Balance Scale and Equations
This year is the first year I have spent this length of time teaching equations using a balance scale. We are on day 3 of only working off of a balance scale concept or model.
Prior to the students working on equations with balance scales we discussed a variable or block as a mystery number. I used the analogy of Mary Poppins magical bag as an example ...though they look small they can hold a lot or a little. Many students hadn't seen Mary Poppins so it also gave them a "Brain Break" to watch a short clip of her pulling out a lamp and various other items when she first meets the children (They also suggested Harry Potter has a similar clip). From there we moved onto chips and blocks to understanding that in any problem it could be blocks, crates, or even in some instances triangles. We also talked about using any letter to stand for these things. (This all built from combining like terms; blocks that are the same color can be grouped together; shapes that are the same can be grouped together etc.)
With the balance scale students conceptualized keeping it even or level, and by the end of day 2 students were quickly solving problems by removing items from the scale. Since they had a basic understanding we moved into what it would look like mathematically. This was an easy transition since they'd done like terms.
By the end of day 3 all students can use blocks and beans to solve equations that involve positive integers with variables on both sides with no mistakes. This progression is faster than rote memorization as I haven't had to go from one step to two step to multi step. Today I gave them the textbook in groups of three to try to solve a problem. I have given them no rules just that a scale has to be balanced (I have never turned students lose on equations with no rules). Problems from the book had no illustrations, but I encouraged them to draw out the picture. Amazingly students were able to solve them. Some even questioned how it would work with negatives. When asked this by a group together we looked over their previous problems to see if they could find a pattern. The students did. They discovered or learned a "trick" as I call it without me telling them the rules.
Below is an example of a couple groups work. Though the mathematical structure could be improved they both have a way to get an answer.
Prior to the students working on equations with balance scales we discussed a variable or block as a mystery number. I used the analogy of Mary Poppins magical bag as an example ...though they look small they can hold a lot or a little. Many students hadn't seen Mary Poppins so it also gave them a "Brain Break" to watch a short clip of her pulling out a lamp and various other items when she first meets the children (They also suggested Harry Potter has a similar clip). From there we moved onto chips and blocks to understanding that in any problem it could be blocks, crates, or even in some instances triangles. We also talked about using any letter to stand for these things. (This all built from combining like terms; blocks that are the same color can be grouped together; shapes that are the same can be grouped together etc.)
With the balance scale students conceptualized keeping it even or level, and by the end of day 2 students were quickly solving problems by removing items from the scale. Since they had a basic understanding we moved into what it would look like mathematically. This was an easy transition since they'd done like terms.
By the end of day 3 all students can use blocks and beans to solve equations that involve positive integers with variables on both sides with no mistakes. This progression is faster than rote memorization as I haven't had to go from one step to two step to multi step. Today I gave them the textbook in groups of three to try to solve a problem. I have given them no rules just that a scale has to be balanced (I have never turned students lose on equations with no rules). Problems from the book had no illustrations, but I encouraged them to draw out the picture. Amazingly students were able to solve them. Some even questioned how it would work with negatives. When asked this by a group together we looked over their previous problems to see if they could find a pattern. The students did. They discovered or learned a "trick" as I call it without me telling them the rules.
Below is an example of a couple groups work. Though the mathematical structure could be improved they both have a way to get an answer.
Lastly, this has been a great feat for this class as previously they have had no manipulative experience and have been reluctant to use or learn with them. I am in the process of equipping this new classroom with more math manipulatives to help students find mathematical patterns to justify the rules/tricks that they discover.
Below are links to two sets of worksheets that have allowed me to teach this way, and still provide HW. (We know math textbooks don't provide these types of problems).
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