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 cards for every two players with all face cards and jokers removed (leave in aces);
One large set of playing cards (can be purchased in the Target game section for $5.00)
Balanced Equations Demonstration Scale (see page 9)
Student Recording Sheet (see page 10)
Independent Practice Sheet (see page 11)
Note: To save money you can make sets of digit cards using the master on page 12. You would need to make 2
sets of 1-9 in Color One (to mimic the red playing cards) and 2 sets of 1-9 in Color Two (to mimic the black
playing cards) for every pair of players. If you are going to have students use the digit cards instead of the
playing cards, you should use digit cards for all of your modeling but enlarge them to a size that will allow
students from the back of your room to easily see the digits. You’ll need to make a ten if you want to do the
demonstration shown in this lesson plan. The lesson plan is written with the assumption that you are using
playing cards. To adjust for using the digit cards, just mentally replace “digit cards” for “playing cards”. Also,
mentally replace “color one” for “red” and “color two” for black.
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. The activity gives them practice with adding numbers, and can assist students in the
transition to working 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 you 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.
Balanced Equations Lesson Plan
2
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 one 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 [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.
Do this again with a 3 and a 2 on the left side and ask what needs to be written on the right side [a 5]. Put a plus
sign between the 3 and 2.
Now write a 1 on the left and a 2 on the right. 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 playing cards to
create balanced equations. The red cards will go on the left and they must balance with the black cards that will
be placed on the right. Tell them you are going to model this for them.
Demonstration (15 minutes):
To ensure a successful beginning to your demonstration, your top 14 cards in your demonstration deck should be
in this order: R2, B7, B6, R8, R10, B2, R1, B7, B5, R5, R3, B5, R9, B9
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). Tell them the ace is always worth one point.
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 red on the left and the black on the right to reinforce that they are
balancing the red on one side of the equation and the black on the other side of the equation.
2 CARDS
3 CARDS
4 CARDS
1 black two
1 red two
1 black two
1 black six
1 red eight
1 black two
1 black seven
1 red ace
1 red eight
5 CARDS
1 black six
1 black seven
1 red ace
1 red two
1 red ten
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. Ask if anyone
can make a balanced equation with six cards (this cannot be done). Ask them why it cannot be done (the black
cards total 15 and there is no way to make 15 with the red cards; if we use 4 red cards the total would be 21 and
there is no way to make 21 with the black cards).
Continue to model how to play the game by competing against one student from the class. Ask that student to
choose which balanced equation they want to claim (hopefully that student chooses to make a balanced
equation with five 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 five card balanced equation has been
universally accepted by the class. Explain that the student earned five points because they created an equation
using five 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).
It is a competition so now it is your turn. Share with the students that the five 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 2 cards that
were not used on the board and deal five additional cards. (Replace the five cards with the top five cards from
the deck. If you put the cards in the suggested order the new seven cards should look like the figure below
though the suits may be different). Tell them you must try to create a balanced equation using these 7 cards.
Balanced Equations Lesson Plan
4
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 that is the only balanced equation that can be made. 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.
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 their recording sheet. Now ask the class if they see other balanced equations (see sample
balanced equations in the figures below).
2 card solution
3 card solution
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 red cards and trying to make that sum with the
black cards or adding all the black and seeing if that will equal the red cards; adding three red and three black 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).
Turn over enough cards so that 7 cards are facing up at the beginning of the new turn. Remind them that you
kept the cards they did not use, and only 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 them on the
two recording sheets that you have created two balanced equations and they have created two balanced
equations. 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 cards from the box.
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.
Determining the Winner: The winner is the player who earned the most points.
Closure/Independent Practice (10 minutes):
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].
Share with students that this game allowed them to use any numbers they wanted to create balanced
equations. However, their math textbooks will oftentimes give them some numbers and they have to figure out
how to balance the equation (similar to what you did when you first introduced what a balanced equation is).
Direct students to a page in their textbook that you want them to do that is related to the game. 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 they are merely moving the
numbers around.
Balanced Equations Lesson Plan
6
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. (There
is an independent practice sheet included on page 11 for you to give students if you choose).
Ensure that students understand that the game they just played involves the same skill set as the textbook is
requiring of them. Assign independent practice.
Extensions:
1. Add the tens and face cards to the deck where the jack is worth 11, a queen is 12, and a king is 13.
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 red 8, red 3, black 4, and black jack in their hand they 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.
3. You can do extension number two but allow the use of other operations.
4. Make this a card game where students keep seven cards in their own hand and work from a draw pile.
See directions on the following page.
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 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
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
+
Extension Activity:
33
=
61
+
n
Use the nine numbers above 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