Number Lines

Routine: Students are presented with a number line that may only have one or two values marked on it. They are asked to locate a particular value on a number line or determine the value of an indicated location and explain their thinking. The teacher should vary the endpoints. Students should be expected to explain their thinking. The number line is a powerful, coherent, and unifying model that may be used across multiple grade levels to develop number sense and computational proficiency.

This routine helps students to understand relationships between numbers (including fractions and decimals-4th/5th only) and the relative magnitude of numbers. Students are asked to determine numbers that could be placed on a given number line. The teacher should vary the endpoints. Students should be expected to explain their thinking.

Present a number line with the endpoints labeled. Ensure that number choices are within the scope of the grade level curriculum.

Additional Suggestions:

• Given a number line with endpoints labeled, have students generate five numbers would fall between the endpoints. Ask students to estimate the location of one of the numbers and justify their thinking.
• Students may use sticky notes for numbers to place them with a follow up discussion as to whether or not any sticky notes placed should be adjusted either left or right. Discuss rationale for adjustment.
• Have students identify the halfway point. Take all responses and allow students to defend or debate.
• Place an arrow and ask what is a reasonable value for the point at which the arrow is pointing?
• As an alternative, place a labeled point on a number line and ask students to generate possible endpoints.
• What is an example of a number that would NOT be placed on this number line? Why?
• Change one of the endpoints and ask students if the placement of a given number will change? Explain.
• Select one of the endpoints and ask students to identify numbers close to that point.

Number Paths is a routine that involves whole-class participation as students use number sense, place value, and reasoning to fill a pathway of numbers from 0-100. This routine gives students the opportunity to create and compare numbers, while reasoning where they should be placed on the "Number Path" in an effort to beat the teacher. Number Paths is a variation of a game taken from Number Sense: Simple Effective Number Sense Experiences, Grades 1-2, by Barbara Reys and Robert Reys.

Materials

• Number Paths Game Board
• Sample Game

NOTE: In first grade, students do a lot of exploration with numbers using the hundred chart. In an effort to minimize confusion, the Number Paths Game Board above has been modified to represent a linear path as opposed to a hundred chart like format.

How to Play

It is recommended that the teacher model the game and discuss number placement strategies before moving into competition mode with the students.

• Either the teacher or a student rolls place value dice or a pair of 10-sided dice and creates a number with the two digits rolled (Use more dice to create a number with more than 2 digits).
• Both the class and the teacher place the number on the path, but don’t show one another which spot on the path they have selected.
• Once the teacher and the students have placed their number on their board, they must show each other where they were placed. (This can be done by the teacher using a document camera).
• Repeat the first three steps with the other participant rolling and creating the next number.
• The next number is placed on the Number Path based on whether it is greater than or less than the original number. It is placed in a box before if it is less than the original number and placed in a box after the original number if it is greater.
• If a number is rolled/created that can’t be placed on the board, because there is no empty box to place it, then a turn is lost for that participant.
• The first player to fill in the path completely and correctly, wins!