found in attachments) teacher will evaluate students' work.

Informal Formative Assessment: As the students are working, the teacher will act as the facilitator and coach. Teacher will ask questions to evaluate students (i.e. How do you know ______? What did you do to get that?) Teacher may pull small groups during investigation on a needs basis.

Acceleration:

The investigation has an included extension on the How Big Should It Be Activity Sheet (found in attachment)

Because this is part of a unit, teacher may develop small groups based on the Ratio 2 Exit Slip or informal questioning as part of the investigative activity.

 

Intervention:

Struggling students should be grouped with a peer tutor and teacher should pay close attention to those groups to assure complete understanding.


View the Special Education resources for instructional guidance in providing modifications and adaptations for students with significant cognitive disabilities who qualify for the Alabama Alternate Assessment.