Download alg1_-_scope_sequence_-_2016-17

Algebra I – Scope and Sequence
Algebra I tends to focus on three major ideas. Order of operations, substitution, and ratio. This version
of the course is a 2015-16 redesign to accommodate the lack of preparation skills observed from a set of
incoming students. Mixture word problems and factor-by-grouping have been cut. A deliberate effort
has been spent on keeping homework at a reasonable length. Many assignments are retyped
combinations of book problems as well as custom ones. The first semester reviews Prealgebra and basic
equation solving. It contains all linear problems, including graphing. Rational expressions are introduced
in applications of fractions, which requires an understanding of rational numbers. The second semester
introduces factoring and quadratics in the form of problems with 𝑥 2 . The second semester also contains
all operations with polynomial, then rational, and finally radical expressions. These variable expressions
reflect a progression from integer, to rational, to irrational operations.
Word problems are sprinkled throughout, and literacy and translation is a major skill in Algebra I.
Function notation is emphasized much less, and students are not introduced to non-linear functions in
Algebra I. Students do not really explore measurement nor are calculator-specific examples ever used.
Linear growth is never contrasted with exponential growth.
There are no statistics in Algebra I.
The Fundamental Theorem of Algebra is briefly touched on as a rationale for quadratics having two
solutions. Complex solutions are held off until Algebra II.
Number sense is problematic. No way my students could do the below problem. Magnitude of
quantities, especially related around multiplying and dividing, is hard on them.
We don't do scientific notation for large/small numbers.
We should explore viewing problems from multiple perspectives, including various representations of
functions and relations as well as word problems.
Finally, JSN suggests we should be collecting student data and collaborating in professional learning
communities (PLCs) more than we are. Perhaps a start would be to go through the final exam at the
start of the school year.
Semester 1
Chapter 1: Introduction to Algebra
Days
Topics
Additional notes
Homework
1
Integers
Factor trees/divisibility/prime
numbers. Basic number sense.
Introduce number line.
Integers WS (need
revisions, easier factoring)
1
Order of Operations with
+ −×÷ and groupings
Signed numbers. Stress
importance of a common
language. Include 𝑃(1 + 𝑛)𝑡
example to highlight
parentheses.
1.1 p.3 wr 1-12 all
1.2 p.8 wr 1-8 all
2.4 p.61 wr 17-31 odd
*2.4 p.61 wr 18-32 even
1
Basic Substitution
1
Substitution with
area/perimeter problems
1
Groupings – evaluate the
expression/simplify
1
1.1 p.3 wr 13-31 odd
1.2 p.8 wr 9-27 odd
*1.1 p.3 wr 14-32 even
*1.2 p.8 wr 10-28 even
Do an example of how to
“leave in terms of 𝜋”.
1.1 p.3 wr 33-40 all
1.2 p.8 wr 29-34 all (on 32
and 33, leave in terms of 𝜋)
Absolute value
Need conceptual number line
representation. Consider
showing a graphical function.
Absolute Value WS
1
Fractions (+/−)
Show example of (−3 ≠ − )
3
3
Mixed numbers, signed, abs
value?
Show fractions on number line.
Compare with mixed numbers.
1
Quiz
1
Review
1
9 days
1
8
Fractions WS. (Consider
plotting on number line.)
Review Packet
Test
Chapter 2: Working with Real Numbers
Days
Topics
Additional notes
Homework
1
Commutative/Associative
Properties
Need to teach mental math tricks Comm/Assoc WS
(incorporates 2.1 WS and
Use p.47 2-28 even as notes?
2.1 p.47 1-27 odd)
2
Distributive Property/Rules More mental math tricks and
for Multiplication
distributing negatives. Might
have to show a little “words to
symbols.”
