A Lesson Plan for ADDverse
By M.W. Penn and Malgorzota Sliczniak-Swirszcz
Gumboot Books, 2009
There are two poems and two themes in ADDverse: Farmer Yercle’s Circles on the geometry of a circle and circle based solids; and Peter Pattern on pattern, the precursor to iterative functions.
Farmer Yercle’s Circles
K.G.1. Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.
K.G.2. Correctly name shapes regardless of their orientations or overall size.
1.G.2. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.1
NCTM Standards (general goals): Pre-K–2 Expectations
In prekindergarten through grade 2 all students should: recognize, name, build, draw, compare, and sort two- and three-dimensional shapes; describe attributes and parts of two- and three-dimensional shapes; investigate and predict the results of putting together and taking apart two- and three-dimensional shapes.
In plane geometry, a circle is a set of all points equidistant from a given point, which is its center. The distance from the center to any point on the circle is the radius of the circle.
In three dimensions: When two circles of the same radius lying in parallel planes are connected by the curved surface formed by the straight lines joining corresponding points on the circles, the enclosed volume is a cylinder. A sphere is the set of all points equidistant from a given point, which is its center. (The definition of a sphere does not restrict the points to a plane, so a sphere is 3 dimensional.)
All three are referenced in Farmer Yercle’s Circles and, after reading the story, you should find each in the text and discuss them:
Circles abound: ‘Hooper-Hooper Hop-along races round a circular race track’ is but one example; Hooper, himself, is quite round of body. The main point is that a shape is either a circle – adhering to the definition – or it is not. You can call any other shape more nearly circular. Ovals are rounded but are not circles.
Cylinders are found in pictures and in the text: Farmer Yercle uses rollers to set his animals’ curly hair; rollers are cylinders. The chicken coops are made of ‘round shaped pegs’, which are cylinders.
The cattle barn is a sphere. So are the balloon shaped chickens.
The Theme of this lesson is exploring the two dimensional shape circle for multi grade levels K-2; it can be used at grade level 3 if the advanced topics are used.
New words: radius; perimeter; point; advanced – intersect; more advanced – tangent.
In lower grades, radius would be simply called the string; perimeter, the circle. Intersect and tangent should be replaced by ‘points they share’ or even ‘child on both circles’.
You will need:
A copy of ADDverse 1. A length of string to use as the ‘radius’ of the largest circle you can easily make on the open space of the floor of your classroom. Informal protractors for each child made from string and pencils and paper to draw circles and (more advanced) cut out disks.
Anticipatory Set and Objective: Tell children you are going to read a poem about a very special shape. Ask them to remember what that shape is.
Input: Read Farmer Yerkle’s Circles. Share the pictures and talk about the shapes you find on each page.
Modeling: Select one child to be the ‘center of the circle’. Have him/her hold one end of the ‘radius’ at a point on the floor in the center of your open space. This point is the center of the circle and cannot move.
Using the string as a guide, place each of the other children on ‘points’ around the circle. Space the children as evenly as possible without being exact about spacing between children but always carefully measuring the distance from the ‘center of the circle’ to their ‘point’ on the circle with the string. Have them describe the shape they form.
Ask a child on the perimeter to move one step but still remain on the circle. Which way can they move? If they move away from the center, will they still be part of the circle? If they move toward the center, will they still be part of the circle?
Now, reduce the length of the radius (string) and remake the circle. The new circle will be smaller. Ask children to move one step but still remain on the circle.
Guided practice: Using your informal protractors, have each child draw a circle of any size they choose that fits on a piece of construction paper. Compare the sizes of the circles they drew. Ask what happens when the string gets longer? Shorter? Children should reach the conclusion that larger radii (longer strings) produce larger circles.
More advanced modeling: (This will take more floor space, so make the initial circle a bit smaller than the space you have available. It is also a good activity for a gym floor.):
Take every second child off the perimeter of the first circle to form a second circle. Choose the center for another circle at a point that will form an intersecting circle with the first. (The center of this second circle can be inside or outside the first circle.) Using the method above, place children around the ‘perimeter’ of the second circle. Examine the two ‘points of intersection’ (or the points where the circles cross). How many children can stand on both circles at the same time? (Only 2, one at each point of intersection.)
You can repeat this using different radii and different placements of the centers: a small circle within a larger circle when they have only one point (one child) in common (tangential circles); a circle outside the first circle that has only one point (one child) in common with the first circle (tangential circles); etc.
