Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Session 10, Grades 3-5, Part D:
More Problems That Illustrate Algebraic Thinking (25 minutes)

At the 3-5 grade level, there are many problems that prepare students for their later work in algebra. Experiences with these problems can also build bridges between arithmetic and algebra, while increasing the students' chances for success in future mathematics. Note 7

In this section we'll analyze problems at the 3-5 level for their algebraic content. For each problem, find a mathematical solution, then answer questions (a)-(f) listed below.

 a. What algebraic content is in the problems? b. What content does it prepare students for later? c. How does this content relate to the mathematical ideas in this course? d. How would your students approach this problem? e. What are other questions that might extend students' thinking about the problem? f. Does your current program in mathematics at your school include problems of this type?

Problem D1

There are red (R) and black (B) marbles in a box in the drawer. Take out some marbles so that there are 3 red marbles for each black marble left in the box.

 a. What is the fewest marbles you must take from the box? b. Draw a picture of the marbles left. Mark B or R to show the color of each marble. c. Tell how you decided how many to take out.

Problem D2

 a. Which block, cube, cylinder, or sphere will balance Scale C? b. List or draw the steps you followed to identify the block.

 c. Which block, cube, cylinder, or sphere will balance Scale F? d. List or draw the steps you followed to identify the block.

 Problem D3 a. Find the weight of each block. cylinder = ________ pounds sphere = ________ pounds cube = ________ pounds b. List or draw the steps you followed to find the weights of the blocks. c. Find the weight of each block. cylinder = ________ pounds sphere = ________ pounds cube = ________ pounds d. List or draw the steps you followed to find the weights of the blocks.

Problem D4

For each Start number, give the End number.

For example, when the start number is 5, here's how you get to 21, the End number.
(Start) 5 + 2 = 7; 7 x 3 = 21 (End)

 a. Start: 0; End: ________ b. Start: 1; End: ________ c. Start: 3; End: ________

When the End number is 24, the Start number is 6.
For each End number, give the Start number.

For example, when the End number is 24, here's how you get to 6, the Start number.
(End) 24 / 3 = 8; 8 - 2 = 6 (Start)

 d. End: 12; Start: ________ e. End: 21; Start: ________ f. End: 36; Start: ________

Problem D5

In the following table, the same shapes are the same numbers. The numbers in the circles are sums, reached by adding across and adding down.

 a. What number is the square? b. What number is the triangle? c. Explain how you found the numbers.

 Problems adapted from Groundworks: Algebraic Thinking, Grade 5, by Carole Greenes and Carol Findell (New York: Creative Publications, Wright Group/McGraw-Hill 1998). The above material may not be reproduced without the written permission of Wright Group/McGraw-Hill.
 Session 10, Grades 3-5: Index | Notes | Solutions | Video