RTI Understanding the Equals Sign: What does it really mean?

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SWBAT demonstrate and explain what an equals sign means in an equation so they can later solve for an unknown.

Big Idea

Using a balance scale helps students make a connection to what equals really means: balance! This essential concept lays down the foundation for many of our 4th grade CCSS standards in regard to equations.

Reconstruction of Understanding the Equals Sign.

20 minutes

Rationale:  My fourth graders have a concept of the equals sign that is linear. I often see them write equations with a trail of continual equals signs: 2+3 = 5 + 6= 11-3=8. This linear thinking creates a lot of problems for them when solving equations with variables! They will solve the equation: 3+4=n +2 as 3+4=7+2 and then add another equals sign and write in the 9. I want them to understand that equals means balancing both sides. I want them to get comfortable with an equation like: 12= 8+4. My students didn't understand that it is OK to write and equation like this! Having the total on the left side completely confused them. Understanding what equals means is essential in order to proceed in supporting all the CCSS standards that expect conceptual understanding of equations, comparison, word problems and standard algorithms. MP1 is being demonstrated within this lesson as we persevere to solve this problem. 

Learn by Watching:

Materials: See-saw type balance scale. 2 objects that are the same weight. 1 of the same objects that weigh more or less. You can use lemons or oranges...they work well...or mini pumpkins.

After placing the balance scale where students could see, I placed a mini pumpkin on one side of the scale.  Tipping the scale caught their attention right away!  I started by asking my students: What do we need to do to make the scale balance? I dropped the other pumpkin on the scale and it didn't balance. I asked:What is wrong? They reacted right away and explained that they need something that weighs "the same." I continued: How do we know when the object weighs the same? We concluded that both objects need to be equal weight or mass and that the scale would then be balanced. Same means "balanced."

I told them that equations work the same way and that we would use this understanding to solve for the unknown. I explained that when I say "unknown" it means the same thing as "what we don't know". I never assume they know how to apply vocabulary to mathematical concepts and always pay attention to clarifying each term.



From See-Saw to Equation: A Lesson in Balance and Understanding Equals.

20 minutes

The Core Lesson:

Materials: Enough connector cubes and balance scales for partners or groups of three.

Learn by Watching and Doing: I asked one student to come up and put three connected connector cubes in one of the trays. Then, another student put two connector cubes in the same tray. I asked my students to write an expression that shows what is in the tray in their notebook. What operation would we use to show what we just did? I asked this question because it sets up their mind for understanding the concept of equals.

I wrote the expression on a note card and placed it on the table near the side with the cubes. I asked them to compare their expression in their notebook. Does it balance? I asked them to correct it if it didn't by looking carefully at one side. I asked: What do we need to do to make it balance? Student's hands shot up at this point and I let students to share their thoughts. A volunteer came up and got the scale to balance by adding five connected cubes.

I then placed the card with a "5" on the other side. I guided their thinking by asking students to show other ways to show 5.

To extend their thinking: I used the connector cubes, breaking them apart 4+1 , (Commutative Property was discussed at this point. I used this teachable moment to talk about it casually and seek out their prior knowledge. From their responses, I knew I needed to review it.) I was sure to bring out  the idea that the total "5" answer 5 is also thought of as 5+0. The other addend is "invisible" . There are no connectors to show 0.

I know I can use this lesson to help students in mastering their fluency in adding math facts, gets them set up in understanding the role in the unknown in an equation created from a word problem, reasoning that equations like 2x3 = 5+1 therefore  100+50 = 30x5 etc. and helps them build the concept. I can always refer back to it throughout the year if the concept is forgotten or needs reinforcement.




Practice, Practice...

25 minutes

In order to master solving for the unknown, students needed to reason abstractly, solve for the unknown, use the tool correctly to demonstrate and attend to precision. This gives them freedom to see common mistakes in their solutions on paper and be able to correct them with the tool. I sent this Educreations movie to all students to watch prior to the lesson to help front load their minds for the next part of the practice.

RTI :My Below Grade Level Achieving Students need to practice using balance scales and blocks.

I created a set of note cards with various equations for them to practice in partners: 5+6 = _ + 7. They used the balance scale and the connector blocks to practice each equation and solve for the unknown. I worked with them and observed their discussions. To help their thinking, I showed them an equation, 5+6=11 + 7. I had one student  show that equation using the balance scale and blocks. Immediately they saw how that looked. I asked them to remember that equals means to balance.

Now, I tried then to have them transfer their thinking to paper. I asked them explain verbally why they thought this common mistake is a mistake. They were more easily able to discuss their understanding, but I could see that it needed practice.

*Teacher Guides towards Mastery of the Goal: I believe this conversation is key to getting them to explain that both sides have to equal each other and that the equals sign means "the same as" or balance. Although they can show it on the scale, transferring it to paper is difficult for some.

