(1) 
What is another name for a solid object? (threedimensional object)

(2) 
What happens to a solid object when small shapes are used to make it? (the new solid object is bigger than any one shape, but still rather small)

(3) 
What happens to a solid object when larger shapes are used to make it? (the new solid object is larger)

(4) 
How many pieces of paper are needed to make a cone? (1)

(5) 
How many triangles does it take to make a square pyramid? [4 triangles and one square (two triangles taped together)]

(1) 
NOTE TO ALL PROJECTS IN THESE LEARNING TIPS: These will be fun activities for most children, but can be frustrating for some because of the smallmotor coordination required. Help your child when assistance is really needed, but let him/her do as much as possible. These activities demonstrate a common feature of mathematical activities: neatness counts! It’s hard to create a useful structure if the pieces were not cut carefully. Small pieces of tape will work much better than big long strips, as satisfying as it is to children to pull off long pieces from the roll! Allow your child to figure out for him/herself how to organize the completion of a project. For the cube, is it best to fasten a bottom (or top) piece in place on one side before the final tape is applied to the 4 pieces that make up the sides of the open box? Or is it easier to complete the 4sided box and then attach the top and bottom pieces? Once the cube is complete, is there a “top” or “bottom” or are all sides the same? Ask similar questions as the other structures are created. Challenge your child to draw new sizes of squares and triangles to make different sizes and shapes of these common threedimensional structures. What happens to your creation if the sides are different dimensions?
Use the activity sheets provided to create these projects; if possible, copy the sheets onto cardweight paper. (Cardboard will be too hard to cut so use something lighter than that.) You will need scissors and tape for these activities. Explain to your child that you are going to use these simple shapes to create new items.
For this first exercise, you will be using small squares and rectangles to create solid objects. Using scissors that are appropriate for your child’s age/skill level, carefully cut out several of the squares and rectangles. Because you want your child to experiment and learn for him/herself what solid objects can be created, allow your child to cut both squares and rectangles. If more are needed later of a particular size your child can come back and cut more. It’s important that the shapes are cut out carefully on the lines as the activity will be very difficult to complete if the shapes are cut offline. Help your child to see why we call these shapes “twodimensional” as we are only considering their width and length. (It’s true that they have some height: the thickness of the paper, but we are not using that property at this time.) Now that your child has some of the cutout squares in front of him/her, remind your child that now he/she will get to make some threedimensional items from the shapes that have been cut out. If your child is not familiar with the terms “twodimensional” and “threedimensional” explain that a threedimensional item still has the width property and a length property (and we can measure those if we wanted to) that define a twodimensional object, but the new objects that will be created will also have a “height” property that’s also measurable.
Allow your child to experiment with both the rectangle shapes and the square shapes and to make solid objects of them. Ask your child to predict whether the item he or she will make could be used as a container (yes). If your child does not know the term, explain that if we use squares to make a threedimensional (or solid) item that it will be called a “cube.” Ask your child to estimate how many square pieces it will take to make a cube. (It will take 6 but you don’t want to supply this information; you want the learner to discover as much as possible for him/herself.) Discuss with your child how best to tape the edges of the squares together. Decide as he/she is working whether or not to close the structure by taping on the final square or to “hinge” it so that it can open like an attached lid.
If your child did not use 4 of the rectangles with 2 squares to create a solid object, encourage him to do that now. These 3dimensional objects are usually just called “boxes” as the term “cube” is generally reserved for items that are the same size on all sides.
Question: What can you store in your cubes and boxes? Which items need a cube/box with a lid that can be opened and which can go into a closed box? (Small food items, dirt so the box can be used as a starterplanter for a seed, small toys; answers about the usefulness of a closed/open box depend mostly on how practical the answer is. Perhaps your child can try some of his/her ideas to see which are most useful.)
Allow your child to play with the creations. Stack them, compare their sizes, explore what items could be made by stacking or taping them together. Some possibilities might include stair steps, little villages made of boxhouses, walls, etc. Discuss when a cube is more or less suitable than a box for a particular task. Can you think of a use for any of your boxes or cubes? (Small things could be stored in them; they can be decorated and used to create a threedimensional art piece, etc.) Look around the house (the kitchen pantry especially!) to find commercial applications of these simple solids.

(2) 
Use the equilateral triangles on the “pyramid” activity sheet for this project’s construction. If your child does not know the term, explain that an item made of equilateral triangles attached together as the squares were done will be called a “pyramid.” Again, ask your child to predict how many triangles it will take to make a pyramid. You have probably realized, but your child probably won’t realize until the triangles are taped together, that the “bottom” of the triangle won’t be the same size as the triangles that were provided for the sides. If you want a closed figure, how can you make the bottom piece? (Place your completed pyramid on a sheet of paper and carefully draw around the structure to make a template for the base. Cut it out, try it for fit, and when it is right, cut it out and attach it.
Question: What items could be stored in a pyramid shape? Would it be best, for each of your ideas, to tape on a bottom piece and stand it on the bottom or find a way to keep the pyramid upright so the items wouldn’t fall out? (Answers will vary, and some of your child’s ideas can be tried to see if the suggestion is practical.)

