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Algebra I β 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 π₯ 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 + β×÷ and groupings Signed numbers. Stress importance of a common language. Include π(1 + π)π‘ 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 β 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 βleave in terms of πβ. 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 (+/β) Show example of (β3 β β ) 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 βwords to symbols.β 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 β 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 β 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 3π₯ β 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 π₯ and π¦. 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 π(0) and solve π(π₯) = 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 π₯ + ππ₯ + π Group with PST? 5.7 WS 1 π₯ 2 + ππ₯ β π This was a helpful buildup. 5.8 WS 2 ππ₯ 2 + ππ₯ + π 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 π(1 + π)π‘ 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 (π¦ = β3π₯ and π₯ 2 + π¦ 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 βπ₯ 2 = |π₯|. 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 β121π2 into β11β11βπβπ 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 β ββ 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))