Unit 3 – Equations, Inequalities, & Systems
Some information on this unit and the suggestions we are making:
We have looked closely at module 1 and found the things we feel you should work on with your students. In the coming
years this would have been integrated into the first unit on linearity or we might create two units between what you did
with linearity and the concepts embedded in this unit. We have outlined below the main outcomes that you will work
towards in the unit. We also enclosed new lessons and lessons from units we have used in the past to help guide you.
Your coach will also work closely with you.
Expressions – Equations – Inequalities – Systems – about 5 weeks
There are 4 sections to this unit:
Equations
- Students can solve and explain the process using properties and the laws of equality.
- Students can solve literal equations and re-arrange formulas
Inequalities
- Solving and graphing inequalities with one variable
- Solving and graphing compound inequalities (And and Or)
Algebraic Problem Solving
- Solving different types of problems using an algebraic approach by building expressions
- These problems include age problems, consecutive integer problems, and mixture problems.
Systems of equations and inequalities
- Solving systems of equations graphically and using substitution
- Solving systems of inequalities graphically.
- Solving problems using systems.
Lesson 1: What is the meaning of equality?
(Note to teacher: The equal sign takes on different meanings depending on how the mathematical statement is
presented. For example, if you are asking is this statement true? √𝑥 + 1 = √𝑥 + 1 . The answer you would have to
say is no, because it is only true when x equals 0. However, if you asked is 2(x+3) = 2x + 6 true? You would have to say
yes, because any value of x would make this true. So we are asking students to get deeper about the meaning of
equality based on the mathematical statement.)
Opening Activity:
(Note to teacher: Students will be given the following equations and they have to figure out what x is in each of these
however they want.)
- 2(x+3) = 2x + 6
- x + 5 = 11
- x+3=x+2
(Note to teacher: This should lead to a discussion about equations where the value of x can be any real number, one
real number, or that there are no values of x that would make the statement true. Share with them the term “True
Equations”, which means that the solution to the equation can be any real number. This term only applies to the first
example.)
1
Activity 2: Create three equations. Have students create one equation that’s always true, one that’s sometimes true,
and one that’s never true.
Activity 3: Revisiting the Blue Border
Revisit the Blue Border problem to look at the different generalizations. Connecting the generalizations to
expressions and when we set to of the expressions equal to each other we make an equation. Have the students
select two of the expressions and set them equal to each other. Then have them try to solve it. When they solve it
they will come to the statement x = x. Help them to see what this means and that we have created a TRUE equation
for all values of x.
2
Lesson 2 – 3: Properties and Laws of Equality
(Note to teacher: The goal of these two lessons is to review with students, in an interesting way, the mathematical
properties and the Laws of Equality. We have included in Appendix 1 lessons we created in the past to give you ideas
about what you might do with your students.
You will need to determine for you and your students how much time needs to be spent revisiting the mathematical
properties and the Laws of Equality.)
Lesson 4: Solving Equations with Justification
(Note to Teacher: You will give the students 3x+3 = 12 with the purpose of leaning about how they can explain what they
did and why they did it.)
Opening Activity:
Solve: 3x + 3 = 12
What did you do?
Why did you do it?
What laws and properties did you use in your process?
(Note to Teacher: Now you can show them how they can put their ideas in two column format)
3x + 3 = 12
-3 -3
3x = 9
3𝑥
3
Given
Equality law for addition or adding the additive inverse
9
=3
Equality law of division or multiplying by the multiplicative inverse
x=3
Activity 2:
Solve and justify using the two column format:
4(x + 2) + 6 = 6x – 4
3
Activity 3: Further understanding of the laws of equality
Part 1: Give the students x + 3 = 12 then ask the students to create a new equation by adding x to both sides of the
equation. Now they’ll have 2x + 3 = x + 12. Have them solve the original equation and the new equation. Ask them: What
happened? And why did it happen?
(Note to teacher: We call these (x + 3 = 12 & 2x + 3 = x + 12) equivalent equations. Also, using the students work for
solving the two equations help the students see that when you maintain equality you will always be creating an
equivalent equation.)
Part 2: Give them an equation, for example 3x + 2 = 4x + 3. Have them create a new equation that will be equivalent to
the original equation.
Activity 4 / Homework
Then have students do more practice with these ideas.
4
Lesson 5: Practice with equations, work with errors
(Note to teacher: You need to gauge where your students are with equations and how much practice you need to give
them. Make sure they can talk about what they did, why they did it, and how they maintained equality by creating new
equivalent equations.)
Lesson 6: Am I flexible thinker vignettes
Flexibility with Equations
Teachers Guide
(To the Teacher: As mentioned in the note to the teacher at the end of the last lesson the opening activity is
based on the students’ journal writing from the end of that lesson. So be sure to read their responses keeping an
eye out for ways to setup a contrast between the ideas communicated in the journal entry. You could setup a
contrast between a flexible and non-flexible solver or between a student who is making the connections to
maintaining equality and the properties and one who is not. The goal in approaching the opening activity in this
way is to give students the opportunity to hear how their classmates are talking about it as well as hear other
ways to think about solving. Be sure to make the journals anonymous in some way so the author of the entry.)
4 Vignettes
(To the Teacher: We are going to use 4 short vignettes to help stretch the students’ ability to think flexibly as problem
solvers. The last one will be an individual performance task. So be sure to allocate time accordingly. After each of the
first three vignettes you should facilitate a discussion on the vignette.)
(To the Teacher: Feel free to edit the way the work is show to coincide with how you want work to be show in your
class.)
Vignette 1: Done two ways, but who’s correct?
Julio and Lilly are both in your group. You all solved the same equation. When Julio looked at Lilly’s work he noticed his
work was different. They both got the same answer. Julio says his way is right, but Lilly just thinks he got lucky.
Lilly’s Work
Julio’s Work
15 = 3 ( x - 2 )
15 = 3 ( x - 2 )
15 = 3(x) - 3(2)
15 3 ( x - 2 )
=
3
3
5= x-2
15 = 3x - 6
15 = 3x - 6
+6
+6
21 = 3x
21 3x
=
3
3
7= x
5= x-2
+2
+2
7= x
5
Help them figure out how to make sense of this situation.
(To the Teacher: Let the students wrestle with this idea. Facilitate a discussion about their ideas. We want to
redefine rightness around the solution and understanding multiple ways to the solution rather than rightness
being only ascribed to one way of solving for a variable. A follow-up question could be: When would Julio’s
approach be less efficient?)
Vignette 2: Can we get a zero while solving?
Rafael and Roman, the twins in the class, solved this equation differently.
Rafael did it this way and got stuck. He’s convinced he right so far, but wasn’t sure how to finish.
7x +10 = 2x
-2x
- 2x
5x +10 = 0
Roman also believes he’s right and has done this so far:
7x +10 = 2x
-2x
- 2x
5x = 10
5x 10
=
5
5
x=2
The other students in the class are yelling, “You’re both wrong”. But the twins are determined and insisted that they
maintained equality.
Help figure out what’s happening here. What are the issues? Did they both maintain equality? How do you know?
Vignette 3: What went wrong?
Natasha needs your help. She did the following work:
3x +9 = 27
3x
27
+9=
3
3
x+9 =9
x+9 =9
-9 -9
x=0
Check:
x=0
3x +9 = 27
3 ( 0 ) + 9 = 27
0 + 9 = 27
9 = 27
This is not equal!
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Natasha said, “What did I do wrong? I thought I could divide first. Was I right or wrong?”
Help Natasha sort out her situation.
Vignette 4: Individual Performance Task – Jose’s Equation
Jose says to Jill, “I can solve for the variable in this equation in at least 4 different ways.” Jill says, “No, you can’t.”
Jose’s equation is: -4(x+2) = 8x +16
Prove if Jose is right or wrong?
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Flexibility with Equations
Student Activity Sheet
Name_______________________
Date________________________
Vignette 1: Done two ways, but who’s correct?
Julio and Lilly are both in your group. You all solved the same equation. When Julio looked at Lilly’s work he noticed his
work was different. They both got the same answer. Julio says his way is right, but Lilly just thinks he got lucky.
