BUILDING CONGRUENT HEXAGONS

Transcription

1 BUILDING CONGRUENT HEXAGONS GEOMETRY Spatial visalization Congrence Eqivalence Getting Ready What Yo ll Need Pattern Blocks, abot 50 per pair Pattern Block triangle paper, page 90 Crayons Overhead Pattern Blocks (optional) Overview Children search to find all possible combinations of Pattern Blocks that can be sed to bild shapes congrent to the yellow hexagon. In this activity, children have the opportnity to: reinforce nderstanding of congrency and eqivalence assess the niqeness of soltions develop spatial reasoning skills The Activity Some children may think they have additional soltions. Show them that other soltions are simply different arrangements of the ble-green combination. Introdcing Have children pt down a red trapezoid. Ask them to se other Pattern Blocks to form a shape that cold cover the red trapezoid exactly. Invite a volnteer to show that his or her arrangement of blocks fits exactly over the red trapezoid. Explain that when two shapes have exactly the same size and shape, they are congrent. Ask if anyone sed a different combination of blocks. Display this second combination. Establish that there are only two different combinations that make a shape congrent to the red trapezoid.

2 On Their Own How many different combinations of Pattern Blocks can yo se to make shapes that are congrent to the yellow hexagon? With a partner, se Pattern Blocks to make shapes that are the same size and shape as the yellow hexagon. Each shape shold se a different combination of blocks. If 2 shapes se only a different arrangement of the same blocks, the second shape shold not cont as a new combination. Record yor shapes on triangle paper. Then color them in. Be ready to explain how yo know yo have fond all the soltions. The Bigger Pictre Thinking and Sharing Invite volnteers to post some of their soltions. Contine ntil children are satisfied that all possible combinations are displayed. Use prompts sch as these to promote class discssion: What were some of yor strategies for finding shapes? How did yo make sre yor soltions were all different? What helped yo decide that yo had fond all the possible combinations? Why didn t yo se the orange sqare and the tan parallelogram? Extending the Activity 1. Have children repeat the activity, bt change the rles so that soltions inclde different arrangements of the same blocks. 2. Direct children to find the different combinations of Pattern Blocks that can be sed to bild eqilateral triangles whose sides are 3 nits long, where 1 nit is the length of a side of a green triangle.

3 Teacher Talk Where s the Mathematics? There are seven different soltions for this activity. Many children may start by trying to make hexagons sing several blocks of the same shape. In their search for additional soltions, children move natrally into the realm of eqivalency and sbstittion, two important concepts in mathematics. Some children may change their same-block soltions by removing one block at a time, then testing whether the empty space can be filled by one or more of another kind of block. For example, a child might take one green triangle away from a hexagon made from six green triangles, then one more green triangle away from the remaining five, to discover that the empty space can be filled by one ble parallelogram. Some children may se strategies based on eqivalences among the Pattern Block shapes. In the following seqence of hexagons, one of the red trapezoids in hexagon A is replaced by a ble parallelogram and a green triangle to form hexagon B. The ble parallelogram in hexagon B is then replaced by two green triangles to form hexagon C. A B C

4 In experimenting with different combinations of blocks, children qickly eliminate the orange sqare and the tan rhombs. They notice that after a sqare or a rhombs is pt down, no other Pattern Block can be added that allows for the formation of a hexagon that is congrent to the yellow block. Children may try to flip or rotate their soltions to identify and weed ot those that are the same. Althogh effective, this strategy is not foolproof in helping children to find dplicate combinations. For example, these two shapes appear to be different soltions bt actally represent the same combination becase each has two greens and two bles: Children may decide that conting the nmber of each kind of block sed in each soltion is a good way to see if they have any dplicates. By recording the nmbers of blocks in a table, children can readily spot combinations that are the same. A table or chart may also help children to consider additional possible combinations of blocks to try. This table shows the seven possible soltions nmerically: Green blocks Ble blocks Red blocks This activity provides an ideal springboard for work with fractions. Specifically, it gives meaning to the addition of halves, thirds, and sixths. For example, the soltion shown can be sed to show 1/2 + 1/6 + 1/6 + 1/6 = 1. It can also be sed to help children nderstand why 1/2 > 1/6 or why 3 x 1/6 = 1/2. Children s familiarity with these kinds of models provides them with concrete ways to think abot and work with fractions.

5 PATTERN BLOCK TRIANGLE PAPER