In this section, students interpret and draw scaled picture graphs and scaled bar graphs to represent data. This builds on their grade 2 experiences with data representation and with skip-counting by 2, 5, and 10.

Students see that each picture in a picture graph and each line or increment in a bar graph can represent more than one object. They work with familiar number scales of 2, 5, and 10.

Students use the information in scaled bar graphs to solve one- and two-step “how many more?” and “how many fewer?” problems within 100. This work allows teachers to formatively assess students’ fluency with addition and subtraction within 100, a grade 2 expectation..

In this section, students make sense of multiplication in terms of equal groups of objects. They use discrete drawings and tape diagrams that show equal groups to represent multiplication. Students then relate these representations to expressions, such as , interpreting them to mean “3 groups of 2.” 

Note that expressions of the form could be interpreted to mean groups of or groups of . Because we tend to say “___ groups of ___” when referring to equal groups, however, in these materials we write multiplication expressions in that order:

It is not necessary for students to use this convention as long as they can explain what each number in their expression represents.

At the end of the section, students write equations to represent multiplication situations and find unknown products or factors. In reasoning about the latter, students begin to make sense of the relationship between multiplication and division, without formally using the language of division..

In this section, students relate the idea of equal groups to the structure of an array, a representation introduced in grade 2.

Students see that the rows and columns of an array represent equal groups. The number of rows (or columns), the number of items in each row (or column), and the total number of objects in an array can therefore be represented with a multiplication equation. The equations may involve an unknown value, be it one of the factors or the product. As students reason about arrays, they also notice that multiplication is commutative.

2 rows with 5 in each row is 10 total.

 or 

5 columns with 2 in each column is 10 total.

or