2.IntroductiontoEquationsTeacherGuide

Name:_____________________
Date:_________
Introduction to Equations: Understanding Equality
Introduction:
Before passing out packets, “Welcome to your new Task group! This Task experience is going to look and
feel a little different. I hope you enjoy it. Before I pass out our learning materials, I am curious, what the
equal sign means to you? Where have you seen it? What comes to mind? Have you ever used it outside of
math class? Think to yourself or talk with the people around you and then lets hear what’s on each other’s
minds.”
● After students share, possibly record different ideas and/or have students relate with similar
experiences or ideas. Then, pass out learning materials and have them begin to think of what
makes things equal.
1. What Does = Mean?
Give examples of some ways you could use an equal sign. When have you seen this symbol used?
Purpose:
This Activity is for students to think about the meaning of equality and that the quantities on either side of
the equal sign have the same value.
Materials:
● Poster Paper: 1 for each group
● Laptop: 1 for each student or group (optional)
● Markers or Colored Pencils: a few for each group
Teaching Tips:
● Chances are students will start with showing equality in a mathematical context (eg. 1+1 = 2).
● Push students to think about equality in other context, (eg. 4 quarters = 1 dollar)
● Some will want to use laptops for conversions they are less comfortable with, this is not essential
for the intended understanding however.
● After 5 minutes, each group shares.
● Make sure to create a MASTER Sheet at the front to record student ideas throughout this Activity.
This should available for them to look at, and add to, throughout the Task experience.
Give an example of two things that are NOT equal. How could you make them equal?
Teaching Tips:
● This Activity can be conducted as a class discussions with students making note of examples they
like in their packet.
● The teacher can take examples from the class master list and alter them so they are not equal eg.
“How could we make 1 + 1 = 3?” “How could we make 3 quarters equal a dollar?”
○ Add another 1 to the left of the equal sign.
○ Take $0.25 away from the dollar.
© 2016 New Classrooms Innovation Partners, Inc. 1

2. Trying to Find a Balance
Directions
1. Access the virtual pan balance: http://tinyurl.com/tto-balance
2. Partner 1 enters a number on one side of the balance.
3. Partner 2 then writes as many math sentences as they can to make the balance even in 1 minute.
4. Switch roles.
An example showing three ways to keep the pans balanced are shown here:
Purpose:
The purpose of this activity is to continue building understanding of what it means for both sides of an
equation to be equal.
Materials:
● Laptops: one for each pair of students (with internet access)
● Additional paper: for each student to show work (optional)
Teaching Tips:
● As students are working, record a few balanced pan pairs showing more than one way to balance
out the same number. Discuss these after students have completed the activity.
● Encourage students to show balance using all four arithmetic operations.
What patterns do you notice?
Possible Student Responses:
● There are A LOT of different ways to make the pans balanced!
● If I’m adding 2 numbers, I can increase 1 and if I decrease the other by the same amount the other
side can stay the same.
How could you tell if a number sentence will make the balance even before typing it in?
Possible Student Response:
● If the number sentence equals the number on the other side I know the balance will be even.
Extensions:
● If one side of the balance is lower than the other, what do you need to do to re-balance?
● How does the balance show equality?

3. Dice Balance Game
Goal: Find a balance with numbers rolled on a dice.
Rules:
1. For each example roll a dice. Write the number you rolled in the
square.
2. Write in your own numbers on the line to find a balance.
Purpose:
The purpose of this activity is to introduce students to asking what needs to be done to a number to keep
an equation balanced.
Materials:
● Dice: One die for each group
Teaching Tips:
● After completing the activity with dice rolls have students explore how rolling a different number
would have changed their responses.
How can you use math operations to make sure that both sides of the “=” are balanced?
● If I do the operations and the numbers come out the same, then the equation is balanced.
Extensions:
● How can you use the operation to predict if the missing number will be larger or smaller than the
number on the right side of the equal sign?
● Put some of these examples into the balances on the computer. What are some things that you
can do that will keep the balance?
Teacher Reflection:
● Communication: Were students asking questions to their peers during Trying to Find a Balance?
What kind of questions?
● Engagement: Which students were not highly engaged? What do you think might draw them in?
● Mathematical Understanding: do students understand that there are many ways to make
equations?
● Teacher Moves: Which students, if any, struggled with the Die Balance Game? How will I create groups that
can help?