Lilly’s Work
Julio’s Work
15 = 3 ( x - 2 )
15 = 3 ( x - 2 )
15 = 3(x) - 3(2)
15 3 ( x - 2 )
=
3
3
5= x-2
15 = 3x - 6
15 = 3x - 6
+6
+6
21 = 3x
21 3x
=
3
3
7= x
5= x-2
+2
+2
7= x
Help them figure out how to make sense of this situation.
8
Flexibility with Equations
Student Activity Sheet
Name_______________________
Date________________________
Vignette 2: Can we get a zero while solving?
Rafael and Roman, the twins in the class, solved this equation differently.
Rafael did it this way and got stuck. He’s convinced he right so far, but wasn’t sure how to finish.
7x +10 = 2x
-2x
- 2x
5x +10 = 0
Roman also believes he’s right and has done this so far:
7x +10 = 2x
-2x
- 2x
5x = 10
5x 10
=
5
5
x=2
The other students in the class are yelling, “You’re both wrong”. But the twins are determined and insisted that they
maintained equality.
Help figure out what’s happening here. What are the issues? Did they both maintain equality? How do you know?
9
Flexibility with Equations
Student Activity Sheet
Name_______________________
Date________________________
Vignette 3: What went wrong?
Natasha needs your help. She did the following work:
3x +9 = 27
3x
27
+9=
3
3
x+9 =9
x+9 =9
-9 -9
x=0
Check:
x=0
3x +9 = 27
3 ( 0 ) + 9 = 27
0 + 9 = 27
9 = 27
This is not equal!
Natasha said, “What did I do wrong? I thought I could divide first. Was I right or wrong?”
Help Natasha sort out her situation.
10
Flexibility with Equations
Student Activity Sheet
Name_______________________
Date________________________
Vignette 4: Individual Performance Task – Jose’s Equation
Jose says to Jill, “I can solve for the variable in this equation in at least 4 different ways.” Jill says, “No, you can’t.”
Jose’s equation is: -4(x+2) = 8x +16
Prove if Jose is right or wrong?
11
Lesson 7: Literal equations and rearranging formulas
Opening Activity:
Solve each equation for 𝒙. For part (c), remember a variable symbol, like 𝒂, 𝒃, and 𝒄, represents a number.
a. 𝟐𝒙−𝟔=𝟏𝟎
b. −𝟑𝒙−𝟑=−𝟏𝟐
c. 𝒂𝒙−𝒃=𝒄
(Note to teacher: Since we are building the relationship between arithmetic ways of working and algebraic ways of
working we want students to see that the way you approach solving an equation with numbers is the same as you do
with variables.)
Activity 2:
Solve for P in the simple interest formula: A = P(1 + rt)
Now here is your challenge try to isolate the t.
Solve 3x + 4 = 6 – 5x for x.
Solve ax + b = d – cx for x.
(Note to teacher: The big step in this one will occur with the need to factor out the x in ax + cx so the student will be able
to isolate the x. The way students can understand this is when he/she solves the equation itself and you combine 3x and
5x it is the same as writing x(3 + 5), which is what must be done in the literal equation. This example is taken from
Module 1 page 222 if you want to view the work.)
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Activity 3:
Vignette: This is confusing. Can you help us?
Michael and Alicia are working together and are complaining about rearranging formulas.
Their teacher wrote the following on the board:
y = 𝑚𝑥 + 𝑏
𝑦
b = 𝑚𝑥
x=
𝑦+𝑏
𝑚
Are all the statements true? Justify your answer. If any of them is wrong correct it and explain
what you did?
Michael said to Alicia, “We better find someone to help us” They come over to you. Help them make sense of
this problem.
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Lesson 8: Inequalities
Inequalities
Teacher Guide
Getting to Know Inequalities
Opening Activity
What do the ideas of “equality” and “inequality” mean to you?
Think about the ideas beyond mathematics. How do you see these ideas within mathematics?
(To the Teacher: The purpose here is for students to think about these ideas first in a global way then within the context
of mathematics. These ideas are inherent to the world students live in so you have a chance to hear how they think about
it. This activity is also a set up for the next activity where students are asked to group different statements. Will students
think about a group called “equality” and a group called “inequality”?
Activity 2
Look at the different statements below. Group the ones you think belong together. Any statement can be in more than
one group.
1)
2)
3)
4)
5)
6)
7)
You need to be more than 5 ft. tall to go on the ride at Great Adventures.
The elevator can hold up to 1500 pounds.
John earned $40,000 last year.
It will take us at least 10 days and no more than 12 days to drive from NY to Los Angeles.
He lost 20 pounds last year.
Lilliana falls in the group of people that earn greater than $50,000 and less than $100,000.
Miguel is 5’5” tall and Jamal is 5’8” tall.
(Note to teacher: The goal of this activity is to see how students observe relationships. Different answers are possible and they all can
be valued. It will be interesting to see if students can separate the statements that represent an inequality from the other statements
that represent equalities. If that doesn’t occur be ready to present those two groups and ask the students why you grouped them in
that manner. The third activity should occur on a separate page so students would not have it to look at when they do the second
activity Also, if you can represent some of the statements in the form of a picture that would be great.)
14
Activity 3
Now let us look at these two groups and I want you to think about how you can represent them mathematically.
Let’s start with the group we called “equalities”. How would you represent each of them mathematically, without any
words?
(To the Teacher: Give students about one minute and then there can be a quick discussion about what makes them
equalities and how we represent them.)
Now let’s look at the other group. With your group mates decide how to represent each of the events mathematically
without any words.
(Note to teacher: Discuss the different results. You can ask questions like, “What is the difference between h>5 and h>5?” or “Why
wasn’t h>5 appropriate for statement 1?” or “what would you have to change in the first statement to make the result h>5?”)
We call all these type of statements inequalities. Can you come up with all the possible types of situations that can
occur? For example we can have “greater than” represented as c>2. (Note to teacher: Have the students share out all the
possibilities they can come up with.)
Activity 4
Let us look at these different symbols. Do any of these have the same meaning? Justify your answer.
<
<
>
>
<
<
<
<
=
(To the Teacher: An interesting discussion can occur about which mean the same. A question that can help kids think
about this is: If x < 4 what values of x make that statement true?)
Activity 5
Now you are going to create two “real world” inequality statements in words that your classmates have to write in
algebraic form. See if you can stump your classmates.
(To the Teacher: Have students present their ideas to each other. An issue can come up if they use ideas that can only be represented
as whole numbers, such as number of people. I would let them think about if there is an issue with this situation. It will be looked at in
an upcoming lesson.)
15
Student Activity Sheet
Getting to Know Inequalities
Name_______________________
Date________________________
Opening Activity
What do the ideas of “equality” and “inequality” mean to you?
Think about the ideas beyond mathematics. How do you see these ideas within mathematics?
Activity 2
Look at the different statements below. Group the ones you think belong together. Any statement can be in more than
one group.
1)
2)
3)
4)
5)
6)
7)
You need to be more than 5 ft. tall to go on the ride at Great Adventures.
The elevator can hold up to 1500 pounds.
John earned $40,000 last year.
It will take us at least 10 days and no more than 12 days to drive from NY to Los Angeles.
He lost 20 pounds last year.
Lilliana falls in the group of people that earn greater than $50,000 and less than $100,000.
Miguel is 5’5” tall and Jamal is 5’8” tall.
16
Activity 3
Now let us look at these two groups and I want you to think about how you can represent them mathematically.
Let’s start with the group we called “equalities”. How would you represent each of them mathematically, without any
words?
Now let’s look at the other group. With your group mates decide how to represent each of the events mathematically
without any words.
We call all these type of statements inequalities. Can you come up with all the possible types of situations that can
occur? For example we can have “greater than” represented as c >2.
17
Activity 4
Let us look at these different symbols. Do any of these have the same meaning? Justify your answer.
<
<
>
>
=
≠
<
<
>
>
Activity 5
Now you are going to create two “real world” inequality statements in words that your classmates have to write in
algebraic form. See if you can stump your classmates.
18
Lesson 9: Inequalities Continued
Inequality
Teacher Guide
Another Representation of Inequalities
Opening Activity
Look at the following number lines and describe the similarities and differences that you notice. What do you think each
of them means? Why?
x
-1
0
1
2
3
4
x
-1
0
1
2
3
4
x
-1
0
1
2
3
4
x
-1
0
1
2
3
4
(Note to teacher: A goal here is to find out if students can transfer their knowledge from the first lesson and transfer it to the new
ideas of number lines. Have students share their ideas and help them to discuss the differences and what they think each of them
means.)