2.4 p.61 wr 37-42, 47-50 all
2.5 p.67 wr 37-65 odd
*2.5 p.67 wr 38-66 even
2.5 p.67 wr 73-78 all
2.6 p.72 wr 19-29 odd, 4355 odd
*2.6 p.72 wr 20-30 even,
44-56 even
2
Definition of division,
reciprocals, multiplying,
dividing fractions
Teach cancelling fractions and
incorporate a couple
subtraction/addition (just to
throw them off)
Mult/Div Fractions WS Day
1 (incorporates
2.8 p.81 wr 1-33 odd plus a
few more)
2nd day, do examples of multiple
1
3
5
fractions in a row (AKA 2 ÷ 4 ÷ 4
or 12 ÷ 2 ⋅ 3)
Mult/Div Fractions WS Day
2 (incorporates 2.9 p.85 wr
1-19 odd, 25-33 all plus a
few more)
Quiz
Review
Review
Test
8 days
Chapter 3a: Solving Equations
Identities are not truly introduced or distinguished from equations. We could feasibly ask them to prove
an identity, similar to trig identities.
Days
Topics
Additional notes
Homework
Addition Property of
Equality
1
Mult/Div Property of
Equality
1
Using Several
Transformation
1
Variables on both sides
1
Transforming formulas
Might want to do actual formulas
here to give real-life application.
4.7 p.166 wr 1-23 odd + WS
(already made with
direction)
1
Solving inequalities
Teach number line scale (even
distribution).
10.2 p.466 wr 11-49 odd
*10.2 p.466 wr 12-50 even
1
Quiz
1
Review
1
9 days
Test
Focus on equal sign as
equivalence vs. operation
3.1 p.97 wr 1-45 odd
*3.1 p.97 wr 45-59 odd (abs
value)
1
3.2 p.104 wr 1-39 odd
3.2 p.104 wr 41-51 odd (abs
value)
Teach identities vs. equations
3.3 p.109 wr 1-57 odd
*3.3 p.109 wr 32-54 all, 5960 all (abs value)
3.4 p.118 wr 28-44 all, 45, 47
Review
Chapter 3b: Solving Problems
This is a make or break chapter. Students who are proficient in literacy will do OK on this. If they don't
do well in this, it spells disaster for later word problems.
Days
Topics
Additional notes
Homework
1
Words to symbols
1
Translating word
problems. Consecutive
odds/evens.
2
More word problems
Could not be more important.
Might want to do more than
one day of this.
3b WS – Words to Symbols
Are consecutive evens/odds
really that important? Should
we axe in favor of mixture
problems later?
2.4 p.61 wr 33-36 all, 43-46 all,
51, 52
2.7 p.77 wr 1-12 all
*1.5 p.20 wr 17-36 all
3.1 p.98 pr 1-6 all
3.2 p.105 pr 1-12 all
3.4 p.113 pr 4, 6, 8-12 all
3.5 p.118 pr 5-7, 9, 11-14, 17,
18 all
1
Cost/Value Problems
1
Quiz
1
Review
1
8 days
3.7 WS
Review
Test
Chapter 7a: Applying Fractions
The conceptual understanding of the ratio is huge here. Perhaps consider exploring the origin of rational
numbers (solve 3𝑥 − 1 = 0). Students also must be able to relate a rational number to a base 10
decimal representation and vice-versa.
Days
Topics
Additional notes
Homework
1
Ratios
Really about simplifying
fractions. Also about
converting units. What about
money and different
currencies?
7.1 p.289 wr 1-14 all, 21-30 all
1
Proportions
Solving by cross-multiplying.
Should this be linked with
Geometry examples?
7.2 p.295 wr 1-35 odd, #30
2
Equations with Fractional
Coefficients
Solving by clearing fractions
with LCD.
Word Problems.
7.3 p.299 wr 5-29 odd, pr 1-2
7.3 p 300 wr 20-30 even, pr 510, 13, 14
1
Fractional Equations
Solving where LCD has variable 7.4a WS
as well.
1
Percents
Changing fractions to percents 7.5 WS
and percents to fractions?
Base 10 connection.
1
Percent Increase/Decrease Word problems with
formulas.
1
Quiz
1
Review
7.6 WS
Review
1
Test
10 days
Chapter 8a: Introduction to Functions
We really do not teach them to rescale the axes. Also, we don't really have units for the axes other than
𝑥 and 𝑦.
Days
Topics
Additional notes
Homework
Graphing on the
Coordinate Plane
1
Graphing by T-chart and T-chart as a representation of a
intercepts
relation/function. Intercepts
have meaning. Introduce now?