(3) 
For this activity you will use the large rectangles on the sheet with the “345” label. Ask your child to cut a straight line, on the diagonal, from one corner to the other on one of the cutout rectangles. To assist you in visualizing this instruction, one rectangle has a dotted line that shows the cutting line, but encourage your child to follow your verbal instructions rather than depending on a predrawn sample. If it would help your child to cut a straight line you could help your child use a straightedge to draw a line along the diagonal that will be cut so your child will have a line to cut on. Once the square has been cut, ask your child to identify what shapes were created by cutting a rectangle into two pieces in this way. Ask, “Are the triangles the same size?” (They should be!) The challenge in this activity is to use the two triangles to create a solid figure by joining the two triangles with a long rectanglestrip of paper. To judge the length of paper strip that will be needed your child needs to know the perimeter of one triangle. Help him/her to measure each side and add the figures together. (The rectangle that your child cut is 3 inches on one side, 4 inches on the other and the diagonal should be 5 inches, so the total perimeter is 12 inches. This is a good size for a child to work with, but is inconvenient because most of the paper you have is only 11 inches long. The easiest solution to this problem is to cut one long rectangle that is 7 inches long, another that is 5 inches long, and join them at one of the corners where the diagonal begins as you create the threedimensional shape. The width of the rectangle strips is up to you; 2 inches is generally easy for a child to handle. Therefore, you will cut 1 rectangle that is 7” long x 2” and one that is 5” x 2”. On the 7” strip, help your child measure where the 3” point is, mark it, and fold the strip at that point, matching the long edges so that the fold will be straight across the strip. With the fold, it should be fairly easy for your child to tape the 3” portion of the long rectangle to the 3” side of one triangle, then the fold will go around the angle and the 4” side of the paper strip can be taped to the 4” side of the triangle. Finally, the 5” rectangle can be taped to the diagonal of the triangle and the unfinished ends of the rectangle strips taped. When the other triangle has been fastened to the opposite sides of the rectangle strips, the little triangular box is complete. Help your child to complete another triangular box exactly like this one. Assist your child in putting the triangular boxes together, diagonal edges together, so that the 2 triangular boxes together form a rectangle. Remind your child that he/she has learned how to calculate the volume of a rectangular box (width x length x height) and the area of the face of the rectangle alone (width x length). Now that you know that 2 triangles take up the same space as one rectangular box, how do you suppose that information could be used to calculate the area or volume of the triangle? (Calculate the area or volume of the box and divide by 2)
Allow your child to make more triangular boxes of different sizes, cutting different shapes of triangles and putting them together to make different structures. Encourage mathematical language (accurate names of the shapes, remembering that the corners are called angles, the sides of the pieces are the “faces” of the object; volume is expressed in cubic measure and area is expressed in square measure, while perimeter is expressed just in the name of the measure (inches, centimeters, feet, etc.).
Question: “What items could be stored in a trianglebox?” (Pieces of candy, vegetables, other food items; marbles, some toys)

(4) 
No specific shape is included for your child to use in structuring a cone. There will be more “learning” in this activity if you ask your child to help you figure out what shape of paper is needed to create a cone. Unlike the other structures, the “rule” for the cone is that it will be made out of only one piece of paper and that it will be taped only along one long edge to hold it together. You may want to cut away areas of your paper that don’t seem to belong in a cone. Thinking about an icecream cone, what are some distinguishing features of a cone? (Like a pyramid but no sharp edges. It looks like it was rolled. There’s a point at one end.) Allow your child to experiment with different pieces of paper to see if he/she can roll a cone. Don’t allow your child to become frustrated with the project; it’s hard to hold the roll and not let the point start to gap while the tape is applied down the edge to hold it. Help where needed. Your child will notice that the bottom edge doesn’t look like the even edge of an icecream cone; there is probably a point of paper protruding. Your child might use scissors to trim the excess paper and then unroll it to see what the flat shape should look like before retaping it into a cone. Ask, “what is the shape of the bottom (or top!) of the completed cone? (Circle) If you wanted to create a closed shape, how could you make the bottom of the cone? (As with the pyramid, set the cone upsidedown on paper, trace the circle, cut out and fasten with small pieces of tape.
Question: “A cone can be a container. What can be put in a cone? (Ice cream, candy, flowers, etc.) How can you modify a cone to make it more useful, since it will not stand up on its own and still be upright to hold items? (Poke a hold on each side of the top, thread in a ribbon or string to make a handle and hang the filled cone on a doorknob, hook, etc.) Could you use your idea to make a pyramid a more useful container?

(5) 
Find some common items around the house that are boxshaped. Possibilities might be a box that contained tea bags, a gift box, a cereal box, etc. Gather up the constructions that your child has made and challenge your child to see how close your child can come to exactly filling one of the household boxes with the cubes, triangles, and cones that have been made. Ask such questions as, “Can you estimate how many cubes/boxes/pyramids/triangular boxes will fit in the chosen box? If you really had to fill this box and you had space left over, how could you fill those spaces? Can you estimate how many of your projectboxes will fit in a chosen box by stacking them next to the box that you chose?
You might wrap up this series of activities by challenging your child to build the most elaborate structure possible, using all of the boxes, cubes, pyramids, triangular boxes, and cones that have survived the activities. Your child might be able to measure how high it is at the highest point, and how wide; how long. Let your child name it! Perhaps you’ll want to take a photo of it!