Activity 2
How do you think you would represent each of the inequalities we just looked at mathematically? Work with each graph
and represent them algebraically.
(Note to teacher: Have students share and explain their reasoning.)
19
Activity 3
Look at the following number lines and describe the similarities and differences that you notice. How do these number
lines differ from the previous set of number lines? How will that affect how you represent them algebraically? Then
represent them algebraically
x
-1
0
1
2
3
4
x
-1
0
1
2
3
4
x
-1
0
1
2
3
4
x
-1
0
1
2
3
4
(Note to teacher: Again it will be important to see how students transfer their knowledge from the previous lesson. It is important
that students can explain their reasoning This is a new idea in representing them algebraically so students will probably need
support)
Activity 4
You are going to be given a set of scenarios. Your job is to represent them both algebraically and graphically.
1) You have to give at least five dollars if you want to join the group.
2) In order for the school to hold the prom more a minimum of $500 is needed and the maximum amount
needed is $2000.
3) At the trip to the ice skating rink we were told we could stay on the ice for at most three hours.
(Note to teacher: Again it would be good to have students share their thinking and reasoning. The last one is interesting. Which is a
better answer x<3 or 0<x<3?)
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Final Activity: Performance Task- Create a Scenario
Create a scenario for each of these inequalities
1) x < 100
2) 11 < x < 19
3) 0 < x < 1000
21
Student Activity Sheet
Another Representation of Inequalities
Name_______________________
Date________________________
Opening Activity
Look at the following number lines and describe the similarities and differences that you notice. What do you think each
of them means? Why?
x
-1
0
1
2
3
4
x
-1
0
1
2
3
4
x
-1
0
1
2
3
4
x
-1
0
1
2
3
4
Activity 2
How do you think you would represent each of the inequalities we just looked at mathematically? Work with each graph
and represent them algebraically.
22
Activity 3
Look at the following number lines and describe the similarities and differences that you notice. How do these number
lines differ from the previous set of number lines? How will that affect how you represent them algebraically? Then
represent them algebraically
x
-1
0
1
2
3
4
x
-1
0
1
2
3
4
x
-1
0
1
2
3
4
x
-1
0
1
2
3
4
Activity 4
You are going to be given a set of scenarios. Your job is to represent them both algebraically and graphically.
1) You have to give at least five dollars if you want to join the group.
2) In order for the school to hold the prom more a minimum of $500 is needed and the maximum amount
needed is $2000.
3) At the trip to the ice skating rink we were told we could stay on the ice for at most three hours.
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Final Activity: Performance Task- Create a Scenario
Name_______________________
Date________________________
Create a scenario for each of these inequalities
1) x < 100
2) 11 < x < 19
3) 0 < x < 1000
24
Lesson 10: AND & OR
(Note to Teacher: We want students to be able to represent solutions to inequalities in multiple ways. For example
-3<x<2 can also be written as (-3,2] or it can be written as x>-3 AND x< 2. If a student has a solution like x>12 you might
want to show them that it can also be written as (12,∞).)
Teacher Guide
Representing sets algebraically and graphically
Opening Activity: Read the following story then answer the questions
Story: Jose, Latoya, Robert and Mai are having a conversation about numbers.
Jose: I am thinking of all the numbers that are greater than 3 and less than 20.
Latoya: I am thinking of all the numbers that are less than 3 and greater than 20.
Robert: I am thinking of all the numbers that are in both of your groups
Mai: I am thinking of all the numbers that are in either Jose’s group or Latoya’s group.
What are the numbers in Robert’s mind? What are the numbers in Mai’s mind?
(Note to Teacher: The groups should share their findings with reasoning and a discussion should arise about the differences between
the ideas of Robert and Mai. How would you represent your solutions mathematically? The solutions would probably be best
represented in sets with Robert’s solution being an empty set.)
Activity 2: What is the difference between and and or?
The teacher has to choose who would be given the candy bar. She placed 10 pieces of paper in a bag, each paper
numbered 1-10.
The two students will pick two numbers. Who has a better chance of winning?
Person A: To win the game he or she must pick a 2 or a 6.
Person B: To win the game he or she must pick a 2 and a 6.
What can you now say is the difference between “and” and “or”?
25
Activity 3:
Given the following information, graph the solution to this problem. Explain what you did and why you did it.
x >3 /\ x < 8 (Note: /\ means and)
(Note to Teacher: It is always important to see how students take a new type of question in which they have the knowledge to
answer it yet have not seen the question in this form.)
Could you come up with another way of writing the given statement algebraically using just one x? Explain why you
think your answer is correct.
Activity 4:
We are going to look at a similar problem from the one you worked on previously except we are going to change one
thing. Try to represent graphically and be ready to discuss how the problem was different and the effect it had on the
solution.
x >3 \/ x < 8 (Note: \/ means or)
Activity 5:
Look at the following problem. How would you graph it?
x < 3 \/ x > 8
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Final Activity
Now come up with the graphical representation of the solution for this problem.
-2 > x /\ 18 < x
(Note to Teacher: This task has no solution. Let us see what the students do with this. For more examples look at the at the Module 1
student materials page S.93 – S.95)
Journal Writing:
Look at the different tasks you experienced today. Write in your words the effect of And and Or on an inequality
statement.
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Student Activity Sheet
Name_______________________
Date________________________
Opening Activity: Read the following story then answer the questions
Story: Jose, Latoya, Robert and Mai are having a conversation about numbers.
Jose: I am thinking of all the numbers that are greater than 3 and less than 20.
Latoya: I am thinking of all the numbers that are less than 3 and greater than 20.
Robert: I am thinking of all the numbers that are in both of your groups
Mai: I am thinking of all the numbers that are in either Jose’s group or Latoya’s group.
What are the numbers in Robert’s mind? What are the numbers in Mai’s mind? justify your answer.
Activity 2: What is the difference between and and or?
The teacher has to choose who would be given the candy bar. She placed 10 pieces of paper in a bag, each paper
numbered 1-10.
The two students will pick two numbers. Who has a better chance of winning?
Person A: To win the game he or she must pick a 2 or a 6.
Person B: To win the game he or she must pick a 2 and a 6.
28
What can you now say is the difference between “and” and “or”?
Activity 3:
Given the following information, graph the solution to this problem. Explain what you did and why you did it.
x >3 /\ x < 8 (Note: /\ means and)
Could you come up with another way of writing the given statement algebraically using just one x? Explain why you
think your answer is correct.
29
Activity 4:
We are going to look at a similar problem from the one you worked on previously except we are going to change one
thing. Try to represent graphically and be ready to discuss how the problem was different and the effect it had on the
solution.
x >3 \/ x < 8 (Note: \/ means or)
Activity 5:
Look at the following problem. How would you graph it?
x < 3 \/ x > 8
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Final Activity
Now come up with the graphical representation of the solution for this problem.
-2 > x /\ 18 < x
Journal Writing:
Look at the different tasks you experienced today. Write in your words the effect of And and Or on an inequality
statement.
31
Lesson 11: How does the domain change our thinking and affect the
solution sets?
Teacher Guide
How does the domain change our thinking and affect the solution sets?
Opening Activity:
Look at the two following questions and explain how they are different including their solutions:
How many whole numbers are there between 3 and 4?
How many numbers are there between 3 and 4?
(To the Teacher: This can lead to a discussion about how solution sets differ based on a given constraint. Here you can
define the notion of a domain as a given constraint. When we move into functions we will expand the definition. The
domains in inequalities affect the solution sets. In activity one we see that in the first example there are an infinite
amount of real numbers between any two real numbers but there are no whole numbers between 3 and 4.. This should
be part of the discussion about the domain affecting the solution set.)
Activity 2:
With your partner look at the next two problems, find the solution set and graph the result. Be ready to explain your
reasoning for each solution
All numbers greater than 4 and less than 7
All whole numbers greater than 4 and less than 7.
