##
* *Reflection: Developing a Conceptual Understanding
Solving Multi-Step problems giving part to whole ratios - Section 3: Whole Group Discussion

As we began to have our whole group discussion, the trend was that students could not understand the relationship between the ratio to the actual amounts. For example, in question 1 the problem stated that 1/3 black cages were sold. It further states that there were 240 black cages sold. The students did not know what to do with this information in order to find the number of silver cages sold. We started with reviewing 1/3 is 1 out of 3. 3 being the whole. If 1 out of 3 were black, that means 2 out of 3 were silver. Once my students were able to use the given ratio in the text to find the missing ratio, they were off for the races. They then understood each cage represents 240. This led to being able to find the number of silver cages. The key here is for students to use the given ratio to find the missing ratio.

*Understanding How to find part to part ratios give light bulb moments!*

*Developing a Conceptual Understanding: Understanding How to find part to part ratios give light bulb moments!*

# Solving Multi-Step problems giving part to whole ratios

Lesson 10 of 10

## Objective: SWBAT solve multi-step problems given part to whole ratios.

## Big Idea: Using MP 1, 2, 3, 4, and 6 students will grapple through and solve rigorous multi-step real world problems given a part to whole ratio.

*40 minutes*

**Pre-Lesson Teacher Guided Notes: **This lesson affords students the opportunity to grapple through word problems that will assess students over their understanding of part to whole relationships. Students will need to have a strong background on identifying ratios. For students who have not been fully immersed in the Common Core, many of them may attempt to use a strategy that involves setting up a proportion and solving for the missing value. This is GREAT! During the bell ringer of this lesson, students will be asked to grapple with the problems on their own for 10 minutes first. This will allow students to truly exercise **MP1**. Learners with strong critical thinking skills will use a variety of strategies. Some students may recognize how to model the problem, some students may understand part to whole relationships and set up and solve equations that will arrive at the final answer. Students with some understanding will be able to recognize the ratio, recognize that there is a total that is represented with the ratio, however not know what to do to solve for the asking total. They will be able to set up a strategy, but not know what to do with the strategy. Students with little to no understanding will not know how to recognize the ratio, and not have a starting point. With these learners you will want to start with identifying the ratio, identifying the numeric value of the order of the ratio, and use the modeling strategy with them. I have found that using illustrations along with equations help students conceptualize what is happening in the problem. For those students who use the strategy of setting up a proportion, do not discourage this. Yes, the common core does ask for us to veer away from this, however this is a great starting point. Using the proportion, you can move into modeling what each part of the proportion represents. This will allow students to practice **MP4**.

Students who are not used to the Common Core, will struggle, give up, and shut down. The majority of my students are having a hard time working through problems, and thinking on their own. It is frustrating for them. They perceive me as a teacher who just simply does not want to help them. I have to consistently refer back to **MP 1.** I empower them by motivating them with constant praise. I use words like, “You are not giving yourself enough credit, you are so much smarter than what you are showing.”, “You do not need me as much as you think you do, I trust in you, all I need you to do is trust in yourself.”

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#### Bell Ringer

*20 min*

Hand students the Bell Ringer as they enter the room. For this bell ringer, students will work on problems, 1 and 3. Problems 2 and 4 will be their homework.

Students will sit in their **Individual Think Time** seats and begin right away using **MP1**, **MP2**, and **MP6** to grapple through two problems. Allow students 10 minutes for **I.T.T. **Students will need to write their thinking strategies in their interactive notebooks. They will use this to share during pair up time. Walk the room to check for understanding.

Once students have worked individually for 10 minutes, have students discuss their work with their **pair up **partners. Students should have 10 minutes to discuss their thinking and compare their responses. Students should be able to guide one another through the process of solving each of these problems. This will put into practice **MP3**.

One common mistake that students may make is to multiply the ratio by the whole number and use the product as the final response. Another common mistake students will make will be to view the ratio as a fraction. It will be important for students to know the difference between fractions and ratios. When walking the room checking for understanding it is important to check that students understand part to whole relationships within a ratio, how to set up the second part to whole ratio from using the given ratio, how to use the total given and which ratio to associate the total to in order to solve for the missing information.

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#### Whole Group Discussion

*10 min*

During this time, students groups should have the opportunity to share out their pair up time discussions, and reveal each of their responses. You may not have time to have each student share. As you filter through the room during pair up time, attempt to identify a group who has understanding, some understanding and little understanding. During the whole group discussion have students debate their responses and defend their thinking. This again will practice **MP 3.** As the facilitator of the discussion, you can head the discussion with open ended questions that will evoke students to defend. For example a student may respond to question 1 with 80. Students may defend their response by stating that they multiplied the ratio by the 240. With this response, ask students, what does this find? What is the problem asking us to find? Does this response answer what the problem is asking us to find? What does the ratio represent? What does the total represent? This will allow students to practice **MP 1** and **MP2**. Students will need to regroup their thinking and attempt to resolve the problem using the new information.

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#### Closing

*5 min*

It is important for true understanding that students are aware of the correct process in solving problems, as well as the correct answer. During the closing take time to go over problem 1. During this time use the modeling method to solve the problem. Show the students how to solve problem 1 then for the exit ticket have students correctly model problem 3. Please see below the directions to solve problem 1 using the modeling method and equations.

**First:** Have students identify that 1 out of 3 cages were black. (not one-third) This totals 240 total black cages. I have my students set up and label the ratios. Have students use any symbol to represent the ratios. Students should recognize the ratio’s numerator represents how many black cage symbols are needed and the remaining out of the whole will be how many silver will be modeled.

Black = 1/3 O = 240

** **

**What do we know? **We know that 1 out of 3 cages total 240 black cages.

**What do we want to know?** We want to know how many cages are silver.

Silver = 2/3 O O = ?

**How do we find out how many cages are silver? **We need to find out the total of each cage. If we use the total number of black cages and divide it by how many black cages are in the ratio, we will find the amount of each cage represented in the ratio.

240/1 = 240

**What does this mean?** Each symbol is worth 240. **How many symbols is represented with silver?** 2

**Finding the amount of silver cages**. 240 + 240 = 480 or 240 x 2= 480 **(MP 7)** ** **

**Final response: 480 silver cages **

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#### Exit Tickets

*5 min*

Have students use a separate sheet of paper to solve problem #3 using the strategy taught during closing. Have students compare their first strategy with the new understanding they gained during the closing.

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##### Similar Lessons

###### Proportional Relationships of Whole Numbers

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- UNIT 1: Introduction to Mathematical Investigations
- UNIT 2: Integers
- UNIT 3: Proportional Reasoning with Percents
- UNIT 4: Proportional Relationships
- UNIT 5: Proportional Reasoning
- UNIT 6: Rational Numbers
- UNIT 7: Number Sense Vocabulary 5 day mini unit
- UNIT 8: Expressions and Equations
- UNIT 9: Expressions and Equations 5 Day Application of Vocabulary Mini Unit

- LESSON 1: Compute the Unit Rate Associated With a Ratio of a Fraction
- LESSON 2: Relate the Graph of a Proportional Relationship to a Table
- LESSON 3: Determine if Two Quantities are in a Proportional Relationship
- LESSON 4: Identify the Constant of Proportionality from a Table
- LESSON 5: Identify the Constant of Proportionality From a Graph
- LESSON 6: Identify the Constant of Proportionality from an Equation
- LESSON 7: Determine the Constant of Proportionality Between the Sides of Right Triangles
- LESSON 8: WP: Identify the Constant of Proportionality
- LESSON 9: Unit Assessment
- LESSON 10: Solving Multi-Step problems giving part to whole ratios