Task 2
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Kyle Clinedinst
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BDM2 Task 2: Understanding and Teaching Ratios and Proportional Reasoning
Grade 4
: 4.NF.1 “Explain why a fraction a/b is equivalent to a fraction (n × a) /(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.” -Ohio Learning Standards/Mathematics Grade 4
Grade 5
:
5.NF.4 “Apply and extend previous understandings of multiplication to multiply
a fraction or whole number by a fraction. a. Interpret the product (a/b) × q as a parts of a
partition of q into b equal parts, equivalently, as the result of a sequence of operations a
× q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) ×
(c/d) = ac/bd.)” -Ohio Learning Standards/Mathematics Grade 5
Grade 6
: 6.RP.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.” -Ohio Learning Standards/Mathematics Grade 6
2. Write a sample problem for each
of the three standards to illustrate the evolution of student understanding.
Grade 4
: 4.NF.1
Tom and Jim each have a cake. Tom split the cake into halves and took ½ of the cake to school for lunch. Jim split his cake into eights. If he wants to take the same amount of cake into school as Tom, what fraction would he need?
a.
Use the equation to help solve.
b.
Fill in the model to prove your answer. Grade 5
:
5.NF.4
McKenna gets to a pizza party late. She opened a pizza box and saw there was ¾ of a whole pizza left. She ate ½ of the remaining pizza. What fraction of the pizza did McKenna eat?
a.
Show a number sentence to solve.
b.
Draw a model to explain your thinking.
Grade 6
: 6.RP.1
The ratio of boys to girls in the Mount Vernon 6th grade class is 3:5. 3 boys to 5 girls. If
there are 45 girls in 6th grade, how many boys are there?
a.
Show your answer with a number sentence. b.
Use the tape diagram to solve.
Boys
Girls
3. Provide a solution for each
problem that demonstrates each step or explains the thinking process involved in determining the solution.
Grade 4
: 4.NF.1
Tom and Jim each have a cake. Tom split the cake into halves and took ½ of the cake to school for lunch. Jim split his cake into eights. If he wants to take the same amount of cake into school as Tom, what fraction would he need?
a.
Use the equation to help solve.
b.
Fill in the model to prove your answer. Thinking- For this problem, students need to build on their prior knowledge of fractions to find equivalent fractions. Students will need to know that if you multiply the top number and the button number by the same integer, the new fraction will be equal to the
first fraction. Next, students will need to know in order to get from a denominator 2 to a denominator of eight, you would multiply it by 4. They then would know to multiply the
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numerator by 4 as well. This would make the second fraction 4/8. The last step is to shade in 4/8 to make it equivalent to ½. Grade 5
:
5.NF.4
McKenna gets to a pizza party late. She opened a pizza box and saw there was ¾ of a whole pizza left. She ate ½ of the remaining pizza. What fraction of the pizza did McKenna eat?
a.
Show a number sentence to solve.
b.
Draw a model to explain your thinking. Thinking- For this problem, students build on their knowledge of fractions and multiplying fractions by whole numbers. Now they will take that a step further and multiply fractions by fractions. First, students will need to know in order to solve this problem, they will be multiplying 2 fractions. This is a difficult task for many students because the word problems don’t have the same key phrases as when they are to multiply whole numbers. In this problem, the key phrase is “of the remaining”. This lets students know they are to multiply. Next, the students will write the equations shown in “a”. The students will multiply the numerators and the denominators. Finally, the students will model the amount of pizza McKenna eats. They will first split a fraction into fourths. If three fourths of the pizza is remaining, that would mean 3 large pieces are remaining. Well, ½ of three is 1.5, and
that can’t be shown as a fraction with fourths as the denominator. So students will need
to change the denominator to eights. Finally, ½ of ¾ is ⅜ as shown in the picture. Grade 6
: 6.RP.1
The ratio of boys to girls in the Mount Vernon 6th grade class is 3:5. 3 boys to 5 girls. If
there are 45 girls in 6th grade, how many boys are there?
a.
Show your answer with a number sentence. b.
Use the tape diagram to solve.
