Cups and Counters – Station 2 - Weebly



Materials:

Overhead Cup & Counters or Elmo

Cups & Counters for each student or one set per pair of students

Every student will need his/her own paper

Strips of “manipulative reference” for each student or pair of students (on the manipulative reference pages, there are pictures of square counters instead of circle counters … in case you use square counters).

Objective:

This is day one of solving equations. This lesson helps students understand how to solve equations in a hands-on manner.

TEKs:

Algebra I – a) Basic Understandings 5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understating underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology … to model mathematical situations to solve meaningful problems.

Algebra I – A.7 Linear functions … B) investigate methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, select a method, and solve the equations and inequalities.

Before beginning, remind students what each manipulative represents

|Unknown value (cup) |Positive numbers |Negative numbers |Equal (String) |

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Modeling:

First, demonstrate some equations and ask students to model them with their cups and counters.

|x + 2 = -4 | |

|x - 5 = 3 | |

|2x = 8 | |

|3x + 1 = 7 | |

Zero Pairs:

Next, remind students that zero pairs (one positive and one negative counter – or cups) reduce to nothing and can be removed.

Ask students to model 4 – 5. Then have them remove the zero pairs to see -1 remains.

This can be modeled in two ways:

|4 – 5 |4 + -5 |

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It is important to model both ways, and have the students notice the result is the same, so the two expressions are equivalent. This should help them understand why we add the opposite when we subtract integers.

Solving Equations Using Addition and Subtraction:

Have students sketch the cups on their paper as they model and solve the following equation with you. x + 2 = -4

➢ Before you begin to solve the equation, remind students that the goal is to determine how many counters (the unknown value) are in the cup. Obviously, there are no counters in the cup, but it represents some unknown amount of counters

➢ They want to get the cup(s) on one side of the equal sign and the counters on the other side.

➢ Also remind students that the only way to move counters is to add the opposite and then remove zero pairs. This is because what we do on one side of the equation we must do to both sides since we cannot “unbalance” the equation.

Model the solution to the equation above asking students at each step what should happen next. Remind them of the goal (1 cup equal to some amount of counters). The algebraic representation is for your use. After you modeled several examples and once you feel the students are ready, you can add the algebraic representation next to the pictorial. Ask them to show both on their paper!

|Algebraic |Manipulative (Cups & Counters) |

|x + 2 = -4 | |

|x + 2 = -4 | |

|-2 -2 | |

|Zero pairs reduce | |

|x = -6 | |

So, in the equation x + 2 = -4, x must equal -6. Ask students to check reasonableness of the solution. If there were negative six counters in the cup and we added positive two counters, would we end up with four negative counters?

Together, model the equation 5 + x = 8, and have students solve the equation. Monitor students to ensure they are drawing the representations on their paper. Have one student model his/her solution for the class. Make sure all students agree with his/her solution. A model is below for your reference.

|Algebraic |Manipulative (Cups & Counters) |

|5 + x = 8 | |

|5 + x = 8 | |

|-5 -5 | |

|Zero pairs reduce | |

|x = 3 | |

Again, have students check the reasonableness of this solution.

Next, have students model and solve x – 3 = -4 on their own. If students are ready, have them add the algebraic symbols next to their cups & counters representations.

Have students model and solve x – ½ = 4½ and 7 – x = 10. These are challenging, and may require some discussion about how to model ½ and –x. Remind students the rules from above still apply … they will need to get one positive cup equal to some amount of counters (so adding a positive cup to both sides will be necessary since multiplying and dividing by a negative has not been discussed yet). Students will not be able to model these two with the cups & counters without destroying the cups & counters, so have them draw the representations on their paper. This is a nice time to ensure students understand the tie between the algebra and the concrete representation.

Solving Equations Using Multiplication and Division:

Remind students to sketch the cups & counters on their paper as they model and solve the equations. Also, remind your students of the “rules”.

➢ The goal is to determine how many counters (the unknown value) are in one full cup.

➢ First, they want to get the cup(s) on one side of the equal sign and the counters on the other side. Then they want to determine how many counters are in one full cup.

