## Number Line Dancing

• Lesson
3-5
1

Project this lesson on a whiteboard and watch your students “number line dance” their way to fluency in estimation of products. The lesson is scaffolded to start students finding basic multiplication facts on a number line and ends with students estimating four digit products on a number line. Use the lesson to teach estimation of products or as a culmination of your own estimation of products lesson.

After a discussion of estimating products and number lines, the class will be divided in half and compete to find products using mental math or estimate products and indicate them on number lines which will be projected on a whiteboard.

To prepare for the lesson:

• The day before the lesson, invite a few students to bring in a CD of their favorite band or choose some songs that you think the students will like. Use different tempos or genres of music for each round.
• Print out Number Line Dancing Cards on card stock, and cut them apart. Or, write the numbers directly on 3 × 5 cards. Sort into piles according to the shape on the cards.
Number Line Dancing Cards
• Take the time to view the Number Line Dancing Presentation a few times before the lesson to get comfortable with the slides.
Number Line Dancing Presentation

Tell the story of a class who won a contest. Their prize was to go to a concert of their favorite band. Twelve classes from schools across the city would be going to the contest. The teacher wanted to know about how many students would be there all together. Most classes had about 27 students. Ask students, "How could you estimate to find the total?" [12 can be rounded to 10, and 27 could be rounded to 30. 30 x 10 = 300 students.]

Ask if anyone can give other examples of when estimation would be needed in real life. [Shopping, planning for parties, sports statistics, etc.]

Explain that estimating is a great time-saver and is also a great way to know whether an answer is reasonable. (Students using calculators sometimes hit a button in error, but can use estimating to see whether their answer is reasonable.)

Put the headings from the table below on your board, and write the first problem.

• How do you estimate products?
[If you round both numbers to the nearest multiple of ten, you will be able to use basic math facts, and then put all the zeroes back in.]
• Is it necessary to round.
[No.]
• What basic fact would you use to solve the problem?
[8 x 3 = 24.]
• Is 240 the exact answer or an estimate? How do you know?
[Exact. We didn’t have to round our original numbers.]
Proceed through the other examples. Add into your list of questions (after asking for the basic math fact), "What about the zero from 30 (or 80)?" [Put it back into the product.]
 Problem Round Basic Fact Put zeroes back Exact Estimate 8 x 30 no 8 x 3 = 24 240 yes 8 x 32 32 to 30 8 x 3 = 24 240 yes 78 x 32 78 to 80 32 to 30 8 x 3 = 24 2,400 yes

Depending on your class’s previous experience, you may want to add more examples to the chart. Ask students to look at the chart and explain what the factors would look like when you need to estimate versus being able to calculate the answer mentally. [If at least one of the factors is 2 or more digits that don’t end in a zero, you will need to estimate. If the factors are one digit or end in zero, you can use mental math to calculate the answer.] As a quick formative assessment, have students pair up. First student should give the second student a 2 digit by 2 digit problem without zero as a digit. They should use paper or a small white board. Second student should estimate the answer and explain each step as it is done. Students should switch roles. Once they feel confident in the skill, they can indicate it to the teacher with thumbs up. You can circulate and help those that still need it.

Explain that the class will be playing a game that will give them a chance to practice quick estimation of products. Students will have to use mental math to estimate products, and then place those products on a number line. Use the Number Line Dancing PowerPoint presentation to talk with students about number lines.

Begin the slide show. When you get to the second slide, ask a student to come to the board and point to where the product of 4 × 6 is on the first number line. The third slide shows a second number line. Have a different student show where the same product is on the second number line. Repeat this with the third number line, which appears on the fourth slide. As each child shows their point, have them explain how they decided where to place it. A good explanation might be “I knew that it had to be between 20 and 30 and 24 is a bit less than half so I went to the middle and put it a little before that.” Accept any answer that is appropriate. If the class needs more practice, give another multiplication problem to practice with all three number lines.

Explain the rules of the game:

• There will be 2 teams standing in a line on each side of the board competing for highest score. However, only one team will go at a time.
• The first person in line will dance to the board, choose a card from each pile needed for that round, and place them on the ledge
• That person will find the product, or an estimate of the factors on the cards, and draw a dot on the number line to indicate the answer.
• Each person will keep the cards with them until the end of the round, but hand the marker to the next person in line for your team.
• After drawing a dot, dance to the end of your line.
• Only the person at the board is allowed to work on the problem.

Depending on the make-up of your class, you may need to change the rules of the game. Here are some suggestions:

• Allow the team to call out answers to help a stumped student at the board.
• Send 2 students up together to solve each problem.
• Give students a lifeline (they can pick one other student to help if they need it). You may want to limit lifelines to 3 per round.
• Allow students to have a clipboard, with paper and pencil, at the board.

Explain that there will be number lines with different intervals for each round. You should use a different song for each round. The first round  will be practice in solving multiplication problems with mental math, but the other rounds will ask students to estimate products.

Move to the next slide.

Demonstrate how students should show their answers on the number line. For example, if the product is 260, put a dot on the number line where you think 260 is, draw a line straight above or below the point, and draw a circle with your initials in it. An example is shown on Slide 5. This will allow the class to check the answers after the round. You may want to have a few students practice recording an answer on this slide to make sure they understand.

