Lesson 1: Mondrian Inspired Art

Pre-Lesson exploration time 

In partners, students take turns drawing small rectangles on graph paper and determining the area for the rectangle using one or more strategies.  While pairs explore, teacher circulates taking note of which students seem most comfortable using which strategies. (Counting strategies, skip counting/repeated addition strategies with attention to rows and/or columns, or multiplication strategies that attend simultaneously to rows and columns).  

Recording Sheet

Artists:  ___________________________   & ____________________________

Dimensions of Canvas:  _________________ x _________________     

(note:  x is said “by”) 

Predicted Area of Canvas: __________________

Actual Area of Canvas: _______________

Design Features: 

Color with the greatest area is: __________________

Color with the smallest area is:  _________________

_______________ has an area of exactly _______________ square inches

               Color                               Number between 10-100

Combined Area for White: _______________

Combined Area for Black: _______________

Combined Area for Red: _______________

Combined Area for Blue: _______________

Combined Area for Yellow: _______________

Sum Total Area for all Colors: _____________     

Perimeter of Canvas: _________________

Predicted Length of Tape to be Used: _____________

Actual Total Length of Tape Used:  ________________

Launch

Whole class discussion format:  Our job today is to begin our work as copy artists, inspired by the work of Piet Mondrian, a Dutch painter who lived from 1877 - 1944.  He is most famous for his paintings that use just 5 or 6 colors and just rectangles and lines. Geometry and measurement are central to his art and we’re going to use his work to inspire our own art and a lot of mathematical thinking.  In particular, we’ll be using multiplication and addition to help us learn about measuring line segments and area.  

Show students a sampling of his work:  (These open-source slide shows are an easy way). 

http://www.slideshare.net/saracerverab/piet-mondrian-6437685?next_slideshow=1 

http://www.slideshare.net/duniwayart/mondrianppt 

Your first job is the same that any painter needs to do.  You need to figure out how big you want your canvas to be.  It needs to be rectangular and at least 15 inches long for both the length and the width.  You’ll need to decide the orientation for your project if it isn’t square. (e.g., is the long side the top to bottom length or the side-to-side length). MATH NOTE:  Squares are rectangles*, and throughout this project, any time a rectangle is called for, students may choose a square in their design.  

Once you’ve determined your dimensions, figure out the area of your canvas in Square Inches.  

That is to say, how many 1” x 1” squares fill up your canvas.  That is the area. We’ll reconvene to discuss results after you’ve completed this part.  

*A rectangle is a quadrilateral (more specifically, a parallelogram) with four 90° angles.  By definition, opposite sides are equal. A square is a special rectangle in which all sides are equal.  

Maker & Math

In pairs, students determine their canvas, and trim it to size, then compute their area. Students then predict what they think the area (in square inches) will be, and record their dimensions and predictions on their Recording Sheet.  Then they compute the actual area measurement. Teacher circulates, observing students’ strategies. Most students will count squares. Some will use grouping strategies such as skip counting by 2s or 5s, grouping into 10s, perhaps using rows and columns.  Some will subdivide into smaller regions, then count those regions. Very few will use multiplication to solve this problem. Note, while it takes some time, it is well worth it for 3rd graders to count large numbers, and being accurate may be a struggle.  

Synthesis

Whole class discussion regarding strategies for determining their area. 

Questions regarding whether students were surprised by the difference between their predictions and actual area measurements.  (Note: generally, students with little experience with area will predict an area measure that is close to the sum of the side lengths.  Most will be very surprised that the areas are all at least 200 in2.)  

Teacher summarizes with a working definition of area:  the number of square units needed to fill up a space.  

Launch

The artists’ plan.  Have students set the personal criteria for their projects (each pair will be unique, but must be pre-determined). 

1. Pre-determine which color will have the greatest area

2. Pre-determine which color will have the smallest area

3. Pick an exact area for one by selecting a number between 10 and 100 

Extra Design Challenge:  Some line segments that don’t go from edge to edge on the paper.  

Pairs will return to their canvases to plan out their design.  In light pencil, draw line segments on graph paper lines (e.g., no partial squares or diagonal lines).  They will first design their layout and color arrangements. (Checking to ensure they’ve fulfilled their 3 design goals). Then they will compute the area for each rectangle in their art. They will then find the combined area for each color, and then the total area for their project. 

Maker & Math

Pairs work to design their projects using their personal criteria in light pencil, noting the colors for each rectangle (Y, W, R, Blu, BLa might be used as initials).  

Pairs then compute the area for each rectangle (by counting, skip counting, repeated addition, and/or multiplication).  They are encouraged to find shortcuts and use repeated reasoning from previous rectangles to solve new ones. (e.g., that one was 5 tall and 3 wide.  It was fifteen. This is 4 wide, so it will be 5 more. 15+5=20.) Students are encouraged to scout computation strategies from classmates. Teacher should circulate and take note of strategies students are using.  NOTE: Time for this part of the task will vary tremendously depending on students’ previous experience with basic multiplication using an area or array model. Adjust according to the needs of your students for this section.  

Once they find the area for each rectangle, then they will find the combined area for each color.  (Check marks to make sure each rectangle is counted once and only once may be useful. Teachers could ask how they are keeping track as they circulate, though shouldn’t offer suggestions.) They should make revisions as needed in order to fulfill their 3 personal criteria.  

Once they have their 3 criteria satisfied and they have totals for each color, they should find the sum total for all colors and compare that to their canvas’ area.  (They should be equal.)  

Teacher should note addition strategies while students work.  

Synthesis

Once all students have computed the area for each rectangle, and have satisfied their 3 personal criteria, you can hold the synthesis discussion, even if not all pairs have finished all final areas.  

Big Ideas for discussion: 

• Strategies for finding areas, visualizing rows and columns, recognizing dimensions, naming rectangles by dimensions (e.g., a 3”x5” rectangle). 

• Connections between skip counting, repeated addition, and multiplication, 

• utility of knowing multiplication facts (note:  this is about efficiency, problems can effectively be solved without memorizing facts, but it takes longer or more steps and there may be greater chance of error)

• Intuitive uses of Commutative, Associative, and Distributive property

• Area measurements are squares or square units.  They’re different from linear measurements, though we can use the linear measurements to help us determine area. 

• Area measurements can be added

• Area is conserved.  (If you have a big space and cut it up into a bunch of smaller regions, the total is still the same.)