(To the Teacher: I would have students share. It is important to see that the work we already did up to this point were all
(real) numbers so they know that already. Will they be able to differentiate between the two problems, how they write
the solution and how they graph the solution?)
Activity 3:
Look at the following solution sets and describe the difference. Describe what the domains could be for each of these
statements.
3 < x < 19
3,4,5,6,…19
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Activity 4:
For the inequality
-3 < x < 8
Graph the solution if the domain is:





all whole numbers {0, 1, 2, 3, …}
All numbers on the number line {All real numbers)
All positive integers less than 5
All negative numbers
All integers less than 8
Activity 5:
Look at the following graphs of solution sets. Jose said that the set of all integers could be the domain for both graphs
while Natasha felt that there is no domain that could have worked for both. What do you think?
x
-1
0
2
1
3
4
x
-1
0
1
2
3
4
Final Activity:
For each graph the following describe the domains and solution sets for each. Then graph the solution sets
1)
2)
3)
4)
5)
6)
You need to be at least 5 ft. tall to go on the ride at Great Adventures.
The elevator can hold up to 1500 pounds.
He has at most 10 tickets to give away.
It will take us at least 10 days and no more than 12 days to drive from NY to Los Angeles.
The store has at least 50 different bicycles to choose from but no more than 75.
Lilliana falls in the group of people that earn between $50,000 and $100,000.
33
Student Activity Sheet
How does the domain change our thinking and affect the solution sets?
Name_______________________
Date________________________
Opening Activity:
Look at the two following questions and explain how they are different including their solutions:
How many whole numbers are there between 3 and 4?
How many numbers are there between 3 and 4?
Activity 2:
With your partner look at the next two problems, find the solution set and graph the result. Be ready to explain your
reasoning for each solution
All numbers greater than 4 and less than 7
All whole numbers greater than 4 and less than 7.
34
Activity 3:
Look at the following solution sets and describe the difference. Describe what the domains could be for each of these
statements.
3 < x < 19
3,4,5,6,…19
Activity 4:
For the inequality
-3 < x < 8
Graph the solution if the domain is:





all whole numbers {0, 1, 2, 3, …}
All numbers on the number line {All real numbers)
All positive integers less than 5
All negative numbers
All integers less than 8
35
Activity 5:
Look at the following graphs of solution sets. Jose said that the set of all integers could be the domain for both graphs
while Natasha felt that there is no domain that could have worked for both. What do you think?
x
-1
0
2
1
3
4
x
-1
0
1
2
3
4
36
Final Activity:
For each graph the following describe the domains and solution sets for each. Then graph the solution sets
1) You need to be at least 5 ft. tall to go on the ride at Great Adventures.
2) The elevator can hold up to 1500 pounds.
3) He has at most 10 tickets to give away.
4) It will take us at least 10 days and no more than 12 days to drive from NY to Los Angeles.
5) The store has at least 50 different bicycles to choose from but no more than 75.
6) Lilliana falls in the group of people that earn between $50,000 and $100,000.
37
Lesson 12: Can you discover how to solve inequalities?
Teacher Guide
Can you discover how to solve inequalities?
Opening Activity:
Look at this statement. What does it mean?
X+3>8
What are possible solutions to this problem? How did you get it?
How many solutions does it have? How do you know?
How does x + 3 > 8 compare with x + 3 = 8? How are they similar and how are they different?
How can you use your knowledge of equations to solve the inequality?
(To the Teacher: A good discussion should arise from these questions. You want students to think about how equations
and inequalities are similar and different.)
How would the solution sets differ if the domain was all whole numbers versus the domain was all real numbers?
Activity 2
Look at these two statements: You know how to solve the equation now see if you can figure out how to solve the
inequality.
5x + 4 = 24
5x + 4 < 24
What is the solution for each of them? How are the solutions similar and how are they different?
38
How can we be sure that the solution(s) are correct?
(To the Teacher: Here it is important to help students to check their answers to inequalities to make sure that their
answer makes sense. This will become very important when we multiply or divide by a negative.)
If you were asked to compare how you solved an equation and how you solved an inequality what would you say? How
are they similar and how are they different?
Activity 3: An Outlier
Solve the following inequality, find your solution and check if your answers make sense?
-3x < 12
(To the Teacher: This should lead to an interesting discussion when students find that their solution did not work? Why
didn’t work? What would be the correct solution? Why does this inequality act differently from the other inequalities?
Can you come up with a possible conjecture about certain inequalities/? Can you test out your ideas on another problem
and see if your idea works? Discuss the different results.)
Activity 4:
Now in your group compare these two inequalities. Can you explain how the solutions differed and why that happened?
-4x > -12
4x > -12
39
Activity 5:
In your group try these two problems and be ready to discuss the solutions
x > 12
3
x > 12
-3
Activity 6: Practice
Solve each of these inequalities. Show your work and explain how you know your solution set is correct. Assume the
domain is all real numbers unless another domain is specified.
3x – 6 > 12
-5x – 3 < 2x – 6
2x-3 > 9 AND -4 > -2x
4 – 2x > 3x – 6 (Domain is all whole numbers)
40
Journal Writing:
You have been asked to teach someone how to solve inequalities and why your approach works. Describe what you
would tell this person.
41
Can you discover how to solve inequalities?
Name_______________________
Date________________________
Opening Activity:
Look at this statement. What does it mean?
X+3>8
What are possible solutions to this problem? How did you get it?
How many solutions does it have? How do you know?
How does x + 3 > 8 compare with x + 3 = 8? How are they similar and how are they different?
How can you use your knowledge of equations to solve the inequality?
How would the solution sets differ if the domain was all whole numbers versus the domain was all real numbers?
42
Activity 2:
Look at these two statements: You know how to solve the equation now see if you can figure out how to solve the
inequality.
5x + 4 = 24
5x + 4 < 24
What is the solution for each of them? How are the solutions similar and how are they different?
How can we be sure that the solution(s) are correct?
If you were asked to compare how you solved an equation and how you solved an inequality what would you say? How
are they similar and how are they different?
43
Activity 3: An Outlier
Solve the following inequality, find your solution and check if your answers make sense?
-3x < 12
Activity 4:
Now in your group compare these two inequalities. Can you explain how the solutions differed and why that happened?
-4x > -12
4x > -12
Activity 5:
In your group try these two problems and be ready to discuss the solutions
x > 12
3
x > 12
-3
44
Activity 6: Practice
Solve each of these inequalities. Show your work and explain how you know your solution set is correct. Assume the
domain is all real numbers unless another domain is specified.
3x – 6 > 12
-5x – 3 < 2x – 6
2x-3 > 9 AND -4 > -2x
4 – 2x > 3x – 6 (Domain is all whole numbers)
45
Journal Writing:
You have been asked to teach someone how to solve inequalities and why your approach works. Describe what you
would tell this person.
46
Lesson 13: Inequality Problems
Teacher Guide
Inequality Problems
(To the Teacher: Here are three problems. They all are inequalities with multiple solutions. How will students
approach these problems. They could use tables or they can attempt algebraic approaches. Let them work one each
problem alone at first. Then after five minutes let them join a partner. There should be a class discussion for each
problem. You can use the third problem as an On Demand Group Performance Task. It can be though about in
multiple ways, and can be done both arithmetically and algebraically. How would students approach it?)
Opening Activity:
You want to join a health club to get back into shape.
The Tone Up Club charges $50 to join plus $9 per visit
Be in Shape has no sign up fee and charges $12 per visit.
You have a maximum of $200 to spend. Which club will you join? Why?
Activity 2:
A) You gave your brother a 50-foot start in a race. If he ran 10 feet every 5 seconds what is the minimum speed you
would have to run to catch him in at least 20 seconds from the time you both started?
B) How long would it take you to catch your brother if you ran as fast as Usain Bolt. Usain runs 300 feet in less than 10
seconds.
47
Student Activity Sheet
Inequality Problems
Name_______________________
Date________________________
Opening Activity:
You want to join a health club to get back into shape.
The Tone Up Club charges $50 to join plus $9 per visit
Be in Shape has no sign up fee and charges $12 per visit.
You have a maximum of $200 to spend. Which club will you join? Why?
48
Activity 2:
A) You gave your brother a 50-foot start in a race. If he ran 10 feet every 5 seconds what is the minimum speed you
would have to run to catch him in at least 20 seconds from the time you both started?
