In this lesson, students add positive integers by counting up and negative integers by counting down (using curved arrows on the number line). Students play the Integer Game to combine integers, justifying that an integer plus its opposite add to zero. Students know the opposite of a number is called the additive inverse because the sum of the two numbers is zero.

In this lesson, students model integer addition on the number line by using horizontal arrows; e.g., an arrow for is a horizontal arrow of length pointing in the negative direction. Students recognize that the length of an arrow on the number line is the absolute value of the integer. Students add arrows (realizing that adding arrows is the same as combining numbers in the Integer Game). Given several arrows, students indicate the number that the arrows represent (the sum).

With this activity, students assemble triangular puzzle pieces by matching the problems and answers on their sides. When the puzzle is complete, the pieces will form a large hexagon.

In this lesson students use a Hot Air Balloon simulation to model integer subtraction. They then move to modeling subtraction on a number line. They use patterns in their work and their answers to write a rule for subtracting integers.

Students practice with pictorial representations and make connections to the abstract concepts and procedural skills of adding and subtracting integers.

In this lesson, students justify the distance formula for rational numbers on a number line: If p and q are rational numbers on a number line, then the distance between p and q is |p – q|. Students know the definition of subtraction in terms of addition (i.e., a – b = c means that b + c = a) and use the definition of subtraction to justify the distance formula. Students solve word problems involving changes in distance or temperature.

In this lesson, students justify the rule for subtraction: Subtracting a number is the same as adding its opposite. Students relate the rule for subtraction to the Integer Game: removing (subtracting) a positive card changes the score in the same way as adding a corresponding negative card. Removing (subtracting) a negative card makes the same change as adding the corresponding positive card. Students justify the rule for subtraction for all rational numbers from the inverse relationship between addition and subtraction; i.e., subtracting a number and adding it back gets you back to where you started: (m - n) + n = m.

Recognize that area represents the product of two numbers and is additive.
Represent a multiplication problem as the area of a rectangle, proportionally or using generic area.
Develop and justify a strategy to determine the product of two multi-digit numbers by representing the product as an area or the sum of areas.

Sample Learning Goals
Calculate the difference between two integers using subtraction
Represent subtraction of integers as the distance between integers
Reason about subtraction of integers in terms of number-locations and directed distances
Show that the distance between two integers on the number line is the absolute value of their difference
Recognize and generate equivalence classes of integer differences (e.g., 5 – 2 = 6 – 3)

Sample Learning Goals
Use positive and negative numbers to represent quantities in multiple contexts.
Explain the meaning of positive values, negative values, and zero, in multiple contexts.
Describe the location of a point on a number line with respect to another number.
Describe the location of a point on a number line with respect to its opposite
Define the absolute value of a number as its distance from zero.
Interpret inequality statements as statements about the relative position of two integers on a number line diagram.

Sample Learning Goals
Represent addition and subtraction of integers on a horizontal number line
Reason about addition and subtraction of integers in terms of number-locations
Use logical necessity to reason that addition has the opposite effect as subtraction or that adding (or subtracting) a negative integer has the opposite effect as adding (or subtracting) a positive integer.
Recognize and generate equivalence classes of integer sums and differences
Show that a number and its additive inverse (opposite) have a sum of 0
Using a net worth context, describe situations that have a positive sum or difference, negative sum or difference, and zero sum or difference
View adding a negative as equivalent to subtracting a positive, and explain why this relationship makes sense in contexts such as net worth.
View subtracting a negative as equivalent to adding a positive, and explain why this relationship makes sense in contexts such as net worth.
Apply a number line model for addition and subtraction to new contexts.

In this lesson, students use number lines to solve addition and subtraction problems involving positive and negative fractions and decimals. They then verify that the same rules they found for integers apply to fractions and decimals as well. Finally, they solve some real-world problems.

For this interactive, students determine which number in the number line best represents the answer.

For this interactive, students determine which number in the number line best represents the answer.

The purpose of this task is to help solidify students' understanding of signed numbers as points on a number line and to understand the geometric interpretation of adding and subtracting signed numbers.

For this interactive, students compete against the computer by adding rational numbers to race to the finish line.

Develop and justify a method to use the area model to determine the product of a monomial and a binomial or the product of two binomials.
Factor an expression, including expressions containing a variable.
Recognize that area represents the product of two numbers and is additive.
Represent a multiplication problem as the area of a rectangle, proportionally or using generic area.
Develop and justify a strategy to determine the product of two multi-digit numbers by representing the product as an area or the sum of areas.

In this hands-on activity rolling a ball down an incline and having it collide into a cup the concepts of mechanical energy, work and power, momentum, and friction are all demonstrated. During the activity, students take measurements and use equations that describe these energy of motion concepts to calculate unknown variables, and review the relationships between these concepts.

In this activity, students review the definition of rational numbers as ratios and as terminating and repeating decimals. They also examine the reciprocal property of rational numbers and operations with rational numbers.