Download Multiplication Table

Monday
Quadratics
Math A Review
Students will lean different methods for solving binomials.
Materials:
-White Boards
-Markers
-Handouts
Lesson Overview:
Students will factor binomials/trinomials and multiply binomials. The students
will use the distributive property do multiply the binomials. The students will
also learn the “box method” to multiply and factor binomials.
Lesson Objectives:
-Students will distinguish whether they will factor or multiply binomials
(trinomials).
-Students will Compare the distributive method to the rectangle method and
determine what method is better for them.
NYS Standards:
2C. Apply the properties of real numbers to various subsets of numbers.
3D. Use field properties to justify mathematical procedures.
Anticipatory Set: (10 min)
Have the students review the distributive property to see if they remember how to
solve problems using this property. Have the students simplify a problem on the
board using the distributive property. 3(x+5) ans. 3x+15, (4+5)2 ans.=18, 328(r+7) ans. 8r-24, x(5+3x) ans. 5x+3x2, 3(x+5)-2x ans. =x+15.
Developmental Activity: (25 min)
1. Have the students multiply binomials by using the distributive property, Give
the students the multiplication chart in order to make finding the factors of
number easier. Also, hand out the worksheet quadratics. The students should
learn how to use the distributive property in order to solve the problems. This
will let them see and understand what mathematics is being done and how to
get the correct answer that answer.
2. After the students solve problems distributing have them factor the
expressions to make binomial(s) This will show the students how to get two
binomials from a trinomial and also help the students check to see if they had
multiplied the binomials correctly because if they factor the trinomial they
should get the same binomials.
3. Show the students the “box” method to distribute the binomials to make one
term. This will show the students’ an alternative way to the distributive
method. The students will multiply terms together and then combine like
terms.
4. Show the students how to factor using the “box method” This method will
show the students how to factor a trinomial by taking out like term. The
students are able to see and understand where the numbers come from that
they get for an answer.
Closure: (5 Min)
Have the students write on a sheet of paper what way they prefer to solve
binomials/trinomials, have the students state the steps that are needed in order to
use this method to get the correct answer.
Assessment: (on going)
Assessment will be on going. Groups will be able to evaluate how the students
are doing by the students’ responses on the worksheets that are given out as notes.
Also a log will be kept when walking around the classroom.
Multiplication Table
Multiplication Table
x 0
1
2
3
4
5
6
7
8
9
10
11
12
0 0
0
0
0
0
0
0
0
0
0
0
0
0
1 0
1
2
3
4
5
6
7
8
9
10
11
12
2 0
2
4
6
8 10 12 14 16
18
20
22
24
3 0
3
6
9 12 15 18 21 24
27
30
33
36
4 0
4
8 12 16 20 24 28 32
36
40
44
48
5 0
5 10 15 20 25 30 35 40
45
50
55
60
6 0
6 12 18 24 30 36 42 48
54
60
66
72
7 0
7 14 21 28 35 42 49 56
63
70
77
84
8 0
8 16 24 32 40 48 56 64
72
80
88
96
9 0
9 18 27 36 45 54 63 72
81
90
99 108
10 0 10 20 30 40 50 60 70 80
90 100 110 120
11 0 11 22 33 44 55 66 77 88
99 110 121 132
12 0 12 24 36 48 60 72 84 96 108 120 132 144
Name______________Answer Key____________
Quadratics
Using the distributive property find the product of the two binomials
Binomials are (x+1), and (x+2)
(x+1)(x+2)= x(x+1)+ 2(x+1)= x2+1x+2x+2= x2 +3x+2
Using the distributive property find the product of the two binomials
Binomials are (x-2), and (x+2)
(x-2)(x+2)=x(x-2)+2(x-2)= x2-2x+2x-4= x2-4
Factor the binomial of x2-9
(x-3)(x+3) This is a perfect square Factors of 9, 9,1 and 3,3 what add or subtract to be
zero
Factor the binomial of c2-8c+15
(c-5)(c-3) what are the factors of 15 {1,15},{5,3} they are going to both be the same
because of the addition sign and they will both be negative.
