Ratios and Scale Factors

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Objective

SWBAT express equivalent ratios by using scale factors to create double number lines and tables

Big Idea

We can create equivalent ratios using a scale factor instead of adding repeatedly.

Think About It

5 minutes

For this Think About It problem, pairs work together to solve the problem.  Pairs will pretty quickly begin to create a ratio table or double number line, and will see that they will need to list many equivalent ratios to get to 36 pounds of rice.  Most groups will not finish their models before I stop the work.

The purpose of this Think About It problem is to have students experience an inefficient method for finding an equivalent ratio.  I model for students how to complete this problem using a scale factor.  

Before moving on to the Intro to New Material portion of the lesson, we discuss where scale factors are used in the real world.  We talk about architects, maps, model cars, and Lego Architecture sets. 

Intro to New Material

15 minutes

Prior to this lesson, students ask why we aren't using multiplication, rather than the additive relationships we've relied on (I let them know that we need to master the models in the first lessons of this unit).  They are excited in this lesson that we are going to now use multiplicative relationships.  

In the Intro to New Material section, I introduce the vocabulary and we discuss strategies for finding the scale factor.  I use a ratio table for problem A and a double number line for problem B, so that students see that scale factors can be used with either.  

For problem A, some students will notice that Myles makes half of his shots.  If they see this relationship, they may complete the ratio table without using a scale factor.  It's important that students can see multiple pathways to answers.  I validate the strategy and commend the students for noticing the relationship between the terms. 

 

Partner Practice and Discussion

15 minutes

Students work in pairs on the Partner Practice.  As they work, I circulate around the classroom.  I am looking for:

  • Are students correctly labeling the double number lines or ratio tables?
  • Are students correctly drawing the double number line? Equal spacing, numbers that correspond to A are along the same line and B along another line?
  • Are students choosing the correct scale factor to multiply their original/known terms by?
  • Are students showing clear, logical work-are they drawing an arrow indicating multiplication?
  • Are students providing an answer to the specific question?
  • Are students checking for the reasonableness of their answer?

I am asking:

  • What does the ratio mean in this problem?
  • What is the value of each part?  How do you know?
  • What is the question asking you to find?
  • How did you find the scale factor?
  • Why did you pick this diagram?

 

After 10 minutes of work time, I bring the class together for a discussion.  We talk about problem 3.  Students will tend to use a ratio of 8:25 for the flowers to seeds.  They'll then get stuck, because they're left with nowhere to go.  I have pairs go back to this problem and underline the sentence that identifies the starting ratio.  I then have a student explain what the ratio should be for the start of the model, and point to the text where this can be found.  

Students complete the check for understanding independently.  I ask for one student to share a double number line and then one student to share a ratio table.  The class offers feedback on the models.  

Independent Practice

15 minutes

Students work on the Independent Practice problem set.  Here's a sample independent practice that shows what student work should look like.  

The wording of problem 4 can be difficult to make sense of for some students, particularly for lower-level readers.  Some students think there is an 'extra' number, because of the word 'one' when referring to the number of rotations the satellite makes in two hours.  If students are struggling during work time with this problem, I find that guiding students through a table helps here, as the parts are labeled.  

 

Closing and Exit Ticket

10 minutes

After independent work time, I bring the class back together.  Problem 9 in the Independent Practice problem set is another that leads to good conversation.  Students learned in a previous lesson how to extend a model using an additive relationship.  Here, they should use a scale factor on the total and on the parts flour.

Students work independently on the Exit Ticket to end the lesson.