Balanced
Equations
Current California Math Standards
Balanced Equations
Grade Three Number Sense
1.0 Students understand the place value of whole numbers:
1.1 Count, read, and write whole numbers to 10,000.
1.2 Compare and order whole numbers to 10,000.
2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division:
2.1 Find the sum or difference of two whole numbers between 0 and 10,000.
Grade Three Algebra and Functions
1.0 Students select appropriate symbols, operations, and properties to represent, describe, simplify, and solve simple
number relationships:
1.1 Represent relationships of quantities in the form of mathematical expressions, equations, or
inequalities.
1.2 Solve problems involving numeric equations or inequalities.
1.3 Select appropriate operational and relational symbols to make an expression true
(e.g., if 4 __ 3 = 12, what operational symbol goes in the blank?).
Grade Four Algebra and Functions
1.0 Students use and interpret variables, mathematical symbols, and properties to write and simplify
expressions and sentences:
1.1 Use letters, boxes, or other symbols to stand for any number in simple expressions or equations (e.g.,
demonstrate an understanding and the use of the concept of a variable).
2.0 Students know how to manipulate equations:
2.1 Know and understand that equals added to equals are equal.
Grade Five Algebra and Functions
1.0 Students use variables in simple expressions, compute the value of the expression for specific values of the variable,
and plot and interpret the results:
1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one
variable by substitution.
Grade Six Algebra and Functions
1.0 Students write verbal expressions and sentences as algebraic expressions and equations; they evaluate algebraic
expressions, solve simple linear equations, and graph and interpret their results:
1.2 Write and evaluate an algebraic expression for a given situation, using up to three variables.
Balanced Equations Standards
1
Balanced Equations
Materials:
One deck of digit cards for every two players (see note below)
One set of digit cards for modeling
Four Copies of the Balanced Equations Demonstration Scale (see page 9)
Student Recording Sheet (see page 10)
Independent Practice Sheet (see page 11)
Note: Using the master on page 12 you can make decks of digit cards. A deck consists of four sets of the 1-9
digits. You would need to make 2 sets of 1-9 in Color One (we are using pink) and 2 sets of 1-9 in Color Two (we
are using blue) for every pair of players. Each deck of digit cards needs to be well mixed before distributing to
students.
Mathematical Practice Standards:
3. Construct viable arguments and critique the reasoning of others.
7. Look for and make use of structure.
Standards:
Grade 3 – NS 1.1, NS 1.2, NS 2.1; AF 1.1, AF 1.2, and AF 1.3
Grade 4 – AF 1.1 and AF 2.1
Grade 5 – AF 1.2
Grade 6 – AF 1.2
Background Information:
This activity provides an opportunity to explore students’ misconceptions about an equal sign denoting the need
for an arithmetic answer. Students play a game where they work to create expressions on each side of the
equal sign to create an overall balanced equation. This helps students internalize the idea that the equal sign
means equivalent. This activity gives students practice with adding numbers, and provides a foundation for
students to work with equations that replace numbers with letters, boxes, or other symbols (see independent
practice).
Introduction (5 minutes):
Share with children that many people believe that an equal sign means that we are looking for an answer to a
math expression. Tell them that it actually means that we are trying to make what is written on the left side of
the equal sign have the same value as what is written on the right side of the equal sign. The expressions need
to be equivalent to one another. To better understand what I mean, I am going model some equations using the
demonstration scale.
Write a 1 on the left side of the scale and a 4 on the right side of the scale. At the top of the page cross out the
numbers 1 and 4 because each digit can only be used once. Ask the class if the scale is balanced. Ask the
students what you need to do to make it balance [they should say write a 3 on the left side of the scale]. Write
the 3 on the left side and then write a plus sign between the 1 and 3.
Balanced Equations Lesson Plan
2
Do this again with a new demonstration sheet. Write a 3 and a 2 on the left side of the scale. At the top of the
page cross out the numbers 3 and 2 because each digit can only be used once. Ask what needs to be written on
the right side [a 5] to balance the equation. Put a plus sign between the 3 and 2 and write a five on the right
side of the scale.
Do this again with a new demonstration sheet. Now write a 1 on the left and a 2 on the right. At the top of the
page cross out the numbers 1 and 2 because each digit can only be used once. Ask them how to balance the
equations. Students should eventually offer up some ideas like putting a 5 on the left and a 6 on the right, or a 6
on the left and a 7 on the right, and etc.