4. Bags and Blocks
Purpose:
The purpose of this activity is to introduce students to variables and how to find the unknown using the
balance.
Materials:
● Student Laptop: One for each student or pair
Teaching Tips:
● Allow students to discover on their own what they have to do to each side of the balance to keep
the pans even rather than tell them a strategy.
● Have students capture the steps they are executing either with manipulatives or by capturing the
process using paper.
● Possible guiding questions are:
○ What would happen if you removed 1 block from the left side of this balance?
○ What would happen if you put 2 bags on the right side of this balance? What would need
to be on the left side to balance 2 bags on the right?
○ Would putting the bag back on the left side make the balance equal? If we are trying to
find out how many blocks are in in one bag why is this not an effective strategy?
○ Even if we don’t know how many blocks are in each bag what do we know that a bag is
equal to?
○ Why is it important to make the balance even after every step?
Your job is to find how many blocks are in each bag using a balance.
Look at the image to the right. How many blocks are in the bag? How do you
know?
● There are 3 blocks in the bag.
The balance below starts off even then a student takes a bag off of the left side. How can you make the
balance even again?
● Remove a bag from the right side.
Using these understandings in pairs complete the activities at http://tinyurl.com/tto-bagsandblocks.
Explain how you can find how many blocks are in a bag using a balance.
● If you get 1 bag on one side and only block on the other side you know how many blocks are in a
bag.

4a. Bags and Blocks (Part 2)
Directions
1. On one side of the balance put a bag on the other roll a die and draw that many blocks.
2. The balance is even so the bag is equal to the number of blocks.
3. On the image below add blocks and bags to both sides so that each side is equal but looks
different.
4. Trade with a partner to see if they can find how many blocks are in 1 bag.
Purpose:
The purpose of this activity is to get students to understand equations better by creating their own using
the principles they just practiced virtually.
Materials:
● Die: One for each student or pair
● Unit Cubes: A handful for each student (optional)
● Paper Bags: A handful for each student (optional)
Teaching Tips:
● It is important that students work systematically to create solvable equations. Encourage students
to solve their own equations before giving them to a partner.
● Students can also continue this exercise using unit cubes and paper bags to create equations with
actual manipulatives.
What do you have to do to make sure the scale is always balanced?
● The balance needs to start off equal, and every addition and subtraction needs to be equal in
order to maintain balance.
Extensions:
● Encourage students to explore different ways they can make equal balances without changing the
number of cubes in each bag.
● Have students consider how they could keep balance with operations other than addition and
subtraction.

5. Secret Numbers
Directions: You are going to think of a secret number, don’t tell anyone what it is!
Follow your teacher’s instructions and write the result below. Only share the result!
Purpose:
The purpose of this activity is to get students to see the inverse relationships between operations. This
teacher-led activity of uncovering a secret number from the result of an arithmetic operation is meant to
be a scaffold to get students to see how to “undo” operations.
Teaching Tips:
● As a class students should write a number they choose. Then teachers should say to apply a “rule”
of a simple arithmetic operation, +3 for example.
● Feel free to do this multiple times with different rules.
Was your teacher able to guess your secret number? How do you think you could use math to guess
someone’s a secret number?
● My teacher was able to guess my secret number because they could undo the rule they gave me
and figure out the original number.
Extensions:
● Are there any secret numbers or rules that wouldn’t allow you to figure out the secret number?
Play the secret number game with a partner. Try to guess their secret number using addition, subtraction
multiplication and division.

6. Cracking the Code
Directions: Now that you have seen some secret number examples explain how you
would use math to figure out the secret numbers below.
n + 7 = 10
n = 3
n - 5 = 9
n = 14
n ✖ 5 = 20
n = 4
n ÷ 6 =3
n = 18
Doing and Undoing
In all of the secret number examples something is being done to the secret number. You can find out
what the secret number is by undoing it. Use the chart below to explain how doing and undoing can help
you figure out secret numbers
Purpose:
This activity has students formally identify the inverse operations.
Teaching Tips:
● Have a few different students explain their thinking, so the class can hear multiple explanations.
● Consider having students come up with their own examples.
When I see What is being done? How can it be undone?
+ Adding Take away (-)
- Take away Adding (+)
✖ Groups of Equal groups (÷)
÷ Equal groups Groups of (✖)