Divide the class into their teams and have them line up in the proper place. For Round One (on Slide 6), place the square and circle piles on a desk near the board. Remind the class that in the first section they will be solving problems using mental math. When you are ready to begin, click on the next slide. Start the music and invite the first student in line to dance up to the board. As the student records his dot on the number line, remind him her to draw a line with a circle and put his/her initials in it. Continue until all members of the team have had a turn. Stop the music. Have each student show their 2 cards. Ask the other team to help determine if the answer is in the correct spot on the number line. Award a point for each correct answer. Collect the square cards, shuffle them, and put on desk for the next team. Do the same with circle cards. Be sure to have a student erase the whiteboard. The second team takes their turn.

For the second round (Slide 7), let the second team go first. For this part of the game, students will round one of the factors, and then use mental math to complete the estimation. Use the square cards and the triangle cards for this round.

The third round (Slide 8) uses the triangle cards and the star cards. Allow the team with the lower score to go first. For this part of the game, students round both of the factors, and then use mental math to complete the estimation.

Give the winning team the option of doing a 30 second celebration dance.

If there is time at the end of the game, have a class discussion using the following questions. If there is no time, discuss on the next day as a follow up.

Here are a few suggestions for discussion questions:

1. Look back at the Slide 2 with the three number lines. Can you tell anything about what happens when you plot the same number on the different number lines as the intervals get larger?

[It’s more difficult to find numbers that are between the larger intervals. The number is closer to the zero mark as the intervals get larger.]

2. Was it important to look at the intervals on the number line to decide where to put your answer?

[I needed to look at the numbers that were marked on the tick lines and decide between which two I needed to put my answer. I needed to see what the middle would be and whether my number should go close to the middle of it or nearer to one tick line or the other.]

3. In Round One and Two, the number line goes from zero to 900. The cards in Pile A went from 2 to 9. Pile B and C had numbers from 10 to 99. Explain why all the products had to fall somewhere on the number line.

[The lowest possible product would be 20 (10 x 2) and the largest possible product would be 900 (9 x 99 rounded to 100)]

4. In Round Three, the number line goes from zero to 10,000. What did the range of numbers in Pile C and D need to be to fit on that number line? All factors were 2 digit numbers.

[They could have been anywhere from 10 to 99. 10 x 10 = 100 for lowest and 99 x 99 rounded to 100 x 100 = 10,000 for highest.]

### Reference

Math Heads CD, Theatrix, 1996 (out of print).

Assessment Options

1. Use the Number Line Dancing Activity Sheet as a formative assessment at the end of the session or at the beginning of the next day’s lesson.
Number Line Dancing Activity Sheet
2. Students write 10 problems of their own whose estimated products fall within a certain range. They create the number line, deciding what the intervals will be, and record the estimations on the number line.
3. Journal Entry – explain to a friend in another class how you would estimate 37 x 49 and where you would put it on a number line that has intervals of 10.

Extensions

1. Introduce other operations into the game.
2. Use the number line dance activity to have students identify mixed numbers (fractions and decimals) on a number line.
3. Have students create a math rap and a line dance to go with it. It must describe how to estimate products or how to find a number on a number line.

Questions for Students

1. Do you think it would be easier or harder to play this game if you were estimating sums, differences, or quotients? Give an answer for each operation.

[It may be easier for sums and differences because I know my addition facts better than my multiplication facts. Quotients would be harder because it takes more time to think of compatible numbers.]

2. Why is it important to round to the greatest place value when estimating?

[It allows you to use basic facts and then do what you need to do with the zeroes.]

3. Is there ever a time when you wouldn’t want to round to the greatest place value?

[If you can find the answer mentally without having to round or want a more accurate answer. 127 × 6 can be rounded to 120 × 6 instead of 100 x 6, because you may know that 12 × 6 = 72 and then put the zero in to give the answer of 720.]

4. Is there any other math topic where number lines and intervals are used? Why are they needed?

[Bar graphs, line graphs, thermometers. The graphs need number lines to show the data and choosing a good interval to use is important for giving the viewer an accurate rendering of the data. Thermometers use number lines because temperature varies.]

Teacher Reflection

• Did having the music increase students’ level of enthusiasm/involvement? Explain why or why not.
• Did you challenge the achievers? How could you have changed the groups of cards to make them more challenging?
• Did you use the suggested changes to help the diverse learners? Did they work?
• How did the students perform in relation to the stated behavioral objectives?
• Would this lesson be better in two different sessions? Why or why not?
• What worked with classroom behavior management? What didn't work? How would you change what didn’t work?

### Learning Objectives

Students will:

• Identify position of whole numbers on a number line with various intervals.
• Increase fluency in multiplication facts.
• Solve multiplication problems using mental math.
• Increase fluency in estimating products.

### NCTM Standards and Expectations

• Develop fluency with basic number combinations for multiplication and division and use these combinations to mentally compute related problems, such as 30x50.
• Develop and use strategies to estimate the results of whole-number computations and to judge the reasonableness of such results.

### Common Core State Standards – Mathematics

• CCSS.Math.Content.3.OA.C.7
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

Grade 4, Num & Ops Base Ten

• CCSS.Math.Content.4.NBT.B.5
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Grade 5, Num & Ops Base Ten

• CCSS.Math.Content.5.NBT.B.5
Fluently multiply multi-digit whole numbers using the standard algorithm.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.