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Alignments to Content Standards: 3.OA.A.3

1. Juanita spent \$9 on each of her 6 grandchildren at the fair. How much money did she spend? 2. Nita bought some games for her grandchildren for \$8 each. If she spent a total of \$48, how many games did Nita buy? 3. Helen spent an equal amount of money on each of her 7 grandchildren at the fair. If she spent a total of \$42, how much did each grandchild get?

IM Commentary

The first of these is a multiplication problem involving equal-sized groups. The next two reflect the two related division problems, namely, "How many groups?" and "How many in each group?"

Sometimes the second type of problem is referred to as a measurement division or repeated subtraction problem. The third type of problem is sometimes called a partitive division or sharing problem. It asks how large is each share when a whole is divided equally into a specified number of pieces. It specifies the size of each share and asks how many of that size are in the whole. The language used in the solution reflects the language in the common core, which also refers to them "Number of Groups Unknown" or "Group Size Unknown," respectively.

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, only one practice connection will be discussed in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.

This particular task helps illustrate Mathematical Practice Standard 4, Model with mathematics. Third graders apply the mathematics they know to solve problems arising in everyday life. They identify the mathematical elements of each situation and determine the solution pathway that is best for them to follow.  For these problems, students may use drawings or tape diagrams to depict each situation. They may also write a multiplication equation to represent how many altogether in Part (a) or write multiplication or division equations to represent how may groups in Part (B) and how many in each group in Part (c). Depending upon their solution pathways, students will have opportunities to explain their solutions to each other and critique each other’s reasoning (Mathematical Practice Standard 3). For example, the teacher could pair students who have used different representations to represent each situation (one with an equation and one with a tape diagram) and have them describe how both representations depict (or do not depict) the same situation. Additionally, students can comment on why ? × 8 = 48 and 48 ÷ 8 = ? are equivalent expressions and can both provide the solution to the question.

Solutions

Solution: Tape diagram

This task needs a tape diagram solution; one is under development.

Solution: Writing multiplication equations for division problems

1. Sandra spent 6 groups of \$9, which is$6 \times 9 = 54$dollars all together. 2. Since the number of games represent the number of groups, but we don’t know how many games she bought, this is a "How many groups?" division problem. We can represent it as $$? \times 8 = 48$$ or $$48 \div 8 = ?$$ So Nita must have bought 6 games. 3. Here we know how many grandchildren there are (so we know the number of groups), but we don’t know how much money each one gets (the number of dollars in each group). So this is a "How many in each group?" division problem. We can represent it as $$7 \times ? = 42$$ or $$42 \div 7 = ?$$ So Helen must have given each grandchild \$6.