Algebra Puzzlers
By Michael Naylor

When children work at solving how many counters are under the cards they are engaging in algebraic reasoning.

Algebra is not a subject just taught in the middle grades and beyond – algebraic reasoning is part of the math standards at every grade level. In the lower grades, we teach children to recognize patterns and find missing parts of quantities; older children build these ideas with increasing sophistication and learn to manage several operations and represent algebraic relationships in many ways. Here are some great activities to engage your students in algebraic reasoning.

What's missing? (Grades K-1)
Place several counters on the overhead and cover some with an index card. Tell your students the total number of counters and ask them to figure out how many are under the card. For example, you may have five counters showing and two underneath a card. Announce there are seven and elicit some answers and explanations. Some students will count up to the missing number (5+2=7), some will subtract (7-5=2).

After several examples, give your students a set of several problems like those shown here. Tailor the numbers to match the level of your students.

The number underneath tells the total. Their task is to figure out how many counters are hidden by the rectangle.

Sticky hidden numbers (Grades 1-2)
Write several addition and subtraction problems on the board in a vertical format and cover some of the numbers with sticky notes. Have students decide what the covered numbers are. Be sure to have students explain their reasoning.

Balances (Grades 3-8)
Building the concept of equality is at the core of understanding equations. A nice way to approach this is with a balance model. Make simple sketches of balances on the board. In each of the two pans, place shapes and numbers.

Simple visual models help move a child's thinking from the theoretical to the practical.

Tell your students that the shapes represent unknown "weights" and each shape weighs the same within each puzzle. Ask students to figure out what the weights are. Start with simple problems using just one shape and build to more complicated puzzles with multiple copies of shapes and numbers on both sides of the balance.

Be sure to have students share their solution methods – these methods are the same students will use when solving equations. You are also using letters instead of shapes, a good introduction to the use of letters for variables. (See above).

Toothpick patterns (Grades 3-8)
Simple manipulatives make a powerful learning tool where kids can build and analyze number patterns. Provide a copy of a growing pattern of toothpicks. Ask your students to extend the pattern and record their results on the T-chart (shown right). Have them describe the number patterns they see with words – an important step in understanding the pattern. You may ask students to determine how many toothpicks are needed to make four or five sections, and how they came to that determination. As an extension, have your class try to determine how many toothpicks would be required to build the pattern at, say, the 100th step.

Below is a more challenging pattern Nearly any manipulative can be used to create these patterns.

Download many more of these patterns from the Bonus Curriculum section of our website here. PDF 101KB

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.