The double number line

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Objective

SWBAT use the double number line to solve problems.

Big Idea

The double number line will help students figure out problems that involve two quantities.

Do Now

10 minutes

The students will be working on a problem from Britannica Mathematics number tools to reinforce their understanding of the number line.  The worksheets shows 4 different ways to get to the same answer and the students will need to explain the strategy used to get to the solution.   Once they have given an explanation, ask the students to decide on which strategy they would have used to solve the problem mentally.

1. This strategy involves starting with 67, counting up to the nearest multiple of 10 (70), and then counting up two sets of 10 (20) to reach 90, and finally, adding the difference between 29 and 23 (6) to reach 96.

2.This strategy involves starting with 67, counting up to the nearest multiple of 10 (70), and then counting up by tens to 100. Since a number greater than 29 was added, the last step is to subtract the difference between 33(the amount added) and 29, which is 4 to reach 96.

3.This strategy involves starting with 67 and counting up by tens. Since 30 was added, the last step is to subtract 1, because the goal is to add 29, not 30 which results in an answer of 96

GREAT STRATEGY TO USE MENTALLY

1. This strategy involves starting with 67, subtracting 1 from 67, and then adding that 1 to 29 to make 30. Subtracting 1 from 67 leaves 66. Adding 30 to 66 results in 96.

GREAT STRATEGY TO USE MENTALLY

Finally, ask the students to think about why each strategy starts with 67 (as opposed to 29)?  Students should say that it is faster to start with the larger number and count up by the smaller number, but that it really doesn’t matter when adding. 

(SMP 1, 2, 6)

Multiple Uses for the Double Number

20 minutes

Begin this part of the lesson by explaining that the double number line is a way to visually represent a comparison of quantities.  The double number line will use the strategies from ratio tables but these quantities will be put on a number line instead of in a table.

Question 1 (Constant Speed):  This problem will allow students to see the double number line with a few missing pieces.  Explain to the students that the double number line is comparing feet to seconds. Ask the students  “how did I know that?”   While students are looking at the number line, ask them to the following questions.  Then ask the students to answer the following questions about this double number line.

If a remote control car is traveling at a constant speed.  The number of feet it can travel for the given number of seconds  is shown on the double number line diagram

- Fill in the boxes in the diagram.  Justify your answer to a partner.

-If the car travels 14 feet at this constant speed, how many seconds did the car travel? Justify your answer to a partner.

-If the car was travelling for 8 seconds at this constant speed, how many feet did it travel? Justify your answer to a partner

-Express, in simplest form, the ratio of feet to seconds as depicted in this problem? (some students may need help with this problem because they may not know what simplest form means.  Explain to students that a simplest form ratio occurs when the comparison can no longer be divided by a common factor)

** By having the students justify (explain) their answer to a partner, the students are thinking about their thinking which supports SMP #3**

Question 2 (Ruler as a double number line):

I chose this problem to do as a whole group because it uses the ruler as a double number line.  The ruler will be millimeters to feet in a scale model drawing.  This problem is rich in concepts, but the idea is for them to be looking only at how to use this double number line.

 

A small display model is being made of a new canoe.  The scale being used for the model is shown on the ruler, where the number of feet in the actual canoe is scaled dow to millimeters for the display model.

Which of the following statements is true in this situation?

a. 3 feet will be represented by approximately 30 mm in the model.

b. 3 feet will be represented by approximately 20 mm in the model

c. 20 mm in the model will represent approximately 5 feet in the canoe.

d. 30 mm in the model will represent approximately 3 feet in the canoe.

Each of these responses has the students looking at the ruler and drawing conclusions about what is being asked.  These type of questions support SMP #1 (making sense of problems) When students have made their decision, have them justify their answer in writing and then share these justifications with a group.  Also, ask students if there is only one right answer.  Allow think time and ask them to share their thoughts with a partner (SMP #2)

 