B) How long would it take you to catch your brother if you ran as fast as Usain Bolt. Usain runs 300 feet in less than 10
seconds.
49
Lesson 14: Final Performance Tasks/Project
Task 1:
When working with inequalities we saw that something unusual happened when we multiplied or divided by a negative
number. Explain what happens and why it is happening. You can use examples to support your thinking.
Task 2:
Ricardo was playing with blocks and created the following three cases.
If Ricardo uses 500 tiles, what is the maximum number of tiles he can have in the first vertical column?
Task 3:
Your class wants to go on a trip to Great Adventures. There are 25 students in your class. Tickets and bus ride cost $35 a
person. You are going to hold a cake sale at school. You were given a donation of cakes, cookie and soda by parents so
everything you sell will go towards the trip. You have to decide how much you will sell each item for so that you make
the minimum amount of money so that everyone could go on the trip. Come up with possible solutions to this problem.
50
Lessons 15 – 19: Algebraic Problem Solving
Algebraic Problem Solving – building expressions from tables working towards constructing equations
In this unit we want teachers to use the students with creating tables to help develop the expressions necessary to solve
algebraic problems…This method can help conceptualize for students how to create equations frol language..
The first problem is simple but is useful as a way of developing algebraic problem solving.
The second problem is a little more complicated. I will share possible problems you can work on with students in this
section of the unit.
1. Jose has twice as many cds as Nadia. If they have 39 cds together, how many cds does each have?
Let students play with this. They can use guess and check or any other method they want and have students present
their thinking.
If a table arose out of the student presentations, use it to get to the algebra. If not, use what is shown below.]
Here’s the problem again you just worked on. We’re going to revisit it in a different way.
1. Jose has twice as many cds as Nadia. If they have 39 cds together, how many cds does each have?
Let’s make a table to represent possible values for the cds of Nadia and Jose. We can pick any values for Nadia to try to
find the answer. If I say, Nadia has 6 cds, how many would Jose have? Does that give us 39? Now fill in the table using
any values for Nadia and calculate the corresponding value for Jose.
Nadia
6
Suppose the value is:
x
Jose
12
What goes here?
Use these variables to make an equation that will help us solve the problem.
Solve your equation. Compare it with the answer you got earlier.
2. Sonia has 4 more pennies than Rafael and Tashawn has three times as many pennies as Rafael. All together they
have 34 pennies. How many does each have? First try to solve this using guess and check. Then solve it by making an
equation.
51
3. Maria was collecting money for the homeless and she lost the can with the money. She is frantic. She is trying to
figure out as best she can how much money she had. She doesn’t want to cheat anyone but she doesn’t have much
money to spare. She knows she had nickels, quarters and dimes in the can the last time her friend checked an hour ago
she had five more dimes than quarters and an equal amount of quarters and nickels. Her friend also checked the total
amount and saw that there was $2.50. Other people have stopped by and put in the same amount of quarters and
nickels that there was when her friend checked earlier. Three people gave dimes. She estimates that at most there was
$4.00 in the can. Is she correct? Can you help Maria and suggest how much she should give to the homeless based on
the information she had?
4. Lulu tells her little brother, Jack, that she is holding 20 coins all of which are dimes and quarters. They have a value of
$4.10. She says she will give him the coins if he can tell her how many of each she is holding. Solve this problem for Jack.
5. There are 50 cows and chickens on the farm . Jose forgets how many of each animal he has but he was told that the
total number of legs of all the animals is 148. How many cows and how many chickens are on the farm?
6. 16 years from now, Pia’s age will be twice her age 12 years ago. Find her present age.
7. Jack is 27 years older than Susan. In five years Jack will be 4 times as old as Susan. How old is Jack and Susan
8. Five years from now, the sum of the ages of a woman and her daughter will be 40 years. The difference in
their present age is 24 years. How old is her daughter now?
9. Find three consecutive odd integers whose sum is 71.
10. Find three consecutive even integers in which twenty more than twice the sum of third is 28 more than the sum of
the first two integers.
11. Donna wants to make trail mix made up of almonds, walnuts and raisins. She wants to mix one
part almonds, two parts walnuts, and three parts raisins. Almonds cost $12 per pound, walnuts
cost $9 per pound, and raisins cost $5 per pound. Donna has $15 to spend on the trail mix. Determine how many pounds
of trail mix she can make.
52
(Note to Teacher: Lesson 20 – 26 are on the following:
2 lessons- Solving linear systems algebraically-review from last year and graphically
2 lesson solving systems of inequalities
2 lessons Application of these ideas in context.)
Lesson 20: Revisiting Linearity
Opening Activity: The Sum of two numbers is 25. What are possible solutions to this question? Graph your solutions.
(Note to teacher: use this activity as an opportunity to revisit Linearity. You can have the students create the equation,
or talk about slope in the context of the situation.)
Activity 2: Look at Module 1, Lesson 21, in the student materials on page 115. This is an opportunity to revisit
relationships between equations and graphs. Add tables to this to show how they are all connected.
Activity 3: Practice if necessary.
Lesson 21: Solution Sets to Inequalities
Opening Activity: Circle each ordered pair that is a solution to the equation y > 4x - 10.
(3,2) (2,3) (-1, -14) (0,0) (1, -6) (5, 10) (0, -10) (3, 4) (6, 0) (4, -1) (0, 8) (-2, 1) (5, 2) (3, 6) (-6, 12) (-8, 14) (-9,0)
Activity 2: Plot all the solutions on the coordinate plane. Then graph the equation y > 4x - 10.
What do you observe about the relationship between the solution points and the graph of the line?
(Note to the teacher: This is your opportunity to have a discussion about solution sets to an inequality and how we
graph them. It is also an opportunity to relate the inequality symbol to the idea of “above” and “below”. Students can
also do a test.)
Activity 3: Practice with graphing inequalities
53
Lesson 22: Solving Systems of Equations
Opening Activity: Graph these two equations. y = 3x - 2 and y = x + 4
Write observations. What do you see?
Activity 2: Using the same two equations, how could you combine them into one equation?
(Note to the teacher: This might be a good question to ask: Since both 3x – 2 and x + 4 both equal to y what can you say
about 3x – 2 and x + 4?)
Practice solving through both graphing and substitution.
(Note to Teacher: Teacher should include problems where students need to rearrange the equation to put it into y = mx +
b.)
(Note to Teacher: We will be working on lessons 23 – 26 and will get it to you within the next
few weeks.)
Lesson 23: Solving Systems of Inequalities
Lesson 24 – 26: Applying Systems in Contextual Situations
1. Find two numbers such that the sum of the first and three times the second is 5 and the sum of second and two
times the first is 8.
54
Appendix 1:
Teachers Guide
Does order matter?
Opening Activity
(To the Teacher: In today’s activities you will be helping students develop a deeper understanding of the
commutative and associative properties. Based on their understanding of the variable students should be able to
generalize arithmetic ideas. The following activities should be performed in groups. You should bring the class
together after each activity to lead a discussion about their findings and understanding.
In the opening activity we’re trying to focus the students’ attention to behavior, because the properties of
commutativity and associativity can be thought of as descriptors of the behaviors.)
Select the question that is most appealing to you:
- Do you think boys behave differently around other boys than they do girls? If so describe the difference in that
behavior and give it a .
- Do you think girls behave differently around other girls than they do boys? If so describe that behavior and give
it a .
- Do you think teenagers behave differently when they are with their friends than when they are with their
families? If so describe that behavior and give it a .
- Do you think high school students behave differently at school than they do outside school? If so describe that
behavior and give it a .
Be prepared to talk about the question you chose.
(To the Teacher: Have students share out their thinking on these questions, how they described the change in
behavior, and how the they gave them. This sequence of observing, describing, naming behavior is basically what
the students will be doing in the rest of the lesson. They will observe how the operations behave in different
situations, describe that behavior, and acquire a for that behavior.
Before transitioning into the next activity you could point out that what they did in this opening activity, generalizing
a human behavior and naming it, is a focus in psychology. Also, if this is something that interests them they could
consider it as a major in college.)
