Classroom Activities
Lesson 1:(language)
We will begin by asking the students to list all the names that they can think of for saying addition. Together we will compile the above list, but if not I will add any omissions to the list. We will do the same for subtraction, multiplication and finally division. Students will be expected to save this list and refer back to it when needed.
I will give several examples of word phrases that use the same language and have a class discussion about why they are the same or why they are different. Here are a few examples of the phrases I will use.
5 more than x. (addition)
The number is increased by 5 (addition)
The sum of the 2 numbers equals 34 (addition)
The first number is 3 less than the second (subtraction)
What is the difference between the two numbers (subtraction)
If I double a number what happens (multiplication)
What is the product of the two numbers (multiplication)
If I split something between 7 people (division)
2/5 of a number (multiplication)
I will have the student just think about the operation. The list that we compiled at the beginning of class will be very useful and students will be encouraged to refer to it.
Lesson 2:(symbols)
The students will change the above phrase into symbols.
The words "5 more than x" will become "x + 5".
The number is increased by five can be shown as "x + 5".
The sum of the two equal number is 34 can be shown as 2x = 34 because if the numbers are equal you are just doubling the two numbers
The first number is 3 less than the second number can be show as (x-3) and (x).
I will continue to go through our list of phrase and help students convert them into symbols. I will continuously stop to ask students if our statements are true and if the symbols match the words.
Lesson 3:(sketch/picture)
Students can connect the word to the symbols, now we will begin making simple sketches to represent the symbols. The scaffolding is evident because we have gone from words to symbols and now to pictures. We will take the mathematical statements and turn them into mathematical sentences. This is just a "picture problem" but I have not introduced them as such to my students yet. An example of a mathematical sentence is "Trish has twice as many books as Paula, together they have 5 five books". To translate this into a sketch we would focus on the fact that two thing together (+), is (=) five. The sketch would be ____ + ____ = 5. Make sure that the sketch is a true representation of the symbols. I would very slowly say the sentence and point to each piece of the sketch to see if there is a representation of each word in my sketch. The sketch must be a true statement.
The statement "three consecutive number whose sum is 138" can be sketched ___ + ___ + ___ = 138.
I will do a few more example to emphasize the connection between the words, symbols and the sketch. I will always ask students to check to make sure that the sketch is representing a true depiction of the statement.
Lesson 4: (identifying the unknown)
Students will be asked to read the following problem.
A full bucket of water weighs eight kilograms. If the water weighs five times as much as the bucket empty, how much does the water weigh?
The students will be instructed to draw a sketch that represents the problem, ___ + ___ = 8 will be a good representation. The problem is talking about water and a bucket and we are given some information about one of them. The problems tells us something about the relationship between the water and the bucket, but we know nothing about the bucket. We will call our unknown (bucket) x. Students will be instructed to put that in the sketch, __X__ + ____ = 8. We now can build on the problem. If the water weighs 5 times as much as the bucket, translating that into symbols would give us "5X", and if we transfer that to the sketch, we would have X + 5X = 8.
Lesson 5: (brushing up on algebra skills)
Students will be called to the board to complete the problem (X + 5X = 8) but only if they are able to explain to the class what and why they are doing to the problem. I will be hoping for "Combining like term" to explain the transition to 6X = 8. Students may have an understanding of this term as just a collection of like terms but may not recognize it as an application of the distributive property so I will demonstrate by showing them that (X + 5X) is X( 1 +5) and that is why we can say 6X = 8.
Teacher: "Should "Combining like terms" be included with your list of terms meaning add?"
Student: "Yes"
Teacher: "Is that the only way to think about it?"
Student: "No, we could add it to the list of multiplication terms."
Teacher: "Great, now what do we want to do?"
Student: "Divide both sides by 6."
Teacher: "Why are you allowed to do that?"
Student: "Because what you do to one side you have to do to the other."
Teacher: "Why?"
Student: "Because the statements are equal or balanced, and if I change one of the sides without changing the other, it will be out of balance."
Teacher: "Why did you divide by 6 though?"
Student: " Because any number divided by itself just becomes one and since I wanted only one x, I can turn six into one by dividing by six."
Teacher: "Great, you really understand what you are doing."