7. The “Undoer”
Directions: For each machine make up an input, and show what the machine does by
writing the output.
Purpose:
The purpose of this activity is to build on students understanding, and comfortability around inverse
operations.
Teaching Tips:
● Have students try different inputs with the same rule and get multiple outputs. Then apply the
“undoing” rule to practice inverse operations.
● Offer students a chance to draw their own machines and “undoers” on a separate sheet of paper,
and then allow students to try each other’s machines.
● As an extension you could have one student create a machine and a partner create the undoer to
that machine.
Now to the right of each machine draw a new machine that undoes what the first machine did. Start
with the output and show the rule that will get you back to your input.
● Machines should be -3, ÷6, -9 and ×2
Extensions:
● “When Jameel looked at the +3 machine, he said that machine that undoes it should be a ÷2
machine because when he put 3 into it, he got 6 out, and 6÷2 is 3.” Do you think he’s right or
wrong? Why?
Teacher Reflection:
● Communication: Were students actively collaborating and conversing on Cracking the Code? What
led to this?
● Engagement: Were there specific times when students were really focused and “in the moment”?
If so, when and why do you think so?
● Mathematical Understanding: Did some students make connections/find patterns faster than others when
learning about inverse operations?
● Teacher Moves: How should I start the next Activity based on where each student is at?

8. Back to the Balance
Directions: Answer these questions about keeping the balance.
Write in numbers and operations that make the balance equal.
Purpose:
The purpose of this activity is to scaffold the understanding of how to use operations on both sides of an
equation to find the value of a missing number.
Teaching Tips:
● Have students answer the questions on this page independently before discussing their answers
with a partner then reviewing as a whole group.
If you added 5 to one side what would happen?
● The balance would be unequal.
What could you do to the other side to make it balanced again?
● Add 5 to the other side
Put numbers in the balances below. Subtract, multiply, or divide one side by a number then do
something to the other side to make it balanced again.
How did you keep the balance? Explain why you think your method works.
● As long as you do the same operation with the same number on both sides of the equation, it will
be always be balanced.

9. Secret Numbers on the Balance
Directions: With a partner look at this student’s thinking. What do you agree with? What would
you change?
I’m trying to find a secret number “x” using math
x + 5 = 12
I know that the = sign means that both sides are balanced so I can put this problem on a balance.
I know that when I see +5, I can undo it with -5.
Does this mean this is balanced?
Purpose:
The purpose of this activity is to further scaffold the understanding of how to use operations to find the
value of a missing number through error analysis.
Teaching Tips:
● This is the first time that students will formally be asked to use inverse operations to balance an
equation and find an unknown.
● Have students answer the questions on this page independently before discussing their answers
with a partner then reviewing as a whole group.
● Teachers may choose to introduce more traditional notations as they review student work as a
class, however the notation is not as important as the understanding of keeping the balance.
● You may want to offer students the chance to revisit the virtual balance at
http://tinyurl.com/tto-balance to show their thinking.
● You may also want to refer back to the understandings developed during activity 4. Bags and
blocks
● In order to keep this balanced, you would have to subtract 5 from both sides.
Extensions:
● Do you think that -5 is the best way to keep this balance even? Explain why or why not.

9a. Secret Numbers on the Balance (Part 2)
Directions
1. The person whose birthday is coming up next is Partner A and the other is Partner B.
2. Partner A writes down the steps they followed to find the missing numbers in the examples on the
first page and partner B does the same on the second page.
3. When you are done switch papers and follow the directions that your partner gave you.
4. Come back together to discuss the directions you received and gave. Together come up with a
set of directions you can use to find the secret number on a balance.
Purpose:
The purpose of this activity is for students to come up with their own method of finding an unknown value
in an equation and solidify that method by working with a partner.
Teaching Tips:
● Have students write the method separately in a way their partner can follow.
● Both partners should be writing 2 methods. One for each of their examples.
● Students should follow their partner's method rather than just trying to solve for the missing
number.
● After both partners have written methods and followed their partner’s method they should
collectively come up with a method for finding an unknown in an equation.
Partner A:
Example A1 Steps:
Example A2 Steps:

9a. Secret Numbers on the Balance (Part 2 - continued)
Partner B:
Example B1 Steps:
Example B2 Steps:
Together, make a balance with a secret number and list the steps for finding the secret number
below.
● Step 1: Undo what is happening to the secret number by using the opposite of what is happening
to it on both sides of the equal sign
● Step 2: udoing should get the secret number by itself. Do the math on the other side to find out
what the secret number is
Teacher Reflection:
● Communication: Which students have worked well together through five sessions? Which have
not?
● Engagement: How was the engagement during the Secret Number on a Balance activity? Were
they ready for it? What could I have done differently to get them ready for it?
● Mathematical Understanding: Am I confident that students really understand how to apply the idea of inverse
operations to solving equations?
● Teacher Moves: Going into the last day, what do I need to make sure to do with them? What should our group
reflection of this task look like and feel like?

