Transforming Formulas

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Objective

SWBAT transform a formula to describe one quantity in terms of the others.

Big Idea

Knowing how to use formulas flexibly can make it easier to solve real world problems.

LAUNCH

15 minutes

In order to access prior knowledge, provide the following task. It not only gets the students’ neurons fired up, but that also relates in some way to the lesson content. Tell the students that solving formulas, or solving literal equations, involves using algebraic properties to justifying the steps taken, just like you’ve done with solving simple one variable equations, only that you may not be able to simplify your answers as much.

Project the following list of algebraic properties on the whiteboard

algebraProperties_APK.docx

Tell students that their task is to provide one algebraic equation example in their notebooks, of each property justification on this list. Motivate them to discuss their work with any of their immediate partners.  When they’re done they should have used all 7 properties. More than one property can be exhibited in one example. Walk around being just the “guide by the side” , and allow them to use any available resource they may have at hand.  

A second option for beginning this lesson is using the following organizer for each student to work on. 

organizer_AlgProperties.docx

 

EXPLORE

40 minutes

New Info / Application

Equation solving is highly conceptual, and procedural as well, since it involves the steps to find solutions to equations. This section takes a structural approach which primarily focuses on procedural understanding. Many times, students, despite being able to solve equations for variable x, have problems using the same deductive reasoning when having to isolate a variable in a formula which contains more variables. This is probably due to a weak comprehension of the algebraic properties, like those of equality, seen in the first section. Some may also memorize the steps when using the same type of equations repeatedly. Other students may not know when they are done solving. This may be due to them expecting a number answer like in regular “solve for x” equations.  Students must be able to isolate any variable in a formula, principally because of the different real world scenarios they can be confronted with.

Group work

Break students up in groups of threes. Try to put students of equal ability level together. Some of the formulas can be solved in various ways, so having trios increases the likelihood that more than one method will come in the group.

Keep the algebraic properties on the board (also on the organizer handout, if used). Hand each group several blank (no line) sheets of paper to work on.

Write the first formula on the board and ask them to solve it for the variable. Tell them to find different ways of solving.

As they work, walk around providing guidance (guide by the side) and listening to their discussions

1) C = 5/9(F - 32)     

{For Formula 1, don’t allow them to convert the fraction into decimal form. Some are tempted to use the distributive property first. Allow this, but guide them to try other routes. Tell students that they are done when they’ve isolated h. State that as it is, the formula is solved for C.}

 Once students are done, call on volunteers to go to the board to solve them. Call on other students who solved the formula differently. Milk the problems a bit by asking any student to solve for another variable in the formula.  

Do the same for each of the following formula.

 2.  V = 1/3 πr2h

 {Students here may not know how to “undo” the squaring of r, once they get to it; Ask “what is the inverse operation of squaring?” Some may say “dividing by two”, which gives you the opportunity of refreshing that division undo’s multiplication, not squaring. Also, tell them that when solving formulas, an answer may contain a radical}

Scaffold:

For students that are having difficulty, have them solve the formulas on this resource. (scaffold_solvingformulas) These are simpler and more resemble equations they’ve seen many times before.

 

3) Have each group invent a formula to solve. It should have 3 variables or more, and a number. 

Again, ask each group to solve for each of the other variables in their formula.

 For example; a formula 2rs – z = y, should be solved for r, s, and z.

 

SUMMARIZE

10 minutes

Summarize:

Students should work alone. Write the following formulas on the board. Have each pair answer the questions on a piece of paper and hand it in as they leave the class for formative assessment.

 

Ask students to describe the approach you would use to solve each equation for “m”, in their own words

A good description for #1 would be something along these terms;

“Use the distributive property on the right side to get 2n = 21 – 3m

Then I add - 21 to both sides using the addition property of equality. Finally, I divide by -3 on either side to solve for m”

 

1. 2n = 7(3 – m)       

2. (m – 5)/6   = – 3x

3. (2/3)(2m + 4) = 9y

4. am + bm = c   (remind students that factoring may sometimes be necessary when the variable to solve for appears more than once in the formula)

 

EXTENSION:

Look up the formula for finding the volume of a sphere.

Rewrite the formula if you were solving for the radius “r”.

 

Homework: Resource;  Formula_homework