➢ Again, remind students that the only way to move counters is to add the opposite and then remove zero pairs.

➢ One additional note in working with cups & counters – if you multiply or divide by a negative number that means you “flip” the cup & counters over (all signs become opposite because to negate something means to consider the opposite).

Have your students model 2x = 8 at their desk, while you model it for the class.

|Algebraic |Manipulative (Cups & Counters) |

|2x = 8 | |

Then ask your students: If 8 counters are in 2 cups, how many would be in one cup? How do you know? What did you do? Students should notice you are dividing the counters by the number of cups. Model the situation as you show the algebra associated with the question.

|[pic] = [pic] | |

Notice each cup would have 4 positive counters noted by separating the sketch with a line to isolate one full cup.

Have students check the answer – does two times four equal eight?

Have students try 3x = -9 on their own. Once most students have finished modeling and solving the equation, have a student demonstrate his/her solution for the class. Be certain the class agrees with his/her solution. Have students check their solution.

Next, have students model and solve –3x = 9. Remind students that multiplying or dividing by a negative means to “flip” all the counters. Another solution would be to add 3 positive cups to both sides, and then move the 9 positive counters by adding 9 negative counters to each side. Find a student who has modeled and solved this equation by each method, and have them model it for the class. If all students used one method, demonstrate the other method and ask if that method works also. Be sure students agree that either method works for solving the equation. You might want to have a discussion about which method is more efficient. Students should be allowed use the method they prefer. Again, have students check their solution. Ask students why this solution is the same as the previous solution while the equations were different.

Before leaving multiplication and division, fractions need to be addressed. Ask students how they would model ½x = 5. Have them draw the representation on their paper – please no cutting cups. (

|Algebraic |Manipulative (Cups & Counters) |

|½x = 5 | |

To solve the equation, remind students they are trying to figure out how many counters would be in one full cup. Ask students if 5 counters are in ½ a cup, how many would be in a full cup? Ask students how they know. Have them think about what action they performed to figure out how many counters were in a whole cup.

|2 (½x) = 2 (5) | |

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Show the algebraic representation that corresponds to the model (above).

Have students check reasonableness of the solution – is five one half of ten?

Once students feel comfortable with this idea, try a more complicated question, [pic]x = 6. First, have students think about how they would model this situation. How many parts is the cup cut into? How many of those parts are shaded?

|[pic]x = 6 | |

If there are 6 counters in [pic] parts, how many is in each [pic]? Then, how many counters would be in one full cup?

|Divide by 2 … or mult. by ½ because we are | |

|considering ½ the counters. | |

|[pic][pic]x = [pic]6 | |

|Multiply by 3 since there are 3 thirds | |

|(3)[pic]x = (3) 3 | |

|So, x = 9 | |

Solving two step equations:

Have students model and solve 3x + 1 = 7. Again, remind them of the “rules”.

➢ The goal is to determine how many counters (the unknown value) are in one full cup.

➢ First, they want to get the cup(s) on one side of the equal sign and the counters on the other side. Then they want to determine how many counters are in one full cup.

➢ Again, remind students that the only way to move counters is to add the opposite and then remove zero pairs.

➢ Multiplying or dividing by a negative number that means you “flip” the cup & counters over (all signs become opposite because to negate something means to consider the opposite).

See if students can solve this without your help. Then have a student model his/her solution. Ask the class if the solution works. Ask if anyone solved it differently. If someone did solve it differently, have him/her demonstrate for the class. Again, have the class decide if the solution works. If the solutions do not work, have the other students help correct the process. If students get stuck, prompt them by reminding them about their goal – one full cup equal to some number of counters.

Have students practice by modeling and solving the following equations:

➢ 5 – 2x = 1

➢ ½x + 3 = 7

If time permits, have students model their solutions for the class. The class should discuss the reasonableness of the solutions as well as the methods.

Adapted from Guerrilla Algebra for the Real World.

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Equivalent Expressions

Zero pair adds no value … but allows 5 positive counters to be taken away.

Once the 5 positive counters are removed, one negative remains.

Remove zero pairs.

Once the zero pairs are removed, one negative remains.

2 of the 3 parts are shaded

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