12. Final Project: Role Playing Adventure
Purpose:
This culminating activity is for students to demonstrate their understanding of equality in an engaging
way. Students are encouraged to use a variety of strategies to find missing numbers and balance
equations.
Materials:
● Extra Poster paper: 1 per child
● Colored pencils, marker, or crayons: a few colors per child
● Scissors: 1 pair per group
Teaching Tips:
● Have students find the missing numbers in the examples, based on those examples students
should create similar problems on their own. On the following (final) day, have students trade
problems and try to solve scenarios created by their peers.
● As an extension, students can try to create a context for the problems they have created.
● Students can create additional problems or continue to decorate their characters or adventures.
● Encourage students to show multiple ways of determining the game score.
Part 1: You are playing a game where characters can pick up items that make them more
powerful or fall into traps that make them weaker. Use what you have learned to explain
what is happening to the different characters in each situation.
Each of the characters below went on an adventure and came back changed. Use what you know
about balancing equations to show how many xp each character had before their adventure.
= =
= =
= =

12. Final Project: Role Playing Adventure (continued)
Part 2: Use the images below, or create your own images to make your own balanced
adventures.
Use the template below to create each adventure.
Teaching Tips:
● Students should refer to the examples and the template to help them create their equations.
● Encourage students to create their own art work the numbers in each scenario represent.
● Have students check their equations by solving themselves before sharing with their classmates.
● Encourage students to be creative!

Part 3: Hide the number that goes with either the character or the event and have a partner find
the missing number by making the balance even. See if you can come up with a story that
explains what happened during each part of the adventure.
Part 4: Come up with a new event that will undo what the first event did, bring your character
back to their original level.

Reflection for Role Playing Adventure Activity
1. How did you make sure each adventure you wrote was balanced?
2. What made finding the missing number in some adventures easier than others?
1. I made sure that the math sentence I came up with matched the number on the other side.
2. Depending on where you put the mystery number some division equations are harder than others.

Understanding Equality: Bringing it All Together
Purpose:
This Activity is here to provide students an opportunity to reflect on their experience: what they learned,
how they learned, and how they worked together.
Materials:
● Hundreds Charts: at least 1 per student
Teaching Tips:
● This assessment should take 15-20 minutes
● As you walk around, here are some possible questions to ask students:
○ Why did you choose that method?
○ Why are you able to do that?
○ What was your first thought?
○ Is there another way to do this?
● Ensure that they finish the Reflection at the end prior to having a class conversation.
● It is important to reflect on what they liked, disliked, wish they would have done differently, are
proud of etc. This is a good time to have students give shout-outs to other students for things they
saw that were awesome or if a peer helped them with something that made a difference. Let them
guide the conversation :)
● Finally, please share some exemplars with your coach so we can share with other teachers!
Question 1: Share your work from the Role Playing Adventure Activity with another partner or
group. Pay attention to how each student found the missing numbers in their partners adventure?
As others share, write down any connections you think of between your strategy and other
people’s strategies. Also, write down any questions you asked or have.
Connections Questions
Question 2: A class is preparing to learn the concept of balancing equations. How would you
explain what a balanced equation is to a student who has never learned about it before?
Question 3: Explain a time in your life that you have (or could) balance an equation.
Question 4: Explain how you would find the missing number in the following balanced equation.
Explain your reasoning for each step.
14 - n = 6

Reflection
Question 1: Now that this task is over, how confident are you that you understand equality?
1 2 3 4 5
NOT AT ALL SOMEWHAT VERY
Why do you think that is?
Question 2: What was your favorite part of this task? Why was it your favorite?
Question 3: What advice do you have for your teacher the next time they work on this task with a
group of students?
Question 4: What did your group do well together? What could you do better together next time?
Question 5: What further questions do you have about equality, equations, or this task?
Teacher Reflection:
● Communication: Which questions/comments by myself or students really pushed the
discussion? Which hindered it?
● Engagement: How was the engagement during the carnival scorekeeping activity? Were they
ready for it? What could I have done differently to get them ready for it?
● Mathematical Understanding: After this series of activities on equality, which students are really grasping
the concept and which are still struggling?
● Teacher Moves: If I were to teach this again, what would I do differently?

2.IntroductiontoEquationsTeacherGuide

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2.IntroductiontoEquationsTeacherGuide