If you feel students understand what is meant by a balanced equation you can move on to demonstrating the
game. If they need further clarification (especially at the lower grades) you can model again with a 3 on the left
and a 6 on the right. [solutions offered could include 4 on the left and 1 on the right, 5 on the left and 2 on the
right, and etc.].
Tell students they are going to use what they just learned to play a game. They are going to use digit cards to
create balanced equations. The pink cards will be placed on the left side of the scale and the blue cards will be
placed on the right. Both sides need to balance. Tell them you are going to model this for them to help them
understand the game.
Demonstration (15 minutes):
To ensure a successful beginning to your demonstration, your top 15 cards in your demonstration deck should be
in this order: B2, P7, P6, B8, B3, P2, B5, P7, P5, B8, B5, P5, P2, P9, and B9 (B = Blue and P = Pink); place the other
cards below these first 15 cards. Order does not matter for these cards. By putting the recommended 15 cards
on top you will be able to model various scenarios and allow students to see numerous possibilities.
Post the first seven cards face up on the white board using magnets (or place the cards under the document
camera or use the Smart Board). Demonstrate to students how to create a balanced equation. (Use the
following cards to ensure a good variety of solutions). Also, group them with the pink on the left and the blue
on the right to reinforce that they are balancing the pink on one side of the equation and the blue on the other
side of the equation.
6
2
7
2
8
3
5
Pink
Pink
Pink
Blue
Blue
Blue
Blue
2 CARDS
3 CARDS
4 CARDS
5 CARDS
1 pink two
1 blue two
1 pink two
1 pink six
1 blue eight
1 pink six
1 pink seven
1 blue five
1 blue eight
1 pink six
1 pink seven
1 blue two
1 blue three
1 blue eight
6 CARDS
1 pink two
1 pink six
1 pink seven
1 blue two
1 blue five
1 blue eight
Balanced Equations Lesson Plan
3
First show students how to create a balanced equation with two cards. Explain that in this game they would
earn 2 points because they created an equation using two numbers. Next show them a balanced equation with
three cards. Explain that they get 3 points because they created an equation using three numbers.
Ask if anyone sees a way to make a balanced equation with four cards. Repeat with five cards and then with six
cards. Ask if anyone can make a balanced equation with seven cards (this cannot be done). Ask them why it
cannot be done (the blue cards total 15 but the pink cards total 18 which would not balance).
Continue to model how to play the game by competing against one student from the class. (Be sure to select a
student that will be able to make some good observations or this demonstration will take a long time and will
not be all that beneficial). Tell the selected student to choose which balanced equation they want to claim
(hopefully that student chooses to make a balanced equation with six cards) and to justify why they selected this
one. Ask students if they agree or disagree with the balanced equation selected. Allow dialogue to continue
until the six card balanced equation has been universally accepted by the class (by having the student select
the 6 point equation you will be able to ensure your next two modeling opportunities provide good examples).
Explain that the student earned six points because they created an equation using six numbers. Use the
document camera or overhead to demonstrate how to record this balanced equation on the recording sheet
(see figure below to see how the balanced equation should be recorded).
6
7
2
Pink
Pink
Pink
5
2
8
Blue
Blue
Blue
It is a competition so now it is your turn. Share with the students that the six cards used in the balanced
equation go into a discard pile and will not be used again this game. Model how you must leave the card that
was not used on the board and deal six additional cards. (Replace the six cards with the top six cards from the
deck. If you put the cards in the suggested order the new seven cards should look like the figure below). Tell
them you must try to create a balanced equation using these 7 cards.
5
2
7
5
3
5
8
Pink
Pink
Pink
Pink
Blue
Blue
Blue
Balanced Equations Lesson Plan
4
5
5
2
7
5
3
8
Blue
Pink
Pink
Pink
Pink
Blue
Blue
Notice how the two fives are moved above the rest of the cards? This is to show that this is one possibility. Think aloud
so students can hear a strategy. “I know that 5 = 5 however, that’s only two points. Is there another balanced equation I
can create to earn more points?” Try a few different combinations, but acknowledge that you can only earn two points
on this turn, because it’s the only balanced equation that can be made. It’s good for students to see that sometimes they
can only earn two points. Use the document camera to record the balanced equation on the teacher’s recording sheet.
