Download Math 1 Support - Coweta County Schools

Fourth Nine-Weeks
2009-2010
Suggested resources for each day are referenced by abbreviations.
Math 1 Support
CZ(AM): Classzone – Animated Math
CZ(G/A): Classzone – Games & Act.
CZ(eWB): Classzone – eWorkbook
Concepts: COPS/COA resources
IG: Informal Geometry workbook
LT: Learning Task
UNIT 2, Lesson 1
Special Products
Algorithms
Review skills needing remediation.
Review multiplying variables.
Review simplifying like terms.
UNIT 2, Lesson 2
Binomial Theorem
Concepts: Lesson 8.1, 10.1, 10.3
Review power of a power rule.
Guided practice on multiplication of
polynomials.
Concepts: Lesson 10.2
UNIT 2, Lesson 3
Modeling Real-Life
Problems
(Polynomial
Arithmetic)
Review exponent rules for division.
Review division by zero and how to
find values that make an expression
equal zero.
UNIT 2, Lesson 3
Modeling Real-Life
Problems
UNIT 2, Lesson 4
Simplify
Polynomial
Expressions
UNIT 2, Lesson 4
Simplify
Polynomial
Expressions
Extra Practice
Review skills needing remediation.
Extra Practice
Review skills needing remediation.
CZ(G/A): Unit 3 – Flying Acrobats
Review skills needing remediation.
PT: Performance Task
Text: McDougal Littell Math 1
WS: Worksheet
Vocabulary: binomial, FOIL,
quadratic
Use algebra tiles to model binomial
multiplication. Practice using FOIL
to find the product of binomials.
Vocabulary: Binomial Theorem,
Pascal’s Triangle
Find powers of the binomial (x + 1).
Identify coefficients. Create Pascal’s
Triangle. Compare.
Vocabulary: rational expression,
expanded form, factored form,
excluded values, complex fraction,
simplest form, least common
denominator.
Concepts: Lesson 11.3
CZ(AM): Modeling Polynomial ÷
Vocabulary: factors, greatest
common factor, grouping
CZ(G/A): Unit 3 – Mummy Chase
Review skills needing remediation.
Concepts: Lesson 10.8
Model quadratic expression with
algebra tiles to find factors.
CZ(eWB): Algebra – Lesson 9.2, 9.5
Review skills needing remediation.
Concepts: Lesson 10.5
Practice for exam.
Practice for exam or begin learning
task: I’ve Got Your Number. (See
2nd 9-Weeks Calendar for more
information on this learning task.)
UNIT 2, Lesson 4
Simplify
Polynomial
Expressions
UNIT 2, Lesson 4
Simplify
Polynomial
Expressions
Concepts: Factors and Multiples, pp.
761-762
Vocabulary: factors, greatest
common factor, grouping, coefficient
CZ(AM): Unit 2 – Modeling
factoring x2 + bx + c
LT: I’ve Got Your Number
Review skills needing remediation.
CZ(eWB): Algebra – Lesson 9.5
Concepts: Lesson 10.6
Review skills needing remediation.
UNIT 2, Lesson 4
Simplify
Polynomial
Expressions
UNIT 2, Lesson 4
Simplify
Polynomial
Expressions
UNIT 2, Lesson 5
Operations with
Rational
Expressions
UNIT 2, Lesson 6
Simplify Rational
Expressions
UNIT 2, Lesson 7
Simplify Radical
Expressions
CZ(AM): Unit 2 – Modeling factoring
ax2 + bx + c
Vocabulary: special products,
difference of squares, binomial square
CZ(eWB): Algebra – Lesson 9.6
CZ(G/A): Unit 2 Bike Racer
Factoring Rummy card game
Review working with fractions –
Concepts: Fractions – pp. 763-766
CZ(eWB): Algebra – Lesson 9.7
Review working with fractions –
Concepts: Fractions – pp. 763-766
Difference of Squares:
http://illuminations.nctm.org/LessonD
etail.aspx?id=L276
Concepts: Lesson 10.7
Concepts: Lesson 10.8
Vocabulary: rational expressions,
expanded form, factored form,
excluded values, complex fraction,
simplest form, least common
denominator
Concepts: Lesson 11.3, 11.4
Concepts: Lesson 11.5, 11.6
CZ(eWB): Algebra – Lesson 9.8
Review fractions/other skills needing
remediation.
