Developing A Relational Understanding of the Equal Sign
Week 10: Assignment 6
Developing A Relational Understanding of the Equal Sign
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The equal sign is used to show a relationship between quantities. When we solve mathematical problems using relational-structural thinking there is no need to calculate. We can simply analyze the relationship between the quantities on each side of the equal sign. |
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Part 1: Consider This...
Elementary students were asked to find the value of the unknown in this problem: 8 + 4 = ___ +5
· What are 3 answers you would expect students to share?
· 1.
· 2.
· 3.
· Read this short article on equality and the equal sign to learn more.
· Describe a common misconception that young children often have about the equal sign.
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· What can teachers do to help students overcome this misconception?
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Part 2: What is a relational-structural view of the equal sign?
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Scenario: Two children were asked to find the missing value in this equation:: 8 + 4 = ___ + 5
· Student 1: 8 plus 4 is 12, I need to find what number plus 5 makes 12. I know 12 minus 5 is 7 so the answer is 7.
· Student 2: The 5 on the right side is one more than the 4 on the left side. That means that the missing number should be one less than 8 which is 7.
Student 1 computes the sum on one side of the equation and then uses the sum to determine the missing part. This student used a relational-computational view.
Student 2 used the relationship between the expression on each side of the equal sign. This student didn’t need to compute but rather relied upon a relational-structural view. This view is much more efficient when working with larger numbers. |
1. How would Student 1 approach finding the missing value in this equation? 328 + 448 = 450 + ____
2. How would Student 2 approach finding the missing value in this equation? 328 + 448 = 450 + ____
*NOTE: The size of the numbers may have made it more difficult for Student 1 to find the missing value while Student 2, who simply looked at relationships between the numbers, would likely find the answer more efficiently (quickly)
Part 3: Watching students
· Watch this video of children sharing their solution to this problem and others.
· How did David go about finding the unknown value in 6 + 2 = ___ +3?
· How did David relate 6+2 = __ + 3 to 57 + 38 = 56 + 39?
· In determining if the statement was true or false, did David utilize a relational-structural view of equality, a relational-computational view of equality or both? Explain your thinking.
Part 4: Many and Ongoing Opportunities
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Students need numerous and ongoing opportunities to develop a more relational-structural view of equality. Posing problems with large numbers that make mental computation more difficult can lead students to try a relational-structural approach.
Now it’s your turn to develop your relational-structural thinking. At first you may find viewing these problems in this way difficult because you may have had little to no experience thinking in this way. I ask that you are patient with yourself as you develop a new way of thinking. Don’t give up, and just keep trying. If you need support, you can see more examples of how to use relational-structural thinking using this link. |
1. Determine if each of the equations are true or false. Explain your thinking using a relational-structural view. (Remember you should not be calculating… that’s using a relational-computational view.)
a. 53 + 86 = 51 + 88
b. 65 + 41 = 68 + 44
c. 95 – 52 = 93 – 54
d. 76 – 37 = 78 – 39
2. Problems like these challenge children, and adults, to think about and describe the effect of mathematical operations on relationships among quantities.
a. Look at equations A-D in #1. Highlight the equations that are true in yellow.
b. Carefully examine the true equations in A-D in #1 that involved addition. Examine numbers being added on the right of the equal sign with the numbers being added on the left of the equal sign. How are the numbers on the right related to the numbers on the left?
c. Carefully examine the true equations in A-D in #1 that involved subtraction. Examine numbers being subtracted on the right of the equal sign with the numbers being subtracted on the left of the equal sign. How are the numbers on the right related to the numbers on the left?
3. Determine the value of the missing number in each equation using relational-structural thinking. Describe how you used relational-structural thinking to find the missing value for each equation.
a. 74 + 84 = 81 + ___
b. 123 + 456 = 223 + ___
c. 71 - 27 = ____ - 30
d. 223 - 167 = 220 - ____
4. The work of Molina and Ambrose (2006) found that asking children to write their own open equations like the ones in #3 were particularly helpful in supporting students in developing their own understanding of the equal sign.
a. Write 2 multi-digit addition and 2 multi-digit subtraction open equations with an unknown.
b. Then write the solution and how it could be found using relational-structural thinking.
5. For each unfinished number sentence below, think about ways you could change the original pair of numbers to create a new pair of numbers that makes the equation a true statement. Show at least 3 possible solutions for each equation.
a. 75 + 43 = __________ + __________
b. 165 – 97 = __________ – __________
Part 5: Putting it All Together
Reflect on your use of relational-structural thinking in the assignment. Respond to these questions to summarize what you have learned.
1. How do you go about finding the missing value in an equation that has similar addends on both sides?
2. How do you go about finding the missing value in an equation that has subtraction on both sides?
This task was adapted from: Stump, Bishop, & Roebuck & Teaching Student Centered Mathematics- Grades PreK-2 by VanDeWalle & Lovin