Emphasize that this type of factoring should be use first before applying any type of factoring. Give examples of this type factoring after the activity. You can use the examples found in learning module. The above activity gave us the idea about the Greatest Common Monomial Factor that appears in every term of the polynomial. Study the illustrative examples on how factoring the Greatest Common Monomial Factor is being done.
Factor 12x3 y5 — 20x5 y2 z a. Find the greatest common factor of the numerical coefficients. The GCF of 12 and 20 is 4. Find the variable with the least exponent that appears in each term of the polynomial.
The product of the greatest common factor in a and b is the GCF of the polynomial. Hence, 4x3 y2 is the GCF of 12x3 y5 — 20x5 y2 z. To completely factor the given polynomial, divide the polynomial by its GCF, the resulting quotient is the other factor. Thus, the factored form of 12x3 y5 — 20x5 y2 z is 4x3 y2 3y3 — 5x2 z Below are other examples of Factoring the Greatest Monomial Factor. Divide the polynomial by 8x to get the other factor.
Divide the given expression by the greatest monomial factor to get the other factor. Complete the table to practice this type of factoring. Why do you think it was given such name? Investigate the number pattern by comparing the products then write your generalizations afterwards. What are the different techniques used to solve for the products? What is the relationship of the product to its factor? Have you seen any pattern in this activity? For you to have a clearer view of this type of factoring, let us have paper folding activity again.
Directions: 1. Get a square paper and label the sides as a. Cut — out a small square in any of its corner and label the side of the small square as b. Cut the remaining figure in half.
Form a rectangle A C G E D F B Before doing the activity for factoring difference of two squares, ask the students why the difference of two squares was given such name.
To start the discussion you can use number pattern to see the relationship of factors to product. You may bring back the students to multiplying sum and difference of binomials in special product to see how factors may be obtained. Students should realize that factors of difference of two squares are sum and difference of binomials.
Ask students to generate rule in factoring difference of two squares. This activity may be done by pair or as a group. Give more examples if necessary. Note: Remind students to use first factoring greatest common monomial factor if applicable before factoring it through difference of two squares 1.
What is the area of square ABDC? What is the area of the cut — out square GFDE? What is the area of the new figure formed? What is the dimension of the new figure formed? What pattern can you create in the given activity? What is the first term of each polynomial? What is the last term of each polynomial?
What is the middle sign of the polynomial? How was the polynomial factored? What pattern is seen in the factors of the difference of two terms? Can all expressions be factored using difference of two squares? Why or why not? When can you factor expressions using difference of two squares? For you to have a better understanding about this lesson, observe how the expressions below are factored and observe the relationships of the term with each other.
Activity 7 Description: This game will help you develop your factoring skills by formulating your problem based on the given expressions. You can integrate other factoring techniques in creating expressions. Create as many factors as you can. Directions: Form difference of two squares problems by pairing two squared quantities then find their factors. Hint: You can create expressions that may require the use of the greatest common monomial factor You have learned from the previous activity how factoring the difference of two squares is done and what expression is considered as the difference of two squares.
We are now ready to find the factors of the sum or difference of two cubes. To answer this question, find the indicated product and observe what pattern is evident. How are the terms of the products related to the terms of the factors? What if the process was reversed and you were asked to find the factors of the products?
How are you going to get the factor? Do you see any common pattern? To check students understanding on factoring difference of two squares, ask them to make pairs of square terms and factor it after.
Students can give as many pairs of difference of two square as they can create. Ask the process question to the students and help them see the pattern in factoring sum or difference of two cubes. Guide them to generate the rule in factoring sum or difference of two cubes. Give more examples of sum or difference of two cubes and factor it to firm — up the understanding of the students in factoring this expression.
Note: Remind the students to use first factoring by greatest common monomial factor before applying this type of factoring if necessary. Is the given expression a sum or difference of two cubes? What are the cube roots of the first and last terms? Write their difference as the first factor. For the second factor, get the trinomial factor by: a. Squaring the first term of the first factor; b. Adding the product of the first and second terms of the first factor.
Squaring the last term of the first factor 4. Write them in factored form. Write their sum as the first factor. Subtracting the product of the first and second terms of the first factor. Represent the volume of this figure. What is the factored form of the volume of given figure? What are the volumes of the cubes? If the cubes are to be joined to create platform for a statue, what will be the volume of the platform?
What are the factors of the volume of the platform? Answers to problem: 1. Activity 9 Directions: Prepare the following: 1. How will you represent the total area of each figure?
Using the sides of the tiles, write all the dimensions of the squares. What did you notice about the dimensions of the squares? Did you find any pattern in their dimensions? If yes, what are those?
The polynomials formed are called perfect square trinomials. Perfect square trinomial is the result of squaring a binomial. A perfect square trinomial has first and last terms which are perfect squares and a middle term which is twice the product of the square root of first and last terms. To start factoring perfect square trinomials, use algebra tiles to model it.