CZ(AM): Unit 3 – Multiplying and
Dividing Rational Expressions
CZ(AM): Unit 3 – Adding and
Subtracting Rational Expressions
Review radicals/skills needing
remediation.
Vocabulary: radical, radicand,
simplify, conjugates
Concepts: Lesson 9.1
CZ(AM): Unit 3 – Simplifying
Radicals
CZ(AM): Unit 3 – Operations with
Radicals
Extra practice as needed to prepare
for Unit 2 test.
Vocabulary: convex/concave polygon
Solve quadratic equations by finding
square roots.
UNIT 2
Extra Practice
Concepts: Selected problems from
p. 792, p. 790, p. 794
UNIT 5, Lesson 1
Application of
Function Families
Review simplifying square roots.
Concepts: Lesson 9.1
Concepts: Lesson 9.2
UNIT 5, Lesson 1
Application of
Function Families
Review vocabulary: zeros, roots, xintercept
Solve quadratic equations by
graphing.
UNIT 5, Lesson 2
Solving Advanced
Algebraic
Equations
UNIT 5, Lesson 2
Solving Advanced
Algebraic
Equations
UNIT 5, Lesson 2
Solving Advanced
Algebraic
Equations
UNIT 5, Lesson 2
Solving Advanced
Algebraic
Equations
Review order of operations and
simplifying square roots.
Concepts: Lesson 9.5
Solve quadratic equations by the
Quadratic Formula.
Concepts: Lesson 1.3
Review skills needing remediation.
Concepts: Lesson 9.6
Solve radical equations.
Concepts: Lesson 12.3
Review skills needing remediation.
Solve radical equations.
Concepts: Lesson 12.3
Review cross multiplying using
common fractions.
Solve rational equations by using
proportions.
WS: Proportions Practice
Treasure map – interactive algebraic
proportion practice:
http://www.themathlab.com/Algebra/
proportional%20thinking%20and%20
probability/algpropmap.htm
Solve rational equations by
multiplying by LCD.
UNIT 5, Lesson 2
Solving Advanced
Algebraic
Equations
UNIT 5, Lesson 2
Solving Advanced
Algebraic
Equations
UNIT 5
Extra Practice
Review finding LCD.
UNIT 5
Extra Practice
Review skills needing remediation.
Collaborate with Math 1 teacher
about unit review.
UNIT 5
Extra Practice
Review skills needing remediation.
Extra practice as needed to prepare
for exam.
Review factoring methods.
Concepts: Lesson 11.7
Solve rational equations by factoring
first then multiplying by LCD.
Review skills needing remediation.
Concepts: Lesson 11.7
Collaborate with Math 1 teacher
about unit review.
One Stop Shop For
Educators
Unit 2
2nd Edition
I’ve Got Your Number
Learning Task
Equivalent algebraic expressions, also called algebraic identities, give us a way to express results
with numbers that always work a certain way. In this task you will explore several “number
tricks” that work because of basic algebra rules. It is recommended that you do this task with a
partner.
Think of a number and call this number x.
Now think of two other numbers, one that is 2 more than your original number and a second that
is 3 more than your original number. No matter what your choice of original number, these two
additional numbers are represented by x + 2 and x + 3.
Multiply your x + 2 and x + 3 and record it as Answer 1 __________ .
Find the square of your original number, x2 ______ ,
five times your original number, 5x ______,
and the sum of these two numbers plus six, x2 + 5 x + 6, and record it as Answer 2 __________.
