TOPIC ANALYSIS Algebraic concepts: Equations, Inequalities, Functions

1. TOPIC ANALYSISAlgebraic concepts: Equations, Inequalities, Functions BY: Kamal, Rashda, and Sana

2. Development of Algebra inGrade 6 Patterns Students will be able to • B1 demonstrate an understanding of the relationships within tables of values to solve problems • B2 represent and describe patterns and relationships using graphs and tables. Variables and Equations • B3 represent generalizations arising from number relationships using equations with letter variables. • B4 demonstrate and explain the meaning of preservation of equality concretely, pictorially, and symbolically

3. represent and describe patterns and relationships using graphs and tables. • Patterns, Input and Output , Data in a table.

4. Algebra concepts appear in grade 6 as empty shapes in an equation. • The answers are by guessing. • 6 + = 8 • × = 12 • So one variable represented by one shape , two variables represented by two different shapes. , . • Students will go through patterns in numbers and started to show balancing between two sides of equations. • Example: How many squares equal one triangle? + = a- Remove the similar numbers from both sides. From right side and from left side b- = c- + = + d- It means each =

5. represent generalizations arising from number relationships using equations with letter variables.

6. Grade 7

7. PLO’s Students will be able to Patterns and Relation • B1 demonstrate an understanding of oral and written patterns and their equivalent linear relations. • B2 create a table of values from a linear relation, graph the table of values, and analyze the graph to draw conclusions and solve problems. Variables and Equations • B3 demonstrate an understanding of preservation of equality by modelling preservation of equality concretely, pictorially, and symbolically applying preservation of equality to solve equations • B4 explain the difference between an expression and an equation • B5 evaluate an expression given the value of the variable. • B6 model and solve problems that can be represented by one-step linear equations of the form y + b = c,concretely, pictorially, and symbolically, where B and C are integers. • B7 model and solve problems that can be represented by linear equations of the form concretely, pictorially, and symbolically, where a, b, and c are whole numbers

8. In Grade 7,students started to have real algebra and it produced into two steps in grade 7 curriculum. • Step 1 • I was surprised that they started at this age to connect students of writing their thinking logs for solving the questions and that was in the beginning of the “Writing to Explain Your Thinking.” This is one of the forms.to solve problems. Thinking LogName ­­­­­­­­­­­-------------- I have been asked to find ……………… Here’s what I’ll try first………………….. To solve this problem I’ll……………………… And then………………………………. And then…………………………….. Here is my solution…………………………………. Think about thinking picture

9. demonstrate an understanding of oral and written patterns and their equivalent linear relations. • create a table of values from a linear relation, graph the table of values, and analyze the graph to draw conclusions and solve problems. We can use any letter, such as n or x, as a variable. The expression n 100 is written as 100n. 100n is an algebraic expression.

10. Then they connected the students with previous knowledge with the length measurements and starting from recalling the variable (n), as representing the quantity that can vary. • Then they gave them algebraic expressionsSuch as • Three more than a number: 3+ n • 7 times a number: 7n • Eight less than a number: n – 8 • A number divided by 20 : • Then they gave them the indicating concepts for an expression: 5k+2 • k: is a variable that represent any number in a set of numbers. • 5: is the numerical coefficientof the variable. • 2: is the constant term • They introduced the relation in patterns. • Building up the tables and input (variable x, n) and the output and graphing the input and the output.

11. They started to lead them for Reading and writing equations. Then solving the equations by using tiles. • Expressing unit tiles and variable tiles. • A unit tile and a variable tile collectively algebra tiles. • Unit tile variable tile • They introduced for a whole chapter operations with fractions. • They introduced more complicated equations with multiple operations on the two sides. • Using models to solve Equations. • Solving Equations Involving Integers. • Solving Equations Using Algebra. • Decoding Word Problems.

