We started another Math inquiry today - The Box Factory.
The focus of this unit is the deepening and extending of students’ understanding of multiplication, specifically the associative and commutative properties and their use with computation, systematic factoring, and the extension of students’ understanding of two-dimensional rectangular arrays to three-dimensional arrays within rectangular prisms.
Student' work after Day 1.
Today begins with a mini lesson on doubling and halving. This is a strategy that students may use to find all possible designs for boxes holding 24 items. Students then continued to work with their math partners to investigate and refine their thinking about this problem, and about how they know that they have found all the possible box configurations for 24 items. They also prepared for Day Three's math congress by creating posters that show their thinking. Developing posters allows students to reflect on their work and may challenge them to generalize their thinking.
Working on the task ....
Our final answers
Day 3
Gallery Walk
Math Congress
Day 4
Mental math string encourages ss to use multiplication facts that they know to figure out more difficult problems. This particular string supports the associative property of multiplication and the use of ten-times. Toward the end of the string students are challenged to make partial products (using the distributive property). For e.g. (4 x 30) + (4 x 9)
Some students also noticed a connection to the previous twp problems.
For e.g. 4 x 39 = 4 x (40 -1) = (4 x 40) - (4 x 1)
This problem requires students to think about the relationship between surface area and volume, to examine how multiplication is related both to surface area and to the total number of cubic units in the box, and to continue to explore arrays (both two-dimensional and three-dimensional).
This group made an important discovery " We saw that there was a pattern - that some box designs were just the numbers switched around. "
Conclusion: When multiplying, factors can be grouped in various ways but the product remains the same (employing the associative and commutative properties)
This group also realized that some of the answers were repeated ....
This group figured out the formula for calculating surface area. Good thinking kiddies!
This group was also on to something ....
This group used the linking cubes to figure out the answers .....
Are you calculating the volume or surface area? Hmmmm ......
Some students were quick to notice that imperial measurements were used in the question. Most also knew that 12 inches = 1 foot. Good job kiddies! Off to a good start!
These boys tried to figure out the answer by using the outline of the 4 feet by 6 feet box on the carpet.
Students' solution
How many 4-inch by 4-inch by 4-inch boxes will fit into the cardboard shipping box that is 4 feet by 6 feet by 4 feet? All the groups were unanimous that 2592 boxes can be shipped but look at the different strategies used ..... Really impressive thinking 😊
Next class, we'll do a debrief of the number of 2-inch by 2-inch by 2-inch boxes that can fit into the same shipping container. We unfortunately ran out of time time during out Math Congress.
Gallery Work before Math Congress
The last thing we'll do is a final reflection of our learning throughout the entire investigation. The aim is to pull out the BIG IDEAS.
So what were the BIG IDEAS?
We ended the investigation on a positive note. In order to figure out the second part of the problem, the students approached the question as division (how many times can this 2-inch by 2-inch by 2-inch box fit into the base of the shipping container that is 4 feet (48 inches) by 6 feet (72 inches) ?)
We came up with an array 24 x 36. The answer (864) was then multiplied by the height of the container (4 feet = 48 inches). Twenty-four boxes can be stacked vertically (height). Therefore the number of boxes that can fit into the entire shipping container was:
( Length x Width) x Height = (24 x 36) x 24
= 20, 736
Conclusion: 20, 736 2" x 2" x 2" boxes can fit into the large shipping container and 2,592 4" x 4" x 4" boxes can fit into the same large shipping container.
Another way to look at the problem:
The large box requires eight times the volume of the small box.
Volume of 2" x 2' x 2" box is 8 cubic inches
Volume of 4" x 4" x 4" box is 64 cubic inches.
Since the large box can fit 2,592 boxes you could figure out the number of small boxes by multiplying 2592 x 8.
Where did the 8 come from? (2 x 2 x 2) - small box
Another BIG IDEA - Square units vs cubic units
Square units are used if only two dimensions are multiplied (e.g Length x Width for Area or (Length x Width) + (Width x Height) + (Length x Height) for Surface Area.
Cubic units are used if three dimensions are multiplied (Length x Width x Height). This, we discovered, is the formula to calculate volume.
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| Sometimes you just need a large visual aid :-) |