Compare Answers 1 and 2. Are they the same number? They should be. If they are not, look for
a mistake in your calculations.
Question: How did the writer of this task know that your Answers 1 and 2 should be the same
even though the writer had no way of knowing what number you would choose for x?
Answer: Algebra proves it has to be this way.
To get Answer 1, we multiply the number x + 2 by the number x + 3: (x + 2)(x + 3)
We can use the distributive property several times to write a different but equivalent expression.
First treat (x + 2) as a single number but think of x + 3 as the sum
of the numbers x and 3, and apply the distributive property to
obtain:
(x + 2)· x + (x + 2) ·3
Now, change your point of view and think of x + 2 as the sum of
the numbers x and 2, and apply the distributive property to each of
the expressions containing x + 2 as a factor to obtain:
x · x + 2· x + x ·3 + 2·3
Using our agreements about algebra notation, rewrite as:
x2 + 2 x + 3 x + 6
Add the like terms 2x and 3x by using the distributive property in
the other direction:
2 x + 3 x = (2 + 3) x = 5 x
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
May 27, 2008
Copyright 2008 © All Rights Reserved
Unit 2: Page 17 of 34
One Stop Shop For Educators
Unit 2
Mathematics I
The final expression equivalent to (x + 2)(x + 3) is:
2nd Edition
x2 + 5 x + 6
The last expression is Answer 2, so we have shown that (x + 2)( x + 3) = x 2 + 5 x + 6 no matter
what number you choose for x. Notice that 5 is the sum of 2 and 3 and 6 is the product of 2 and 3.
The above calculations are just an example of the following equivalence of algebraic expressions.

 
x a
x


2

-54a b x ab
-68(Pattern 1)
b x
For the remainder of this task, we’ll refer to the above equivalence as Pattern 1. Note that we
used a = 2 and b = 3 is our example, but your task right now is to show, geometrically, that a
could represent any real number and so could b.
1. (a) Each of the diagrams below illustrates Pattern 1. Match each diagram with one of the
following cases for a and b.
Case 1: a positive, b positive
Figure _____.
Case 2: a positive, b negative Figure _____.
Case 3: a negative, b positive Figure _____.
Case 4: a negative, b negative Figure _____.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
May 27, 2008
Copyright 2008 © All Rights Reserved
Unit 2: Page 18 of 34
One Stop Shop For Educators
Unit 2
Mathematics I
2nd Edition
(b) Thinking of the case it represents, for each diagram above, find the rectangle whose area is

x a
x b and use a pencil to put diagonal stripes on this rectangle. Then explain
how the diagram illustrates the pattern. Note that when a is a negative number, a  ,
and when a is a positive number, then a ; and similarly for b. Tell two of your
explanations to another student and let that student explain the other two to you.
(c) What happens to the pattern and the diagrams if a is 0? if b is 0? if both a and b are 0?
2. Pattern 1 can be used to give an alternate way to multiply two digit numbers that have the
same digit in the ten’s place.
For example, (31)(37) can be thought of as (30 + 1)(30 + 7) so we let x = 30, a = 1, and b = 7.
Since 302 = 900, 1 + 7 = 8 and 1·7 = 7,
3137
= 30 +1
30 + 7 = 900 + 8 30 + 7 = 900 + 240 + 7 =1147 .
Use Pattern 1 to calculate each of the following products.
(a) (52)(57) =
(b) (16)(13) =
(c) (48)(42) =
(d) (72)(75) =
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
May 27, 2008
Copyright 2008 © All Rights Reserved
Unit 2: Page 19 of 34
One Stop Shop For Educators
Unit 2
Mathematics I
2nd Edition
3. Look at your result for 2(c). You can get the answer another way by doing the following.
Take 4, the common ten’s digit, and multiply it by the next integer, 5, to get 20 as the number
of hundreds in the answer. Then multiply the units digits 4 x 6 to get 24 for the last two
digits.
