# A Story of Ratios Tape Diagrams TIME ALLOTTED FOR THIS SLIDE:

## Presentation on theme: "A Story of Ratios Tape Diagrams TIME ALLOTTED FOR THIS SLIDE:"— Presentation transcript:

A Story of Ratios Tape Diagrams TIME ALLOTTED FOR THIS SLIDE:
Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X A Story of Ratios Tape Diagrams

Session Objectives Experience how proficiency in the tape diagram method can be developed in students and colleagues. Let’s begin by looking at our new objective. Experience how proficiency in the tape diagram method can be developed in students and colleagues.

Opening Exercise If you have any tape diagramming experience, try to solve this problem using tape diagrams. If not, try to solve it algebraically. 88 children were in swimming camp. One-third of the boys and three-sevenths of the girls wore goggles. If 34 students wore goggles, how many girls wore goggles? If you finish before time is up, re-write the problem as a ratio problem. .

Opening Exercise If you have any tape diagramming experience, try to solve this problem using tape diagrams. If not, try to solve it algebraically. 94 children are in a reading club. One-third of the boys and three-sevenths of the girls prefer fiction. If 36 students prefer fiction, how many girls prefer fiction? If you finish before time is up, re-write the problem as a ratio problem.

Using Tape Diagrams Promote perseverance in reasoning through problems. Develop students’ independence in asking themselves: “Can I draw something?” “What can I label?” “What do I see?” “What can I learn from my drawing?” Before we begin, consider that the use of tape diagrams goes hand in hand with word problems and that our delivery should promote perseverance in problem solving and develop independence in students’ ability to work through problems. We want the process and the questioning to be naturally internalized by students.

Forms of the Tape Diagram
8 5 ? 5 8 ? There are two basic forms of the bar diagram model. The first form is sometimes called the part-whole model; it uses bar segments placed end-to-end. The second form, sometimes called the comparison model, uses two or more bars stacked in rows that are left-justified; in this form the whole is depicted off to the side. We will reflect on the nuances of the two forms when we have finished this section. For now, you can use whichever works best for you with any given problem.

Foundations for Tape Diagrams in PK–1
Sara has 2 apples. Jon has 5 apples. How many apples do they have altogether? How many more apples does Jon have than Sara? In the very early grades, we count out objects, and do comparisons of quantities (e.g., Who has more? Who has fewer? How many more? How many fewer?). It is important that students see groups of objects in many arrangements and learn to instantly recognize quantities up to 5 or 6. But it is equally important that we begin modeling for students the laying out of objects in an organized fashion that previews bar modeling, in both the end-to-end fashion and the comparison fashion. This is especially appropriate when working with word problems of addition and subtraction. As the teacher, model a comparison of two quantities in the manor that makes the comparison easiest to see. It is not recommended to become overly structured in forcing students to model it a certain way every time. The suggestion, ”This time can you lay yours out like I have mine” will serve to build this habit of setting up items in rows to support counting, comparison, and the model of using rectangular bars.

Example 1: Sara has 5 stamps. Mark brings her 4 more stamps

Example 2: Sara has 16 stamps. Mark brings her 4 more stamps
Example 2: Sara has 16 stamps. Mark brings her 4 more stamps. How many stamps does Sara have now? We have many of these problems to do, so while I will demonstrate quite a bit of delivery as I just did, I am not going to go through the entire delivery of every problem. And, in particular we will not be taking time to write out the number sentence and the answer in a complete sentence as shown here. Those steps are very important with students, but to make most efficient use of our time, we will focus on the tape diagram leading to the answer and then move on to the next example. You would work at least one other problem with the class in this same way where individual stamps are represented, perhaps even spend a few days working in this fashion before moving to an example like this next one where you have a quantity large enough that it makes drawing discrete segments inconvenient. Simply suggest using a plain rectangular bar to represent the entire quantity. (EXAMPLE 2) (Continue to demonstrate with a flip chart and marker.) Read the first sentence with me. “Sara has 16 stamps.” I want to draw something but 16 is a lot of boxes; I’m going to just draw this long rectangle and make a note here that this is 16. Is that okay? Can you imagine that there are 16 stamps in this row? (Add the label, “Stamps Sara has.”) Read the next sentence with me. “Mark brings her 4 more stamps.” If this is 16 stamps, can you imagine how long of a rectangle I should make to show 4 more stamps? Can you see it? I’m going to start drawing, and you tell me when to stop. (Begin to draw the second bar slowly waiting for participants to say, “stop.” Add the label, “Stamps Mark brings.”) So this is how we get students to model using the simple, rectangular bar. The approach of imagining the length of the bar, and ‘tell me when to stop’ should be used often until students begin to demonstrate independence in that judgment process. (CLICK TO REVEAL SOLUTION.)

