Task 2
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Western Governors University *
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AVA2
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Mathematics
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Feb 20, 2024
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Kyle Clinedinst
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BMM2- Task 2: Understanding and Teaching Solid Geometry or Measurement of Solid Figures
Part A:
Study the solid geometry or measurement of solid figures content standards for your state and do the following:
1. List three content standards from your state that apply to solid geometry or measurement of solid figures for grades K–6. The three selected standards must represent three different grade levels.
Grade 3: “
3.MD.7 Relate area to the operations of multiplication and addition. a. Find the area of a rectangle with whole number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.”
-Ohio Learning Standards/Mathematics Grade 3
Grade 5:
“5.MD.5 Relate volume to the operations of multiplication and addition and solve real-world and mathematical problems involving volume. a. Find the volume of a right rectangular prism with whole number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole number products as volumes, e.g., to represent the Associative Property of Multiplication.” -Ohio Learning Standards/Mathematics Grade 5
Grade 6:
“6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = ℓ⋅w⋅h and V = B⋅h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.” -
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 3: What is the area of the rectangle?
a.
Write an equation to represent the area. _________
b.
Solve the equation above to find the area. ________
Grade 5: Using your counting blocks, make a model of the picture shown below. a.
What is the equation you would use to find the volume? ________
b.
What is the volume of the solid? __________
Grade 6: What is the volume of solid below?
a.
What equation would you use to solve for the volume? __________________
b.
Show your work. 3. Provide a solution for each
problem that demonstrates each
step or explains the thinking process involved in determining the solution.
Grade 3: What is the area of the rectangle?
c.
Write an equation to represent the area. ____
6 x 4 = _____
d.
Solve the equation above to find the area. ___
6 x 4 = 24 square inches
_____
Thinking- For this problem, students must know that area means the number of square units it takes to make up a plane shape. To start, students simply count the number of square units. Then, students start thinking of the square units like a multiplication array.
Next, they relate that array to length times width. Finally, they learn to multiply length times width to solve for area.
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Grade 5: Using your counting blocks, make a model of the picture shown below. c.
What is the equation you would use to find the volume? ___
4x3x5
_____
d.
What is the volume of the solid? ______
60 units cubed
____
Thinking- For this problem, I wanted students to start by modeling the number of cubes needed to find the volume. They would do this with counting blocks shown below. This would show the students that when we find volume, it’s not units squared like area, they
are counting the number of cubes. For this problem, students could model the problem and simply count the number of cubes to find the volume. But I want them to also use the associative property of multiplication to write an equation to solve the number of unit
cubes. Grade 6: What is the volume of solid below?
c.
What equation would you use to solve for the volume? ___
3 ¼ x 8 x 2 ⅛ ___
d.
Show your work. (3 ¼ x 8) x 2 ⅛
26 x 2 ⅛
55 ¼ Thinking- For this problem, students need to first know how to find volume. This builds on the 5th grade standard (LxWxH). Next, students will need to know how to multiply mixed numbers. Students can use the associative property of multiplication to solve this
equation. For the problem above, I started with 3 ¼ multiplied by 8. 3x8= 24 and ¼ of 8= 2. 24 + 2 = 26. Next, students will take 26 multiplied by 2 ⅛. 26 x 2 = 52 and ⅛ of 26 is 3 ¼. 52 + 3 ¼ = 55 ¼ . 4. Discuss how the chosen standards and problems build student understanding of solid geometry or measurement of solid figures across the three K–6 grade levels selected in part A1.
I chose these standards and questions because they build on prior knowledge of finding
the area of a plane shape to eventually finding the volume of a solid shape. I chose to start with a 3rd grade standard because this is when students start finding area using length times width. To find the volume of a solid shape, students first need to know how
to find the area of a plane shape. Next, in 5th grade, students start to find the volume of
solid shapes. I chose to start by having the students’ model how to find the volume using counting blocks to model the solid shape. Then, they use their model to write an equation, length times width time height. Finally, in 6th grade, students find the volume of solid shapes, but take it a step further. In 6th grade, students use mixed numbers to find the volume of solid shapes as shown in part c.
