Students have learned the basics of equations, and in fact have already been solving some in previous lessons. This lesson provides more information on the theory behind the addition and subtraction properties of equality. This topic was addressed in Unit 10 for seventh grade.
This lesson contains more problems that have fractions to provide more practice. Fractions will be emphasized in subsequent lessons as a including multi-step equations with fractional coefficients so that students have that under their belt so they are better prepared for more general fractional equations later in the course.
This lesson also goes deeper into how the principles of addition and subtraction equality work and provides an opportunity for students to try their hand at a proof.
Warm-Ups.
1. Is the sentence below true if k=1, m=6, and n=0?
Answer: Yes; 1(6-0) = 6(1) - 0
2. Write an equation for this problem and solve. The sum of two numbers is 46. The smaller number is 10. Find the larger number. Answer: 10 + x = 46; x = 36
3. Solve. x – 32 = 2 Answer: x=34
4. -16 + 20 = ? Answer: 4
5. -16 – 20 = ? Answer: - 36
Problems 4 and 5 provide review for working with negative numbers. It is assumed that students have mastery of working with negative numbers from seventh grade, but some reminders may be necessary in the beginning of the course.
Addition and subtraction properties of equality. Problems 2 and 3 of the Warm-Ups require solving simple equations which students should know how to do.
“You’ve learned how to solve these equations in seventh grade, so today is essentially a review. And the basic rule for today is that if we add the same number to equal numbers, the sums are equal.”
I write on the board and have them write this in their notebooks:
Addition Property of Equality: For each a, b and c, if a = b, then a + c = b + c
“This is called the Addition Property of Equality. Now I want you to write what the Subtraction Property of Equality is, using symbols like I just did on the board.”
I have them write in their notebooks or on mini-whiteboards. This is fairly easy, but some students may be hesitant. I use this exercise primarily to see where students are mathematically.
They should have written: For each a, b, and c, if a = b, then a – c = b – c.
“This should be somewhat familiar since you’ve been working with it and you’ve already solved equations using these properties. Here is one more important principle, the substitution principle.”
On board and in their notebooks:
Substitution Principle: For any numbers a and b, if a = b, then a and b may be substituted for each other.
“Now we’re going to go a little deeper and actually prove that the addition property of equality is true.”
Proof of Addition and Subtraction Properties of Equality. “In the previous lesson I said we would see a proof of a property. This is that time and this is that proof.”
I have them write in their notebooks:
Given: a = b; To prove: a + c = b + c
“The word ‘Given’ means that we are told what is being assumed. Let’s start by writing something we know is true: a + c = b + c.
“Anyone know why this is true? Look at your notes from yesterday if you need to.”
I give them a minute. A hint, if needed is “It’s the same reason for a =a”. This will usually do it, and some students see that it is the reflexive property.
“Right, any number is equal to itself. a is a number and so is a + c.”
I write a = b. “How do I know that’s true?”
If no answer, a prompt is “What did we say in ‘Given’?” A chorus of OHH is usually heard. “Right. We are told it’s true.” They are to write “Given” after “a = b” as a step in the proof.
“Can we substitute b in for a on the right hand side of the equation a + c = a + c ? Why?”
They really should know this but this is all new and they may not be keeping track of what they know. I’m looking for “substitution principle” and someone will usually say it.
“So if I do that what do we have?” I check notebooks and I will ask a student who has the following to state it for the class. (Usually most students have it right).
The answer is: a + c = b + c. “And that’s what we were proving. So we’re done!”
The steps now look like the following in their notebooks and I also write it on the board:
1. a + c = a + c (Reflexive Property)
2. a = b (Given)
3. a + c = b + c (Substitution Property)
“Now when we finish a proof, we can either say ‘Which was to be proven’ or we write QED, which stands for ‘quod erat demonstrandum’ which is Latin for “Which was to be proven”.
I now instruct them to take out a sheet of paper, and using what we just did, write a proof for the Subtraction Property of Equality. I give them five minutes, and they may use their notes and what is on the board. They are to hand it in, and correct answers will receive 2 extra credit points on the next test or quiz. Many students get it but there are still those who will struggle. I do this to see who is able to reproduce it (even by copying which takes some reasoning to know what it is one is copying) and, again, to get a hint of an indication of where students are mathematically.
Solving Equations. Students have been solving equations using these properties for a while, so this is not exactly new. I give a few examples, but focus on equations which have fractions and decimals in them to ramp it up a bit from what they’ve been used to—although we covered fractions and decimals in equations in seventh grade as well.
Examples.
1. d + 3.2 = 7.8 Answer: d = 7.8 – 3.2 = 4.6
2. ¾ = 0.75 +r A prompt if there are questions is “Can you convert ¾ to a decimal, or 0.75 to a fraction? Answer: r = ¾ - ¾ = 0, or r = 0.75 - 0.75 = 0
3. b – 2/3 = 3 Answer: b = 3 – 2/3; b = 9/3 – 2/3 = 7/3. I tell students to leave answers in fraction form, not mixed number. If it makes it easier they can write 3 as 3/1 so they can see more easily that the equivalent fraction is 9/3.
4. b -5 = 2/5 Answer: b = 5 + 2/5 = 25/5 + 2/5 = 27/5
5. 3/10 = x – 13/10 Answer: x = 3/10 + 13/10 = 16/10 Some students will write 1.6 which is correct, but I want them to get used to writing things in fraction form.
6. x -2/3 = 7/6 This may require a prompt reminding them to convert fractions so they have the same denominator. Answer: x = 7/6 + 4/6 = 11/6
Homework: Some of the examples above may be taken from the homework so the students are familiar with the more difficult problems and have the confidence to do the problems. The homework should be a mix of integer and fractions in the equations. The variable should not have a coefficient for these problems—that will be covered in the next lesson on using division and multiplication to solve equations.