Example 3: Sara brought 4 apples to school
Example 3: Sara brought 4 apples to school. After Mark brings her some more apples, she has 9 apples altogether. How many apples did Mark bring her? Before we solve this problem, let’s review quickly how we transitioned the students into using this bar to begin with. The transition into bar diagrams requires transitioning from sets of actual objects to pictures of objects, to bar-shaped pictures that still depict individual objects, and then to rectangular bars with no distinct markings of individual items. A benefit of using rectangular bars without the markings of individual items is that students can now model non-discrete quantities – like measurements of distance or weight – as well as being able to represent unknown quantities. The next jump in complexity is in moving from a problem where both bar segments represent known quantities and the unknown is the total or the difference, to a problem where the total or difference is known and the bar is representing an unknown. (EXAMPLE 3) Go ahead and try depicting this problem. (Allow a moment for participants to work.) (CLICK TO REVEAL SOLUTION.) How does your depiction compare to this one? Are we all on the same page? (Address any questions or concerns.)

Example 4: Matteo has 5 toy cars. Josiah has 2 more than Matteo
Example 4: Matteo has 5 toy cars. Josiah has 2 more than Matteo. How many toy cars do Matteo and Josiah have altogether? (EXAMPLE 4) Work example 4. (Allow a moment for participants to work.) (CLICK TO REVEAL SOLUTION.) Does your diagram look like this one? What complexities are added here, that were not present in Example 3? (Call on a participant to answer.) So this example required two computations in order to answer the question. This is an example of a two-step problem as called for in the standards beginning in Grade 2.

Example 5: Jasmine had 328 gumballs
Example 5: Jasmine had 328 gumballs. Then, she gave 132 gumballs to her friend. How many gumballs does Jasmine have now? (EXAMPLE 5) Read and work Example 5. (Allow a moment for participants to work.) Did this problem lend itself to a part-whole model or a comparison model? Did anyone present it this way? Is it wrong to present it this way? Is this problem more or less complex that the previous problem? (Allow for group response.) So we have removed the two-step complexity, but we’ve added computational complexity of working with 3-digit numbers. What else added complexity to this problem? (Allow participants to comment. Some may have found it difficult to address being given the whole first, and thereby feeling forced into starting with a part-whole model.) (CLICK TO REVEAL SOLUTION.) Before we move to the next example, let’s take a poll. The question is, “Was the use of the tape diagram model in Example 5 an example of descriptive modeling or analytic modeling?” Raise your hand if you think it is descriptive? Analytic? Have no idea? (Allow for hand-raising and summarize the result.) This is subtle, and there is no clear-cut answer, but here is the key: If the student is using the diagram to reveal to them what operation should be applied, then the model is analytic. If they are using the diagram to simply provide more clarity of visualization, then it is purely descriptive. Let’s move now into some multiplication and division problems. As with addition and subtraction, the ‘compare to’ situations are the ones that benefit most from use of the tape diagram. So, that is where we will begin.

Example 6: Jose has 4 paper clips
Example 6: Jose has 4 paper clips. Harry has twice as many paper clips as Jose. How many paper clips does Harry have? (EXAMPLE 6) Read and work Example 6. (Allow 2 minutes for independent work.) (CLICK TO REVEAL SOLUTION.) With multiplication and division problems, we introduce use of a consistently shaped bar to represent equal parts in the problem. We refer to this quantity as a unit, and then reason through the problem with this language: 1 unit is 4 paper clips, so 2 units would be 8 paper clips. Often times the reasoning applies in a division context. We might see that, “4 units is 28 paperclips, so 1 unit would be 7 paperclips.”

Example 7: Jose has 4 paper clips
Example 7: Jose has 4 paper clips. Harry has twice as many paper clips as Jose. How many paper clips do they have altogether? (EXAMPLE 7) Read problem 7. (Participants do not need to work this problem.) (CLICK TO REVEAL SOLUTION.) How is this problem more complex than the previous? (Expected response – by asking how many paper clips they have altogether, it becomes a two-step problem, requiring you to first calculate how many Harry has, and then combine it with Jose’s to get the total.)

