Looking at Combinations B
Anna and Dale are going to remodel a workroom. They want to put new cabinets along one wall of the room. They start by measuring the room and drawing this diagram.
Anna and Dale find out that the cabinets come in two different widths:
45 centimeters (cm) and 60 cm.
17. How many of each cabinet do Anna and Dale need in order for the cabinets to fit exactly along the wall that measures 315 cm?
Try to find more than one possibility.
Looking at Combinations
B
Section B: Looking at Combinations 10T
Hints and Comments
Materials
calculator;
Student Activity Sheet 1, optional
Overview
Students now find ways to determine the possible combinations of long and short cabinets that will fit in a given wall space.
About the Mathematics
In most of the problem situations in this section, students know the value of each item and find the value of the combination. Later in this unit, students use the values of combinations to find the value of one item of each kind. This involves solving systems of equations. In the early sections of this unit, students are encouraged to make their own
combinations so that they can later use combinations to solve other problems.
Solutions and Samples
17. There are two possible answers: seven cabinets 45 centimeters wide, or three 45-centimeter and three 60-centimeter cabinets. Strategies will vary.
Sample student strategies:
• Some students may make a chart to look for all possibilities.
• Some students may develop a combination chart.
Number of 45-cm Cabinets
Number of 60-cm Cabinets
Total Length (in cm)
Result
7 6 5 5 4 4 3 3 2 2 1 1 0 0
0 1 1 2 2 1 2 3 3 4 4 5 5 6
315 330 285 345 300 240 255 315 270 330 285 345 300 360
exact fit too long too short too long too short too short too short exact fit too short
too long too short too long too short too long
Number of Long Cabinets Lengths of Combinations (in cm)
Number of Short Cabinets
270 315 360 405 450 495
330 375 420 465 510 555
390 435 480 525 570
450 495 540 585
510 555
570 6
7 8 9 10 11
0 1 2 3 4 5
0 45 90 135 180 225
0 60 105 150 195 240 285
120 165 210 255 300 345
180 225 270 315 360 405
240 285 330 375 420 465
300 345 390 435 480 525
2 3 4 5 6 7 8
360 405 450 495 540 585
420 465 510 555
480 525 570
1
Looking at Combinations B
Anna and Dale wonder how they can design cabinets for the longer wall.
The cabinet store has a convenient chart. The chart makes it easy to find out how many 60-cm and 45-cm cabinets are needed for different wall lengths.
18. Explain how Anna and Dale can use the chart to find the number of cabinets they need for the longer wall in the workroom.
Looking at Combinations B
Window
Door
315 cm
330 cm
Number of Long Cabinets Lengths of Combinations (in cm)
Number of Short Cabinets
270 315 360 405 450 495
330 375 420 465 510 555
390 435 480 525 570
450 495 540 585
510 555
570 6
7 8 9 10 11
0 1 2 3 4 5
0 45 90 135 180 225
0 60 105 150 195 240 285
120 165 210 255 300 345
180 225 270 315 360 405
240 285 330 375 420 465
300 345 390 435 480 525
2 3 4 5 6 7 8
360 405 450 495 540 585
420 465 510 555
480 525 570
1
Notes
Be sure to discuss the combination chart on this page with students.
Have students find combinations of cabinets for 240 centimeters (four short cabinets and one long cabinet) and 405 centimeters (nine short cabinets, five short cabinets, and three long cabinets, or one short cabinet and six long cabinets). Make sure they understand how to use the chart correctly before they work on problem 18.
Reaching All Learners
Extension
Ask students to write a paragraph in their journals to explain why the combination 495 appears three times on the chart.
Assessment Pyramid
18 18
Investigate and extend patterns.
Interpret and use combination charts.
11 Comparing Quantities
Section B: Looking at Combinations 11T
Hints and Comments
Materials
transparency of the combination chart on Student Book page 11, optional (one per class);
Overview
Students investigate a new combination chart to find out how many short and long cabinets will fit on a wall.
Comments About the Solutions
18. This problem assesses students’ ability to interpret and use combination charts, notebook notation, and equations; to discover, investigate, and extend patterns; and to informally solve problems that involve systems of equations.
Solutions and Samples
18. Explanations will vary. Sample explanation:
The longest wall measures 330 cm. This number appears in the chart twice.
• four cabinets of 60 cm and two cabinets of 45 cm, or
• one cabinet of 60 cm and six cabinets of 45 cm.