48
x
4 2
2 0 1 6
4 · 5 = 20
8 · 2 = 16
Use this scheme to calculate the products below and verify the answers using Pattern 1.
(a) (34)(36) = _____.
(b) (63)(67) = _____.
(c) (81)(89) = _____.
(d) (95)(95) = _____.
(e) In each of the products immediately above, look at the pairs of units digits: 8 and 2 in the
example, 4 and 6 for part (a), 3 and 7 in part (b), 1 and 9 in part (c), and 5 and 5 in part (d).
What sum do each of the pairs have?
(f) Now let’s use Pattern 1 to see why this scheme works for these products with the property
you noted in part (e). To represent two-digit numbers with the same ten’s digit, start by
using n to represent the ten’s digit. So, n is 4 for the example and 3 for part (a). What is n
for parts (b), (c), and (d)?
(g) Next, represent the first two-digit number as 10 n + a and the second one as 10 n + b.
In part (a):
(32)(38) = (30 + 2)(30 + 8) = (10n + a)( 10n + b) for n = 3, a = 2, and b = 8.
List n_____. , a_____. , and b _____. for part (b).
List n _____. a _____ and b _____ f for part (c).
List n _____, a _____, and b _____ f for part (d)
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
May 27, 2008
Copyright 2008 © All Rights Reserved
Unit 2: Page 20 of 34
One Stop Shop For Educators
Unit 2
Mathematics I
2nd Edition
(h) Use Pattern 1 to multiply (10 n + a)( 10 n + b) where we have numbers like the example
and parts (a), (b), (c),and (d). We are assuming that each of n, a, and b is a single digit
number. What are we assuming about the sum a + b? a + b = _____.
Write your result in the form 100 k + ab where k is an expression containing the variable n
and numbers.
Write the expression for k .
Explain why k represents the product of two consecutive integers.
(i) Create three other multiplication exercises that can be done with this scheme and
exchange your exercises with a classmate. When you both are done, check each other’s
answers.
(i) __________
(ii) __________
(iii) __________
(j) Explain the quick way to calculate each of the following:
(15)(15)
(45)(45)
(85)(85)
Why do each of these products fit the pattern of this question?
(k) Summarize in your own words what you’ve learned in the parts of question 3.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
May 27, 2008
Copyright 2008 © All Rights Reserved
Unit 2: Page 21 of 34
One Stop Shop For Educators
Unit 2
Mathematics I
2nd Edition
4. The parts of question 3 explored one special case of Pattern 1. Now, let’s consider another
special case to see what happens when a and b in Pattern 1 are the same number. Start by
considering the square below created by adding 4 to the length of each side of a square with
side length x.
(a) What is the area of the square with side length
x?
(b) The square with side length x + 4 has greater
area. Use Pattern 1 to calculate its total area.
When you use Pattern 1, what are a, b, a + b?
(c) How much greater is the area of the square
with side length x + 4? Use the figure to show
this additional area. Where is the square with
area 16 square units?
(d) How would your answers to parts (b) and (c) change if the larger square had been created
to have side length x + y, that is, if both a and b are both the same number y?
(e) At the right, draw a figure to illustrate the area of a square with side length x + y assuming
that x and y are positive numbers. Use your figure to explain the pattern below.
x y2x22
y
(Pattern 2 - the Square of a Sum)
5. This pattern gives a rule for squaring a sum. Use it to calculate each of the following by
making convenient choices for x and y.
(a) 3022 =
(b) 542 =
(c) 652 =
(d) 2.12=
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
May 27, 2008
Copyright 2008 © All Rights Reserved
Unit 2: Page 22 of 34
One Stop Shop For Educators
Unit 2
Mathematics I
2nd Edition
(e) Look back at question 3, (k). Why will the rule for squaring a sum also work on those
exercises? Can the method of Question 3 be used for the square of any sum?