I had these students  practice several problems I had written on the white board, instructing them to  solve the one side and then the other to balance. I encouraged them to use the balance scale to see if they were correct.

At and Above Grade Level Achieving Students:  This group of students was able to transfer their understanding to paper using simple problems very easily. I had these students work in partners to create equations using multi-digit numbers.  I wrote these on my other white board.  They  worked with equations that look like this and I introduced "variable" to them to further their mastery.

 234 + 100 = n+ 134, then build

2,345 + 100 = n + 445

234,567 + 100 = 234,000 +n   

Students easily solved these and so then I kept going to challenge them.

*Differentiated in partners for really advanced thinkers: I created inequalities like this for two student to solve. They were ready for a challenge. I had them then exchange the work and answer one another. They worked on white boards or on their iPad "Show Me" app to practice. 43+2> ___ x 9 It was set up to "Choose:  a.  5   b. 6" to imitate standardized testing format and this gets them to think about solution choices. I think for the higher level thinker, it becomes meaningful work and they get excited to see that they were moving ahead as well as having to think a little bit harder this time ! This truly was a time of discovery and deepening of understanding as we simply worked to understand the meaning of equals.

Science Extension and Connection: Equilibrium and Flight

40 minutes

Looking for an extension for those who have mastered this concept? This "extra" lesson is a great way to incorporate the understanding of what balance and equals and equilibrium all mean. And it's just plain fun!

School Community Connection: Recently, I  able to support and counsel a Girl Scout Senior who is earning her Gold Award through creating and teaching summer school lessons to middle school children about flight. Elizabeth Bullock is a former student at Brookwood and currently a senior at Badger High School in Lake Geneva. This particular lesson is a very modifiable lesson that connects science with the concepts learned in the math lesson about what the equals sign really means. This lesson puts the whole concept on a higher level and would be a great way to differentiate learning about equals for different levels of students. 

Equilibrium and Flight

Materials and Prep: 

Helium filled balloons. Enough for your class either as individuals or in partners. 

A small paper cup (4 oz bathroom size) with 4 holes punched equidistant around the perimeter of the lip of the cup. 

Different types of candy: Various sizes like M&M's, Nerds, Skittles or you can use different size paper clips if you don't want to use candy.

One yard of a piece of yarn per student so they can loop it around the balloon and through the cup holes to fasten the balloon ( the cup is like the basket on a hot air balloon).

Opening: 5 minutes.

She began the lesson by introducing the vocabulary word: Equilibrium. We broke apart the word for meaning and connected to math, balance and idea of the balance scale. 

Activity: 10-15 minutes

Materials were given to students at this point.

In order to adhere to the concepts of discovery based science, she allowed them to construct the balloon with minimal guidance. When the balloons were all put together, the classroom was filled with balloons that looked like real hot air balloons with different length strings, but students soon figured out how to balance the paper cup basket through trial and error. As this was going on, I visited students and asked higher order thinking questions to help them think through and understand strategies of achieving equilibrium. 


They were encouraged to use the candy to load the paper cup up until the balloon simply hung in mid air between the floor and the ceiling. Students worked at using different types, some mixing the Nerds and other choices, while some students simply chose one kind of candy. They could observe how each other interpreted what would work and what would be too light or too heavy. 

The classroom was filled with lots of laughter, discussion and engaged children as they sometimes had to reach up to grab their balloons. 

When all had seemed to succeed or get close to the idea of what was going on, students raced their balloons by blowing them. The person closest to equilibrium won the race because they didn't worry about blowing theirs up or down. 

Post Discovery Discussion: 

Students were led by Elizabeth in discussion as to why the person with the best point of equilibrium won the race.

She used questions like:

What did you notice?

Why do you think (Name) won the race?

What would it have to do with the point of equilibrium? 

 Students continued to engage themselves in working on their balloons to create the point of equilibrium as they played with the candy and their balloons. 

Building Upon Scientific Concepts:

She settled them down to introduce understanding the concept of "lift." In order for them to discover how "lift" can work. She directed them to blow across the top of the balloon. She drew an airplane on the white board and compared how that works to how lift on a hot air balloon works. 

The children raced their balloons again using this concept. 


Post Discovery Discussion:

Students gathered around to share what they had observed this time. As Elizabeth led them with similar questioning, they continued to talk about how they noticed now that some students who had heavier baskets were affected by the lift. They concluded that a person with a slightly too heavy basket had won the race and not the person with equilibrium. 

They did discuss understanding what the concept of "false lift" is and that with their helium balloon, there is the concept that if it isn't there, the heavier balloon will fall. But, with a hot air balloon, the heat comes into context in keeping it in the air.