8.2 WS
1
Slope
Slope is a rate of change.
Introduce units? Perhaps a
simple velocity problem?
8.3 WS (too long)
1
Graphing with Slope
Simple graphing. Have them
show all points on the graph.
8.4a p.368 wr 1-30 all
8.4a Worksheet
1
Quiz
Changing into slope-intercept
form and then graph as before.
8.4b Worksheet
1
Linear Inequalities
Transforming inequality to SI
10.7 p.492 3, 5, 6, 7, 9, 12, 13form and graphing. Test a point. 33 all
1
Quiz
1
Review
1
9 days
Equations in two variables.
Solutions are ordered pairs.
Also introduce vertical and
horizontal lines.
8.1 p.351 oral 1-12 all, 41-42
all
8.2 p.356 oral 1-26 all
1
Test
Chapter 8b: More Concepts with Functions
Review #1 and Review #2
Most students have not seen point-slope form before. Most domain/range problems are in Algebra II.
We also don't do much qualitative graphical analysis (increasing, decreasing, max/min). We do have
multiple representation of functions, but not really a lot.
Days
Topics
Additional notes
Homework
First day standard form, point- 8.5a Worksheet
8.5b Worksheet
slope form.
Second day, all three forms.
2
Writing Equations of a
Line
1
Parallel and Perpendicular Graphing and equationLines
writing.
2
Domain and Range, Zeroes Function notation. Find the
8.7a Worksheet
8.7b Worksheet
of a Function
range.
Find 𝑓(0) and solve 𝑓(𝑥) = 0.
Simple operations on
functions.
Should we have contextual
function examples?
2
Linear Equations Word
Problems
1
Review
1
9 days
Modeling linear equations.
Consider modifying to use
function notation? Maybe
give them equation and have
them figure out the parts.
(Build to the phone plan
example.)
8.5c Worksheet
8.8a Worksheet (consider
giving equation and having
them interpret)
8.8b Worksheet (move the
previous WS to the second day)
Review
Test
Chapter 4b/7b: Word Problems Involving Rate
I'm worried about not including this. At least DRT. We will need feedback from Algebra II.
Days
Topics
Additional notes
Homework
2
Distance, Rate, Time
4.8a p.170 1-13 odd
4.8b Worksheet
1
Investment/Wages
7.7a p.324 pr 7, 16, 21
7.7a p.347 pr 13
7.7a p.673 Investment and
Wages 1-4 all
1
Grocery
7.7b p.324 1, 5, 12, 13, 14, 20
7.7b p.672 Grocery 1-4 all
1
Chemistry
7.7c p.324 2, 3, 4, 9, 10, 11,
17, 18, 19
7.7c p.672 Chemistry 1-4 all
(cut some of these problems)
1
5 days
Review
Planned for 62 days
We have 65 days in the first semester for 2015.
Semester 2
Chapter 4: Polynomials
This chapter must ultimately connect to operations on integers (closed with respect to addition,
subtraction, multiplication). In that respect, dividing monomial and negative exponents are out of place
since they break those themes. Maybe those belong in rational expressions (Chapter 6).
Days
Topics
Additional notes
Homework
Commutative/associative/distributive 4.1-4.2 - HW p.143-144 wr
3-48 thirds; p.148-149 wr
don't apply.
Is degree of multivariable polynomials 1-7 odd, 31-55 odd
important?
Focus on polynomials having the same
properties of closure as the integers.
1
Exponents
1
Multiplying Monomials Introduce multiplying with the same
base.
Area, perimeter, surface area.
Consecutive even/odds again.
1
Powers of Monomials
Power of powers rule. How do we not 4.4 - HW p.156-157 wr 1get them to confuse this with previous 53 odd
day?
1
Dividing Monomials
This breaks the idea of polynomials as
integers. So do the next two sections.
5.2 - HW p.191-193 oral
13-35 odd, wr 1-29 odd,
53-59 odd
1
Negative Exponents
Popcorn, confused with
addition/subtraction signs.
7.9a - HW p.333 wr 1-57
odds
1
Quiz
7.9b - HW WS
1
Multiplying Polynomials This returns to the theme of what
by Monomials
multiplying actually is. Distributive
property used.