Note: Sometimes a balanced equation cannot be made with the seven cards that are showing. Inform students that if
this happens, they should turn over one additional card and then make a balanced equation. If there is still no way to
make a balanced equation another card would be turned over. Once a balanced equation is made you would replace
enough cards so that the opponent starts with 7 cards.
9
2
7
5
3
9
8
Pink
Pink
Pink
Pink
Blue
Blue
Blue
Deal again, but this time there will be more options for balanced equations, which will give an opportunity for
students to come up with multiple solutions. Ask the student to identify their balanced equation. Move those
cards above the other cards so the class can see that balanced equation. Use the document camera to model
how to record this on the student recording sheet. Now ask the class if they see other balanced equations (see
sample balanced equations in the figures below; there are additional solutions).
9
9
Pink
9
Pink
2
7
Pink
Pink
2
7
Pink
Pink
5
Blue
3
Blue
Pink
Blue
2 card solution
9
3
5
Pink
Blue
3 card solution
Blue
8
Blue
9
Pink
8
2
Pink
7
5
3
9
Pink
Pink
Blue
Blue
8
Blue
4 card solution
Balanced Equations Lesson Plan
5
Eliciting Student Thinking: Once 2, 3, and 4 card solutions have been shared, ask students if they can create a
five, six, or seven card balanced equation. What strategy are they using to try to find a balanced equation with
more cards? Possible strategies could include adding all the pink cards and trying to make that sum with the
blue cards or adding all the blue and seeing if that will equal the pink cards; adding three pink and three blue to
make a balanced equation; or taking the low numbers from one color and making them equal a higher number
of the other color (among others).
Completing the Demonstration: Turn over enough cards so that 7 cards are facing up at the beginning of the
new turn. Remind them that you discarded the cards they used in their equation and kept the cards they did
not use. You then added enough cards to have seven cards showing. This is done at the beginning of each turn.
Make your balanced equation and record it on the teacher sheet. Show the class that you have each created
two balanced equations on your recording sheets. Tell them you are going to end the game here. Who won?
How do you know? Share with them that they will play against a partner and the game ends when they have
each created four balanced equations.
Students Play the Game (15 minutes): Tell students they are going to do the following:
1)
2)
3)
4)
Remove the digit cards from the baggie.
Determine who is going to go first by seeing who turns over the highest card.
Deal seven cards face up.
Take turns making and recording balanced equations until each player has created four balanced
equations.
Note: When the game first begins circulate to make sure everyone is playing the game correctly. Students may
accidently deal each player seven cards because they are confused about the directions. They may also try to
reuse the discard pile. Look for things of that nature. Be careful not to get stuck with a group of students
before you have circulated throughout the class. If you have not checked to see if all students are playing the
game correctly, chances are some of the teams will not be playing correctly.
Determining the Winner: The winner is the player who earned the most points.
Closure/Independent Practice (10 minutes):
Closure should tie together the game and the standard you have chosen to focus on. Part of the focus of the
game was to help students understand what an equals signs signifies (the expressions on either side of the
equals sign name the same quantity). Ask students to explain the meaning of an equals sign [look for them to
share that it does not necessarily signify an answer to an expression]. You also had a standard specific to the
grade level you were teaching. Be sure to highlight this standard in your closure. See below for ideas.
Share with students that this game allowed them to choose from a variety of numbers to create balanced
equations. However, their math textbooks will oftentimes give them some numbers and they have to figure out
a specific number that will balance the equation (similar to what you did when you first introduced what a
balanced equation is). For example, at third and fourth grades, textbooks often have problems similar to 37 + 25
= 25 + □ and ask students to identify what number should go in the box. This illustrates the commutative
property. Students will soon realize this is much easier than figuring out how to balance the equation because
Balanced Equations Lesson Plan
6
they are merely moving the numbers around. At fifth and sixth grades students will encounter problems like x +
46 = 25 + 37 where they have to solve for x. This is exactly what they have been doing. They find a sum on the
right side and have to see what number can be added to 46 to get that sum. This is easier because they will
always be able to balance the equation, whereas with the game they could only use the cards turned over so
sometimes the equations would not balance.
Direct students to a page in their textbook that you want them to do that is related to the game, or if
appropriate, use the independent practice sheet included on page 11. Ensure that students understand that the
game they just played involves the same skill set as the textbook/independent practice is requiring of them.