This activity will give the students picture of perfect square trinomials. See to it that the students will produce a square. Discuss the answers of the students on process questions. Point out that the result of squaring binomial is a perfect square trinomial.
At this point students should see the pattern of factoring perfect square trinomials and be able to generate the rule in factoring such polynomials. Discuss when an expression is a perfect square. Do the perfect hunt activity to check the students understanding in identifying perfect square trinomials.
Answers might be in diagonal, horizontal or vertical in form. Get the square root of the first and last terms. Examples of factoring perfect square trinomials should be given to ensure mastery. Answers on exercise: a. The square root of the first term is 2r and the square root of the last term is 3 so that its factored form is 2r — 3 2. What is the effect of removing 3t? Exercises Supply the missing term to make a true statement.
Are all trinomials perfect squares? How do we factor trinomials that are not perfect squares? In the next activity, you will see how trinomials that are not perfect squares are factored. Ask them to factor it. This will make the students realize that there are some trinomials that are not factorable using perfect square trinomials.
Use this as springboard before proceeding to activity Note: Make sure to it that the students will form rectangle as their figure. Ask them to compare the dimensions of the figure formed in this activity and activity 9.
Activity 11 Description: You will arrange the tiles according to the instructions given to form a polygon and find its dimensions afterwards. Directions: Form rectangles using the algebra tiles that you prepared.
Use only tiles that are required in each item below. Cut — out 4 pieces of 3 in. Cut — out 8 pieces of rectangular cardboard with dimensions of 3 in. Cut — out another square measuring 1 in. What is the total area of each figure? Using the sides of the tiles, write all the dimensions of the rectangles. How did you get the dimensions of the rectangles? Did you find difficulty in getting the dimensions? Based on the previous activity, how can the unknown quantities in geometric problems be solved?
If you have noticed there are two trinomials that were formed in the preceding activity, trinomials that contains numerical coefficient greater than 1 in its highest degree and trinomials whose numerical coefficient in its highest degree is exactly 1. Let us study first how factoring trinomials whose leading coefficient is 1 being factored. List all the possible factors of 6. Find factors of 6 whose sum is 5. List all the factors of — 21 Factors of -3 7 -7 3 1 -1 21 b. Find factors of whose sum is 4.
Listing all the factors of — Give examples of general trinomials whose leading coefficient is 1. You can use trial and error in factoring these examples. Use the examples found in learning module. Giving more examples is highly suggested. You can ask the students to generalize how factoring of this trinomial is attained. Remind them again that they should use factoring by greatest common monomial factor using this type of factoring, if applicable. Write on a strip the polynomials below and place them on container.
Draw the strip and read it in class, give the students time to factor the polynomials. Activity 12 Description: Bingo game is an activity to practice your factoring skills with speed and accuracy.
Pick 8 different factors from the table below and write the in the grid. As your teacher reads the trinomial, you will locate its factors and marked it x.
The first one who makes the x pattern wins. How did you factor the trinomials? What did you do to factor the trinomials easily? Did you find any difficulty in factoring the trinomials? What are your difficulties? How will you address those difficulties? Are trinomials of that form factorable? There are many ways of factoring these types of polynomials, one of which is by inspection.
Trial and error are being utilized in factoring this type of trinomials. Here is an example: Factor 6z2 — 5z — 6 through trial and error: Give all the factors of 6z2 and — 6 Write the all possible factors using the values above and determine the middle term by multiplying the factors.
How was inspection used in factoring? What do you think is the disadvantage of using this? Factors of: 6z2 -6 3z 2z 3 -2 6z z -3 2 1 -6 -1 6 Give polynomials whose numerical coefficient of the leading term is not 1. Factor this using trial and error. Allow the students to stress out the disadvantages that they have encountered in using this technique.
Introduce the factoring by grouping or the AC method after. Ask them to compare the process. Provide for more examples. Factoring through inspection is a tedious and a long process, thus, knowing another way of factoring trinomial would be very beneficial in your study of this module. Another way of factoring is through grouping or AC method. Closely look at the given steps and compare it with trial and error. Factor 6z2 — 5z — 6 1.
Find the product of the leading term and the last term. Find the factor of — 36z2 whose sum is — 5z. Rewrite the trinomial as four — term expressions by replacing the middle term by the sum factor. Group terms with common factors. Factor the groups using greatest common monomial factor. Factor out the common binomial and write the remaining factor as sum or difference of binomial.
Multiply the first and last terms. Find the factors of 24k2 whose sum is k. Find the factors of 12h2 whose sum is h. Factor out the common binomial, and write the remaining factor as sum or difference of binomial. Activity 13 Description: This game will help you practice your factoring skills through a game. Instruction: Form a group of 5. Your task as a group is to factor the trinomial that the other group will give. Raise a flaglet if you have already factored the trinomial and shout, We have it!