6. We can extend the ideas of questions 4 and 5 to cubes.
(a) What is the volume of a cube with side length 4? _____
(b) What is the volume of a cube with side length x? _____
(c) Now determine the volume of a cube with side length x + 4. First, use the rule for
squaring a sum to find the area of the base of the cube. __________
Now use the distributive property several times to multiply the area of the base by the
height, x + 4. Simplify your answer.
(d) Repeat parts (b) and (c) for a cube with side length x + y. Write your result as a rule for
the cube of a sum.
area of the base of the cube __________
area of base multiplied by the height, x + y:
_________________________ – (Pattern 3 – the Cube of a Sum)
(e) Making convenient choices for x and y, use Pattern 3 to find the following cubes.
113 =
233 =
1013 =
Use the rule for cubing a sum to cube 2 = 1 + 1. Do you get the same number as (2)(2)(2)?
(f) Use the cube of the sum pattern to simplify the following expressions.
(t + 5)3 =
(w + 2) 3 =
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
May 27, 2008
Copyright 2008 © All Rights Reserved
Unit 2: Page 23 of 34
One Stop Shop For Educators
Unit 2
Mathematics I
2nd Edition
7. (a) Let y represent any positive number. Go back to Pattern 1 and substitute – y for a and for
b to get the following rule for squaring a difference.
x  y 2x2 2
y 
2
(Pattern 4 - The Square of a Difference)
(b) In the diagram below find the square with side x, the square with side y, and two different
rectangles with area xy. Now, use the diagram to give a geometric explanation of the rule
for the Square of a Difference.
(c) By making a convenient choices for x
and y, use the Square of a Difference
pattern to find the following squares.
Note that 99 = 100 – 1, 38 = 40 – 2,
and 17 = 20 – 3.
992
382
172
8. (a) Find a rule for the cube of a difference.
(b) Check your rule for the Cube of a Difference by using it to calculate:
the cube of 1 using 1 = 2 – 1 and the cube of 2 using 2 = 5 – 3.
13
23
9. Now let’s consider what happens in Pattern 1 if a and b are opposite real numbers. Use
Pattern 1 to calculate each of the following. Substitute other variables for x as necessary.
(a) Calculate (x + 8)( x – 8). Remember that x – 8 can also be expressed as x + (– 8).
(x + 8)( x – 8)
(b) Calculate (x – 6)( x + 6) . Remember that x – 6 can also be expressed as x + (– 6).
(x – 6)( x + 6)
(c) Calculate (z + 12)( z – 12) . Remember that z – 12 can also be expressed as z + (– 12).
(z + 12)( z – 12)
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
May 27, 2008
Copyright 2008 © All Rights Reserved
Unit 2: Page 24 of 34
One Stop Shop For Educators
Unit 2
Mathematics I
2nd Edition
(d) Calculate (w + 3)( w – 3) . Remember that w – 3 can also be expressed as w + (– 3).
(w + 3)( w – 3)
(e) Calculate (t – 7)( t + 7) . Remember that t – 7 can also be expressed as t + (– 7).
(t – 7)( t + 7)
(f) Substitute y for a and – y for b in Pattern 1 to find an pattern for the product (x + y)(x – y).
(x + y)(x – y) ____________________
(Pattern 6)
10. Make appropriate choices for x and y to use Pattern 6 to calculate each of the following.
(a) (101)(99)
(b) (22)(18)
(c) (45)(35)
(d) (63)(57)
(e) (6.3)(5.7)




(f)3 1

2 
11.
(a) In Question 10, you computed several products of the form (x + y)(x – y) verifying that
the product is always of the form x2 – y2. Thus, if we choose values for x and y so that x =
y, then the product (x + y)(x – y) will equal 0. If x = y, what is x – y?
(b) Is there any other way to choose numbers to substitute x and y so that the product
(x + y)(x – y) will equal 0? If so, what is x + y?
(c) In general, if the product of two numbers is zero, what must be true about one of them?