Example 8: William’s weight is 40 kg
Example 8: William’s weight is 40 kg. He is 4 times as heavy as his youngest brother Sean. What is Sean’s weight? (EXAMPLE 8) Read problem 8. Work the problem and share your tape diagram and answer with a partner. (CLICK TO REVEAL SOLUTION.) What mistake are students most likely to make when solving this problem? (Allow someone to share – we are looking for them to say that students might take the information ‘4 times as heavy’ and interpret that as Sean is 4 times as heavy as William, leading to an answer of 160 kg.) This reason right here is a case in point of why teachers want students to internalize a specific habit from the RDW process. Whenever a second quantity is introduced in any of the comparison styles, ask the students, ‘who has more’ or, in this case, “Who weighs more, William or Sean?” That simple reflection should be a standard part of reading a word problem with a comparison. Once internalized, students will be much less likely to make these mistakes of misrepresenting the relationship stated. They will instead have a habit of reflecting on who has more, and when asked directly they are much more likely to make a thoughtful reply, double checking the wording if they are unsure.

Example 9: Jamal has 8 more marbles than Thomas

Example 10: The total weight of a football and 10 tennis balls is 1 kg

Example 11: Two pears and a pineapple cost \$2
Example 11: Two pears and a pineapple cost \$2. Two pears and three pineapples cost \$ Find the cost of a pineapple. (EXAMPLE 11) In the spirit of that thought, try Example 10 on your own. (Allow participants 1-3 minutes to work the problem.) Compare your model with a partner at your table. (Allow participants 1 minute to compare their work.) Who has answer? (Allow for 1 or more people to answer.) Is he/she right? Did anyone get something different? (If there is any difference of opinion, allow 2 participants with different answers to draw their solutions on flip charts. Allow each participant a chance to explain their reasoning.) (CLICK TO REVEAL SOLUTION.) Notice that again in this situation, length of the bar did not represent the quantity of pears or pineapples, but rather their cost. We used multiple bars of the same length to show when we had 2 pears and to show we had 1 or 3 pineapples. The use of the length to represent something other that quantity of items is another form of complexity. Would you agree that this added complexity is a fairly significant one relative to the others? Let’s move now into word problems involving using the tape diagram as a visual fraction model.

Example 12: David spent 2/5 of his money on a storybook

Example 13: Alex bought some chairs

Example 14: Jim had 360 stamps
Example 14: Jim had 360 stamps. He sold 1/3 of them on Monday and ¼ of the remainder on Tuesday. How many stamps did he sell on Tuesday? (EXAMPLE 14) Let’s read the first sentence of the problem. “Jim had 360 stamps.” Can we draw something? What can we draw? Can we add a label to our drawing? Is there anything else that we can draw or label? Let me read you the next sentence. “He sold a fraction of them on Monday and a fraction of the remainder on Tuesday.” What fraction did he sell on Monday? Can we draw something to show what he sold on Monday? How should I label this? What fraction did he sell on Tuesday? One fourth of the remainder. Where is the remainder shown in this diagram? Can you imagine what one fourth of the remainder looks like? How can I show one fourth of the e remainder. If I mark this like so. Is the whole of the stamps still partitioned into equal parts? What can I do to make sure I am partitioning my whole into equal parts? Does anybody know? (Allow for contributions). Oh, ___ is suggesting that I partition the whole into sixths. (Demonstrate partitioning the whole into sixths.) Can I still see one third. Is this still one third? Can I see one fourth of the remainder? Is there anything else I can label or draw? What can I see when I look at my diagram? Let’s read the final sentence. “How many stamps did he sell on Tuesday?” Where can I see this on the diagram? There are two ways to model this for students, one way is within the existing bar, another way is to redraw ‘the remainder’ just below and then partition only the remainder. (CLICK TO REVEAL BOTH DIAGRAMS.)

Example 15: Max spent 3/5 of his money in a shop and ¼ of the remainder in another shop. What fraction of his money was left? If he had \$90 left, how much did he have at first? (EXAMPLE 15) (If time allows, have participants try Example 15 on their own. Otherwise, move on to example 16.)

