Source:  Wiggins, G., & McTighe, J. Understanding by Design. Merrill Prentice Hall: 1998.
                For further information about Backward Design refer to


Title: __Ratios                             _ Subject/Course: __Math__________________________ Topic: ____Algebra______________ Grade(s): __6_____ Designer(s): __Kevin Smith _____________________

Stage 1 Desired Results

Established Goal(s)

M6A1  Students will understand the concept of ratio and use it to represent quantitative relationships.

Understanding(s) Students will understand that...

1.      Ratios are quantities that are shown in relationship to each other. 

2.  Ratios are used in recipes, sports, grades that teachers give, and in many other life activities.

3.  Ratios are written as follows:  X : Y ; X to Y; and/or X/Y

4.  Ratios are similar to fractions in many ways.


Essential Question(s)

1.      What is a ratio?

2.  How does one determine a ratio?

3.  How are ratios used?

4.  How will knowing how to use ratios help me? 



Students will know...

 the definitions of ratio and equivalent ratio. 


Students will be able to...

1.  determine a ratio as visibly seen.

2.  determine a ratio from specific information not able to be seen. 

*This will be explained below.


Stage 2 Assessment Evidence


Performance Task(s) Summary in G.R.A.S.P.S. form

1.      I will place students in groups of three or four and give each one a bag of colored chips (red, blue, and white).

2.  Students will place all chips in front of them.  There will be one red chip, two white chips, and three blue chips. 

3.  Students will be asked how many red chips there are and how many chips there are total (1:6). 

4.  The teacher will explain that there are one red chip and six chips total giving a ratio of 1:6.

5.  The students will be asked the "ratio" between red chips and blue chips (1:3).

6.  The students will respond based on what they see.

7.  The students will be asked how many "non-blue" chips there are in relation to the blue chips (3:3).

8.  The students will respond based on what they see.

9.  The students will be asked the "ratio" of non-blue chips to total number of chips. (This is to help them differentiate between ratios that involve 2 or more totally separate quantities (for example,  five red marbles and three blue marbles) and ratios that involve a portion and the whole (for example, five red marbles and eight total marbles).

10.  The students will respond based on what they see.

*Thus far the students are simply speaking on what is visible.

11.  The teacher will then present the students with a problem:  If the ratio of red chips to blue chips  (1:3) is doubled, that is to say all the chips were doubled how many of each would there be.  Answer:  2 red, 6 blue. 

12.  Using the board the teacher will show the correlation between ratios and fractions 1:3 and 2:6 to 1/3 and 2/6 - "There are one third as many red chips as there are blue chips."

13. Students will then be asked to figure out how many blue chips there would be if there were 7 red chips.  How many total chips? 



Key Criteria:

  Classroom observation, homework, quizzes, tests.

Other Evidence

Out of class project - Students discover ratios in their home and document them.  Examples would be cassettes to CD's or jeans to khakis




Stage 3 Learning Plan

Learning Activities Consider the W.H.E.R.E.T.O. elements.


1.       The teacher will recap the batting performance of a baseball player.  For example:  Andruw Jones batted sixty times last month.  He hit six homeruns.  What is his ratio of "at bats" to homeruns for the month (60:6).  What is the ratio if you "reduced" the ratio?  (10:1). 


2.  A chef is preparing a desert for forty people.  He has a recipe that is set up to make a desert for five people.  The recipe calls for one stick of butter and three eggs.  How many sticks of butter and how many eggs will the chef need to feed his forty guests?  (8:24)



Concept extension


The teacher will explain that baseball managers will try to put together batters in a lineup that are hitting well in order to have the highest probability of having runners score.  Example:  Andrew Jones bats 3rd and Chipper Jones bats 4th.  If Andruw Jones is getting on base one out of four times and Chipper Jones is getting on base one out of three times, their ratios are 1:4 and 1:3, but the probability of both getting on base together is 1/4 x 1/3.  What is the probability or the "odds" or chance that they will get on base right after the other?



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