Factor the binomial of p2+p-6
(p+3)(p-2) What are the factors of 6 {6,1},{3,2} The signs will be opposite because of
the negative sign and the larger number will be positive.
Find the products of the binomials using the “box method” (y+3) and (y+9)
y2
y
+3
3y
9y
27
y
+9
Combine like terms to get y2+12y+27
Find the products of the binomials using the “box method” (x-5) and (x-2)
x
x
-5
x
2
-5x
-2x
+10
-2
Combine like terms to get x2-7x+10
Find the products of the binomials using the “box method” (x-5) and (x+3)
x
-5
x
x2
-5x
+3
+3x
-15
Combine like terms x2-2x-15
Factor the expression using the “box method” z2-4z-12
Z
+2
Z
z2
+2z
-6
-6z
-12
(z-6)(z+2) What are the factors of 12 {12,1},{4,3},{6,2}
what add or subtract to give you 4
the signs will be different and the larger number will be negative Take out like terms to
get the numbers on the side
Factor the expression using the “box method” a2-17a-38
a
a
+2
a2
+2a
-9a
-38
-9
a2 and 38 goes in the first and last square. Factors of 38 and sign w/ a go in the other 2
boxes take out like terms.
Name________________________
Quadratics
Using the distributive property find the product of the two binomials
Binomials
are (x+1), and (x+2)
Using the distributive property find the product of the two binomials
Binomials are (x-2), and (x+2)
Factor the binomial of x2-9
Factor the binomial of c2-8c+15
Factor the binomial of p2+p-6
Find the products of the binomials using the “box method” (y+3) and (y+9)
Find the products of the binomials using the “box method” (x-5) and (x+3)
Factor the expression using the “box method” z2-4z-12
Factor the expression using the “box method” a2-17a-38
Lesson: Factoring Harder Quadratics
Grade Level: 10
Materials: Pencil and Notebook
Lesson Overview: This lesson will continue what the students have learned previously
about factoring and finding roots or quadratic equations, but will proceed to harder
examples, i.e. quadratics with coefficients that are not one.
Lesson Objectives: By the end of the lesson, students will be able to how to factor
quadratic equations that have a coefficient other than one. They will demonstrate their
understanding by being able to factor equations and find the roots of the equations.
NYS Standards: 4A. Represent problem situations symbolically by using algebraic
expressions, sequences, tree diagrams, geometric figures and graphs.
Anticipatory Set: (5 minutes) Place (x – 7)(x + 2) on the board. Ask the students to
multiply the expressions. They should get x2 – 5x –14. This is a review of yesterday.
Now place (3x – 7)(x + 2) on the board. Ask the students to multiply these expressions.
They should get an answer of 3x2 – x –14. Tell them that today, we are going to learn
how to factor expressions that look like this one and how to find their roots.
Developmental Activity: (30 minutes) Hand out Factoring and Roots worksheet to the
students. Go through the first two or three problems with them. Have them complete the
remainder of the sheet. They may work together if they want to.
Closure: (5 minutes) Go over the correct answers to the worksheet. Ask the students if
they have any further questions or if there is anything else that seems unclear.
Assessment: (ongoing) Monitor the students’ progress as they complete the sheet. Give
necessary guidance as needed.
Name:_____________________
Factoring and Roots
Factor the equation and find the roots.
1.)
3x2 – x –14 = 0
Step 1: Break the equation down into 3 terms: A, B, and C. A = the coefficient of the x2
term. In this case, 3. B = the coefficient of the x term. In the case –1. C = the final term
without a variable. Here it is –14.
Step 2: Multiply A and C. In this problem AC = 3(-14) = - 42.
Step 3: Find the possible factors of AC. In this case, AC = - 42. The possible factors of
– 42 are 1 and 42, 2 and 21, 3 and 14, and 6 and 7. One of the numbers in the pair must
be negative. We will use these pairs to see if any of them would add up to give us B,
which in this case B = -1. We see that –7 + 6 = -1.
Step 4: Draw a two-by-two grid, and put the first term in the upper left-hand corner and
the last term in the lower right-hand corner.