Example 16: Henry bought 280 blue and red paper cups

Example 17: A club had 600 members. 60% of them were males

Example 18: Meagan had \$1780 and Lisa had \$1910

Example 19: Ingrid is mixing yellow and green paint together for a large art project. She uses a ratio of 2 pints of yellow paint for every 3 pints of green paint. Option 1: ____________________________________________________________________________________________________________________________________________________________________________________

Example 19: Ingrid is mixing yellow and green paint together for a large art project. She uses a ratio of 2 pints of yellow paint for every 3 pints of green paint. Option 2: ____________________________________________________________________________________________________________________________________________________________________________________

Example 19: Ingrid is mixing yellow and green paint together for a large art project. She uses a ratio of 2 pints of yellow paint for every 3 pints of green paint. Option 3: ____________________________________________________________________________________________________________________________________________________________________________________

Example 20: The ratio of the length of Tom’s rope to the length of Jan’s rope was 3:1. The ratio of the length of Maxwell’s rope to the length of Jan’s rope was 4:1. If Tom, Maxwell and Jan have 80 feet of rope altogether, how many feet of rope does Tom have? (Allow students to work the problem independently, or scaffold with the following script.) Let’s read the first sentence, “The ratio of the length of Tom’s rope to the length of Jan’s rope was 3:1.” Can you draw something? What labels can you add? Is there anything else you can see? Let’s read the next sentence. “The ratio of the length of Maxwell’s rope to the length of Jan’s rope was 4:1. Can you add something to your drawing? What else can you label? Is there anything else you can see? Let’s read the next sentence. “If Tom, Maxwell and Jan have 80 feet of rope altogether, how many feet of rope does Tom have? Can you add something to your drawing? What else can you label? Is there anything else you can see? How would changing the total feet of rope from 80 to 32 change the complexity of the problem? Would it be more or less difficult? (Facilitate a discussion; there is no wrong or right answer.)

Example 21: Lena finds two boxes of printer paper in the teacher supply room. The ratio of the packs of paper in Box A to the packs of paper in Box B is 4:3. If half of the paper in Box A is moved to Box B, what is the new ratio of packs of paper in Box A to Box B?

Example 22: Sana and Amy collect bottle caps
Example 22: Sana and Amy collect bottle caps. The ratio of the number of bottle caps Sana has to the number Amy has is 2:3. The ratio became 5:6 when Sana added 8 more bottle caps to her collection. How many bottle caps does Amy have?

Example 23: The ratio of songs on Jessa’s phone to songs on Tessie’s phone is 2 to 3. Tessie deletes half of her songs and now has 60 fewer songs than Jessa. How many songs does Jessa have?

Opening Exercise: 88 children were in swimming camp
Opening Exercise: 88 children were in swimming camp. One-third of the boys and three-sevenths of the girls wore goggles. If 34 students wore goggles, how many girls wore goggles?

Opening Exercise: 94 children are in a reading club
Opening Exercise: 94 children are in a reading club. One-third of the boys and three-sevenths of the girls prefer fiction. If 36 students prefer fiction, how many girls prefer fiction?

July 2013 Network Team Institute 88 children were in swimming camp. One-third of the boys and three-sevenths of the girls wore goggles. If 34 students wore goggles, how many girls wore goggles? 88 Children at swim camp Boys Girls 34 54 Wore goggles Did not wear goggles Wore goggles 20 14 34

Children in reading club
July 2013 Network Team Institute 94 children are in a reading club. One-third of the boys and three-sevenths of the girls prefer fiction. If 36 students prefer fiction, how many girls prefer fiction? 94 Children in reading club Boys Girls 36 58 Prefer fiction Prefer non-fiction Prefer fiction 22 14 36

Key Points When building proficiency in tape diagraming skills start with simple accessible situations and add complexities one at a time. Develop habits of mind in students to reflect on the size of bars relative to one another. Part-whole models are more helpful when modeling situations where:_____________________________ Compare to models are best when: ____________________________________ Let’s review some key points of this section. When building proficiency in tape diagraming skills start with simple accessible situations and add complexities one at a time. Develop habits of mind in students to continue to ask, ‘is there anything else I can see in my model’ before moving on to the next sentence in the problem. Develop habits of mind in students to reflect on the size of bars relative to one another, by asking, ‘who has more’ type questions. Part-whole models are more helpful when modeling situations where you are given information relative to a whole. Compare to models are best when comparing quantities.