7th Grade Math : Unit #4 - Proportional Relationships : Lesson #10

Solving Multi-Step problems giving part to whole ratios

Objective: SWBAT solve multi-step problems given part to whole ratios.
Standards: 7.RP.A.2 7.RP.A.3 7.EE.B.3 7.EE.B.4 MP1 MP2 MP3 MP4 MP6
Subject(s): Math
60 minutes
1 Teacher Guided Pre-Lesson Notes - 0 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.”  

MP 1, the one mathematical practice you will read a lot about!
Routines and Procedures

A must share MP 1 victory.  This reflection is added after a reteaching of this lesson to students who struggled.  With this particular student, she stayed for our after school tutoring program with me and it pays off.

As stated at the bottom of this section, if you are truly allowing students to practice MP 1 properly, they will hate you! (SMILE) I know in time my precious little ones will appreciate the time they are given each day to try and solve, figure out, manipulate, and think on their own with the goal of solving problems using prior knowledge and critical thinking with little help from me.  At this point, my students are up in arms. 

I am slowly but surely having few "AH HAAA" moments.  One student struggle with the same objective over, and over, and over.  I kept sending her back with similar but different problems within the same objective.  We had one on one sessions, she could not figure out what the heck her difficulty was.  She could understand what to do during our one on one sessions, but when left on her own she kept making mistakes.  As we identified her mistakes, and began to log the mistakes, she used the log to help her not repeat that particular mistake, she did this until she finally got it.  I did nothing but go over her mistakes and made the suggestion for her to log what she was doing incorrect and for her to then go back and figure out what she should do next.  When she got it, it was like magic. She felt so good.  She cried after the fourth visit, thinking she would never get it. But once understanding set in, she was overjoyed.  The reward was much sweeter because of the work she put in to get there.

2 Bell Ringer - 20 minutes

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. 

Student reaction with mathematical practice 1

My students are new to the full implementation of the mathematical practices embedded in the Common Core.  Our students have had a lot of practice with having mathematical discussion with their peers.  MP 3 is one of the strongest of the 8 practices with our students.  The toughest of the 8 practices for my students is MP 1.  My students crave help when they are stuck.  Finding starting points is their toughest struggle.  My students tend to read a word problem once and automatically give up.  I have had many students shut down completely when I tell them to keep thinking, or continue to struggle through, or I am unavailable for 10 minutes.  I see students sit and stare off, or moan, or even throw their pencils.  The best reactions that I find are those who moan, and throw their pencils.  This is a sign that they are truly trying.  Those who stare off worry me.  These are the students who simply just give up. For these students I tell them to begin working.  Stop staring.  The usual reaction is "I don't get it."  "You won't help me." When I tell them to continue thinking they respond with, "But I don't get it, so I am not doing it."  At this time is when my super teacher cape is put on and I give them consistent motivation.  I give some examples in my teacher guided note section of this lesson.  

With this particular lesson, I had one female student who refused to continue to think through. She was so extremely frustrated that she threw her pencil on to the desk, placed her hand on her face, crossed her legs, and rolled her eyes over and over.  Yes, honestly this provoked frustration in me.  My first reaction was to tell her she had no choice but to continue to work, or she would get no credit for today's lesson.  After 5 minutes went by and she still sat and did no work at all, I called her to have a one on one with me.  The learning lesson for me, was that I can not expect struggling learners to jump into these mathematical practices with no transition at all. Students will not gain from these practices without having a guided hand into full implementation.  After asking her guided questions, pointing out information from the problem that she knew, she gained a starting point.  We worked through problem 1 together enough for her to begin to work on her own.  During pair up time she felt she had enough to share with her peers, and the remaining information she needed to solve the problem was gained through the discussion with her peer group. 

3 Whole Group Discussion - 10 minutes

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.  

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

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.  

The importance of teacher engagement during Individual Think Time
Shared Expectations

It is so important that when you are giving your students the opportunity to practice MP 1 and 2 that you are monitoring student thinking.  It is easy to get caught up in, ok they are working on a task for 10 to 20 minutes silently, let me get some odds and ends accomplished.  In oder for MP 1 to be highly effective you must immerse yourself into their thinking by asking opened ended questions.  "Why did you do that?"  "How will that help you meet the goal of the task?"  "What is the task asking you to do?"  "What will be your first step?"  etc.  This will allow you to begin to see trends within the thought process of your individual students.  You will get the true insight and inner workings of the child's brain.  It is fascinating what you can do with the data you collect from the engagement with them during critical thinking moments.  You will start to see who your strong learners are vs. your lower level thinkers.  Not only will you be able to identify these students to group them appropriately, you will be able to understand why certain students answer the way that they do, what they are lacking, and what to do to build mastery.  Please seek out the open ended example questions from my strategy folder, for those of you who may need a starter kit.  Happy Engaging!

4 Closing - 5 minutes

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                     

They must know what is right, and what is wrong!
Connection to Prior Knowledge

Students really appreciated the use of illustrations for this particular problem.  Some comments made were. "Oh man, I get it." " I can see why now!", "Ahhhh this is easy now that you showed us that."  The importance of showing students the correct answer and a way to get there is crucial.  For some students who already understand, this may validate their thinking, and show them another strategy to use in addition to what they already are doing, for others it may be a direct instruction moment that allows students to understand a new objective that they did not understand at the start of the lesson.

5 Exit Tickets - 5 minutes

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.  

Exit tickets a great formative assessment over the lesson of the day!
Checks for Understanding

I use exit tickets to help me check for common mistakes, misconceptions, what students understand, why they understand, what they do not understand, why don't they understande, etc.  I am a big stickler over students showing their work.  Students must share their thinking either through explanantion or through showing their work.  I may ask for a combination of both.  When students have multiple choice them must justify their responses with work shown.  There is never an opportunity be it formative assessments, classwork, summative assessments, etc, that students will be able to just circle a response.  When this happens, the student will get no credit even if the response is correct, or they are given back the assignment or assessment and must justify their response, ALWAYS.

6 Homework - 0 minutes

Have students solve problems 2 and 4 for homework. Make sure they show their work. 

To Give Homework, or Not to give? That is the Question.
Checks for Understanding

I will expound on my philosphy of homework at another time.  I know I have said this in several of my lessons, but this lesson has assigned homework, and I want to speak to this assignment.  I am definitely QUALITY over QUANTITY.  When I was a rookie educator, I was definitely the one who assigned 1-30 all, show your work, and make sure it is done or else, typ of teacher.  I began to see a trend in myself, the homework would not be assessed appropriately, the students who did do it, would not get the proper feedback from me to see gains, and it was me being immature in my career.  Now, that was 16 years ago, math has changed drastically.  Could you imagine giving a student 30 questions that encompassed all we need from them.  Boy oh boy.  I choose two questions that are going to show me what the students gained from the lesson of the day, or a combination of recall from previous lessons taught.  I love to use 4 rich assessment questions for the entire lesson, one for the lesson, one for the exit ticket, and two for the homework if it is condusive to do so.  This builds consisitency in the lesson, good lesson flow for the students, and provides an example for the parents to help with the child once home.