Boys
9
9
9
Girls
9
9
9
9
9
Thinking- For this ratio problem, students can solve it a couple of ways. First, they can think of the ratio as a fraction. Knowing that there are 3 boys to 5 girls, it’s the same as . And if there are 45 girls, they can multiply the denominator 5 by 9 to get the number
⅗
of girls. Then, they would just need to multiply the numerator 3 by 9 to get the number of boys. The second way to solve this problem is by making a tape diagram. They know there are 3 boys to 5 girls, so they can make 3 boxes for the boys and 5 boxes for
the girls. Then, they would use their knowledge of multiplication to know that 5 boxes of
9 gives them the number of girls. Lasty, they would use the same factor 9 for the boys and get 3 boxes of 9 which is 27. 4. Discuss how the chosen standards and problems build student understanding of ratios and proportional reasoning across the three K–6 grade levels selected previously.
I chose these standards because they build on each other from grades 4-6. In order to understand ratios and proportional reasoning, students first need to understand fractions. This skill starts in 3rd grade in my state. The 4th grade standard I chose teaches children that equivalent fractions can be found by multiplying the numerator and
denominator by a whole number. This is a simplified way of finding ratios. The 5th grade standard now really deals with proportional reasoning. When multiplying a fraction by another fraction, the product will be a smaller fraction. This can be difficult for students to understand because up until this point, multiplication meant finding a number that is bigger. The proportion of a fraction being multiplied now means the fraction will be split more, making the denominator larger and pieces smaller. The 6th grade standard turns these fractions into ratios. For the example given, students need to take the ratio 3:5 and multiply it by an unknown number to find the 5x?=45. Once they rely on their previous knowledge about multiplication and division to solve ?=9, then they can apply it to the first number of the ratio, 3x9= 27. Part B:
Watch the “Introduction to Ratios & Proportional Relationships” video and do the following:
1. Describe one example from the video that demonstrates how concrete representations are
used to model the concept of ratios and proportional relationships.
In this video, the students used concrete representations or manipulatives to model the concept of ratios and proportional relationships. The students used counting blocks to model the ratio 3 cups of red paint to 2 cups of blue paint or 3:2. The students were asked to model 2 cups to 3 cups, but make it out of 20 cups. So they had to use their concrete representations to model the cups of paint, but take it a step farther. Instead of doing just 2 cups to 3 cups, they had to show 20 cups. Some students did a model that showed a total of 20 cups, some did a model that showed 3:2 as one cup and multiplied it by 20. a. Explain why the use of concrete representations is an effective instructional strategy.
This concrete representation was an effective instructional strategy because the students got to investigate and explore ratios through hands-on activities. As the teacher said in the video, “the students could visually see it, but they could also put their
hand on something that made sense to them and really share their thinking.” This really
gets students excited to share their findings. The teacher also says that when they create something of their own, they are more apt to start talking about their creativity. An example of this is when two girls showed their thinking by drawing a diagram of the ratio they created.
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2. Explain how the teacher effectively integrates cooperative learning into the lesson using specific examples from the video.
The teacher used cooperative learning by doing a “team consensus.” She first discussed how a team consensus works. 1. Students share ideas and thoughts. 2. Students are to ask questions if they don’t understand or need more explanation. 3. All students decide the process as a group. This helps by making sure every student gets a voice. In the video, one group had 3 different representations of the question. The teacher pointed this out to them and told them to work together to figure out which one makes sense. The students then had to work together to see which idea works best. This represents “Peer-assisted learning” -
(Van De Wallie & Karp & Bay-Williams, 2012, p.100).
a. Describe the role of the teacher and the students in this lesson using specific examples from the video.
The role of the teacher was that of the facilitator. As described by -(Van De Wallie & Karp & Bay-Williams, 2012, p.54) “You can support student thinking without evaluation.”
In this video, the teacher gave the students a ratio scenario, a problem, and manipulatives to solve. The teacher didn’t tell the students how to solve, but rather walked around the classroom encouraging students to share ideas and asking open ended questions when needed. The students were physically doing the work and sharing ideas and strategies as they worked. The students were in charge of their own learning. At times, they were also the teachers. In the video, the students were the ones exploring ideas and trying different representations for the question. This was done a couple of different ways. Some students made plans first, thinking and explaining how to answer the question. It looked
like some students were doing trial and error by putting some blocks together then seeing if it made sense. Also, the students were teaching others. They were working together and explaining their ideas and giving support to each other as needed. 3. Explain how the teacher identifies student misconceptions and redirects student thinking using specific details from the video.
