## Setting the Pace

3-5
1

Students continue their investigation of modeling multiplication on the number line using the Distance-Speed-Time Simulation from the NCTM E-Examples.

This lesson, as well as the remaining three lessons in this unit, make significant use of the Distance, Speed and Time Simulation from NCTM's E-Examples.

Before attempting this or any of the following lessons, use the applet on your own to ensure that you are familiar with all of its buttons and features.

Begin this lesson by having students take out the individual Fact Mastery Record that they began in the last lesson.

 Fact Mastery Record

Have them choose two facts that they will learn next. Ask them to draw a number line model of each fact on a file card and describe to a partner how each product was obtained. On the back of each file card, they should write the corresponding equation. Tell students to review these facts several times during the day, and encourage them to take the cards home and practice with their family. (The next day, test each student privately on the two facts. If the answers are given rapidly and correctly, direct the student to blacken in these facts on their Fact Mastery Record. Continue this strategy until all the facts have been mastered.)

Project the Distance, Speed and Time Simulation, one of the NCTM E‑Examples, using an overhead projector, or allow students to bring up this simulation on computers, if Internet connections are available. Call on a volunteer to choose whether the boy or the girl will be used to model a multiplication fact. If the girl was chosen, click on the picture of the boy just above the "starting position" box; this will remove him from the simulation. (If the boy was chosen, then click on the pitcure of the girl instead.)

Establish the girl’s step size (i.e., stride length) using the arrows at the bottom of the applet. The step size can be any number from 1 to 15. Hit the PLAY button, and let students watch the demonstration.

 Play Step Forward

Then ask, "How many strides do you think she took to get to the tree?" Allow students enough time to make a prediction; then, hit the STOP button, and run the simulation again, this time using the STEP FORWARD button. Each time you click on the STEP FORWARD button, the runner will move ahead one stride; students should count aloud each time the girl takes a step forward. (Note that for some stride lengths, you will not want to continue all the way to the end. For stride lengths of 2, 4, and 5, the runner will end exactly at 100. For other stride lengths, the runner will move all the way to 100, but the last stride will not be a full stride; however, this is not reflected in the applet. For instance, if a runner moves with stride length 6, she will move to 6, 12, 18, ..., 90, 96; on her next step, however, she will move to 100, which is only a stride of length 4. From the applet, however, there is no way to know that this last step was shorter than the others, and a student may get the false impression that 100 is a multiple of 6.)

At some point in the middle, pause the runner, and ask students to generate an equation that represents the situation. The number of steps will be the first factor, and the runner’s stride length will be the second factor. (Note that the applet records the number of strides as "time." For instructional purposes, the unit of time in lessons 2-4 will be defined as "strides." The designation will switch to "seconds" in lesson 5.) For instance, if the runner took 7 steps of length 6, the applet will show the total distance as 42. The corresponding equation would be 7 × 6 = 42.

Reset the runner to the start, and run the simulation again, setting a different length stride and using a different number of strides. Continue making changes, and allow students to predict the results. Ask questions like, "How far will she go if she takes three strides of length 8?" [24] Or, "How many strides of length 6 will she need to take to get to 36?" [6] Call students’ attention to the graph after each run of the simulation.

Now pair the students and distribute two dice to each pair. Also give them two copies of the Number Lines activity sheet.

 Number Lines Activity Sheet

Ask them to take turns rolling the dice, using the two numbers as factors in a multiplication number sentence. They should model the multiplication sentence on the number line, and then say the fact aloud. For instance, if a student rolled a 2 and a 6, he would draw, write and say the following:

2 × 6 = 12

"Two hops of 6 equals 12."

As some students work on this task, assign other pairs to use the applet to model and then record multiplication facts.

Assessments

At this time in the unit, students should be able to do the following:

• Model multiplication on the number line
• Predict the product of two factors

The guiding questions above may assist you in assessing your students’ level of understanding, but others may suggest themselves as you talk with your students. You may find it helpful add to your recordings on the Class Notes recording sheet that you began earlier in this unit. This data will be helpful as you plan strategies for regrouping students in future lessons.

Questions for Students

1. How would you model three strides of 2 on the number line? What equation does this represent? How would you model two strides of 3? What equation does this represent? What is alike about these equations? What is different?

[Three strides of 2 and two strides of 3 both end at 6 on the number line. The first situation represents 3 × 2 = 6, and the second situation represents 2 × 3 = 6. The only difference between these two equations is the order in which the factors appear. The final product, however, is the same.]

2. What numbers do you land on when you set the stride length to 2? ...when you set it to 3?

[For 2, you land on 2, 4, 6, 8, 10, ..., and all even numbers.
For 2, you land on 3, 6, 9, 12, 15, ..., and all multiples of 3.]

3. Which multiplication facts are you sure of? Which two facts will you learn for tomorrow?

[Answers will vary, but it is important to have students answer this question and commit to learning two new facts each day.]

Teacher Reflection

• Which pairs worked most effectively together? Which pairs were less effective?
• Which students used the applet with ease?
• Which students could model multiplication on the number line and record the product? What instructional experiences do they need next?
• Which students could not yet model multiplication on the number line? What instructional experiences do they need?
• Which students have learned two multiplication facts by heart since yesterday? Which students did not? How can I motivate these students to learn the products for the chosen pairs of factors?
• What adjustments should I make the next time I teach this lesson?

### Number Line Journeys

3-5
In this lesson, students generate products using a number line model. Students are encouraged to predict the products and to answer puzzles involving multiplication.

### The First Race

3-5
Again using the E-Example simulation, students will model multiplication facts on the number line and compare various representations.

### Telling Racing Stories

3-5
In this lesson, students model races in which runners start from various positions. They enter numbers in a table of values, model races on a coordinate grid, and compare the results. Students begin to develop an understanding of linear relationships.

### Running Races

3-5
Students generate and compare paths which model given problem situations on graphing grids.

### Common Core State Standards – Mathematics

• CCSS.Math.Content.3.OA.A.1
Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 x 7.

• 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.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP6
Attend to precision.