Proportional Relationships The connection of Unit rates, lines, and linear equations

How important is the =sign? = sign means there is a relationship. – Numbers can be expressed as sum: 6=1+5 – 6x7=5x7+7 – Distributive property: 7(1+5)+7=7(1)+7(5)+7

Promoting relational thinking Study the picture and the chart. Describe the relationships you see.

Promoting relational thinking cont. Talk with a partner to find two distinctly DIFFERENT explanations for why 5 should go in the box: 7- = 6 – 4. How could we use relational thinking to solve: =174 + Hint: Don’t compute!! Think…

Exit Thinking Use relational thinking to determine if the relationships below are true or false. Discuss the 1 st two problems with a partner; together you must decide if the relationships are true or false. For the last two, work independently to explain your choice of true or false.

Challenge the teacher Groups of 3 or less Use today’s date to make up your own true/false sentences and have the teacher choose the correct response. You can choose any operation combination, but YOU must be able to prove the answer choice as well. Assignment worth 10 points

Exit ticket

Mathematical relationships: Equations & Graphs “=“ tells us that there is a relationship 7x+2=2x+2-9x f(x)=x f(x)=mx f(x)=mx+b what is the relationship between the expressions listed above? Which of the “relationships” represent an equation, and which represents a linear equation?

Solving Equations with one variable Do all equations result in a solution? Explain the basis for your answer? Try these: -7x+2=2x+2-9x -7x+3=2x+2-9x -7x+3=2x+2 W.W.K.D??

y=x: “y is x” y=mx: y is the product of “m” and “x” y=mx+b: y is the sum of the product “mx and b” Thus: Y is a function of x

Relationship of functions

Relationship of functions, cont.. The rule does NOT have to be mathematical… The rule could be a direction of what will happen after “a” for h(a) i.e: h is the rule that says: the next largest integer that starts with the same letter as a(this is the rule) so for a=2 what would h(a)=? What if a=8?

What do the graphs of functions look like? We have already used functions such as y=x+1. This says y is a function of x, such that y =x+1. f(y)=x+1. f(0)=1 f(1)=2 f(3)=? The graphs produce a “LINE”. The lines represent a linear equation. The equation is linear because its variables are solutions that represent CONTINUOUS ordered pairs when graphed on a coordinate plane. All equations don’t produce a line, but all functions produce lines, some straight some not….

Linear Functions explorations f(x)=x f(x)=mx f(x)=mx+b Do these linear equations (functions) represent proportional equations? Why or Why not?

Linear functions(equations) of non-proportional relationships Linear equations can be written in the form y = mx + b. When b ≠ 0, the relationship between x and y is non-proportional. “the constant is “interrupted by the addition of a (+) or (-) value…”

The mathematician’s Rule “If the same symbol or letter appears more than once in an equation, then it must stand for the same number every place it occurs.” = +17 Use relational thinking to decide on a number that would make the statement true.

Conversely More than one variable in a sentence will represent more than one value. a+6=10-b

What do proportional relationships look like? Unit rates, constant of proportionality, straight lines that rise and run at a constant rate, and run through the origin of the graph. Graphically- y=mx