4.5 - HW p.159-160 wr 143 odd, skip 9&11
1
Multiplying Polynomials Distributive property only.
by Polynomials
4.6 - HW p.162-163 wr 947 odd, skip 27 (maybe
make a similar problem)
2
Review
Review
1
Test
4.3 - HW p.153-154 wr 330 multiples of three, 3339 odd, p.150 pr 5-13 odd
(word problems)
11 days
Chapter 5a: Factoring
There is an argument made to introduce factoring alongside graphing and roots. Also, PST might just as
well be grouped with other quadratics. I didn't find a good benefit to factoring PSTs separately since
they seem so artificially created.
Days
Topics
Additional notes
Homework
5.3 WS
1
GCF (Monomial Factors of
Polynomials)
Long notes and long
assignment. Need to do
examples of evaluating by
factoring first.
1
FOIL and DOTS multiply
More emphasis on recognizing 5.4 p.201 wr 1-37 odd
DOTS pattern. What do we end 5.5 p.206 wr 1-17 odd
up with?
1
DOTS factoring with GCF
Could not spend enough time
on this. Maybe have them
check by re-multiplying? That
would be a pain but useful.
5.5b WS
1
PST multiplying and
factoring
Should we be doing PST as a
separate day, or just launch
into quadratics? I'm not
convinced this is helpful.
5.6 WS (PST/DOTS)
1
Quiz
1
2
𝑥 + 𝑏𝑥 + 𝑐
Group with PST?
5.7 WS
1
𝑥 2 + 𝑏𝑥 − 𝑐
This was a helpful buildup.
5.8 WS
2
𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Homework asked them to
5.9a WS
check by multiplying. Do this in 5.9b WS
notes.
1
Quiz
1
Review
1
13 days
5.3-5.6 Review
Review
Test
Chapter 5b: Solve by Factoring
We should examine if "harder dots" and "group GCF" are truly necessary if this population is going to
Precalc (non-Acc). More time on word problems with multiple representations could replace that. Or
this could be where graphing quadratics is introduced (at least shape and with the calculator).
Days
Topics
Additional notes
Homework
2
Harder Dots with Group
GCF
This is again about pattern
5.11a WS
recognition. Not explicitly stated 5.11b WS
in standards, but necessary for
calculus.
5.12
Do a better job of splitting up
the harder problems at the
end of the first 26 problem
WS
2
Solve by Factoring
This is where showing a visual of
what they are doing (roots of a
polynomial) may be helpful.
Calculator? Desmos?
2
Word Problems
Not sure that border problems 5.13
Check the WS, kill the border
are as important as DRT
problems
problems or optimization
modeling with a quadratic.
Consider replacing with
functions where finding roots is
valuable as a visual.
1
Quiz
1
Review
1
9 days
Test
Chapter 6: Rational Expressions
The connection here is that we are now relating operations on rational expressions to the same
operations on rational numbers. Students really need to know what rational numbers are (see Chapter 7
in 1st semester).
Days
Topics
Additional notes
Homework
1
Simplifying Rational
Expressions
Factor and cancel. Emphasis on 6.1 p.248 wr 1-27 odd, 39-47
not cancelling terms. Connect
odd
back to 𝑃(1 + 𝑛)𝑡
1
Multiplying Rational
Expressions
Relate to multiplying fractions.
Perhaps connect to an
understanding again of what a
rational number is.
6.2 p.253 wr 13-55 odd, skip 33
1
Dividing Rational
Expressions
The skill of factoring is more
problematic than students
remembering to "flip" the
fraction.
6.3 p.256 wr 11-43 odd
1
LCD
Does the equal sign throw them 6.4 Worksheet on LCD + Review
off? Should we replace with +
and - and just tell them not to
simplify?
1
Quiz
Maybe one of us should have a
review session for fraction
review before this.
Review of Adding/Subtracting
Fractions
2
Adding and Subtracting
Rational Expressions
Homework really needs to be
scrutinized in this section.
6.5a p.267 wr 5-29 odd
6.5b p.268 wr 35-55 odd
1
Mixed Expressions
Relate to mixed numbers and
number line.