Step 5: Then we take our factors –7 and 6, and put them, complete with signs and
variables, in the diagonal corners. (It doesn't matter which way you do the diagonal
entries; the answer will work out the same anyway!)
Step 6: Factor out the columns and rows. (The signs for the bottom-row entry and the
right-column entry come from the closest term that you are factoring from. Do not forget
your signs!)
3x2
6x
-7x
-14
Step 7: Set these new expressions equal to 0 and solve them to find the roots.
3x – 7 = 0
or
x+2=0
Notice that we use the conjunction "or," because x takes on only one value at a time.
2.)
2x2 + x – 6 = 0
Step 1: Break the equation down into 3 terms:
A:
B:
C:
Step 2: Multiply A and C.
A*C=
Step 3: Find the possible factors of AC.
Step 4: Draw a two-by-two grid, and put the first term in the upper left-hand corner and
the last term in the lower right-hand corner.
Step 5: Then we take your factors and put them, complete with signs and variables, in the
diagonal corners. (It doesn't matter which way you do the diagonal entries; the answer
will work out the same anyway!)
Step 6: Factor out the columns and rows. (The signs for the bottom-row entry and the
right-column entry come from the closest term that you are factoring from. Do not forget
your signs!)
Step 7: Set these new expressions equal to 0 and solve them to find the roots.
=0
3.) 4x2 – 19x + 12 = 0
4.) 4x2 – 15x + 9 = 0
=0
5.) 2x2 + 7x + 3 = 0
6.) 3x2 + x – 2 = 0
7.) 3x2 + 4x = 0
8.) 2x2 – x = 0
9.) 3x2 + x = 10
10.) 2x2 = x
Name:_____________________
Factoring and Roots (KEY)
Factor the equation and find the roots.
1.) 3x2 – x –14 = 0
Step 1: Break the equation down into 3 terms: A, B, and C. A = the coefficient of the x2
term. In this case, 3. B = the coefficient of the x term. In the case –1. C = the final term
without a variable. Here it is –14.
Step 2: Multiply A and C. In this problem AC = 3(-14) = - 42.
Step 3: Find the possible factors of AC. In this case, AC = - 42. The possible factors of
– 42 are 1 and 42, 2 and 21, 3 and 14, and 6 and 7. One of the numbers in the pair must
be negative. We will use these pairs to see if any of them would add up to give us B,
which in this case B = -1. We see that –7 + 6 = -1.
Step 4: Draw a two-by-two grid, and put the first term in the upper left-hand corner and
the last term in the lower right-hand corner.
3x2
6x
-7x
-14
Step 5: Then we take our factors –7 and 6, and put them, complete with signs and
variables, in the diagonal corners. (It doesn't matter which way you do the diagonal
entries; the answer will work out the same anyway!)
Step 6: Factor out the columns and rows. (The signs for the bottom-row entry and the
right-column entry come from the closest term that you are factoring from. Do not forget
your signs!)
x
+2
3x
3x2
6x
-7
-7x
-14
Step 7: Set these new expressions equal to 0 and solve them to find the roots.
3x – 7 = 0
x = 7/3
or
x+2=0
x = -2
Notice that we use the conjunction "or," because x takes on only one value at a time.
2.)
2x2 + x – 6 = 0
Step 1: Break the equation down into 3 terms:
Step 2: Multiply A and C.
A: 2
B: 1
C: -6
A * C = -12
Step 3: Find the possible factors of AC.
1 and 12, 2 and 6, 3 and 4. One must be negative.
Step 4: Draw a two-by-two grid, and put the first term in the upper left-hand corner and
the last term in the lower right-hand corner.
Step 5: Then we take your factors and put them, complete with signs and variables, in the
diagonal corners. (It doesn't matter which way you do the diagonal entries; the answer
will work out the same anyway!)
Step 6: Factor out the columns and rows. (The signs for the bottom-row entry and the
right-column entry come from the closest term that you are factoring from. Do not forget
your signs!)
2x
-3
x
2x2
-3x
+2
4x
-6
Step 7: Set these new expressions equal to 0 and solve them to find the roots.