Assign independent practice.
Extensions:
1. Allow students to use any operation to create the balanced equation. For example, a student may use
6-3 = 1 + 2 or 5 x 5 = 9 + 9 + 7.
2. For further practice, deal out 6 cards to each player. Turn the next two cards face up. This is the "target
expression." Players must use cards in their hands to find an expression with a sum equivalent to the
sum of the two cards turned over. If a player cannot make the “target expression”, they do not lay any
cards down and miss the opportunity to score on this round. Once all members of the group have
created their problems or passed, deal 2 new cards to each player (including the people who may not
have been able to play their cards) and turn over a new “target expression”. After 4 rounds, the players
total the points from all of the balanced equations they made. Then they subtract the value of any cards
remaining in their hands from that total.
For example, if the player has a pink 8, pink 3, blue 4, and blue 7 in their hand they determine value of
the cards in their hand, which in this case would be 8 + 3 + 4 + 7, for a total of 22 points. If they had laid
down balanced equations totaling 58 points they would take 58 – 22 to determine that their overall
score is 36. The player with the most points wins.
3. You can do extension number one but allow the use of other operations.
4. Use a deck of playing cards instead of the digit cards. Have students use the tens and face cards where
the jack is worth 11, a queen is 12, and a king is 13.
5. Students can play the Balanced Equations Card Game described on page 8.
Balanced Equations Lesson Plan
7
Balanced Equations
A Card Game for 2-4 players
Materials: Use a standard 52-card deck. Choose a dealer by drawing for the high card.
Note: Aces are worth one point, number cards are worth face value, jacks are worth 11, queens are
worth 12, and kings are worth 13.
Setup: Shuffle the deck and deal 7 cards to each player. With the remaining cards, form a draw pile in the
middle of the table. Place the top card face up next to the draw pile to start a discard pile.
Gameplay: Each turn, a player must follow this sequence.
1. Draw one card, either from the top of the draw pile or the top of the discard pile. A player may choose to
draw more than the top card from the discard pile but must take all the cards on top of it and must play that
bottom card in a balanced equation during that turn.
2. Create a balanced equation if you can.
Example 1: A red 6 and a black 6
Example 2: A black king, a red 2, a red 3, and a red 8
3. Whether you were able to make a balanced equation or not, discard one card, adding it (face up) to the
top of the discard pile. The card should be placed so that the cards below can still be seen. The previous
discards are still available for play. If a player chooses to draw only the top card on the discard pile in step 1,
that card may not be discarded during the same turn in step 3.
Going Out: A player goes out when the last card in his or her hand is played as a discard.
Scoring: When a player goes out, the hand is scored. Players total the points from all of the balanced
equations they made. Then they subtract the value of any cards remaining in their hands from that total.
For example, if the player has a red 8, red 3, black 4, and black jack in their hand they need to determine
value of the cards in their hand, which in this case would be 8 + 3 + 4 + 11, for a total of 26 points. If they
had laid down balanced equations totaling 58 points they would take 58 – 26 to determine that their overall
score is 32. The player with the most points wins
Balanced Equations Lesson Plan
8
Use each number only once
1
2
3
4
5
6
7
8
9
=
Balanced Equations Demonstration Scale
9
Name
Balanced Equations
Number of Points for Equation One #1 _________
=
Number of Points for Equation #2 _________
=
Number of Points for Equation #3 _________
=
Number of Points for Equation #4 _________
=
On the back, explain how you know when an equation is balanced. Explain what you were thinking
as you were working to make a balanced equation.
Balanced Equations Student Recording Sheet
10
Name
Balanced Equations Independent Practice Page
Using the digits below to balance these equations.
5
12
19
26
33
40
47
54
61
x=
1.
40
+
n
=
26
+
33
x=
2.
5
+
33
=
n
+
12
x=
3.
n
+
54
=
33
+
61
x=
4.
54
+
12
=
40
+
n
x=
5.
47
+
5
=
33
+
n
x=
6.
40
+
5
=
n
+
19
x=
7.
61
+
n
=
54
+
33
x=
8.
47
+
33
=
61
+
n
Extension Activity: Use the nine numbers at the top of the page to create your own balanced
equations.
Balanced Equations Independent Practice
11
1
2
3
4
5
6
7
8
9
Balanced Equations Digit Cards
12