55
Second Activity
(To the Teacher: As human beings we all behave differently in different situations. How do we behave in the
three situations below? In two of them we will always do one before the other. In one of them it could differ each
day.)
1) You’re going to be given three events. You need to determine if the order in which you do these activities will affect
the overall result. In other words can you do either a first or b first and still get the same result.
a. Scrape the dinner dishes
b. Wash the dinner dishes
a. Wash your face
b. Brush your teeth
a. Drain the used oil from the car’s engine
b. Add four quarts of new oil
1) Which of these events would work best as a model for addition of two numbers? Explain your answer.
2) If a and b represent any two numbers is it always true that a + b = b + a? Explain.
(To the Teacher: You need to give this property its . Consider asking the students, “Why would you need to this
characteristic of addition?” Communication is an important and often over looked aspect of mathematics.
Through the discussion help them understand that mathematical vocabulary is used to generalize whole
mathematical concepts for increased efficiency in our communication. This way we can use the vocabulary
instead of non-specific references like, “that thing” or “that behavior of addition”. Additionally, giving things a
acts in a similar fashion to the variable. It allows us to speak generally of every instance of that specific concept.
Talking about these ideas will help answer the perpetual question of “Why?” and hopefully help expand their
understanding of the variable. Show them the Commutative Law for Addition: a + b = b + a.)
56
Third Activity
Fill in the following table:
(To the Teacher: Feel free to change the values in the table as you see fit. However, with the values in the table
it’d be good for students to have the opportunity to realize when commutativity holds true for subtraction and
division as well as what’s special about those cases. This will also set you up to ask some additional questions to
students as you are entering into their investigation.)
a
1
2
3
4
4
1.5
6
5
5
b
2
2
3
3
4
3
-3
-5
10
a+b
b+a
a-b
b-a
a*b
b*a
a÷b
b÷a
Observe your table. Write down what you notice.
(To the Teacher: A way to help students focus their observations is to ask them, “What’s the focus of in filling out
this table?” Some students will have astute answers to this question, while other may need some more help. We
want to help students to grow in their ability to make observations that are meaningful to the task at hand. This
is where the focusing on the idea of behavior in the opening activity comes in handy, because here we are really
wanting to make observations about how the operations are behaving when we switch the numbers being
evaluated.)
What is your conjecture about the different operations when you switch the order of the numbers? Will your conjecture
work with fractions? Decimals?
To be able to prove a law doesn’t work in mathematics, you just need one example. Give an example for each of the
operations for which this law doesn’t work.
Make a general statement for when the commutative property (switching the order of the numbers) holds for a given
operation, e.g. addition, subtraction, multiplication, division?
Fourth Activity
The commutative property says that you can switch the order of the numbers for a given operation and still get the
same result. Now let’s look at another property.
Are these true?
57
?
4 + (5 + 6) = (4 + 5) + 6
?
1 + (8 + 3) = (1 + 8) + 3
?
9 + (7 + 2) = (9 + 7) + 2
(To the teacher: The above is formatted using a borderless table.)
1) If you replaced the numbers with different numbers would it still be true? Justify your answer?
2) Can you find an example that won’t work?
3) This called the associative property. It says, we can group numbers differently and still get the same result. Describe
how this definition of the property is related to the three examples you worked with.
Try this property with the other three operations.
Subtraction: Do you think the associative property is true of subtraction? Why or why not?
Test it out:
?
4 – (5 – 6) = (4 – 5) – 6
Multiplication: Do you think the associative property is true of subtraction? Why or why not?
Test it out:
?
4 ∗ (5 ∗ 6) = (4 ∗ 5) ∗ 6
Division: Do you think the associative property is true of subtraction? Why or why not?
Test it out:
?
4 ÷ (5 ÷ 6) = (4 ÷ 5) ÷ 6
What connections do you see among the operations that the associativity property holds true for and among those that
it doesn’t? How would you explain this connection?
(To the Teacher: Students could make any number of observations about the connection between addition and
multiplication and the associative property holding true for them, but not for division and subtraction. However,
what’s most interesting to consider here is the connection between the commutative property and the
associative property. Could the associative hold true if the commutative property doesn’t hold true for a give
operation? It would be a wonderful question to consider, but not resolve with students.)
58
What general statement can you make now about the associativity property and the four operations?
(To the teacher: Have the students share out their thoughts on the connections as well as their general
statements. Memorialize some of their general statements to keep up in the classroom.)
59
Does order matter?
Student Activity Sheet
Name_______________________
Date________________________
Opening Activity
Select the question that is most appealing to you:
- Do you think boys behave differently around other boys than they do girls? If so describe the difference in that
behavior and give it a .
- Do you think girls behave differently around other girls than they do boys? If so describe that behavior and give
it a .
- Do you think teenagers behave differently when they are with their friends than when they are with their
families? If so describe that behavior and give it a name.
- Do you think high school students behave differently at school than they do outside school? If so describe that
behavior and give it a name.
Be prepared to talk about the question you chose.
60
Second Activity
1) You’re going to be given three events. You need to determine if the order in which you do these activities will affect
the overall result. In other words can you do either a first or b first and still get the same result.
a. Scrape the dinner dishes
b. Wash the dinner dishes
a. Wash your face
b. Brush your teeth
a. Drain the used oil from the car’s engine
b. Add four quarts of new oil
1) Which of these events would work best as a model for addition of two numbers? Explain your answer.
2) If a and b represent any two numbers is it always true that a + b = b + a ? Explain.
61
Third Activity
Fill in the following table:
a
b
1
2
2
2
3
3
4
3
4
4
1.5
3
6
-3
5
-5
5
10
a+b
b+a
a-b
b-a
a*b
b*a
a÷b
b÷a
Observe your table. Write down what you notice.
What is your conjecture about the different operations when you switch the order of the numbers? Will your conjecture
work with fractions? Decimals?
62
To be able to prove a law doesn’t work in mathematics, you just need one example. Give an example for each of the
operations for which this law doesn’t work.
Make a general statement for when the commutative property (switching the order of the numbers) holds for a given
operation, e.g. addition, subtraction, multiplication, division?
63
Fourth Activity
The commutative property says that you can switch the order of the numbers for a given operation and still get the
same result. Now let’s look at another property.
Are these true?
?
4 + (5 + 6) = (4 + 5) + 6
?
1 + (8 + 3) = (1 + 8) + 3
?
9 + (7 + 2) = (9 + 7) + 2
1) If you replaced the numbers with different numbers would it still be true? Justify your answer?
2) Can you find an example that won’t work?
3) This called the associative property. It says, we can group numbers differently and still get the same result. Describe
how this definition of the property is related to the three examples you worked with.
64
Try this property with the other three operations.
Subtraction: Do you think the associative property is true of subtraction? Why or why not?
Test it out:
?
4 – (5 – 6) = (4 – 5) – 6
Multiplication: Do you think the associative property is true of subtraction? Why or why not?
Test it out:
?
4 * (5 * 6) = (4 * 5) * 6
Division: Do you think the associative property is true of subtraction? Why or why not?
Test it out:
?
4 ÷ (5 ÷ 6) = (4 ÷ 5) ÷ 6
65
What connections do you see among the operations that the associativity property holds true for and among those that
it doesn’t? How would you explain this connection?
What general statement can you make now about the associativity property and the four operations?
66
Are both methods equivalent?
Teachers Guide
Opening Activity
You and your friend Shawn are selling boxes of candy on the bus/subway. You begin your speeches by saying, “I’m not
going to lie to you. I’m not selling candy for my team, my church or any other organization. I’m selling this for me, to
keep me off the street.” At the end of the week, your plan is to share the profits equally. For each box of candy sold,
you and Shawn make $7. You sold 18 boxes and Shawn sold 23 boxes.
Shawn’s method:
Shawn chose to calculate the amount you and he will earn by summing the total number of boxes sold and
multiplying 7.
Your method:
You figured out your own earnings before you figured out Shawn’s earnings. Then you added your
earnings and Shawn’s earnings together.
Compare your results with Shawn’s. What do you notice?
Which method makes more sense to you? Why does it make more sense?
1) Let m stand for the number of boxes you sold, s stand for the number Shawn sold and let c stand for the amount you
earn on each box.
Write an algebraic expression showing the method that Shawn used.