2x - 3 = 0
x = 3/2
or
x+2=0
x = -2
3.) 4x2 – 19x + 12 = 0
x
-4
4x
4x2
-16x
-3
-3x
12
A=4
B = -19
4x – 3 = 0
x=¾
AC = 48
or
or
x–4=0
x=4
or
or
x–3=0
x=3
2x + 1 = 0
or
x+3=0
x = - ½ or
x = -3
C = 12
4.) 4x2 – 15x + 9 = 0
x
-3
4x
4x2
-12x
-3
-3x
9
A=4
B = -15
4x – 3 = 0
x=¾
AC = 36
C=9
5.) 2x2 + 7x + 3 = 0
A=2
B=7
C=3
x
+3
2x
2x2
6x
+1
x
3
AC = 6
6.) 3x2 + x – 2 = 0
x
+1
3x
3x2
3x
-2
-2x
-2
A=3
AC = -6
B=1
3x – 2 = 0
or
x+1=0
x = 2/3
or
x = -1
C = -2
7.) 3x2 + 4x = 0
x (3x + 4) = 0
x=0
or
3x + 4 = 0
x = -4/3
8.) 2x2 – x = 0
x (2x – 1) = 0
x=0
or
2x – 1 = 0
x=½
9.) 3x2 + x = 10
x
+2
3x
3x2
6x
-5
-5x
-10
A=3
AC = -30
B=1
C = -10
10.) 2x2 = x
2x2 – x = 0
x (2x – 1) = 0
x=0
or
2x – 1 = 0
x=½
3x – 5 = 0
or
x+2=0
x = 5/3
or
x = -2
Wednesday’s Lesson
Finding quadratic equations from a graph and learning to use the
quadratic formula.
Geometry Grade 10
Materials needed:
Worksheets
Writing Materials
Graphing Calculator
Overview:
Through the use of the quadratic formula students will find the roots
of quadratic equations. The students will also learn how to find the
quadratic equation from the solutions, by them having the solutions in x =
form and by seeing the parabola on a graph.
Objectives:
Students shall use the quadratic formula to find roots (solutions) of a
quadratic equation.
Students shall write an equation for a parabola from a graph and from
solutions.
NYS Standards:
7A. Represent and analyze functions, using verbal descriptions, tables,
equations, and graphs.
7E. Apply axiomatic structure to algebra.
Anticipatory Set: (3-5 minutes)
On the board have the following quadratic equations on the board and
have the students solve for x. (the solution is in blue)
a2 – 2a + 1 = 0
12x2 – 2x - 2
(a-1)(a-1)
(3x+1)(4x-2)
1
a=1 or a=1
x= or x = ½
3
Developmental Set: (30-35 minutes)
Introduce the standard form
of the quadratic equation to the students.
ax2 + bx + c = 0
Give them the following equation and have the students find the value for
a, b and c.
(1/2)x2 – 6x-14
Once the students have found a, b and c introduce the quadratic
formula to the students.
x = -b  b2 – 4ac
2a
Then have the students find the roots by substituting the values for a, b
and c into the formula. The solutions for the above quadratic are x = 14 or
-2. After the students have found the roots have them check their solution
by setting the quadratic equal to zero and then substituting the values for
x.
Next have the students complete the first 2 questions on the worksheet
using the quadratic formula. You can either work with the students give
them about five minutes to complete the questions.
Next on the board write the solution set {1/2, -3} ask the students what
they think they need to do to find the quadratic equation.
If the students are stumped talk the students through the question by
setting each value equal to x. Then find the binomial and then the
trinomial.
Next have the students complete the next two questions on the worksheet.
This should take less than 5 minutes.
Finally have the students look at question 5 on the worksheet. They will
see a graph with a parabola. Have the students find the value of x when
it crosses the x axis. Once the students have found the values of x have
them find the quadratic equation using the same method from the
previous equation.
Closure: (2-3 minutes)
The last page of the worksheet asks the students to tell us as teachers
what was their favorite method to solve a quadratic equation and why?
Have them hand this in.
Assessment:
There is a homework assignment where the students have to solve
quadratics using the quadratic formulas and find the quadratic formula
from the solutions.
Name: ___________________________
________________________
Date:
Quadratic Formula and solutions
Don’t forget to check your solutions to see if they work.
