targeted content standard(s): student friendly learning...

Grade: 6 Lesson Title: Ratios and Rates Unit 2: Ratio, Rate and Proportions (Lesson 1 of 5) Time Frame: 9-11 Days Essential Question:

When is it useful to be able to relate one quantity to another?

What is the connection between a ratio and a fraction?

Targeted Content Standard(s): Student Friendly Learning Targets 6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. 6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b≠0, and use rate language in the context of a ratio relationship. 6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagram, double number line diagrams and equations.

I can…

Represent ratios and unit rates with diagrams.

Represent ratios and unit rates with expressions and/or equations.

Identify which situations are ratios and which situations are not.

Identify which situations are unit rates and which situations are not.

Use ratio and rate reasoning to represent and solve real-world problems.

Targeted Mathematical Practice(s): 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 an express regularity in repeated reasoning

Supporting Content Standard(s): (optional)

Purpose of the Lesson: Students understand how to represent real world situations involving ratios and rates using equivalent fractions, tape diagrams, double number line diagrams or tables.

Explanation of Rigor: (Fill in those that are appropriate.) Conceptual: Students are developing the concepts of ratio and rate by attending to the language of the real world situations and representing the situations with diagrams and algebraic symbols. (6.RP.1, 6.RP. 2)

Procedural: Students will use the procedures of translating situations into diagrams and/or algebraic expressions or equations. (6.RP.1, 6.RP.2)

Application: Ratio and Proportions will be used in the context of real world problems to build mathematical models. (6.RP 3)

Vocabulary: ratio rate unit rate per

Evidence of Learning (Assessment):

Pre-Assessment: Unit 2 Vocabulary Pre and Post Assessment, Equivalence & Scaling Pre Assessment (Segment 1)

Formative Assessment(s): Matching Ratios (Segment 2), Exit Slip RP.1

Summative Assessment: Lesson 1 Pre and Post Assessment

Self-Assessment: Radar (Part of Lesson 1 Pre and Post Assessment), Skeleton 6.RP.1, 6.RP2, & 6.RP.3 Lesson 1 Assessment




Lesson Procedures: Segment 1 Assessing prior learning

Approximate Time Frame:

Unit 2 Vocabulary Pre and Post assessment (3 min)

Equivalence & Scaling (30 min)

Lesson Format: Whole Group Small Group Independent

Modeled Guided Collaborative Assessment

Resources:

Lesson 1 Pre and Post Assessment

Equivalence & Scaling

Focus:

The focus of this segment is to assess students’ prior knowledge of vocabulary and prior learning of equivalent fractions and scaling.

Modalities Represented: Concrete/Manipulative Picture/Graph Table/Chart Symbolic Oral/Written Language Real-Life Situation

Math Practice Look For(s): MP1: Look for students to make sense of the problem in their own way. MP 6: Look for students to pay attention to the precise language of “times as much” to guide their model development. MP7: Look for students to notice that multiple structures can be represented in a problem. (For example: multiplicative comparison and putting together.)

MP4: Look for how accurately students’ models reflect the multiple structures of the problem.

Differentiation for Remediation:

Differentiation for English Language Learners:

Real context problems will be rephrased as needed.

Differentiation for Enrichment:

Potential Pitfall(s):

Independent Practice (Homework):

Steps:

1. Give students the Equivalence & Scaling Pre-Assessment. This is an assessment of students’ understanding of concepts that are to be taught at previous grade levels. (Grades 4 and 5) 4.NF.1 and 5.NF.5 are foundational expectations for 6.RP.1 and 6.RP.2 standards on ratios and proportions.

2. Give students the Unit 2 Vocabulary Pre and Post Assessment. Students rank their understanding of the vocabulary terms on a scale of 1 to 5 by writing a point on each spoke of the radar that corresponds to their level of understanding of each term. They then connect the dots to show their current “web of understanding.”

Teacher Notes/Reflections:




Segment 2 Introducing the concept of a ratio and using ratio language


90 minutes (2 days)



Resources:

Unifix cubes, 2 color counters

Matching Ratios

Matching Ratios Observation Checklist

Focus:

Introduce diagrams to represent real-world situations.


Math Practice Look For(s): MP1: Look for students to make sense of the problem in their own way. Note who chooses to use manipulatives, diagrams, equations, table, etc. MP 6: Look for students to pay attention to the precise language of “times as much” to guide their model development. MP7: Look for students to notice that multiple structures can be represented in a problem. (For example: multiplicative comparison and putting together.)

MP4: Look for how accurately students’ models reflect the multiple structures of the problem.


Teacher would work in small groups, using visual models to represent ratios.


Students will be paired with an English-speaking study buddy. Real context problems will be reread and rephrased as needed. Visual pictures will be used to depict the context of the story.


Students create their own matching ratio cards.

Problem of the Month: First Rate Level E

http://insidemathematics.org/problems-of-the-month/pom-firstrate.pdf


Some students may not be familiar with using visual representations to represent and solve word problems.


