Learning Resource Type

Lesson Plan

Painter Problems

Subject Area





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.

    Mathematics (2019) Grade(s): 6


    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



    • Ratio
    • Ratio Language
    • Part-to-Part
    • Part-to-Whole
    • Attributes
    • Quantity
    • Measures
    • Fraction


    Students know:
    • Characteristics of additive situations.
    • Characteristics of multiplicative situations


    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).


    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.
    Mathematics (2019) Grade(s): 6


    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



    • 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


    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.


    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.


    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.

    Primary Learning Objectives

    I CAN identify and develop ratios in real world situations.

    I CAN identify equivalent ratios.



    1. The teacher will conduct a math discussion on ratios, using the Discussion Cards. 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. 

    Assessment Strategies

    Formal Formative Assessment: Ratio Exit Slip 

    Formal Assessment: Using the Investigative Activity Rubric, the 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.



    The investigation has an included extension on the Painter Problem Activity Sheet.


    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.

    Approximate Duration

    Total Duration

    61 to 90 Minutes

    Background and 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 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; each student will need one Exit Slip.

    The students must have prior knowledge of fractions and how to develop equivalent fractions. 

    Materials and Resources

    Materials and Resources

    Discussion Cards 

    Ratio Exit Slip 

    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)