The first group to get 10 correct answers wins the game. We can use factoring by grouping technique in finding the factors of a polynomial with more than three terms. Group terms with common factor. Factor out the greatest common monomial factor in each group. Do you find difficulty in playing the game? What hindered you to factor the trinomial? What plan do you have to address these difficulties?
To practice the factoring skills of the students, do Activity 13 in class. Group the students and distribute flaglet on each group. Ask one group to give a factorable polynomial then let the other group factor it.
Extend the concept of factoring by grouping by applying it to polynomials with four terms. You can use the examples on learning module. Perform Activity 14 as a group after.
You can use the graphic organizer for this activity. This may serve as bring home activity. Make sure to it that in every group there is one responsible student. You may facilitate the mentoring or you can give the group free hand in doing this activity. Factor out the common binomial and write the remaining factor as sum or difference of two terms. Instruction: With your groupmates factor the following expressions by grouping and write a four - letter word using the variable of the factors to reveal the 10 most frequently used four - letter word.
Direction: Together with your group mates, discuss your thoughts and queries regarding factoring. What different types of factoring have you encountered? What are your difficulties in each factoring technique?
Why did you face such difficulties? How are you going to address these difficulties? What techniques have you used to answer the questions? What things did you consider in factoring? Did you find difficulty in the factoring the polynomials? You may add box if necessary.
Activity 16 Description: This is a flash card drill activity to help you practice with speed and accuracy your factoring technique. Instruction: As a group you will factor the expressions flashed by your teacher, each correct answer is given a point. The group with the most number of points wins the game. Ask the students to go back to IRF sheet and answer the Revise part.
Discuss the answer of the students in class. Instructions: Do as directed. Your classmate asserted that x2 — 4x — 12 and 12 — 4x — x2 has the same factors. Is your classmate correct? Prove by showing your solution. Can the difference of two squares be applicable to 3x3 — 12x? If yes, how? If no, why? Make a generalization for the errors found in the following polynomials. The following activities will check your mastery in factoring polynomials.
What to UnderstandWhat to Understand This part provides learners activities to further validate and deepen their understanding on the applications of factoring and to check their knowledge against misconception.
Difference of two squares is only applied if the middle operation is minus. Are all polynomial expressions factorable? Cite examples to defend your answer. What have you observed from your answers in your initial column? Is there a big difference? What realization can you make with regard to the relationship of special products and factors? You will need the guidance of your teacher in doing such.
What different factoring techniques have you used to arrive at the solution? What was your realization in this activity? After performing activity 19 allow the students to revisit IRF worksheets and discuss their answers as a group. You can ask them their thoughts in this lesson. Challenge the students by doing Activity Guide them in doing this activity and help them realize that there is an error in this process.
Cite the mistake in the activity given. Increasingly, you will tackle problems which require several steps to solve them. Basically problems can be categoried into different types which needs different approaches. The most problematic is the word problems. These problems should be the most interesting ones to solve.
This step is usually the most challenging part of an applied problem. If possible, start by drawing a picture. Label it with all the quantities mentioned in the problem. If a quantity in the problem is not a fixed number, name it by a variable. Identify the goal of the problem. Then complete the conversion of the problem into math, i.
Solve the math problem you have generated, using whatever skills and techniques you need. As a final step, you should convert the answer of your math problem back into words, so that you have now solved the original applied problem. Did You Know That.. Everything you can do with a ruler and a compass you can do with the compass alone. Among all shapes with the same perimeter a circle has the largest area.
Much as with people, there are irrational, perfect, complex numbers You are wrong if you think Mathematics is not fun. Some numbers are square, yet others are triangular. One can cut a pie into 8 pieces with three movements. Among all shapes with the same area circle has the shortest perimeter. What happens?
It finally comes to zero. Try it again with different starting numbers, and see how it always comes down to 0. Try it on your friends. Let them choose the numbers. Views An interesting problem for Geometrists acron shared this question 10 years ago. Geometry Inventor dynamic software of Logal Company seems to be defunct at this time.
Oldest Newest Popular. Files: Tri2p. Reply URL. I think there is a problem with your construction Tony or was that intentional? I think the ratio of the areas is exactly 7. Maybe I misunderstood the drawing specification? Hi, Agree with mm - now to prove it Mine match Daniel's!
I see that several of the Geometricians have done an excellent response. Thanks for the great work. Mathmagic speaks of the solutions found in the article. Thanks, Tony PS I found the article while working on toward some material toward my dissertation.
Hi again! A satisfactory end to my weekend! I would be interested to see your proof Kathryn. Mine is a bit different, I think. The large triangle DEF is thus partitioned into 7 triangles with equal areas. Files: triangle7. Files: Triangles. This is a nice problem, and I am still investigating; fun! I am glad that I can still recognize problems that challenge others fun!
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