Find a real world example of a ratio either in the news or in a newspaper and describe the relationship between the two quantities.

Steps: (Concept of ratio discussion) 1. Pose the following problem: Dan, Dave and Steve collect baseball cards. Dan has twice as many cards as Dave and four times as many cards as Steve. If Dave has 12 cards, how many cards do the boys have altogether? How might you represent this problem with Unifix cubes? Build it with blocks then draw it on a piece of paper and







label numbers. Select students to share their representations. Explain that the representation of the Unifix cubes can be drawn to represent a tape diagram. Ask what language in the problem caused them to create their representations.

2. Pose the next question: The ratio of the number of men to the number of women in a bus is 3:1. The ratio of the number of women to the number of children is 3:5. If there are 20 children, how many people are there altogether? Select students to share their representations. Ask what language in the problem caused them to create their representations. Have students model the problem with Unifix cubes while working in pairs. After they have created a model with Unifix cubes have students record it as a tape diagram to represent the problem. If students have not created tape diagrams to represent the ratio relationship, model how to draw a tape diagram for this problem. If they have, discuss the representation of the tape diagram.


3. Matching Ratios Students work in pairs using the Matching Ratios cards to match equivalent representations. Alternatively, students could play memory with the matching cards. Teachers can use the Matching Ratios Observational Checklist to formatively assess students’ learning of the concept of ratios.


4. Possible exit slip:

What do ratios represent? How are they used?




Segment 3 Use ratios to solve problems


90 min (2 days)



Resources: Thinking Blocks website

http://www.mathplayground.com/thinkingblocks.html

Ice Cream Parlor Ratios

Lesson 1 Exit Slips 6.RP.1 & 6.RP2

Focus:

Given a situation, use ratios to solve real-world problems. Given a model of a ratio create a situation to represent the real-world.


Math Practice Look For(s): MP6: Attend to precision. Students will identify ratio relationships referenced in this poem.

MP3: Students will construct arguments about the appropriateness of the comparisons.

MP2: Students understand the relationship between two quantities in order to express them mathematically. They use ratio and rate notation as well as visual models and contexts to demonstrate reasoning.


Teacher would work in small groups, using visual models to represent ratios and prompting students for possible context. Teacher would work in small groups, using models such as cuisenaire rods along with the computer program.


Students will be paired with an English-speaking study buddy. Real context problems will be rephrased as needed. Students may use visual pictures to depict the context of the real world situation.



Some students may have difficulty creating a context for a visual representation.

Independent Practice (Homework)

Steps: 1. Introduce the Thinking Blocks Website for modeling

ratio problems.










2. Allow students use of this site for practice. Students may work independently or in partners.



3. Ice Cream Parlor Ratios: Have students work in partners on the Ice Cream Parlor ration problem. Once students have completed the task, have them share their responses with the whole group and discuss their thought processes for their responses.


4. Give students a ratio (3:2) and have them describe a real-world context for that ratio.

Have students share their contexts. Facilitate discussion around the contexts created by students. Ask students to comment on the contexts, pointing out the ratio language used.


5. Debrief the activity by discussing the difficulty levels of their problems and strategies used to solve them.

Exit slip :

What precise mathematical language was used to create

or solve problems involving these ratios?

Examples:

“Times as much,” “Ratio of 3 to 2,” “Each”


Segment 4 Reasonable Ratios


30-40 minutes



Resources: One Inch Tall – By Shel Silverstein “One inch” ratio relationships

Focus: Students will identify which situations are realistic ratios and which situations are not.







Math Practice Look For(s):

MP1. Students understand the problem context in order to translate them into ratios/rates.

MP2. Students understand the relationship between two quantities in order to express them mathematically. They use ratio and rate notation as well as visual models and contexts to demonstrate reasoning.


Actual visual models of an inchworm, an inch, something that is the actual size of an ant.


Visual pictures will be used to depict the context of the story.


Students will change unreasonable situations to reasonable ratios or create an alternative poem.


Students may struggle with the idea of scaling. Students may benefit from visual models to interpret scale.


Steps: 1. Read One Inch Tall by Shel Silverstein Have students work in small groups to discuss the ratio relationships in this poem. Students’ record which examples are appropriate estimates and which are unreasonable. For example: If you were only one inch tall, you'd ride a worm to school. This could be reasonable size-wise (although it would depend upon the worm). The teardrop of a crying ant would be your swimming pool. Would not be reasonable because the crying ant is much smaller than 1 inch. So, it would not produce enough tears to fill your pool.


2. As an extension, some students can research the actual sizes for some of the ratio relationships described in the poem to represent the ratio relationships mathematically.





Segment 5 Tape Diagrams and Double Number Lines


30-40 minutes



Resources:

4 6.RP.3 tape diagrams and double number line

5 6.RP.3 tape diagrams and double number line

Focus:

Students are able to take their knowledge of ratios and create tape diagrams and double number lines to solve problems.