12. Grade 8

13. PLO’s • PATTERNS AND RELATIONS • Patterns • B1 graph and analyze two-variable linear relations • Variables and Equations • B2 model and solve problems using linear equations of the form • , a ≠ 0 concretely, pictorially, and symbolically, where a, b, and c are integers

14. Using the a balance scale model to solve an equations

15. They use to solve equations using tiles by that they connect them with grade 7 Example

16. Using algebra to solve equation. • Then they started to teach them algebraic steps to solve the equation. • Solving Equations involving Fractions • We still in linear equation but they added the distributive property as a new property to in their problem solving. • Then they use Graphing linear relation.

17. Decoding Word Problems. Solving Equations Involving the Distributive Property Solving Equations Involving Fractions

18. Consistent development from grade 6 to 8 I found that there is a great consistency through all transitions from grade 6 to grade 8 and through out the grade itself. The curriculum were highly developed gradually through out the level itself and while moving from level up to the other. They started to establish algebraic concepts in earlier grades even from grade one. We can’t consider dealing with algebraic concepts a way from numbers, operations, patterns and graphing etc. So, by establishing a concrete base of teaching numbers, patterns and operations to build algebraic concepts was highly taking care of it. Demonstration the equation concept from real life was obvious in the curriculum and solving has several steps before reaching to the point using algebraic concepts. ( using the scales’ pictures and tiles then they use algebraic operation.) The repetition and the connection from level to the next level has clearly existence in the curriculum. I like it.

19. Grade 9 PLO’s • PATTERNS AND RELATIONS – use patterns to describe the world and solve problems • Patterns • • generalize a pattern using linear equations • • graph linear relations for interpolation and extrapolation • Variables and Equations • • linear equations • • single variable linear inequalities with rational coefficients • • addition, subtraction, multiplication, and division of polynomials

20. Linear Relations • Use equations to solve problems using patterns • Analyze graph of linear relation

21. Linear Relations • Recognize equations of horizontal, vertical and oblique lines and graph them • Use interpolation and extrapolation to estimate values on a graph

22. Polynomials • Model polynomials using algebra tiles and also learn about like and unlike terms • Add and subtract polynomials using various strategies

23. Polynomials • Multiplying and dividing by a constant • Multiplying and dividing by a monomial

24. Linear Equations and Inequalities • Model a problem with a linear equation, use balance strategies to solve equation pictorially and record the process symbolically • Write and graph inequalities

25. Linear Equations and Inequalities • Use addition and subtraction to solve inequalities • Use multiplication and division to solve inequalities

26. Inconsistencies • Linear inequalities may overwhelm students as it is an entirely new topic for them. • There is a huge chunk of linear equations which is a review from previous classes plus new forms which are added in grade 9. • Polynomials has lot of new vocabulary to learn and use. Students need lot of practice in performing 4 fundamental operations with polynomials as they need to use the integer rules and also exponent laws. • Linear relations also has new concepts but all these will form a base for grade 10 math.

27. Grade 10 PLO’s • Identify and correct errors in a simplification of an expression that involves powers. • Demonstrate an understanding of the multiplication of polynomial expressions (limited to monomials, binomials and trinomials), concretely, pictorially and symbolically. • Demonstrate an understanding of common factors and trinomial factoring, concretely, pictorially and symbolically. • Demonstrate an understanding of relations and functions. • Describe and represent linear relations, using: words,ordered pairs, tables of values, graphs and equations. • Relate linear relations expressed in: slope–intercept form (y = mx + b), general form (Ax + By + C = 0),slope–point form (y – y1 = m(x – x1)) to their graphs. • Solve problems that involve systems of linear equations in two variables, graphically and algebraically.

28. Polynomials • Multiply polynomials • Factorize various polynomials using different methods

29. Roots and Powers • Simplify algebraic expressions with rational exponents

30. Relations and Functions • Identify and represent linear relations in different ways • Use intercepts, rate of change, domain and range to describe graph

31. Linear Functions • Graph of linear function to its equation in slope-intercept form. • Graph of linear function to its equation in slope-point form.