1. x2 – 9x = -18
What is are first step?
a=
b=
c=
The formula is?
Solve for x.
2. 2x2 + 5x – 3 = 0
3. {3/4, -2}
4. x = -3.5 and x = 2
5. Find the solutions for the following parabola. Then find quadratic
equation.
1
-4
-2
2
-1
-2
-3
4
Name: _________________________________
What is your favorite method for finding the roots to a quadratic equation?
Why?
Name: _____________________________Date: ___________________________
Homework for quadratic formula
For the following 3 questions solve using the quadratic formula.
1. 9x2 + 3x – 2 = 0
2. 0 = x2 –x -30
3. (2x+3)(3x-1)-2 = 0 *first you must multiply the binomials and then
subtract the two, then use the quadratic equation to find the roots*
4. Find the quadratic equation for the solution set. {1,-9}
5. Find the quadratic equation for the following solution set: {-3/5, 2/3}
Name: _____answer key___________
________________________
Quadratic Formula and solutions
1. x2 – 9x = -18
What is are first step?
a=1
b = -9
c = 18
Date:
The quadratic formula is?
x = -b  b2 – 4ac
2a
Solve for x.
x = 6 and 3
2. 2x2 + 5x – 3 = 0
x = ½ and -3
3. {3/4, -2}
x2 + (5/4)x – (6/4)= 0
4. x = -3.5 and x = 2
x2 + 1.5x – 7 = 0
5. Find the solutions for the following parabola. Then find the quadratic
equation.
1
-4
-2
2
-1
-2
-3
x is equal to 2 and 1
-x2 – 3x + 2 = 0
Name: _____answer key____________
Date: __________________
Homework for quadratic formula
For the following 3 questions solve using the quadratic formula.
1. 9x2 + 3x – 2 = 0
x = 1/3 and –(2/3)
2. 15 = x2 –x -15
x = 6 and -5
3. (2x+3)(3x-1) = 2
x = ½ and –(5/3)
4. Find the quadratic equation for the solution set. {1,-9}
x2 + 8x – 9 = 0
5. Find the quadratic equation for the following solution set: {-3/5, 2/3}
15x2 – x – 6 = 0
Thursday
4
Math A Practice with Quadratics
10th grade Geometry
Solve problems on the Math A
Materials:
-Markers
-Worksheet
- Pen/Pencil
Lesson Overview:
-Students will solve problems that of quadratics. Students will find roots and
factor the quadratics. Students will also write the equation for a quadratic from
looking at graph.
Lesson Objectives:
-Students will analyze math questions from previous exams and justify their
answers using the correct mathematical reasons.
-Students will also be able to identify types of questions that they are going to
solve on the Math A exam.
NYS Standards:
2C. Apply the properties of real numbers to various subsets of numbers.
3D. Use field properties to justify mathematical procedures.
Anticipatory Set:(10 min)
Have the students’ state the different ways that they can use to solve problems
dealing with quadratic equations Have the students tell you what the roots of the
equation are. Can distribute, Use the “box method” graph the problem.
Development Activity:(25 min)
Hand the worksheet (Math A problems) out to the students. All of the questions
are off past regents exams. Have the students start to solve the problems
individually or with a partner. As the students solve the problems make sure that
they write the reason for the answer they chose. Reasons should be mathematically
sound. After 10 minutes or sooner depending if the students are having problems,
solve any remaining questions as a group. This will help the students see the
correct ways to approach questions when they take the Math A. Make sure
students’ state the reasons for the answers. The students should be able to explain
how they came up with a particular answer. The reasons will let the teachers know
if the students understand the material. The multiple choice questions are all part
one questions off of the exam. The graph questions are from part three questions.
The questions are three points a piece. For the rubric look at
http://regentsprep.org/Regents/math/teachres/LinkPage.htm. That will tell you how
the question should be graded and what the students should do to get points. The
second part of question 1 I would consider that to be a part II. Work problems out
on the board so that the students are able to see the mathematics done correctly.
Closure: (5 min)
Ask the students how they felt about answering the questions today. Highlight
these are some examples of what they will see on the exam. Have students write down
anything that confused them or was a challenge? Try to address their concerns.
