The problems on this page involve comparison multiplication (How many times larger... how many times younger something is as compared to something else). This concept is abstract and can be difficult for third graders. Ms. Frey wanted to introduce it to her entire second grade class.

We introduced questions about height: Juan is 3 feet tall. His brother is two times taller. How tall is his brother.

Often, students do not use multiplicative reasoning to solve this problem. They simply add 3 + 2. We used unifix cubes to show Juan's height: 3 cubes. Then we discussed what two times means: Juan's height, two times. So, the brother's height is two sets of 3 feet. We showed the brother's height with 6 cubes. We placed Juan's cubes next to the brother's cubes. We made a point of showing that two of Juan's height equals the brother's height. Students then drew pictures. Finally, students wrote a number sentence: some used addition; some used multiplication; some used division.

In the example above, students are confronted with what would typically be thought of as a division problem. It is important to expose students to situations that typically use the inverse operation. Students used the same strategies to solve this problem as they did other problems typically thought of as multiplication. They made a Unifix tower to show the mother's age. When they tried to create three times more for Billy's age, they were confronted with the contradiction that Billy would be older than his mother. They eventually figured out the answer using the manipulatives and drawings.

To follow up on the activity, students created a pictograph reflecting the ages of Billy's family members. The pictograph clearly shows that Lucille is two times as old as Billy, and that Bert is 9 times as old as Billy.

Students worked in groups of three to solve problems that were quite challenging for second grade. By insisting that students use the unifix cubes each time and to draw a picture, students were successful with little teacher prompting. The octopus problem was given last. Students had to discuss what information they knew already before answering. These student examples show two approaches to solving the problem. The group on the left used a more symbolic picture to represent how many boots were needed. They then related their pictorial model to multiplication. The group on the right drew a more representational picture to solve the problem.

By using manipulatives, pictures and cooperative groups, students were very successful exploring an above grade level objective.