ALEX | Alabama Learning Exchange

Painter Problems

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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:	33035
Title:	Painter Problems
Overview/Annotation:	This lesson will allow students to become familiar with ratios. In this investigative lesson students will compare ratios and determine equivalent ratios. This is an introductory lesson to be used as part of a unit. 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.

Local/National Standards:

Math Practice Standards:

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. 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.

Additional Learning Objective(s):

Preparation Information

Total Duration:	61 to 90 Minutes
Materials and Resources:	Discussion Cards (provided in the attachment section) Ratio Exit Slip (provided in the attachment section) Painter Problems Activity Guide Investigative Activity Rubric Chart paper Math Toolbox which include the following: pencil, paper, graph paper, markers, scissors, glue, calculator, and sticky notes
Technology Resources Needed:	Interactive Whiteboard (Optional) with required software Document camera or projector Access to search engine (individually or whole group)
Background/Preparation:	The teacher must prepare the appropriate number of math tool boxes for the class; several students can use one tool box. The teacher must make the appropriate number of copies of the Painter Problems Activity guide; to promote student collaboration several students may use one guide. The teacher must prepare Ratio Discussion Cards (found in attachments) or ratio models may be created on the interactive whiteboard software or the application of paint. The teacher must make the appropriate number of Ratio Exit Slips (found in attachments); each student will need one Exit Slip. The students must have prior knowledge of fractions and how to develop equivalent fractions.

Procedures/Activities:

1. The teacher will conduct a math discussion on ratios, using the Discussion Cards (found in attachments). The teacher will display Ratio card 1. The teacher will ask students to give the fraction of the red tiles, and ask the students "What does a fraction tell us?" The students will give feedback on the significance of the numerator and denominator. The teacher will then introduce “ratio” as a math word. The teacher will identify a ratio as a number that compares two quantities and provide the three ways to write a ratio (a to b; a/b; a:b). The teacher will ask, “What is the ratio of red tiles to white tiles?” “White to red?” “Red to the total amount?” The teacher will continue this discussion with the remaining cards. The teacher will ask the students the main difference between a fraction and a ratio.

2. Once the discussion subsides, the teacher will allow students time to search "Ratios in Advertisements." Students will discuss the different ratios they view on the Web (an example: 2 out of 3 people choose us). If every student cannot individually search, students may work in groups or teacher can lead a whole group search.

3. The teacher will transition students into the investigative activity, Painter Problems. To build background knowledge the teacher will explain how paint is mixed at the local hardware store. "Using a white base, workers must provide the appropriate drops of dye to get the desired color. Today there are computers for this, but many times computers fail." To introduce the activity the teacher will tell the students that they have a summer job at a paint store where the computer does not work. Using the ratios provided, they must fulfill the orders for the customers.

4. The students will begin the investigation. Students may work individually or collaboratively.

5. Once adequate time (30-45 minutes) is given, the students will share their finding 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 do you know that ratio is equivalent to the first ratio?" "How did you know to do _______?" "Did someone do this differently or find a different answer?"

6. Toward the end of class, the teacher will distribute the Exit Slip (found in attachments).

Assessment

Assessment Strategies

Formal Formative Assessment: Ratio Exit Slip (found in attachments)

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 Painter Problem Activity Sheet (found in attachments).
Intervention:	Because this is part of a unit, teacher may develop small groups based on the Ratio Exit Slip or informal questioning as part of the investigative activity.

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.

ALEX Lesson Plan

Painter Problems