Assessment:
Assessment is ongoing throughout the activity. Students should receive positive
feedback when they complete a question correctly. If they complete a question
incorrectly it is a perfect opportunity to correct any misconceptions or problems that the
students are having about the material. Remember have a student put their work on the
board and explain what they are doing as they do it. We need to have the students
verbalize what they have written on paper. Teachers should record data in the logs of
how the students are doing. Teachers can look at students responses to the questions that
are being asked then write down in their journals how each student is doing.
Name______________Answer Key____________________
Math A Review
1. Factored completely, the expression 2y2 + 12y – 54 is equivalent to
(1) 2(y + 9)(y – 3)
(3) (y + 6)(2y – 9)
(2) 2(y – 3)(y – 9)
(4) (2y + 6)(y – 9)
What are the roots of the expression 2y2 + 12y – 54=0
-9, 3
2. The solution set of the equation x2(1) {-6,2}
(2) {-4,3}
x-
=
(3) {-2,6}
(4) {-3,4}
3. The length of a side of a square window in Jessica’s bedroom is represented
by 2x – 1. Which expression represents the area of the window?
(1) 2x^2 + 1
(3) 4x^2 + 4x – 1
(2) 4x^2 + 1
(4) 4x^2 – 4x + 1
4. What is a common factor of x^2 – 9 and x^2 – 5x + 6?
(1) x + 3
(3) x – 2
(2) x – 3
(4) x^2
5. What are the factors of x^2 – 10x – 24?
(1) (x – 4)(x + 6)
(3) (x – 12)(x + 2)
(2) (x – 4)(x – 6)
(4) (x + 12)(x – 2)
6.The graph of a quadratic equation is shown in the accompanying diagram. The scale on
the axes on the axes is a unit scale. Write an equation of this graph in standard form.
[This question is not off the test but one like it is on
the Jan 03 exam likewise for #7]
.F[x]=a(x-h)2-k
F[x]=x2
7. The graph of a quadratic equation is shown in the
accompanying diagram. The scale on the axes on
the axes is a unit scale. Write an equation of this
graph in standard form.
Vertex is at (-1,-4), the roots are -3,1
F[x]=a(x-h)2-k
-3=a(0+3)(0-1)
-3=a*-3
1=a
f[x]=1[x-(-1)]2-(-4)
8. One of the roots of the equation x^2 + 3x – 18 = 0 is 3. What is the other root?
(1) 15
(3) –6
(2) 6
(4) –21
Name__________________________________
Math A Review
1. Factored completely, the expression 2y2 + 12y – 54 is equivalent to
(1) 2(y + 9)(y – 3)
(3) (y + 6)(2y – 9)
(2) 2(y – 3)(y – 9)
(4) (2y + 6)(y – 9)
What are the roots of the expression 2y2 + 12y – 54=0
2. The solution set of the equation x2(1) {-6,2}
(2) {-4,3}
x-
=
(3) {-2,6}
(4) {-3,4}
3. The length of a side of a square window in Jessica’s bedroom is represented
by 2x – 1. Which expression represents the area of the window?
(1) 2x^2 + 1
(3) 4x^2 + 4x – 1
(2) 4x^2 + 1
(4) 4x^2 – 4x + 1
4. What is a common factor of x^2 – 9 and x^2 – 5x + 6?
(1) x + 3
(3) x – 2
(2) x – 3
(4) x^2
5. What are the factors of x^2 – 10x – 24?
(1) (x – 4)(x + 6)
(3) (x – 12)(x + 2)
(2) (x – 4)(x – 6)
(4) (x + 12)(x – 2)
6.The graph of a quadratic equation is shown in the accompanying diagram. The scale on
the axes on the axes is a unit scale. Write an equation of this graph in standard form.
.
7. The graph of a quadratic equation is shown in the accompanying diagram. The scale
on the axes on the axes is a unit scale. Write an equation of this graph in standard form.
Vertex is at (-1,-4)
8. One of the roots of the equation x^2 + 3x – 18 = 0 is 3. What is the other root?
(1) 15
(3) –6
(2) 6
(4) –21