Using the Double Number Line

20 minutes

During this section, I’m going to focus on percents.  The double number line is an excellent resource to use when trying to find the percent of a number.  Begin by asking the students the following question:  If you had $30, what percent of the money do you have?  Give students time to think about the solution to this problem?  The students should say you have 100% of your money.  Draw a number line from zero to 30 on the bottom and 0 to 100% on the top.  Show the students that 100% = $30.  Next, ask the students if you had $0 what percent of your money would you have? (0).  Show them where that is on the number line.  Next, ask the students if you had 50% of your money, how much would you have ($15).  Place 50% and $15 on the number line.  Ask the students how much money they would have if they had 10% of $30.  (Allow students time to think and come up with how to find this number).  I’m anticipating that most students will see that if you divide 50 by 5 you get 10 so if you divide 15 by 5 you get $3.  Some students may divide 100 by 10.  Give the students time to share their strategies.  Next, have the students fill in the rest of the number line (20%, 30%, 40%, 60%, 70%, 80%, 90%).  Have students share their answers with a partner.  While students are working through their double number line, be sure to ask them if their answer makes sense?  Do your numbers follow a pattern on the number line?  Once students have processed through this problem, it is time to have them work on one independently.

Ask the students the following question? 

Marilyn saves 30% of the money she earns each month.  She earns $350 each month.  How much does she save?

Students should know to use a double number line to help them solve this problem.  If they are having trouble understanding how to start, ask them if there is a visual representation they could use to help them.  When students make the number line, I’m anticipating that they will be challenged by where to put the numbers.  Allow students time to think about where the numbers go.  Continue to ask them if there process makes sense. (SMP 1 and 4)

You should see the students putting 0 and 100% and 0 and $350 on the number line.  After that, students can use any of the strategies learned from ratio tables (halving, doubling, multiplying, dividing) to get to their solution.  As students finish the problem, have them write down ” what did they do” and “what were you thinking”.  This will get students thinking about their mathematical decisions. (SMP #1)

Numbered Heads Together

25 minutes

I will be using 4 problems for this activity using Numbered Heads Together.  I have chosen the problems because they can be used on the double number line and I will be encouraging the students to support their answer with this tool.

Question 1: 

 It took Mary 2 hours to complete 3 pages of math homework. Assuming she works at a constant rate, if she works for 8 hours, how many pages of math homework will she complete? What is the average rate at which she works?

I’m looking for students to put this on a double number line.  If they use a ratio table, that’s ok too. Students should be able to work either tool with ease.  The part they will struggle with is the average rate.  Ask students to think about that statement, what do you think it means?  If they continue to struggle, ask them how many pages can she read in an hour? Explain that this is the average rate (per hour).  Students should come up with Mary can do 12 pages of homework in 8 hours and she can do 1 ½ pages in 1 hour

Question 2

Octavio has to buy 12 tickets for a play.  One ticket costs $14.  Calculate the price for 12 tickets.  Represent your answer with a double number line.  Explain how you got to your solution.

Students will need to set up a double number line to solve this problem.  Then I will be looking at their thinking.  I’m anticipating some students will multiply straight up (12 x 14), while others will take intermediate steps to get to their answer.

Question 3:

Two out of five students have read The Hobbit.  What percent is equivalent to two out of five. 

In order for students to figure out this problem, they will have to know that the 2 and 5 are on the same mark of the number line.  The goal will be to find the number equal to 100 on the number line.  Students should model their thinking with a ratio table or a double number line.  Again, have the students write down the “ what they did” and the “why they did it” to think about their thinking.

 

Question 4:

The school has 250 students, but five are not in school today.  What is the percent equivalence for the percent of students not in school.

The students will struggle with the wording in this problem.  Allow them time to think through what the problem is asking them to find.  Students should realize that 250 is equal to 100% of the students.  A double number line would be the best tool to use for this, but a ratio table will work as well. 

(SMP 1,2,3,4,6)

Closure

10 minutes

To get students to think about their thinking, I’m going to have them choose their favorite method to solving ratio problems (Ratio table vs double number line).  I want the students to choose the method they like the best and explain in writing why they chose that method.  I always make sure to let students know that their explanation needs to be supported with mathematical language and reasoning, not “just because”. When they have had enough time to write down their strategy of choice, I’m going to have them do a HUSUPU but they must find someone who chose a different strategy.  Once in groups, the students will each share their strategy of choice and the support for their vote.  Students should take turns sharing.  Once done, I’m going to have student give a little “high five” and say, “thanks for sharing with me today”!