Task 3
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Western Governors University *
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Mathematics
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Jan 9, 2024
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Kyle Clinedinst
Student ID-001029234
BDM2 Task 3: Understanding and Teaching Equations and Inequalities
Part A:
Study the equations and inequalities content standards for your state and do the following:
1. List three content standards from your state that apply to equations and inequalities for grades K–6. The three selected standards must represent three different grade levels.
Grade 4: 4.OA.3 “Solve multistep word problems posed with whole numbers and having
whole number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter
standing for the unknown quantity. Assess the reasonableness of answers using mental
computation and estimation strategies including rounding.” - -Ohio Learning Standards/Mathematics Grade 4
Grade 5: 5.OA.1 “Use parentheses in numerical expressions and evaluate expressions with this symbol. Formal use of algebraic order of operations is not necessary.” -Ohio Learning Standards/Mathematics Grade 5
Grade 6: 6.EE.5 “Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set
makes an equation or inequality true.” -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: Mr. Clinedinst is buying pencils for his class. He buys 4 packs of 10 pencils and 5 packs of 12 pencils. He gives each of his 20 students the same number of pencils. How many pencils (P) does each student get? Use the equation below to solve. ((4X10) + (5x12)) ÷ 20 = P
Grade 5: Sam and Jodi go to the ice cream shop, and each get a cone. Their total bill is $15. If they have a $3 off coupon, how much does each child spend on ice cream?
a.
Write an equation to represent this word problem.
b.
Solve your equation.
Grade 6: Anne was holiday shopping for her friends and her mother. She has $500 dollars to spend. She wants to have at least $200 to spend on her mom. Each gift she buys for her friends is $50. How many gifts can she buy for her friends and make sure she has enough money for her mom’s gift. Use the inequation to solve.
500 - 50x ≥ 200
3. Provide a solution for each
problem that demonstrates each
step or explains the thinking process involved in determining the solution.
Grade 4: Mr. Clinedinst is buying pencils for his class. He buys 4 packs of 10 pencils and 5 packs of 12 pencils. He gives each of his 20 students the same number of pencils. How many pencils (P) does each student get? Use the equation below to solve. ((4x10) + (5x12)) ÷ 20 = P
( 40 + 60 ) ÷ 20 = P
100 ÷ 20 = P
5 = P Thinking- For this problem, I wanted to show the students a real-world situation where we would use the order of operations to solve an equation. I could even buy the pencils
for my students to manipulate as a class.
In order to solve the equation, students would have to know that they should start with the parentheses. Multiply 4x10 and 5x12. Then add the products to find the total number of pencils. We could do this with real pencils to model. Then, they would split up the pencils equally among all the students to get the answer, 100 ÷ 20 =5. For this problem, I also wanted to provide the equation because many students wouldn't know how to write this equation using the order of operations at this level. The above level students may be able to do this as an extension. Grade 5: Sam and Jodi go to the ice cream shop, and each get a cone. Their total bill is $15. If they have a $3 off coupon, how much does each child spend on ice cream?
a.
Write an equation to represent this word problem.
(15-3) ÷ 2 =
b.
Solve your equation.
(15-3) ÷ 2 =
12 ÷ 2 = 6
Thinking- For this problem, I wanted to take it a step further than the 4th grade order of operations and solving equations problem. I wanted to give a real-world example of an equation, but I want students to write the equation themselves. To do this, they need to
understand that “the equal sign is like a balance.” The two girls in this problem will both pay the same amount, so the “balance” is level.
-(Van De Wallie & Karp & Bay-Williams,
2012, p.263) To do this, students will first need to subtract the total amount for the ice cream by the $3 coupon. Then, they will be able to take the difference and divide it by 2
to find the answers. Grade 6: Anne was holiday shopping for her friends and her mother. She has $500 dollars to spend. She wants to have at least $200 to spend on her mom. Each gift she buys for her friends is $50. How many gifts can she buy for her friends and make sure she has enough money for her mom’s gift. Use the inequation to solve.
500 - 50x ≥ 200
Thinking- For this problem, there is more than 1 possible answer. I really liked how Van
De Wallie & Karp & Bay-Williams described equations and inequalities as a balance. Students would need to understand that x doesn’t have to equal 6 making the balance level. For this problem, the left side 500-50x could equal 200 to make the balance level, or it could be more than 200 making this an inequality. Anne could buy 1-6, $50 presents to make sure she has at least $200 for her mother. Technically, x could equal 0 as well.
4. Discuss how the chosen standards and problems build student understanding of equations and inequalities across the three K–6 grade levels selected previously.
I chose these standards and problems because they are vertically aligned. The 4th and
5th grade questions deal with equations with the equal sign acting like a balance. -(Van De Wallie & Karp & Bay-Williams, 2012, p.263) For the 4th grade question, I would provide the equation for the students. They would need to understand the order of operations to solve. For the 5th grade question, students would need to understand the
order of operations. However, they are asked to take it a step further and provide the solution. To solve inequalities in 6th grade, students first need to have a full understanding of equations. Then, they can be taught that there are some order of operations problems that don’t “balance” and this is an inequality. The 6th grade problem above also deals with order of operations like the 4th and 5th grade problems. But as I explained in A3, there could be more than 1 answer for x in the 6th grade problem to make the statement
true.
