Number Line Journeys

3-5
1

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

To begin the lesson, assess prior knowledge by drawing a long line on the floor or sidewalk with a piece of chalk. Demonstrate how the line can be marked off in equal intervals, and then assign a volunteer to fill in the numbers 0-30 on the line.

Ask a volunteer to name a number between 2 and 4. Then, starting at 0, have the volunteer hop a counter down the number line as the rest of the class skip counts by that number (for example: 3, 6, 9, 12, 15, ...). Repeat with other volunteers using other numbers.

Inform students that they will find products using a number line model. Write the following equation on the board: 4 × 5 = ____. Then, demonstrate with a counter how hops of five can be taken on the number line. You may wish to encourage students to count aloud as the hops are made. When 20 is reached, complete the equation (4 × 5 = 20) and encourage the students to translate it as, "When you take four hops of five, you land on 20." After several examples with five as a factor, ask a student to determine what size hop to use next. Work out and record several multiplication equations. When students are ready, encourage them to predict the products and then to verify their predictions by moving a counter along the large number line that was drawn on the floor or sidewalk.

After allowing time for exploration, give each student a copy of the Number Lines Activity Sheet and ask them to draw five examples of "hops" and then trade them with a friend. (Note that the activity sheet contains eight number lines. Students should use the first six of them for this part of the lesson; an extra one is provided in case they make a mistake. The last two number lines on the sheet will be used later in the lesson, and students should not write on them yet.) The friend should record the multiplication fact modeled on the number line using an equation of the form n × s = ___, where n is the number of hops and s is the size of each hop. When all pairs have finished, call the class together to model what they did and to display their equations.

Then say, "Show on your blue number line five hops of 3. Where did you land?" [15] "What equation is that?" [5 × 3 = 15] Then ask them to show three hops of 5 on the orange number line, record the equation, and compare it to the equation from the blue number line. Encourage them to explore other examples of the Order Property and state their findings. [The Order Property states that the order in which two numbers are multiplied does not affect the result. For instance, 3 × 5 = 15, and 5 × 3 = 15.]

Now give each student a copy of the Fact Mastery Record. Ask them to use a dark crayon to cover each product that they are sure of.

You may wish to talk about the properties of multiplying by 1 and by 0. This will assure that they cover at least the two rows and two columns shaded pink on the chart. You might also want to point out that, because of the order property, learning the products in the unshaded portion will mean that they have also "learned" the products in the cells that are shaded light blue. By noting these observations, students will be able to color in many of the cells in the chart, making the task of learning the multiplication facts seem less overwhelming. Because this unit is highly dependent on proficiency in naming products, you may wish to establish individual plans with students who have many facts left to memorize.

As a concluding activity for the day, pose puzzles such as, "I am a number between 14 and 19. You say my name when you hop by 5’s. Who am I?" [15.] Encourage students to create and share similar problems. You may also want to have them locate the products on the Fact Mastery Record.

Assessment Option

Record your observations about student progress by using the Class Notes sheet. At this point in the unit, students should be able to:

• Model multiplication on a number line;
• Use the number line model to find products; and,
• Identify known multiplication facts on a chart.

Make notes about student progress toward each of these goals. Recording the information in this manner may be useful when discussing student progress with parents, administrators, and colleagues.

Extension

Move on to the next lesson, Setting the Pace.

Questions for Students

1. How would you model four hops of 6? What equation does this represent? Where would you find the product on the Fact Mastery Record?

[Four hops of 6 represents the equation 4 × 6 = 24. This result occurs on the Fact Mastery Record in the fourth row and sixth column; because of the Order Property, it also occurs in the sixth row and fourth column.]

2. What numbers did you land on when you hopped by 5 to 30?

[5, 10, 15, 20, 25, 30.]

3. What numbers did you land on when you hopped by 3 to 30?

[3, 6, 9, 12, 15, 18, 21, 24, 27, 30.]

4. Did any of your hops end on the same number? What equations did those hops represent?

[When hopping by 3 or by 5, you land on both 15 and 30. The equations for 15 are 3 × 5 = 15 and 5 × 3 = 15, and the equations for 30 are 10 × 3 = 30 and 3 × 10 = 30.]

Teacher Reflection

• Which pairs worked most effectively together? Which pairs were less effective?
• Which students could easily model multiplication on the number line? What instructional experiences are appropriate now?
• Which students could easily identify the multiplication facts that they knew by heart? Which students could not? What instructional experiences do they need next?
• How can you ensure that all students will be able to quickly give the product for any pair of factors 0-9?
• What adjustments should be made the next time you teach this lesson?

Setting the Pace

3-5
Students continue their investigation of modeling multiplication on the number line using the Distance-Speed-Time Simulation from the NCTM E-Examples.

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.

Learning Objectives

Students will:

• Model multiplication on the number line.
• Use the number line model to find products.
• Identify known multiplication facts on a chart.

NCTM Standards and Expectations

• Describe, extend, and make generalizations about geometric and numeric patterns.
• Represent and analyze patterns and functions, using words, tables, and graphs.
• Model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions.

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.

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.