Advanced Guided practice: Have each child draw and cut out two different size circles (disks) from a piece of construction paper. Compare the sizes of the circles they drew. They should reach the conclusion that larger radii produce larger circles. Now have them place the smaller circle inside the larger circle and make them tangential; intersecting.
Check for Understanding:
With one circle
What happens to the circle when the center of the original circle moves one step to the right? One step forward? Etc. What happens when you keep the same center but make the radius (string) longer? Shorter?
When you are working with two circles:
What happens if the center of the second circle in inside the ‘perimeter’ of the first circle and the ‘radius’ is longer? What happens if the center of the second circle in inside the ‘perimeter’ of the first circle and the radius is shorter? Can you make two different circles using the same center? What would you have to change? Can you make two different circles using the same radius but a different center?
What happens if the center of the second circle is outside the ‘perimeter’ of the first circle? When will the two circles no longer ‘intersect’? When will they no longer even ‘touch’? How far away do the circles have to be before they no longer touch? (over two radii)
The CCSS state that good math students look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts.
NCTM Standards (general goals): Pre-K–2 Expectations
In prekindergarten through grade 2 all students should: recognize, describe, and extend patterns such as sequences of sounds and shapes or simple numeric patterns and translate from one representation to another; analyze how both repeating and growing patterns are generated.
Teacher notes: A knowledge of patterns supports the development of skip-counting and algebraic thinking. In Math Matters: Understanding the Math You Teach, Chapin and Johnson state: When students explore and generalize relationships among numbers they are developing informal understanding of one of the most important topics of high school and college algebra – functions. (Chapin, Suzanne H. and Johnson, Art. 2000. Sausalito, California: Math Solutions Publications.)
Pattern is inherent at every level of Peter Pattern: the verse itself can be examined for the pattern of the rhyme and meter; the patterns described by the verse can be explored in the physical world; the illustrations can be studied, seeking pattern ‘everywhere’, rather like a ‘hidden picture’ game. ‘Do you see it?’ ‘Can you hear it?’
Lesson Plan for multi grade levels K-2
The theme of this lesson is finding patterns and making patterns.
You will need:
As many copies of ADDverse 1 as possible to have children explore the patterns in groups, or an overhead projector to have all the children view the patterns on each page; simple small shapes cut from colored paper. (You can also use different objects such as paper clips, pennies, erasers, etc.) Two small mirrors.
New words: meter; rhyme; sequence; repeat.
Anticipatory set: Read the poem.
Input and Modeling: If you have multiple copies of the book, group the children so that each group shares a copy or project the page on a screen. Read the first two pages again and ask each group to find a pattern in the illustration on that page. Then ask each group to find an example of that pattern in the classroom. EG, the threads of warp and woof lines of woven fabrics or rugs with the intricate pattern of over under weaving; braided hair; or braided string.
Read the next two pages. Ask questions about the pattern of the poems meter. Can they count the beats in each line on these pages? (7) Ask each group to compose a short rhythm pattern using beats and to repeat the rhythm three times. Choose one member of each group to present the rhythm they composed.
Read the verse ‘Sequenced numbers growing fast….’ What is the pattern between the numbers the clown juggles? Are there other patterns on the page? (In his clothing or in his shoes?) Which group can find the most patterns on this page? Ask the same questions for the next page with the numbers above the ducks. Is there also a pattern to the line of ducks?
Follow the same method for the ‘set of numbers shrinking’. What is the next number in the sequence above the gloves? Is there a pattern to the spacing of these numbers and the gloves? (A pair of gloves between each number.) Ask the same questions for the sequence below and those on the following page.
On the last pages, ask about the pattern of ‘the picture of Peter reading a book which has a picture of Peter reading a book which has a picture of Peter reading a book ….’ Use two mirrors to demonstrate this infinite diminishing pattern.
Guided Practice: Distribute a cup of different shapes to each group; be sure each cup contains several of each shape. Ask each group to form a repeating pattern using some of the shapes. EG, 3 circles, 2 squares, a triangle, then 3 circles…. The pattern they make must be repeated at least twice.
Checking Understanding: Next, have groups switch tables and continue the pattern begun by the group they replaced.
Independent Practice: Group the children in pairs. One child is asked to make a ‘growing’ number pattern. The second child should discover the rule and continue the pattern with the next number.