# T4T Exploring Equality

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Lesson excerpt:

NC Mathematics Standards:

Analyze addition and subtraction equations within 20.

NC.1.OA.7 Apply understanding of the equal sign to determine if equations involving addition and subtraction are true.

NC.1.OA.6 Add and subtract, within 20, using strategies such as:

·      Counting on

·      Making ten

·      Decomposing a number leading to a ten

·      Using the relationship between addition and subtraction

·      Using a number line

·      Creating equivalent but simpler or known sums

Standards for Mathematical Practice:

3. Construct viable arguments and critique the reasoning of others

6. Attend to precision

7. Look for and make use of structure

Student Outcomes:

●     I can determine if equations involving sets and/or numbers are true using the equal sign.

●     I can use the equal sign appropriately to show when groups or expressions are equal.

●     I can explain why equations are true by using addition and subtraction.

Math Language:

What words or phrases do I expect students to use as they talk during this lesson?

equal, equation, true, add, addend, subtract, more, fewer, more of, less of, same as

Materials:

●     Manipulatives to use for making groups or counting, such as cubes or counters

●     Math journals or student activity pages for students to show their thinking

●     Exit tickets

●     Gather materials

●     Copy exit tickets and perhaps also student activity page(s)

Launch:

1. Introduce the task (2-3 minutes)

Tell students you are buying balloons for a birthday party.  You want to have an equal amount of blue and red balloons.  Present the task with sets to students and tell that this shows the red and blue balloons that you have bought for the party.  You need them to tell you if this is true, that you indeed do have an equal amount of red and blue balloons.  (Avoid using the word “same” when introducing the task; hopefully students will use in their explanations.)

Explore:

1. Solving the Problem (5-7 minutes)

Have students work individually to determine if there is an equal amount of red and blue balloons.  Encourage them to use their math journals or student activity pages to explain why it is true, that there are equal amounts, or if it is not true.  They should use numbers and/or words to show their thinking.

Notice how students are determining if the sets are equal and also how they are explaining their thinking and reasoning.  Consider a few different examples of student work (could be accurate or inaccurate) and in what order you would like students to share their thinking to spark discussion once you bring the students back together as a class.

Discuss:

1. Discussion (8-10 minutes)

Bring students back together as a large group and take a quick tally of who says Yes, it is true because the sets of balloons are equal and who says No it is not true because the sets are not equal.  Respond based on how students respond.  (If all say No, respond with something such as Wait!  I really don’t have an equal amount of red and blue balloons?!  How do you know?  Or if some say Yes and some say No, respond such as: Uh-oh, it doesn’t sound like we agree and I definitely need an equal amount of red and blue balloons.  Let’s work together to make sure I end up with equal amounts.)

Ask a few students (whose work you identified while students were exploring) to share what they were thinking and how they determined if the sets were equal.  Ask questions to encourage students to explain their thinking in more depth, such as:

·         What did you do first?  Then what did you do?

·         How can you prove the amounts are not equal?

·         Did you show what you were thinking with numbers?

·         What would need to happen for the amounts (or groups) to be equal?

As students talk, make note of the words they are using; you may even want to jot them on a chart or on the board.  You are specifically listening for same, same as, same amount/number, more, less, fewer, etc.  As you or your students recap what is determined (that the sets are not equal), use and reiterate this vocabulary (e.g. The amounts are not the same because there are more blue balloons than red balloons.  To make the amounts equal, we need the same number of blue and red balloons.)

Launch:

1. Introduce the task (2-3 minutes)

Tell students they have been so helpful that you would like their help again.  Explain that your dad is making cupcakes for the party and he wants to make sure there is an equal amount of chocolate and vanilla cupcakes.  As he is baking, he uses the following equation to show how many chocolate and vanilla cupcakes he has.

chocolate                 vanilla

4   +   4        =        5   +  3

Is my dad’s equation true?  Does he have an equal amount of chocolate and

vanilla cupcakes?

Explore:

1. Solving the Problem (8-10 minutes)

Have students work in partners or groups of three to determine if the expression is true and there is an equal amount of chocolate and vanilla cupcakes.  Try not to prompt or guide, but allow students to use any method of their choosing to determine equality.  Use questions such as the following as needed:

·         How did you determine they are/are not equal?

·         How could you draw this situation?

·         Can you explain your thinking to me?

Consider a few different examples of student work (could be accurate or inaccurate) and in what order you would like students to share their thinking to spark discussion once you bring the students back together as a class.

Discuss:

1. Discussion (10-15 minutes)

Bring students back together as a large group and again take a quick tally of who says Yes, it is true and who says No, it is not true.  Respond accordingly, then proceed with selected students sharing their thinking and their work.  As students explain their process of determining if the equation is true, draw/write what the students share so the thinking is visible for all.

If not demonstrated by a student, you may say I saw someone drawing groups or sets to decide if it was true, like in our balloon situation.  Proceed by showing with drawings or manipulatives.  This will allow students to make connections and will allow for those struggling with the expressions to see the equation and equality more clearly.

As the discussion flows, be sure to draw out the idea of equality – how what is on one side of the equal sign must be the same, or have the same value, as what is on the other side of the equal sign.  Explain this is the purpose of the equal sign, to show amounts as being the same, not to indicate an answer or a total. You may even consider showing the same equation, but with the expressions switched (5+3 = 4+4).  Ask students:  Is this equation true?  Hopefully they will immediately respond Yes since all you did was switch the sides the addends are on.

1. Practice and Application (10 minutes)

In order to allow students to extend this idea and practice, one option is to give students another equation and ask them to determine if it is true.  This will allow some students to try another method for determining or proving/disproving equality that they did not use previously but was highlighted during group discussion.  An example could be:

2  +  4   =   3  +  2

To provide an opportunity for students to apply what they are learning, show them the following numbers and ask them to write a true equation using only these numbers and symbols.

8              11            3

The resulting true equations could be 11 = 8 + 3 or 8 + 3 = 11.  Be sure to highlight both ways to reinforce that the equal sign is a symbol meaning the same amount and not just preceding the total.

If students are grasping this idea of equality with general ease, consider using four numbers for this activity instead of three. Have students write a true equation with the four numbers.

2              6              7            3

These numbers could result in an equation using addition or subtraction.  It would be interesting to see how students respond.  2 + 7 = 6 + 3  or   6 – 2 = 7 – 3

Evaluation of Student Understanding

Informal Evaluation:

Observe students to see what strategies they use to determine and prove or disprove equality with each equation.  Students will likely fall into the following categories:

-those that lack an understanding of equality in general and/or the symbols used (=, +)

-those that grasp equality but use pictures and counting to check whether equations are true

-those that understand equality and have a good grasp of computation so they are able to

see equality through mental computation or using what they know about the operations to

arrive at a total to check for equality

As needed, use questioning to further understand students’ thinking.  Make note of the category students tend to be in and also strategies they are using when solving or computing.

Formal Evaluation/Exit Ticket:

Give the students the following task and ask them to show and justify their thinking.

Yasmine says the equation 12 = 7 + 4 is true because 12 is the same amount as when you