I really liked how the teacher identified students' misconceptions and redirected them. It
was a great example of the teacher being the facilitator. When the teacher saw a group
that had 3 different representations of the ratio 2:3 and only one being the correct representation, she did not just tell them who was correct. First, she referred to the question. Then, she told them “I will be back in a couple of minutes to see which one makes sense to your group.” She basically told them that only one representation was
correct but allowed the students to discover who was correct on their own. This makes the students do the thinking and discovering by not just giving them the answer. 4. Explain how the lesson activities and related student conclusions could be used to further
extend student thinking.
I’m not sure if this question is asking how did the lesson activities and conclusion extend their thinking
? Or is it asking how these could be used with another activity to further student thinking
? I will attempt to answer both thoughts. 1.
The lesson activities and the conclusion both extended the students' thinking. First, the activities allow students to create their own strategies and ideas when solving the ratio problem. In some cases, students were not successful at first. This is ok because it makes them extend their thinking and ask themselves “why was this not successful.” Also, in the lesson students had to share ideas with each other. As the teacher says, “students have to be good leaders and share their ideas, but they also have to be really good listeners and see it as an opportunity to learn from their peers.” This makes students extend their thinking because it’s one thing to solve the problem, but explaining the steps needed to solve is an extension. The conclusion allowed students to bring models and share with the whole group. This allows students to expand their thinking by elaborating on what they have created. Also, one student made a model with 3 pink blocks and 2 green blocks. He said that each block stands for 4 cups, so that would make 12 cups of red paint and 8 cups of blue for a total of 20 cups. The next student showed a model that looked the same, 3 blocks of one color and 2 of another. But instead of each block representing 4 cups like the previous student, all 5 blocks represented 1 cup. She explained it as 3 cups of red and 2 cups of blue to make 1 “round”. This model represents a different overall answer, but both students showed a ratio of 3:2. Finally, the students were able to share their ideas to extend their thinking. 2.
The lessons activity and related student conclusion allowed students to show their understanding by using concrete examples and sharing ideas with their peers. The next step to extend students' thinking could be to have them draw a model of what their group presented and write about the process involved in solving. This would allow them to take their concrete representations and see if they can translate it using a model. Next, the students would have to write their thinking and mathematical processes in words. This is difficult for many students
and would really extend their thinking.
C. Prepare to create an original lesson plan on ratios and proportional reasoning by doing the following:
1. Describe an evidence-based instructional strategy that will be incorporated into your original lesson plan.
The evidence-based instructional strategy that I will be using is the “Walkabout” strategy. “4-5 problems/questions are printed on card-stock, one problem/question per card. Place one card pre table around the room. Groups of students are given a prearranged amount of time at each table to collaboratively work on each problem. Each group is assigned to display and explain one of the problems for the class.” -(Buffington, 2007, p.64)
There will be 4 cards with 4 ratio problems. One will be solved using a number line. One will be solved using the bar model. One will be solved using manipulatives, counting blocks. The last will be the same ratio problem as the one in the manipulatives
station, but students will be asked to draw a picture of the ratio. a. Explain why the chosen instructional strategy would be beneficial in a lesson on ratios and proportional reasoning using evidence from a credible source to support your selection.
I chose to use this strategy because it gets the students moving, allows students to work together and learn from each other, and has differentiation built in. First, “Walkabout” allows students to move around the classroom. Many students learn better
when they are mobile. If we can incorporate movement into our lessons, it will help them learn and stay on task. Next, this strategy allows students to work together and benefit from peer assisted learning. Finally, this lesson has diversity built in. The grouping of students would have at least one student on grade level, one below grade level, and one above grade level. The students that are below grade level can lean on their peers for assistance and ideas. As stated above, at the end of the lesson, students will share their thinking of one “station” as the culminating activity. The students who are above grade level will work on explaining their ideas to extend their thinking. Stations
1.
Complete a ratio using a number line.
2.
Complete a ratio using a bar model or table.
3.
Complete a ratio using manipulatives, (counting blocks.)
4.
Draw a model of a ratio using paper and colored pencils. (Extension station)
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Part E
:
Acknowledge sources, using in-text citations and references, for content that is quoted, paraphrased, or summarized.
Ohio Learning Standards Mathematics
. (2017). Ohio Department of Education. https://education.ohio.gov/getattachment/Topics/Learning-in-Ohio/Mathematics/
Ohio-s-Learning-Standards-in-Mathematics/MATH-Standards-2017.pdf.aspx?
lang=en-US
Buffington, B. B. (2007). Strategy Ring
. Knox County ESC.