1. DRAW A PICTURE OR DIAGRAM We use a picture or a diagram to solve a problem when there are many pieces of information in a problem that need to be organized and can best be organized in pictorial form. For example, look at this problem: 28 students were surveyed and asked what modes of transportation they used to commute to school and back. All 28 used either an automobile (A), a bicycle (B) or a cab (C), some of them used two modes of transportation, and some of them used all three. Use the information below to figure out how many students used both an automobile and a bicycle. A total of 16 people chose automobiles. 5 people chose both automobiles and cabs but not bicycles. Of the people who used only one mode of transportation, 1 used an automobile, 2 used bicycles, and 4 used cabs. The greatest number of students used all three modes of transportation.

2. DRAW A DIAGRAM OR PICTURE TO SOLVE EACH PROBLEM 1)CARLA’S HOUSE IS 4 MILES DUE NORTH OF SAM’S HOUSE. LILY’S HOUSE IS THREE MILES AWAY FROM CARLA’S HOUSE AND TWO MILES AWAY FROM SAM’S HOUSE. SHOW THE POSSIBLE LOCATIONS FOR LILY’S HOUSE. 2)DRAW A TRIANGLE. CONNECT EACH VERTEX WITH THE MIDPOINT OF THE OPPOSITE SIDE. HOW MANY TOTAL TRIANGLES DOES THIS PRODUCE? 3)FARMER BOB WANTS TO BUILD A RECTANGULAR PEN USING 36 FEET OF FENCE. ONE SIDE OF THE PEN WILL BE THE BARN. FIND THE DIMENSIONS AND AREA OF THE LARGEST SUCH PEN HE CAN MAKE, PROVIDED THE LENGTHS OF THE SIDES WILL BE INTEGERS. 4)HOW MANY DIFFERENT LINE SEGMENTS DO WE NEED TO CONNECT EACH VERTEX OF A HEXAGON WITH EVERY OTHER VERTEX?

3. 1) CARLA’S HOUSE IS 4 MILES DUE NORTH OF SAM’S HOUSE. LILY’S HOUSE IS THREE MILES AWAY FROM CARLA’S HOUSE AND TWO MILES AWAY FROM SAM’S HOUSE. SHOW THE POSSIBLE LOCATIONS FOR LILY’S HOUSE. TO SOLVE THIS PROBLEM, WE WILL BEGIN BY MARKING CARLA’S HOUSE AND SAM’S HOUSE, MEASURING OUT A DISTANCE OF 4 BETWEEN THEM. THEN, WE WILL USE A COMPASS SET TO 3 TO DRAW A CIRCLE AROUND CARLA’S HOUSE. FINALLY, WE WILL USE A COMPASS SET TO 2 TO DRAW A CIRCLE AROUND SAM’S HOUSE. THESE CIRCLES WILL INTERSECT IN TWO POINTS, WHICH WILL GIVE THE TWO POSSIBLE LOCATIONS OF LILY’S HOUSE. c s Lily’s house

4. 2) DRAW A TRIANGLE. CONNECT EACH VERTEX WITH THE MIDPOINT OF THE OPPOSITE SIDE. HOW MANY TOTAL TRIANGLES DOES THIS PRODUCE? WE JUST FOLLOW THE DIRECTIONS TO DRAW THIS SERIES OF TRIANGLES. NEXT, WE NEED TO COUNT THEM. THERE IS 1 LARGE TRIANGLE, 6 SMALL TRIANGLES, 6 TRIANGLES THAT ARE EXACTLY HALF THE SIZE OF THE LARGE TRIANGLE, AND 3 TRIANGLES THAT USE THE CENTER VERTEX AND A COMPLETE SIDE. THIS GIVES A TOTAL OF 16 TRIANGLES.

5. 3) FARMER BOB WANTS TO BUILD A RECTANGULAR PEN USING 36 FEET OF FENCE. ONE SIDE OF THE PEN WILL BE THE BARN. FIND THE DIMENSIONS AND AREA OF THE LARGEST SUCH PEN HE CAN MAKE, PROVIDED THE LENGTHS OF THE SIDES WILL BE INTEGERS. TO SOLVE THIS PROBLEM, WE WILL BEGIN BY DRAWING THE DIAGRAM AND THEN LABELING THE SIDES. AFTER THAT, WE WILL TRY THE POSSIBILITIES UNTIL WE ARRIVE AT THE CORRECT ANSWER. THE TOTAL AMOUNT OF FENCING IS 36, SO IF WE LET THE SMALL SIDES BE X, THE LARGE SIDE WILL BE 36 – X – X, OR 36-2X LIST THE POSSIBLITIES Side of barn xx 36 – 2X x36 – 2xArea 134 526130 722154 918162 1016160 820160 As we can see from the chart, the dimensions will be 9 feet by 18 feet with a total area of 162 square feet.

6. 4) HOW MANY DIFFERENT LINE SEGMENTS DO WE NEED TO CONNECT EACH VERTEX OF A HEXAGON WITH EVERY OTHER VERTEX? TO SOLVE THIS PROBLEM, WE DRAW LINE SEGMENTS WHILE BEING VERY CAREFUL TO COUNT THEM. WE NEED 5 TOTAL SEGMENTS FROM THE FIRST VERTEX, 4 NEW SEGMENTS FROM THE SECOND VERTEX, 3 NEW SEGMENTS FROM THE THIRD VERTEX, 2 FROM THE FOURTH, 1 FROM THE FIFTH, AND 0 FROM THE SIXTH. THIS GIVES A TOTAL OF 15 LINE SEGMENTS.

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