Then write an algebraic expression method what you used.
Write an equation that includes the information from both of your methods.
(To the teacher: You’re looking for students to write:
c(m + s)
cm + cs
c(m + s) = cm + cs
As you know, this third statement is the general equation for the distributive property of multiplication over
addition. How do you ensure that students understand this important idea? The following activity might help)
2) Experiment with some other numbers to see if your equation holds true.
67
Let:
m = 4, s = + 6 , and c = 9
Let:
m = 3, s = -2, and c = -4
Do you think this will hold true for any kind of numbers we use? What makes you think this?
4) Do you think this property is true for multiplication over subtraction? Try several examples and report your
conclusion.
Second Activity
(To the Teacher: In this activity we are developing the identity property. It is another idea that students have
seen before but one that is important in algebraic understanding and its relationship to the inverse property.)
1) Pick a number, any number. What would you add to this number so that the result does not change?
Try this with a couple other numbers.
1a) Write an equation with your number that shows this to be true. Is this true for any number you could choose? How
do you?
1b) Write a general equation. Use x represent to the numbers you and your classmates chose.
(To the Teacher: Let the students know that this is called the additive identity property and zero is the additive
identity element.)
2) Pick a number, any number. What would you multiply this number by such that the result does not change?
Try this with a couple other numbers.
2a) Write an equation with your number that shows this to be true. Is this true for any number you could choose? How
do you?
2b) Write a general equation. Use x represent to the numbers you and your classmates chose.
(To the Teacher: Let the students know that this is called the multiplicative identity property and one is the
multiplicative identity element.)
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Third Activity
Questions to ponder:
1) What can you add to 3 to get the additive identity as your answer?
2) What can you add to -4 to get the additive identity as your answer?
3) What can you add to x to get the additive identity as your answer?
4) Write a general statement and equation that describes this property.
5) Why do you think this is an important idea to know about addition?
(To the teacher: Answers could be of the form, “a number plus its opposite gives the additive identity element.”
x + (-x) = 0
x and (-x) are called additive inverses. A question you might ask for discussion is, “Why is this
property important for the understanding of addition of signed numbers?”)
Fourth Activity
Questions to Ponder
1) What can you multiply 3 by to get the multiplicative identity element?
(To the Teacher: As a heads up, students often struggle with making sense of multiplying to get 1. How can you
help them make sense of the notion of the getting to the identity.)
2) What can you multiply -4 by to get the multiplicative identity element?
3) What can you add to x to get the multiplicative identity element?
4) Write a general statement and equation that describes this property.
5) Why do you think this is an important idea to know about multiplication?
(To the Teacher: Answers could be of the form, “a number times its reciprocal gives the multiplicative identity element.”
x * (1/x) = 1
x and 1/x are called multiplicative inverses. Be sure to think about “Why is this an important idea for
students to think about?” beforehand.)
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Are both methods equivalent?
Student Activity Sheet
Name_______________________
Date________________________
Opening Activity
You and your friend Shawn are selling boxes of candy on the bus/subway. You begin your speeches by saying, “I’m not
going to lie to you. I’m not selling candy for my team, my church or any other organization. I’m selling this for me, to
keep me off the street.” At the end of the week, your plan is to share the profits equally. For each box of candy sold,
you and Shawn make $7. You sold 18 boxes and Shawn sold 23 boxes.
Shawn’s method:
Shawn chose to calculate the amount you and he will earn by summing the total number of boxes sold and
multiplying 7.
Your method:
You figured out your own earnings before you figured out Shawn’s earnings. Then you added your
earnings and Shawn’s earnings together.
Compare your results with Shawn’s. What do you notice?
Which method makes more sense to you? Why does it make more sense?
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1) Let m stand for the number of boxes you sold, s stand for the number Shawn sold and let c stand for the amount you
earn on each box.
Write an algebraic expression showing the method that Shawn used.
Then write an algebraic expression method what you used.
Write an equation that includes the information from both of your methods.
3) Experiment with some other numbers to see if your equation holds true.
Let:
m = 4, s = + 6, and c = 9
Let:
m = 3, s = -2, and c = -4
Do you think this will hold true for any kind of numbers we use? What makes you think this?
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4) Do you think this property is true for multiplication over subtraction? Try several examples and report your
conclusion.
Second Activity
1) Pick a number, any number. What would you add to this number so that the result does not change?
Try this with a couple other numbers.
1a) Write an equation with your number that shows this to be true. Is this true for any number you could choose? How
do you?
1b) Write a general equation. Use x represent to the numbers you and your classmates chose.
72
2) Pick a number, any number. What would you multiply this number by such that the result does not change?
Try this with a couple other numbers.
2a) Write an equation with your number that shows this to be true. Is this true for any number you could choose? How
do you?
2b) Write a general equation. Use x represent to the numbers you and your classmates chose.
73
Third Activity
Questions to ponder
1) What can you add to 3 to get the additive identity as your answer?
2) What can you add to -4 to get the additive identity as your answer?
3) What can you add to x to get the additive identity as your answer?
4) Write a general statement and equation that describes this property.
5) Why do you think this is an important idea to know about addition?
74
Fourth Activity
Questions to ponder
1) What can you multiply 3 by to get the multiplicative identity element?
2) What can you multiply -4 by to get the multiplicative identity element?
3) What can you add to x to get the multiplicative identity element?
4) Write a general statement and equation that describes this property.
5) Why do you think this is an important idea to know about multiplication?
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Two Coupons Mystery
Teachers Guide
(To the Teacher: Today’s lesson will be a group performance task. That means you will form groups of two or
three and ask them to work on the problem by themselves without your support. You want to make sure they
understand the problem so you can answer clarifying questions, but they must think about this with each other.
When there are 15 minutes left in the class you should bring the class together to begin to discuss the problem. It
should not be about answering all the questions, but around student thinking and process. This problem really
focuses on whether (50 * .20) – 5 ?=? (50 – 5) * .20. You could put this up on the board and ask them to explain
how it connects to the problem.)
Group Performance Task: Mystery of the Two Coupons
Stephanie and Julio go jeans shopping. They each have a 20% off coupon and a $5 off coupon. They each found a pair of
jeans that they really like for $50. When they compared receipts after going through the register, they saw that
Stephanie had paid less than Julio even though they had the same coupons and they each paid 10% in tax.
What happened at the register that made the difference? Support your answer with mathematical evidence.
What occurred here and why did it make a difference?
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Two Coupons Mystery
Student Activity Sheet
Names _____________________________________________________________________________
Date ________________________
Group Performance Task: Mystery of the Two Coupons
Stephanie and Julio go jeans shopping. They each have a 20% off coupon and a $5 off coupon. They each found a pair of
jeans that they really like for $50. When they compared receipts after going through the register, they saw that
Stephanie had paid less than Julio even though they had the same coupons and they each paid 10% in tax.
What happened at the register that made the difference? Support your answer with mathematical evidence.
What occurred here and why did it make a difference?
77
Four Laws of Equality
Teachers Guide
(To the Teacher: Our goal throughout the rest of the unit is twofold. First, we want students to have a meaningful
understanding of what equality is and how it is linked to solving. This is about understanding that as we make
changes through the operations, the two sides are still equal. So though we change the way they look, they still
remain the same in their equality. This powerful point often gets lost in student thinking. Secondly, we want
students to develop flexibility in their ability to approach solving equations. Developing both of these in our
students takes time and often teachers rush them to efficient means of solving an equation rather than letting
them work inefficiently for a while. Remember in the mess of inefficiency is where the long-term understandings
will be forged, so resist the urge to correct an inefficient method and stay focused on developing an
understanding of equality and flexibility. If you rush your students towards efficiency you will have enabled them
to repeat your methods, but will have stunted their ability to think flexibly about solving.)
Opening Activity:
Suppose we write: 3 + 8 = 4 + 7
Is this true? How do you know?
What does the symbol = mean, without using the word equal?
(To the Teacher: One of the first things to have a conversation about is how to refer to each side of the equal sign
and how you are going to talk about it as a class. This may sound trivial as it seems intuitive to refer to each side
of the equal sign using directional language such as left and right, but developing this language with the
students gives you the opportunity to get everyone in the room on the same page. Having this conversation up
front will save you from the natural consequences of assuming that everyone is with you on this one.