6.6 Worksheet
1
Complex Fractions
Relate back to equivalent
6.8 p.278 wr 1-19 odd, 27-31
operations (left and right side of odd
equal sign).
1
Fractional Equations
Struggle to understand when it 7.4b Worksheet
is OK to clear fractions and
when it is not. Maybe make
them tell us for a change. Add
problem on homework? In your
own words, when it is OK to
clear fractions?
1
Quiz
1
Review
1
13 days
6.x Review
Test
Chapter 9: Systems of Linear Equations
This is the hardest chapter of the year to this point. The smallest algebraic mistake can throw them off.
They must also recall graphing lines, which we have not done since first semester. Enrichment problems
could ask for a solution to a simple system of a linear equation and a conic (𝑦 = −3𝑥 and 𝑥 2 + 𝑦 2 = 3),
but without graphing technology or a provided visual they will not have a conceptual understanding of
what is happening. Our examples of systems are rather dry. Look at p.223 of NCTM for an example
involving cell phone plans.
Days
Topics
Additional notes
Homework
1
Solve by Graphing
Include examples where the
9.1 p.414 oral 1-9 all, p.415
lines do not cross exactly and
wr 1-12 all
make them approximate (guess)
the solution.
1
Solve by Substitution
Error correction is huge here.
Consider teaching technology
usage to check for steps.
1
Solve by Elimination
Same as above. Error correction 9.4 p.427 wr 1-17 odd
and self-checking is huge.
9.5 p.432 wr 11-23 odd, 31-35
odd
1
Review
2
Word Problems with
Systems
9.2 p.419 wr 1-39 odd
9.5b Harder Systems WS 1-19
odd
First word problems in awhile.
9.5c WS Word Problems
This is a huge standard and may 9.5d WS Word Problems (just
benefit from problems with pre- odds)
existing visuals. Or some with
prebuilt equations as
scaffolding.
1
Linear Inequalities
1
Quiz
1
Review
1
10 days
We are missing linear inequality 10.8 p.496 wr 1-18 all
word problems.
9.x Review
Test
Chapter 11: Rational and Irrational Numbers
This chapter is overly long and had the lowest scores on the test. Next year, I may test before solving
equations and then incorporate equation-solving and quadratic formula into the final. It does help to
see the QF before Algebra II. This year, I did not teach √𝑥 2 = |𝑥|. Also is rationalizing binomials really
necessary anymore for skills coming down the pipeline?
Days
Topics
Additional notes
Homework
1
Rational and Irrational
Square Roots
This is the first exposure of
some to irrational numbers.
Have them check with
calculator the validity of their
answers.
11.3 p.519 wr 1-47 odd
11.4 p.522 wr 1-29 odd
1
Square Roots of Variable
Expressions
Maybe break down √121𝑎2
into √11√11√𝑎√𝑎
11.5 Worksheet
1
Pythagorean Theorem
Do more with word problems
on this, making them draw the
triangle on each problem.
11.6 Worksheet (Acces)
1
Multiplying, Dividing,
Simplifying Radicals
Again, might help to separate
all pieces on this one.
11.7 p.538 wr 1-53 odd
1
Large Quiz or MiniTest
Need HW of easy problems
OR consider 11.8 p.541 1-18
all with little to no instruction
1
Adding/Subtracting Radicals Not as intuitive as I thought.
The struggle was in simplifying
radicals. Really struggled with
11.8 p.541 wr 1-33 odd
2
5
√ −√
5
2
2
Multiplication of Radical
Binomials
This is the straw I'd consider
11.9a p.545 wr 1-22 all
moving to Algebra II in favor of 11.9b p.546 wr 23-40 all
more word problems. This
makes the chapter long.
2
Simple Radical Equations
Solving equations really should 11.10a p.548 wr 1-24 all
be its own chapter.
11.10b p.548 wr 25-48 all
1
Quadratic Formula
Divorced of completing the
square, this makes little sense
and meaning.
1
Quiz
1
Review
1
Test
13 days
69 days planned
72 days in 2nd semester
Translating: p.68 67-70
Worksheet (just simple
example (ABC))