B. Watch the “What’s Fun About Surface Area” video and do the following:
1. Describe two instructional strategies that are used to maximize instruction time or improve communication.
Two instructional strategies used in this video were think-pair-share as part of the mini lesson and explicit strategy instruction as the whole group lesson. Think-Pair-Share: The first instructional strategy I noticed was the think-pair-share strategy. This was a part of the mini lesson where the teacher was introducing the concept of finding surface area. She showed 2 different shapes, a simple shape and a complex shape. Then she had the students think about the differences that they noticed. When she said “precision” she had the students discuss what they noticed with
a partner. (I like how she used precision, reminding the students to share their ideas in complete sentences and give details when explaining.) As the students were discussing, the teacher was listening to conversations. This let her listen as a form of formative assessment. Explicit Strategy Instruction: As described by (Wallie & Karp & Bay-Williams, 2012, p.98), “teacher-led instruction on a specific strategy. The teacher does not merely model
the strategy and have students practice it but attempts to illuminate the decision making
that may be troublesome for these learners.” The teacher in this video was doing a whole group activity or a tier 1 activity. She was doing this by showing the students how
to split a complex shape to make 2 simple shapes and then find the area of both shapes
to solve. I’m assuming this is a newer concept for the children because she was not really working as the facilitator, but rather modeling and showing students the steps needed to find the area of complex shapes. 2. Observe the students in the video. Describe two examples of student engagement.
The teacher does a good job keeping the students engaged and redirecting them after they talk with their partners. The first student engagement strategy I see is when she has the students copy her hand motions to regain their attention. This was after the students were working together, and the teacher was ready to have a whole group discussion. She says, “follow me” and makes some hand motions that the students copy. This quickly gets the class quiet and redirects their attention to the teacher. The second student engagement I see is when the students use a hand signal to show that they agree or disagree. This happens throughout the video. One student gives an answer and explanation, and the students are doing a hand motion to show they agree.
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Later, the teacher models how to find the area of a complex shape by finding the area of
2 simple shapes. She multiplies both quadrilaterals and says, “that’s it, I’m done.” The students make a different hand motion to show that they disagree. This is a great way for the teacher to see who understands and for each student to show whether they agree or not without having everyone talk at the same time. Also, this helps keep students engaged. 3. Describe a follow-up activity that you would use after this lesson to deepen the students’ understanding of solid geometry. Include a description of any supporting material needed for
the follow-up activity.
After this activity, I would do some sort of hands-on surface area activity where the students are the ones exploring and discovering. This lesson had a lot of teacher-led activities which is good for new content. An example of this would be to have students find the area of complex shapes or 3 dimensional shapes. They could first fold 3 dimensional shapes as shown below. Then, they could measure each part of the 3-
dimensional shapes to find the surface area. Part C: Prepare to create an original lesson plan on solid geometry or measurement of solid figures by doing the following:
1. Describe an evidence-based instructional strategy that will be incorporated into your original lesson plan.
For this lesson, I will use explicit instruction combined with visual representations. I will be making a lesson plan to find the volume of solid rectangular prisms. As described by
Wallie & Karp & Bay-Williams, 2012, p.98, explicit instruction is when “the teacher does not merely model the strategy and have students practice it but attempts to illuminate the decision.” I will lead the discussion, but the students will aid in the discussion and use visual representations to explain their thinking. “Visual representation is a way for students to see math. You can visually represent math using number lines, tape
diagrams (also known as bar models), pictures, graphs, and graphic organizers.” (Greene, 2014)
I will have the students make models of rectangular prisms with the counting cubes and have the students help lead the discussion on how to use an equation to find the volume of a rectangular prism. a. Explain why the chosen instructional strategy would be beneficial in a lesson on solid geometry or measurement of solid figures using evidence from a credible source to support your selection.
I chose the explicit instruction combined with visual representations to teach finding the volume of a rectangular prism because “visual representations help all students understand abstract math concepts and solve problems.” (-Greene, 2014) To me, volume is an abstract concept because it’s not something that is easily visualized. Most of the real-life volume examples deal with measuring liquid, yet the standard in 5th grade deals with finding volume with cubes. This is something that we do not see in our daily lives and will be abstract for the students. This is also why I chose to do explicit instruction. This lesson will be teacher led, allowing the students to use manipulatives to find volume, then turn that into using an equation to find volume. Part E:
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
Bay-Williams, J.V.D.W.K.S.K.J. M. (2012). Elementary and Middle School Mathematics: Teaching Developmentally, VitalSource for Western Governors
University (8th Edition). Pearson Learning Solutions. https://wgu.vitalsource.com/books/9781256957669
Greene. (2014). Evidence-based math instruction: What you need to know
. Understood. https://www.understood.org/en/articles/evidence-based-math-
instruction-for-struggling-students
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