Once the language about how you are going to talk about the contents on each side of the equal sign is
established, it’s important to explore what the students think the meaning the equal sign means. You’ll learn a
lot about their concept of mathematical equality simply by asking.
We want to prepare them for solving equations and you can only understand the process of solving if you
understand the full meaning of equality.)
Now look at:
Is this still equal?
3+9
+4
=
4+8 +4
Why?
Try this with some other numbers, can you add any number to both sides and still maintain equality? Explain.
Can you come with a general statement to describe this law?
(To the Teacher: Have students share their statements. Pick one that best describes this property of equality,
write on big paper and place on the wall. Name it after the student who developed it, e.g. Jose’s Law.)
78
(To the Teacher: Through all of these examples the focus is not on getting through the examples, but seeing the
maintenance of equality as well as developing a sense of flexibility in the students. So have fun with it. Do some
nontraditional examples with really big or really small numbers or exponents. For instance with multiplication
you could do 1,000,000(3+9) = 1,000,000(4+8) or with subtraction use the same -9, but write it as -32.
Your students will pick the options they are comfortable with, but give them a larger perceptive on the power of
equality and the flexibility we do have when maintaining equality. You’ll need to exhibit the kind of flexibility you
want to see in your students understanding and you can help build some of that by encouraging them to think
outside of what’s comfortable or obvious.)
Now if we look at
3+9 -9 = 4+8 -9
Is this still equal?
Why?
Try this with some other numbers, can you subtract any number to both sides and still maintain equality? Explain.
Can you come with a general statement to describe this law?
(To the Teacher: Have students share their statements. Pick one that best describes this property of equality,
write on big paper and place on the wall. Name it after the student who developed it, e.g. Nadia’s Law.)
Now if we look at
5(3 + 9) = 5(4 + 8)
Is this still equal?
Why?
Try this with some other numbers, can you multiply any number to both sides and still maintain equality? Explain.
Would this work for a multiplication by a negative number? Give an example.
Can you come with a general statement to describe this law?
(To the Teacher: Have students share their statements. Pick one that best describes this property of equality,
write on big paper and place on the wall. Name it after the student who developed it, e.g. Richard’s Law.)
Now if we look at
(3 + 9)/2 = (4 + 8)/2
Is this still equal?
Why?
Try this with some other numbers, can you divide by any number on both sides and still maintain equality? Explain.
Would this work for division by a negative number? Give an example.
79
Can you come with a general statement to describe this law?
(To the Teacher: Have students share their statements. Pick one that best describes this property of equality,
write on big paper and place on the wall. Name it after the student who developed it, e.g. Jackie’s Law.)
Let’s look at a few more equalities.
Suppose:
4+10 = 3 + 11
Now look at the following:
4 + 10 -10 = 3 + 11 -10
We know from a previous law that this is still equal.
Look at the left side of the equation after you performed the operations. Notice that we’re left with 4, the first number
in the equality. Why did that happen? Can you recall the idea we learned that predicts this?
(To the Teacher: We want students to connect the additive identity with this operation.)
If I had
8–5=2+1
What can I do to both sides of the equation so that each side equals 8?
Let’s do another one. If I had
-7 + 11 = 1 + 3
What would you do to have both sides of the equation equal 11?
(To the Teacher: We want students to come to the conclusion that you have to add the additive inverse to each
side of the equation to get a particular number in the equation by itself.)
Now you should create equality similar to the previous two and decide which number on either side you want to isolate
(remain by itself). Be ready to present yours to the class.
Suppose we have:
5(6) = 30
What can you do to isolate the 6? Explain.
Now if we have:
-2(-4) = 8
What can you do to isolate the -4? Explain.
Make a general statement that will help us to isolate a particular number in an equation that involves multiplication.
Now let’s look at one final equation:
𝟏𝟎
=2
5
80
What can you do to isolate the 10? Explain
Make a general statement that will help us to isolate a particular number in an equation that involves division.
Practice: Isolate the darkened number in each of the following equations, using the ideas you have just learned. Show
all your steps and prove that you’ve maintained equality.
1.
3.
5.
7.
4 + 12 = 9 + 7
8 – 4 = 15 – 11
8(4) = 32
𝟐𝟒
6
=4
2.
4.
6.
8.
-6 + 9 = 2 + 1
6 + 9 = 20 – 5
-5(3) = -15
24
𝟔
=4
(To the Teacher: As we transition into solving be sure to communicate to the students that their work that
allowed them to arrive at their answer and their explanation of that process is just, if not more, important than
the answer itself. You will, if you haven’t already in your career, have students who push back on having to show
work or ask for justification for why they should show work. It is very important that your reflection on this
question goes beyond, because I said so. That response is not sufficient to anyone no matter what the age. This
conversation is an opportunity to help expand your student’s vision of mathematics. Help them to understand
that mathematics is a human endeavor, thus a major component of mathematics is the ability to communicate
what we are doing and why we are doing it. Additionally, just like text, emails, and voicemails, enables us to
communicate when we aren’t around so showing our work enables us to communicate to each other what we
are doing and how we are thinking about something when the reader is not around. Any time you have the
opportunity to humanize and expand your students’ perceptions of mathematics take advantage of it. Hopefully,
as a result of taking advantage of opportunities like this, your students will leave your class with a more robust
view of mathematics than the one they came in with.)
81
Four Laws of Equality
Student Activity Sheet
Name_______________________
Date________________________
Opening Activity:
Suppose we write: 3 + 8 = 4 + 7
Is this true? How do you know?
What does the symbol = mean, without using the word equal?
(A) Now look at:
Is this still equal?
3+9
+4
=
4+8 +4
Why?
Try this with some other numbers, can you add any number to both sides and still maintain equality? Explain.
Can you come with a general statement to describe this law?
82
(B) Now if we look at
3+9 -9 = 4+8 -9
Is this still equal?
Why?
Try this with some other numbers, can you subtract any number to both sides and still maintain equality? Explain.
Can you come with a general statement to describe this law?
(C) Now if we look at
5(3 + 9) = 5(4 + 8)
Is this still equal?
Why?
Try this with some other numbers, can you multiply any number to both sides and still maintain equality? Explain.
Would this work for a multiplication by a negative number? Give an example.
Can you come with a general statement to describe this law?
83
(D) Now if we look at
(3 + 9)/2 = (4 + 8)/2
Is this still equal?
Why?
Try this with some other numbers, can you divide by any number on both sides and still maintain equality? Explain.
Would this work for division by a negative number? Give an example.
Can you come with a general statement to describe this law?
(E) Let’s look at a few more equalities.
Suppose:
4+10 = 3 + 11
Now look at the following:
4 + 10 -10 = 3 + 11 -10
We know from a previous law that this is still equal.
Look at the left side of the equation after you performed the operations. Notice that we’re left with 4, the first number
in the equality. Why did that happen? Can you recall the idea we learned that predicts this?
84
(F) If I had
8–5=2+1
What can I do to both sides of the equation so that each side equals 8?
Let’s do another one. If I had
-7 + 11 = 1 + 3
What would you do to have both sides of the equation equal 11?
Now you should create equality similar to the previous two and decide which number on either side you want to isolate
(remain by itself). Be ready to present yours to the class.
85
(G) Suppose we have:
5(6) = 30
What can you do to isolate the 6? Explain.
Now if we have:
-2(-4) = 8
What can you do to isolate the -4? Explain.
Make a general statement that will help us to isolate a particular number in an equation that involves multiplication.
86
(H) Now let’s look at one final equation:
𝟏𝟎
=2
5
What can you do to isolate the 10? Explain
Make a general statement that will help us to isolate a particular number in an equation that involves division.
87
Four Laws of Equality
Student Activity Sheet
Name_______________________
Date________________________
Practice: Isolate the darkened number in each of the following equations, using the ideas you have just learned. Show
all your steps and prove that you’ve maintained equality.
1.
4 + 12 = 9 + 7
2.
-6 + 9 = 2 + 1
3.
8 – 4 = 15 – 11
4.
6 + 9 = 20 – 5
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5.
7.
8(4) = 32
𝟐𝟒
6
=4
6.
8.
-5(3) = -15
24
𝟔
=4
89
90