Background

Students often are thinking additively rather than multiplicatively.  For example, if you present the scenario, "One puppy grew from 5 pounds to 10 pound.  Another puppy grew from 100 pounds to 108 pounds." and ask, "Which puppy grew more?" someone who is thinking additively will say that the one who now weighs 108 grew more because he gained 8 pounds while the other gained 5 pounds.  Someone who is thinking multiplicatively will say that the one that now weighs 10 pounds grew more because he doubled his weight while the other only added a few pounds.  While both are correct answers, multiplicative thinking is needed for proportional reasoning.  If your students are thinking additively, you can nudge them toward multiplicative thinking with an activity such as this.

The ratio cards used in this activity are visual rather than numerical so that students have the option to use informal methods to compare ratios or identify unit rates.  What is meant by visual and informal?  In this example of Preschool R, a student may see that the staff-to-child ratio 6:18 is equivalent to a staff-to-child ratio of 1:3.

Preparation:

1)  Cut out a set of Child Care Ratio Cards and/or Preschool Ratio Cards.  Choose one set for smaller classes, both sets for larger classes.  Do not laminate the cards so that students have the option to use informal methods to compare ratios or identify unit rates.  You might consider printing in black and white so they feel free to write on them. 

There are Child Care cards equivalent to a staff-to-child ratio of:

·         1:2                      

·         3:4 (unit rate 1 : 1.3̅3)

·         2:3 (unit rate 1 : 1.5)

There are Preschool cards equivalent to a staff-to-child ratio of:

·         1:3

·         2:5 (unit rate 1 : 2.5)

·         2:7 (unit rate 1 : 3.5)

2)  Read and make sense of the Activity Directions, especially the connections to be made in the Debrief.

3)  Print the Student Practice and try it yourself with the cards and the digital version if you will be sharing that with students.

Activity Directions:

1)  Give a ratio card to each student when they walk into class. 

2)  Set up the sort in a way that works for your class.  For example, tell students you are looking for a child care and a preschool for your children and you collected this information about the child cares / preschools in your area.  There are so many to choose from, that you need their help deciding where to send your children.  You are concerned about how much attention your children will get.  They can help you by first finding the student(s) with a match for their staff-to-child ratio. 

3)  Encourage visual proofs, such as the Preschool R example above.  Make sure students know that you cannot fire a staff member or cut children in pieces to make the groupings work!  Ask students, “How do you know you are a match?” and listen to the explanation of their reasoning.  Push back with questions such as, "I don't remember that procedure.  Convince me that that works." and "Will that always work?"  Introduce ratio language as needed, and listen for students to describe the relationship between staff and children with ratio language that lists the number of staff first, no matter the order on their card, when they are talking about the staff-to-child ratio.  If needed, ask if the ratio one-to-two is the same as two-to-one.

4)  Ask the groups to create another ratio that would match their ratios.  They may choose to use symbols, such as Xs and Os instead of drawing heads and stick figures.  

5)  Next, ask students to identify or find the unit rate.  [Unit rates are rates where you have exactly one of either component in a ratio.  In this case, the unit rates are all written with one staff and the number of students.]  Note:  Depending on the class' experience, you may choose to wait on this until after debriefing the individual work. 

6)  Debrief the activity, relating the visual methods and thinking processes students used to numerical and multiplicative thinking in order to help students make connections.  For example, Preschool R has 6 staff to 18 preschoolers.  Use student thinking about the visuals and what the numbers have in common (common multiples) to create numerical representations.  In the process, make the properties of numbers explicit, not hidden.  For example, 2/2 is 1 (not mysteriously 'cancelled out') because a number divided by itself is one.  Also, when you multiply something by one, you have one of them - the result is what you started with.

7)  Follow up with individual practice below. 

Student Practice

Follow up by giving the sort as a classwork or homework assignment.  Along with the Student Practice Sheet, include either a set or sets of ratio cards, or this digital version.

[The teacher view of the digital version can be found at Desmos.]