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Part B:
Watch the “True or False Equation Routine: Fourth Grade” video and do the following:
1. Explain how the questioning strategies used in the video lesson promote student understanding of equations.
In this video, Kristin Gray uses a questioning strategy called “True/False equation routine”. This is where 2 expressions are set equal to one another. The students are asked to look at the expressions and decide if true, they are equal, or false, they are not
equal. As Kristin describes, students can do this by solving both sides or by looking at it
relationally. An example of a student looking at it relationally was the second equation she put on the board, ⅜ + 4/8 = ¼ + ¼ + ⅛ + ⅛. One student said it was false because the left side is odd, and the right side is even. He noticed that no matter how you solve the right side, the numerator would be an even number. The left side numerator is an odd number. (I thought this was a great observation from that student.)
During this strategy, students were thinking first, then sharing with a partner whether they believe the equation to be true or false and explaining how they know. This allows children to share their thinking and listen to others who may solve it differently. Finally, Kristin asked multiple students “how do you know” allowing them to share their thoughts
with the class. 2. Describe how the independent practice work is designed to deepen student understanding
of equations.
The independent practice is designed to deepen students' understanding of equations by having them find as many solutions as possible to 6 x ½ = ___. Students are so used to an equation having only one possible solution. For this problem, 6 x ½ = ___, first the students shared a couple different possible solutions as a whole group. Then, there were many kids who had other ideas but didn’t get to share. This lets the students
know there are many other possible solutions. Kristin asked the students to go back and journal as many as they can. Doing this really makes the students think about how there are endless amounts of solutions. This really drives home the point that some expressions have more than one possible answer.
3. Describe one student misconception that occurred during the lesson.
One misconception was when the little girls said ⅔ + 3 + ⅓ = 3 x ⅓ + 9/3 was false. The girl starts by explaining that ⅔ + ⅓ is a whole. Then, 1 + 3 = 4. She was correct so
far. Then on the other side 3 x 1 = 3/3. Then she stopped herself and figured out that the expression was true.
a. Describe one strategy the instructor used to address the student misconception.
I really liked how Kristin addressed the little girl's misconception. She did not give her any evidence that she was wrong. A lot of times, students look for facial or body language from the teacher letting them know that their answer is wrong. This doesn’t let
the student go through the thinking process needed to solve. In the video, the teacher asked someone who said “false” to explain their thinking first. She did this so that they don’t realize it’s true and then the student wouldn’t have corrected herself. The little girl started by correctly explaining that the left side of the expression ⅔ + 3 + ⅓ was 4. Then she said for the other side, 3x1= 3/3…. Then she realized that it was true. This lets the student go through the thinking process needed and find where exactly she made the mistake the first time. The student was able to correct her own mistake which
is more valuable than a teacher correcting a student mistake. 4. Discuss how the lesson could be further developed to incorporate unknown values.
To incorporate unknown values, Kristin could continue the true false question activity, but incorporate an unknown either as a whole number or part of the fraction. I’ll give two examples. (
X stands for multiply in the examples. Y is the unknown.
)
Example 1- ¼ + ¼ + ¼ + Y/4 = ¼ x 2 + 2/4 + ¼ For this example, Kristin could ask if Y=2 to follow the true/false format. Then, give the students chances to explain how they solved it. True: Y= 2 (each side equals 5/4)
Example 2 - ⅛ + ⅛ + ⅝ + ⅜ = Y x ⅛ + ⅝ Kristin could ask if the whole number Y= 4. False: The left side is equal to 11/8. If Y equals 4 then the right side would equal 10/8. For each example, Kristin could follow the same format as in the video. Show the expression, give the unknown value, and have the students think-pair-share. Furthermore, a guided practice activity in which the teacher was the facilitator using unknown values could be added after the whole group examples. Kristin could have split the children into groups and given each group two fraction expressions, where there was an unknown. They could work as a group to determine the unknown value to make the statement true. For example, ½ + ½ + ½ + ½ + 2 = ½ x Y, where Y is the unknown. Y=8. (x stands for multiply in this example.) Then they would discuss the strategies used to determine the unknown as a group. Finally, as a group, they would pick one strategy used and share their ideas with the class.
Part C:
C. Prepare to create an original lesson plan on equations and inequalities by doing the following:
1. Describe an evidence-based instructional strategy that will be incorporated into your original lesson plan.
An evidence-based instructional strategy that I will incorporate into this lesson is the “CSA (concrete, semi-concrete, abstract) teaching sequence”. As described by Van De Wallie & Karp & Bay-Williams, 2012, p.98, this strategy originates from Bruner’s reasoning theory (1966). This strategy starts with concrete representations or manipulations. Then, students think about math solutions in a semi-concrete manner using pictures or drawings. Finally, they move to abstract thinking using numbers. I will
provide a lesson where students transition through all 3. a. Explain why the chosen instructional strategy would be beneficial in a lesson on equations and inequalities using evidence from a credible source to support your selection.
I have chosen this strategy because first I really like how Van De Wallie & Karp & Bay-
Williams describe equations and inequalities as a balance. I will incorporate this into my
6th grade inequalities math lesson. As a whole group, we will discuss and decompose one real world expression. Then, in groups, students will use a balance scale and gram(s) weights to determine if the expression is equal or not. Next, students will be asked to draw a picture of their findings and share them with their group as the culminating activity. Finally, students will be given 4 expressions to solve as an exit ticket. Some students will solve strictly using the numbers or in an abstract manner. Some students may use pictures to solve the expressions. 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
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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