Modalities Represented: Concrete/Manipulative

Picture/Graph Table/Chart Symbolic Oral/Written Language

Real-Life Situation


MP1: Students understand the problem context in order to translate them into ratios/rates.


Teacher would work in small groups, using unifix cubes to represent ratios. The students would then draw a representation of the unifix cubes.




Students create a matching diagrams game. The students would create a verbal description and then construct a tape diagram and a double number line for the situation.


Tape diagrams are best used when the two quantities have the same units. Students may try to use the tape diagram even though in the context of the problem, the quantities have different units. Students may not realize that a double number line would work best in this case.


Steps:

Have students work in partners on the Tape Diagrams and Double Number Line problems. Once students have completed the task, have them share their responses with the whole group and discuss their thought processes for their responses.

Note: Analyze the results of the Equivalence & Scaling pretest. If the students did not do well, the students may





need more practice using a double number line.




Segment 6 Write ratios from a graph


30-40 minutes



Resources:

Recess Choice

Focus:

Ratio understanding.

Modalities Represented: Concrete/Manipulative

Picture/Graph

Table/Chart Symbolic Oral/Written Language

Real-Life Situation


MP1: Students understand the problem context in order to translate them into ratios/rates.


Teacher would work in small groups and ask the students specific questions about the scale of the graph, what do the labels mean, etc.




Students can research the actual ratios for some of the relationships described in the graph. They would then create their own graph to represent these ratios.


Students will need to be able to read and interpret the graph appropriately.


Steps:

1. Students working in pairs will look at a graph and create ratios according to that graph. Discuss the ways in which they can show the ratio as a whole group as they finish. Students will also show ratios using tape diagrams as well.


2. Teacher puts on soccer, Frisbee, flag football and basketball on the board.

a. Students go up and put a tally mark by their favorite activity at recess.

b. Students create a bar graph of the class





Segment 7 Rates, Ratios and Unit rate


150-200 minutes (3-4 days)



Resources:

Exploring Unit Rate

Origami Yoda

Fastest Texter in the World

Focus:

Students will be introduced to the concept of unit rate.



MP1: Look for students to make sense of the problem in

their own way. Students understand the problem context in

order to translate them into ratios/rates.

MP7: Look for students to notice the structure in the table. For example the number of seconds are multiples of 10.


The students might need to use unifix cubes in the Exploring Unit Rate to demonstrate the increase in price for each snack.

Teacher would work in small groups and ask the students specific questions about interpreting the meanings of unit rate such as miles per hour, heartbeats per minute, rate of pay, etc.


Students will be paired with an English-speaking study buddy. Real context problems will be rephrased as needed.


Students actually find their own unit rate. The students research the number of words that a teenager text in a year. The students then calculate how much time they will spend texting if they use their unit rate.


Students may not realize that unit rates such miles per hour, mph, is actually the number of miles that you travel in one hour.


data. c. Students then have to answer the same

questions about the class data from the Recess choice document.




1. Exploring Unit Rate

The students should be put in groups of 2 or 3 to discuss the problem. After reading the problem posed, the students should first examine the grocery store price labels to come up with a definition for unit rate.

The class should then come together to discuss the definition.

The students should then work together answer Kendra’s and Isabel’s questions on the sheet.

The students should then present their conclusions to the rest of the class and as a class determine who is correct.


2. Brainstorm with students, examples of rates they have encountered (heart rate, mph, pay rate, etc.). Write them on the board and highlight the units. Refer to these as they encounter examples in future problems to distinguish between unit rates and other ratios.


3. Pose the following problem: Sam reads 30 pages in 10 minutes. How many pages can he read in 1 minute? Identify the ratio language in this problem. How does the language of this problem compare with the previous examples?


4. Put the following on the board: Ratio Rate Unit Rate Ask the students, “How do the definitions compare?”. Students should discuss their answers with a peer and then discuss as a class.


5. Origami Yoda

Ask students what they know about origami.

Introduce the problem of the origami Yoda.

Show the students the video.

Have students work on the problem filling in the table and answering questions 1-4.

Discuss answers as a class.


6. Fastest Texter in the World Teacher Notes/Reflections:




Segment 8 Lesson Assessment


50 min



Resources:

Lesson 1 Problem Solving with Ratios and Rates

Focus:

Problem Solving with real-world problems.



MP1: Look for students to make sense of the problem in

their own way.

MP2. Students understand the relationship between two

quantities in order to express them mathematically. They

use ratio and rate notation as well as visual models and

contexts to demonstrate reasoning.

MP7: Look for students to notice the structure in the table. Students should discuss how the structure of tables relates to the contexts of the situations.



Real context problems will be read and rephrased as needed.




Steps:

1. Give students the Post-Assessment.


2. Students will use the Skeleton to self reflect on their learning.


targeted content standard(s): student friendly learning...

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