Standards-based assessment and Instruction

# Math 6-8

## Making Squares

A piece of wire 63 inches long is cut into 2 parts. The 2 parts are then bent to form 2 different-size squares. The difference between the measures of the perimeters of the 2 squares is 5 inches. What is the sum of the areas of the 2 squares?

#### More Accessible Version:

A piece of wire 64 inches long is cut into 2 parts. The 2 parts are then bent to form 2 equal-size
squares. What is the sum of the areas of the 2 squares?

#### Making Equilateral Triangles

A piece of wire 30 inches long is cut into 2 parts. The 2 parts are then bent to form 2 different-size equilateral triangles. The difference between the measures of the perimeters of the 2 equilateral triangles is 6 inches. What is the sum of the areas of the 2 equilateral triangles?

### NCTM Content Standards and Evidence

#### Geometry Standard for Grades 6–8

Instructional programs from Pre Kindergarten through grade 12 should enable students to ...

• Analyze characteristics and properties of two- and three-dimensional geometric shapes and
develop mathematical arguments about geometric relationships.
• NCTM Evidence: Understand relationships among the angles, side lengths, perimeters, areas and volumes of similar objects.
• Exemplars Task Specific Evidence: This task asks students to find the lengths of squares knowing their perimeters and then finding their areas.

#### Number and Operations for Grades 6–8

Instructional programs from Pre Kindergarten through grade 12 should enable students to ...

• Compute fluently and make reasonable estimates.
• NCTM Evidence: Work flexibly with fractions, decimals and percents to solve problems.
• Exemplars Task Specific Evidence: This task requires students to work with parts of an inch using decimals or fractions.

### Time/Context/Qualifiers/Tip(s) From Piloting Teacher

This is a short-length task and can be completed in one class period. Students should be familiar with finding area and perimeter of rectangles and be able to work with decimals. Calculators should be made available to students. Squaring the decimals and then finding the sum is too cumbersome.

### Common Strategies Used to Solve This Task

Many students used guess, check and refine to find the perimeters of the two squares. They then found the length of one side, squared the sides to find the area and added the two areas.

### Possible Solutions

#### Original Version:

The smaller square measures 7.25 inches by 7.25 inches and has an area of 52.5625 square inches. The larger square measures 8.5 inches by 8.5 inches and has an area of 72.25 square inches. The total area of the two squares is 124.8125 square inches.

#### More Accessible Version:

Both squares measure 8 inches by 8 inches and each has an area of 64 square inches. The total area of both squares is 128 square inches.

#### More Challenging Version:

The smaller equilateral triangle measures 4 inches on each side and has an area of 4√3 or approximately 6.92822032 square inches. The larger equilateral triangle measures 6 inches on each side and has an area of 9√3 or approximately 15.588457 square inches. The total area of both equilateral triangles is 13√3 or approximately 22.51666 square inches.

General Notes: Be sure that the measurements in the student work are labeled correctly.

Click on a level for student example.
Novice The Novice will fail to understand all the parameters of the problem. Although they may appear to draw two squares, there will not be enough evidence to show a successful strategy. No reasoning or justification for reasoning will be present. The diagrams will not be labeled adequately. This Novice student seems to be drawing pictures of ways to coil wire, but there is no evidence that the parameters of the problem are understood except that the two lengths, 13 inches and 50 inches, do add to 63 inches. There is no attempt to find the lengths of each side of the squares or in finding the areas of the squares. No appropriate representation is made nor any connections or observations.
Apprentice The Apprentice may solve only part of the task correctly, show understanding of area and perimeter but will make a mathematical error, or solve all parts of the task but fail to make a connection, observation or verify the solution. There may be a flawed representation. This Apprentice uses a flawed strategy for the first part of the task. S/he divides the wire in half and then adds the five inches to one length of wire, making the original total length more than 63 inches. Because the communication is quite clear, her/his flawed strategy is easy to follow. The strategy and communication of the second half of the task is correct and quite clear. The solution to the second half is not correct because this student is using the incorrect square lengths from the first part of the task. This student uses some correct mathematical reasoning in using the area formula, finding the sum of the two areas, and correctly labeling the work. There is a weak attempt at constructing a representation of the two labeled squares, although it really is only a drawing of two squares. No connections or observations are made.
Practitioner The Practitioner will have achieved a correct solution. There will be an understanding of all parts of the task. An understanding of area and perimeter will be demonstrated. There will be some use of mathematical representation. A connection or observation will be made. This Practitioner’s well-communicated solution starts with making a chart and using a guess check and refine strategy to find the lengths of the two wires. This student calls on their previous knowledge of finding the area of squares to calculate the areas and then find their sum. S/he uses formal math language throughout the presentation such as “multiple,” “factor,” “dimensions,” “formula” and appropriate use of exponents in the area formula and the label for area. This student makes observations along the way and also connects this task to similar tasks and compares and contrasts the different tasks.
Expert The Expert will find a correct solution. The solution will show a deep understanding of the problem. They may use formulas and/or algebra to generalize the solution. There will be connections or observations made. This Expert uses their knowledge of algebra to find the length of the two wires, quite an efficient strategy. Their solution is well documented throughout the presentation. Precise mathematical language and symbols are used such as “equation,” “multiple,” “constant,” “formula,” “algebra notation” and “exponents.” There are observations and conjectures about the lengths of wire being multiples of four and connections between this task and other tasks done using some of the same mathematics. The representation, a graph, of perimeters of squares with various length sides helps illustrate the student’s discussion on multiples of four.

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