Number of Long Cabinets Lengths of Combinations (in cm)
Number of Short Cabinets
270 315 360 405 450 495
330 375 420 465 510 555
390 435 480 525 570
450 495 540 585
510 555
570 6
7 8 9 10 11
0 1 2 3 4 5
0 45 90 135 180 225
0 60 105 150 195 240 285
120 165 210 255 300 345
180 225 270 315 360 405
240 285 330 375 420 465
300 345 390 435 480 525
2 3 4 5 6 7 8
360 405 450 495 540 585
420 465 510 555
480 525 570
1
Looking at Combinations B
19. Can the cabinet store provide cabinets to fit a wall that is exactly 4 meters (m) long?
Explain your answer.
If cabinets don’t fit exactly, the cabinet store sells a strip to fill the gap. Most customers want the strip to be as small as possible.
20. What size strip is necessary for cabinets along a 4-m wall?
The chart has been completed to only 585 cm because longer rows of cabinets are not purchased often. However, one day an order comes in for cabinets to fit a wall exactly 6 m long. One possible way to fill this order is 10 cabinets of 60 cm each.
21. Reflect What are other possibilities for a cabinet arrangement that will fit a 6-m wall? Note that although you do not see 600 in the chart, you can still use the chart to find the answer. How?
Looking at Combinations
B
0 45 90 135 180 225 270 315
60 105 150 195 240 285 330 375
120 165 210 255 300 345 390 435
180 225 270 315 360 405 450 495
240 285 330 375 420 465 510 555
300 345 390 435 480 525 570
0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7
360 405 450 495 540 585
420 465 510 555
Number of Long Cabinets Lengths of Combinations (in cm)
Number of Short Cabinets
On the left is a part of the cabinet combination chart.
22. What is special about the move shown by the arrow?
23. If you start in another square in this chart and you make the same move, what do you notice? How can you explain this?
Strip Wall
Notes
After students have finished working on these problems, you may discuss the chart using a
transparency of the chart from problem 18. Ask, Where do you see five short and two long cabinet combinations? (In the row for 5 and the column for 2.) What does the 345 entry mean? (It is the total number of centimeters needed for five short cabinets and two long cabinets.) What patterns do you see in the chart?
(Answers will vary.)
Reaching All Learners
Intervention
Students struggling with problem 21 may need to continue the chart to find the answer. Remember that your eventual goal is for them to be able to find the pattern.
English Language Learners
English language learners may need continued support to understand the context of building and hanging cabinets.
Assessment Pyramid
21
Interpret a mathematical solution in terms of the problem situation.
12 Comparing Quantities
Section B: Looking at Combinations 12T
Hints and Comments
Materials
transparency of the combination chart on Student Book page 11, optional (one per class);
overhead projector, optional (one per class)
Overview
Students investigate more patterns in the
combination chart and determine combinations of cabinets for lengths that do not appear in the chart.
About the Mathematics
Students may realize that making a chart large enough for all possible combinations is not practical and that they can use patterns to find new combinations.
Comments About the Solutions
21. Encourage students to explain how to find combinations of cabinets that are 600 cm long without extending the chart.
Solutions and Samples
19. No. Four meters equals 400 cm, and this number does not appear in the chart.
20. 10 cm. The number closest to 400 cm and less than 400 cm on the chart is 390 cm. So the strip must be 10 cm long.
21. Answers will vary. Sample responses:
• 0 short cabinets and 10 long cabinets
• 4 short cabinets and 7 long cabinets,
• 8 short cabinets and 4 long cabinets
• 12 short cabinets and 1 long cabinet Strategies will vary. Sample student strategies:
• Some students will extend the chart to include 600 cm.
• If the lengths of two cabinets of 60 cm and four cabinets of 45 cm is 300 cm, then four 60-centimeter cabinets and eight 45-centimeter cabinets will fit a 600-centimeter wall.
22. Decreasing the number of short cabinets by four (180 cm) and increasing the number of long cabinets by three (180 cm) produces the same result.
23. No matter where you start, the end result is the same. Explanations will vary. Sample explanation:
Since 4 45 cm 3 60 cm, exchanging four short cabinets for three long cabinets does not change the total length.
Number of Long Cabinets Lengths of Combinations (in cm)
Number of Short Cabinets
270 315 360
330 375 420
390 435 480
450 495 540
510 555
570 6
7 8
0 1 2 3 4 5
0 45 90 135 180 225
0 60 105 150 195 240 285
120 165 210 255 300 345
180 225 270 315 360 405
240 285 330 375 420 465
300 345 390 435 480 525
2 3 4 5 6 7 8
360 405 450 495 540 600
600 585
420 465 510 555
480 525 570
1
Looking at Combinations B
24. Complete the puzzles on Student Activity Sheet 2.
Looking at Combinations B