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
May 27, 2008
Copyright 2008 © All Rights Reserved
Unit 2: Page 25 of 34
One Stop Shop For Educators
Mathematics I
Unit 2
2nd Edition
(d) Consider Pattern 2 for the Square of a Sum: x y x 2 y
2
2
2
. Is there a way to
choose numbers to substitute for x and y so that the product xy equals 0?
(e) Is it ever possible that (x + y)2 could equal x2 + y2? Explain your answer.
(f) Could (x – y)2 ever equal x2 + y2? Could (x – y)2 ever equal x2 – y2? Explain your answer.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
May 27, 2008
Copyright 2008 © All Rights Reserved
Unit 2: Page 26 of 34
Factoring Rummy card game
Learning Task
Essential Question: How can you find the factors of a trinomial in the form x2 + bx + c?
To Be Used With: Unit 2 – Algebra Investigations
Standards Addressed: MM1A2 c, f
Materials Needed: Factoring Rummy Cards sheet, 40 index cards per group (or poster board,
scissors, and ruler to make cards), pencil or markers
Teaching Strategies: Groups of 4
Teaching Notes: This learning task uses a game format to reinforce multiplying a monomial
and a polynomial, multiplying two binomials, and factoring.
Here is an easy way to get your original sets of cards made.
1. Pass out the Factoring Rummy Cards sheet to each group.
2. Pass out 40 blank index cards to each group. (An alternate plan is to have students
measure and cut out the cards from poster board.)
3. Two people in each group will write expressions from column A from the Factoring
Rummy Cards sheet, one expression per card.
4. Two people in each group will write expressions from column B from the Factoring
Rummy Cards sheet, one expression per card.
5. When the index cards are complete, each group should have a deck of 40 cards.
Instructions for game:
1. A student deals four cards to each player.
2. The rest of the deck is placed face down in the middle of the players.
3. Students take turns asking for the match to a card in their hand. Example A: If a student
has x2 – 10x + 25 in his hand, this student would ask for (x – 5)(x – 5). Example B: If a
student has (x + 3)(x – 2) in her hand, she would ask for x2 + x – 6. If there is no match,
the student draws a card from the deck. Play then goes to the player to the left. If there
is a match, the matched pair is laid on the table in front of the students so all can check it.
Then the same student asks for another match to a card in his/her hand. This player
keeps playing until he/she does not receive a match. Then the play goes to the player on
the left.
4. Play continues until all cards are gone.
5. Scoring: One matched pair equals one point. The winner is the student with the most
points.
6. Play can continue for several rounds, as the teacher directs. The person with the most
points wins whenever time is called.
Suggestion: Create another cards sheet with additional types of factoring problems.
Adapted from materials, © Lawrence R. Burke 1999
Estimated Time: As appropriate
Rubric: None required
Factoring Rummy Cards
Column A
x2 + 8x + 16
x2 + 16x + 15
6x2 + 6x
x2 – 5x + 6
x2 + x – 12
x2 + x – 20
9x2 + 6x
x2 – x – 6
x2 – 18x + 81
x2 – 9x + 20
x2 – 12x + 32
x2 – 4x – 32
x2 – 15x + 54
x2 + 3x – 54
x2 + 7x + 6
x2 + 9x – 22
x2 + 12x + 36
12x2 + 8x
x2 – 3x – 18
x2 – x – 12
Adapted from materials, © Lawrence R. Burke 1999
Column B
(x + 2)(x – 3)
(x + 3)(x – 4)
(x – 6)(x – 9)
(x + 4)(x – 8)
(x + 1)(x + 6)
(x + 6)(x + 6)
4x(3x + 2)
(x + 15)(x + 1)
(x – 2)(x + 11)
(x – 6)(x + 9)
(x – 4)(x – 8)
(x – 9)(x – 9)
(x + 3)(x – 6)
(x – 4)(x – 5)
(x – 2)(x – 3)
3x(3x + 2)
(x – 3)(x + 4)
(x – 4)(x + 5)
(x + 4)(x + 4)
6x(x + 1)