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Essential Questions: What are the different kinds of ratio relationships…
Essential Questions: What are the different kinds of ratio relationships that we can see between two numbers? How do ratio relationships show up in the real-world?
RELEVANCE
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A father and his young toddler walk along a sidewalk. For every 3 steps the father takes, the son takes 5 to keep up. What is the ratio of the number of steps that the father takes to the number of steps that his son takes? DOK2 / Think Pair Share (engageny.org)
Work with your group to make a list of ways you see ratios in daily life or that you know about in the world DOK3/Group Task (Meilin)
Lemonade Experiment - test out different ratios of lemon juice/sugar vs. water to figure out the best-tasting recipe DOK4 / Performance Task (Meilin)
CCSS.MATH.CONTENT.6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
CCSS.MATH.CONTENT.6.RP.A.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. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."
CONNECTIONS
Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? CCSS.MATH.CONTENT.6.RP.A.3.C
CCSS.MATH.CONTENT.6.RP.A.3.B Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
When one flower is three times taller than another flower, can you write the relationship between the two flower heights as a ratio? DOK1 (Meilin) Thin-Pair-Share
Is saying we have five pennies the same or different from saying we have five cents of a dollar? DOK3 (Meilin) Small Group
A sixth-grade math teacher can grade 25 homework assignments in 20 minutes. Can you divide up a pie, showing how many 20 minute segments are in one hour? Then, can you write in how many homework assignments he can complete in the whole hour? DOK2 (Meilin) Performance Task
Is it important to understand ratios when thinking about fairness in society? If we say, one person has one thousand times more money in his pocket than another, how does that affect not only how we look at math but other subjects such as history and economics? What about music? When you make art, is it interesting to draw things that are different sizes when you compare them to one another? DOK4 (Meilin) Small Group Discussion followed by full class discussion
CCSS.MATH.CONTENT.6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."
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PERSPECTIVE
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What does it mean to believe something, versus proving something? (Meilin) Small Group discussion followed by full class
SUPPOSITION
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Here is a theorem: If 𝑨: 𝑩 with 𝑩 ≠ 𝟎 and 𝑪: 𝑫 with 𝑫 ≠ 𝟎 are equivalent, then they have the same value: 𝑨 𝑩 = 𝑪 𝑫 . This is essentially stating that if two ratios are equivalent, then their values are the same (when they have values).
Can you provide any counterexamples to the theorem above? .
Observe the progress of students and question students’ counterexamples. Ask for further clarification or proof that the two ratios are equivalent but do not have the same value. (engageny.org) Think-Pair-Share followed by full class discussion
What if someone did not believe that 1/2 is the same value as 50/100? What would happen to our abilities to manage big numbers? (Meilin) Think-Pair-Share followed by full class discussion
what happens if we did not have the concept of ratios - what would we MISS? (Meilin) Small Groups (and my observations) followed by full-class discussion
EVIDENCE
CCSS.MATH.CONTENT.6.RP.A.3.A
Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
On the news broadcast, a chef from a local Italian restaurant demonstrated how he makes fresh pasta daily for his restaurant. The recipe for his pasta is below: 𝟑 eggs, beaten 𝟏 teaspoon salt 𝟐 cups all-purpose flour 𝟐 tablespoons water 𝟐 tablespoons vegetable oil. Determine the ratio of the number of tablespoons of water to the number of eggs. DOK1 (engageny.org) Think-Pair-Share
Using this relationship, predict how many views the website will have 𝟒, 𝟓, and 𝟔 hours after the end of the broadcast. DOK3 (engageny.org) Performance Task
If, in addition to the story about pets, the station ran stories about basketball and grocery stores on other days, and then checked the website two hours after each story to see how many views the website had, how would this help them figure out what kinds of stories their audience liked best? DOK4 (Meilin) Think-Pair-Share
At the end of this morning’s news segment, the local television station highlighted area pets that need to be adopted. The station posted a specific website on the screen for viewers to find more information on the pets shown and the adoption process. The station producer checked the website two hours after the end of the broadcast and saw that the website had 𝟐𝟒 views. One hour after that, the website had 𝟑𝟔 views. Create a table to determine how many views the website probably had one hour after the end of the broadcast based on how many views it had two and three hours after the end of the broadcast. DOK2 (engageny.org) Small Group
You created a new playlist, and 100 of your friends listened to it and shared if they liked the new playlist or not. Nadhii said the ratio of the number of people who liked the playlist to the number of people who did not like the playlist is 75: 25. Dylan said that for every three people who liked the playlist, one person did not. Do Nadhii and Dylan agree? Prove your answer using the values of the ratios. DOK2 (engageny.org) Performance Task
FA Moves
While students are working, circulate and ask students to share their solving strategies. It is important to also ask students to prove their claims. If a student has simply written that one beverage has a higher amount of water per juice than the other, ask the student to prove or explain how the answer was determined. Students share how they have compared the values in the table. (engageny.org, math-g6-m1-topic-b-lesson-9) Performance Task
Students create the three ratio tables on the student pages. Have index cards ready that say Table 1, Table 2, and Table 3 to hand out to students so that students can place the assigned table on chart paper (15 minutes). After the charts are created, have students focus on how they created the tables and discuss the structure of the tables with a partner or small group. After students have had a chance to work, pull the class together as a whole group for a discussion about the structure of the tables and how the tables are related. Use the questions below to guide the discussion. (engageny.org, math-g6-m1-topic-b-lesson-10) Think-Pair-Share followed by full class discussion
In pairs or small groups, students complete the following problems. After students are given time to work, have groups explain their answers The ratio of the number of people who own a smartphone to the number of people who own a flip phone is 𝟒: 𝟑. If 𝟓𝟎𝟎 more people own a smartphone than a flip phone, how many people own each type of phone? (engageny.org, math-g6-m1-topic-a-lesson-5) Think-Pair-Share
ACADEMIC LANGUAGE: unit rate, value of a ratio, ratio table, equivalent ratios, mixture, constant rates