32. Systems of Linear Equations • Solving linear system graphically, using substitution and elimination strategy. • Determine the number of solutions of different types of linear systems

33. Inconsistencies • Lot of factorization in polynomials may overwhelm students • Lots of graphing in linear relations • No inequalities in grade 10 • New vocabulary to learn in functions and relations • Students have to show their work

34. DEVELOPMENT OF TOPICS IN GRADE 11 AND 12

35. GRADE 11 & 12 FOUNDATION GENERAL OUTCOME • Develop algebraic and graphical reasoning through the study of relations

36. Grade 11 curriculum covers two major chapters under relations and functions: • Systems of Linear inequalities: It is expected that students will model and solve problems that involve systems of linear inequalities in two variables. • Graphing linear inequalities in two variables • World problems • Explore graphs of systems of Linear Inequalities • Use of graphing calculator • Optimization problems: students may find difficulty in understanding these problems. .

37. Solving Systems of Linear Inequalities The area between the green arrows is the region of overlap and thus the solution. a: 3x + 4y > - 4 b: x + 2y < 2

38. Solve problems by modelling systems of linear inequalities. • A company makes two types of boats on different assembly lines: aluminum fishing boats and fibreglass bow riders. • When both assembly lines are running at full capacity, a maximum of 20 boats can be made in a day. • The demand for fiberglass boats is greater than the demand for aluminum boats, so the company makes at least 5 more fiberglass boats than aluminum boats each day • What combination of boats should the company make each day?

39. Quadratic Functions and Equations • It is expected that students will demonstrate an understanding of the characteristic of quadratic functions including vertex, intercepts, domain and range, and axis of symmetry. • Explore quadratic relations • Explore properties of Graphs of Quadratic Functions • Solve Quadratic Equations by Graphing • Learn Factored form of Quadratic Function • Solve Quadratic Equations by Factoring • Solve Quadratic Equations using the Quadratic Formula • Solve problems using Quadratic Models.

40. World problem to solve quadratic equation by factoring • The entry to the main exhibit hall in an art gallery is a parabolic arch. The arch can be modelled by the function h(w) = - 0.625w^2 + 5w where the height, h(w), and width, are measured in feet. Several sculptures are going to be delivered to the exhibit hall in crates. Each crate is a square based rectangular prism that is 7.5 ft high, including the wheels. The crates must be handled as shown, to avoid damaging the fragile contents. What is the maximum width of a 7.5 ft high crate that can enter the exhibit hall through the arch?

41. Grade 12 Foundation curriculum cover three major functions: • Sinusoidal functions: It is expected that students will represent data, using sinusoidal functions to solve problems. • Polynomial functions: It is expected that students will represent data, using polynomial functions to solve problems. • Exponential and logarithmic function: It is expected that students will represent data, using exponential and lograthmic functions to solve problems.

42. Examples • Sinusoidal function: Ferris wheel problems are nice to explain sinusoidal function and equation.

44. Exponential and Logarithmic Function • Explore characteristics of exponential and logarithmic functions • Use graphing calculator • Students learn to model data using exponential and logarithmic function

45. HOLES OR INCONSISTENT DEVELOPMENT

46. TRANSITION FROM GRADE 10 TO 11 JUMP OF INEQUALITY FROM GRADE 9 TO 11 • There is nothing under inequality in grade 10 and grade 11 foundation take students directly to the system of linear inequalities for which they need a good revision of the inequality stuff from grade 9. Some teachers prefer to revise the basic stuff and some don’t, so there should be at least some revision part in curriculum particularly for these kind of concept gaps.

47. Good Revision is necessary • Students should be given a good opportunity to revise concepts of inequalities. In addition, to understand functions better a good review from grade 10 should also be given. ‘Simple to Complex’ strategy is recommended for teachers to explain functions and related world problems better. Sometime concept maps make things clear in students mind.

48. Transition from grade 11 to 12 Math Foundation • Transition from grade 11 to 12 Math foundation is pretty consistent where the focus is on exploring relations and function. Students start from system of linear inequalities and explore quadratic, Sinusoidal, polynomial, logarithmic and exponential functions.

49. How three strands cover algebraic concepts in grade 11 & 12

50. Questions