Student: "Yes, I had a great teacher."
This conversation may be far fetched but it is important to check for understanding. The concept that should be stressed is the idea of doing the same thing to both sides and the math magic that allows you to turn 6x's into one x, and why you are allowed to do it.
Lesson 6 (Identifying an Equal Problem)
Students will read the problem.
The number of wild horses at the Lazy Z Dude Ranch could be found by counting them, but when Hank visits the ranch, he suggests that 23 fewer than five times the number of horses at the ranch is the same as 58 more than twice the number of horses on the property. If Hank is right, how many horses does the Lazy Z have?
Teacher: "Can you draw a simple sketch for this story?"
Students: "No."
Teacher: "Do you see any math terms than can be changed to math symbols?"
Students: "Yes, 23 fewer than 5 times the number of horses."
Teacher: "How would you change that to symbols?"
Students: "5x - 23"
Teacher: "How do you represent the same as in symbols?"
Students: "Equal sign (=)."
Teacher: "Are there any more math terms we can change to math symbols?"
Students: "58 more than twice the number of horses, and we can change that to 2x +58."
Teacher: "Do you see where we are going with this?"
Students: "Yes, we can now just make an equation that tells our story. The equation will be 5x - 23 = 2x + 58.
The algebra steps that are needed to solve the equation are very important and should be emphasized. It is another opportunity to not only stress the steps but to check for understanding. I will ask my students why they are doing what they are doing and why they are allowed to do it. The following steps are used to solve the problem.
We will add -2x to both equations because whatever you do to one side of the equation you must do to the other, we chose -2x because we are trying to get all the x's on one side of the equation and -2x will make the +2x disappear on the one side. I like to call this "math magic" because it makes one of the variables disappear and creates a new equation (3x -23 = 58). Students can use the same step from the Picture Problem to finish the problem, and will discover that x = 21 or that there are 21 horses at the ranch. Students will be expected to check the answer by plugging the value of x into the original equations to check to see if our statements are true.
I will continue this lesson by doing several of the same problems and emphasizing why I have named them Equal Problems. I will stress the skills needed to not only solve these types of problems but to also recognize one when we see it. I will finish this lesson by pointing out the "red flags" to look for to spot a equal problem and talk about the differences between a equal problem and a sketch problem.
Lesson 7 ( recognizing Two Step Problems)
The students will be asked to read the problem aloud.
Fifty more "couples" tickets than "singles" tickets were sold for the dance. "Singles" tickets cost $ 10 and "couples" tickets cost $ 15. If $ 400 was collected, how many of each kind of ticket was sold?
Students will be asked if this problem has any of the "red flags" from the first two example problems. It will be hard to draw a picture of this problem so we can not call it a Picture Problem, and since there is no equal sign any where in the problem we can not call it an Equal Problem. Students will asked what two things we are talking about in the problem (tickets/money). I have already give a very detailed example in my Strategies portion so I will not go into details for solving this problem. The lesson is recognition first and the solving will just fall into place with some teacher instruction.
Lesson 8 ( recognizing different problems)
Students will be given one example of the above types of problems on a sheet of paper and asked to name the type of problem. They will them be asked to go to the group that has the same problems and talk with the other students to see if they are all similar. Once the group is in agreement that they are all the same, they will be asked to explain what helped them decide. They will determine the "red flags" from each problem and share their finding with the rest of the class. Students will state the mathematical terms or symbols that helped them decide what category their problem fell into. I will use this bank of problem for future lessons and will have them as warm-ups to use through out the year. I will add problems and titles as the year progress and will continue to expect students to identify the problems by name. The complete list of problems is attached.
This lesson has been broken down into eight mini lesson because of the different scheduling throughout the country. I think that several of these lessons can be taught in one day depending on the skill level of the students but each idea addressed is an important point that should not be overlooked. Having a solid base will assure success for the more complex problems that students will face as the year progresses. The students will cover Age Problems, Distance Problems, Mixture Problems, Work Problems, Finance Problems and Quadratic Problems throughout the course of this class. I have designed this first lesson of the year in the hopes of creating success and understanding. I will compare and contrast the problems through out the year and continue to assist students in recognizing problems and giving them the tools they need become problem solvers.
Comments: