Home |
## Fractions, Ratios, Money, Decimals and Percent
If no one knows, we teach the name to use. Teacher: There are two parts. We write the fraction for one part like this:The bottom number is how many equal pieces together make up the whole. The top number is the number of pieces that we have. (illustration 11-1-13) (S-5 square with four S-3 squares next to it formed into a square the same shape.) Teacher: How many equal parts make up this whole?Students: Four. Teacher: Does anyone know what we call each of the equal parts? Student: Fourths. If no one knows, we teach the name to use. Teacher: Here's the fraction for a fourth.What does the bottom number mean? Student: How many pieces to make the whole. Teacher: What does the top number mean? Student: That's for the one little piece. Teacher: Let's try another one. We do not worry about reducing fractions to their lowest denominators. We leave 2/4 as 2/4 for now. Proving blocks...High school students learn about axioms, theorems and proofs in geometry. An axiom is a statement that is accepted as true without proof. A theorem is a statement that can be proved. Axiom: The S-1 square has an area of one. Theorem: Find the areas of the other shapes. Keep a record of your proofs. Each proven area is a theorem that follows from the axiom. New axiom: The T-1 triangle has an area of one. When the axiom is changed, do the theorems change as well? Power Blocks provide our students a background for
understanding fractions regardless of the child's
(illustration 11-2-1) (Show a few different ways of making two and one-half square units. Put both the square and triangle paper and different size Power Block proofs along side each geoboard shape.) In Beginning Number, Lesson Eight, (page 000), we asked our students to find ways to make shapes on their geoboards with areas of twos, threes and more. We find areas on geoboards by adding or subtracting wholes and halves. Our students added and subtracted fractions and whole numbers, even though we had not reached the point in this book where fractions were taught. Chapter titles and lesson numbers are not meant to limit what or when we teach. Teacher: Please record the shapes you discover, so that you can keep images
of the shapes you areinventing, even when you run out of room on your geoboard. (illustration 11-2-2) (Student recorded geoboard designs on paper. Label each shape "2 1/2".) As our students work, we roam around the room asking individual children to prove the areas of their shapes to us. They also prove their shapes to one another. Teacher: Prove the area each shape that you think has an area of two and
one-half to yourneighbor before you record it on your paper. If your neighbor agrees with your proof, then your neighbor must sign his or her name next to your drawing. If your neighbor does not accept your proof, then raise your hand and I will help you both decide if your proof is sufficient. Students checking students means thirty teachers for every
child in the room. Finding areas for any kind of shape is a fraction lesson automatically. (illustration 11-2-4) (Student recorded geoboard designs for a variety of shapes on paper. Label each shape with its proven area.) As our students work, we walk around the room, sharing everyone's strategies for proofs. Right triangles...Teacher: Today I want you to see how many different right triangles you can find areas for. Right triangles are triangles with a right angle in them. (Geometry, Lesson Seven, page 000.) The rule for the right triangles is that the base for each new triangle you make has to be on the bottom row of nails on your geoboard. Please record the triangles you make and the areas that you find. (illustration 11-2-5) (All the right triangles are made with their bases on the bottom row of the geoboard. Give one example of a triangle made with its base n o t on the bottom row and add in the caption that this is not allowed. Demonstrate with the Power Blocks that right triangles are half of rectangles or squares. Then show on the geoboard right triangles as halves of rectangles or squares. Show a recording sheet with the areas written inside the triangles. All of the triangles on the recording sheet should have their bases on the bottom line of the recording area.) Teacher: Let's see if we can find a pattern in the areas of the triangles you
have made that mighthelp us predict the areas for triangles not yet made. For each area that you find, please write numbers for the height, the base and the area of the triangle on your chalkboard and hold it up for me to see. You already know how to find the number for the area. Find the number for the base and number for the height by counting the spaces and not the nails. (illustration 11-2-6) (Show geoboard examples of what is meant by base and height and counting spaces.) Teacher: I'll record each new set of numbers on the overhead. Please look to see if I already have the numbers for your triangle written on the overhead before you write your numbers on your board.
Will our students see the pattern? Will they see that area
is found by multiplying the base times the
(illustration 11-2-7) (Show one acute triangle and one obtuse triangle. Show that, for the acute triangle, the height is the perpendicular line from the base to the highest point. Show that, for the obtuse triangle, the height is also the perpendicular line from the base to the highest point, but we need to extend the base beyond the triangle to find that perpendicular point. The height is how tall the triangle is, straight up from the line the base is on.) |