(1) A rectangular prism is cut by a vertical plane through a diagonal of its base into equal triangular prisms, and a chain of blanks turns the box's volume formula into the triangular prism's. (2) Joining the point where a cube's diagonals meet to the vertices of each face makes congruent square pyramids, and a second chain turns the cube's volume into the pyramid's. Fill in every blank.
Your answer
Watch outA sixth of the cube suggests 1/6, but the matching prism is half, so it is 1/3.
How to solve
StrategyCreate a Physical Representation — Both derivations are cutting arguments, and the honest way to see a cutting argument is to hold the solid and cut it: take a block of clay, slice it top to bottom through a diagonal of its base, and lift the two pieces apart; take a cube and cut from its centre out to every face. Once the pieces are in front of you, the only question left is 'are these pieces really copies of each other?' - and that is what tool 17 is for, because the answer is a turn: rotate the block half a turn and the front piece lands on the back piece, rotate the cube a quarter turn and one pyramid lands on the next. Everything after that is the easier related problem I already know how to do (the volume of a box), plus regrouping a product so the 1/2 sits next to the base area instead of at the end.
1STEP 1
Part (1): make the cut and count the pieces
One cut gives the box 2 pieces.
one cut → 2 pieces
A single straight cut through a solid can only make two pieces, and slicing a box along a base diagonal keeps the cut face flat and rectangular, so both pieces really are prisms and not some odd shape.
7.G.A.3Create A Physical Representation
2STEP 2
Why the two pieces are congruent, not just 'about the same'
A half turn matches them, so they are congruent.
half turn about the centre: front piece ⟼ back piece
Two shapes are the same size and shape exactly when one can be moved onto the other without bending or stretching it. Here that motion is a plain half turn, so no measuring is needed to be sure the two halves match.
8.G.A.2Visualize Spatial Relationships
3STEP 3
Turn 'half the box' into a formula
So the prism is half the box.
(triangular prism) = (box) × 1/2 = (area of the base of the box) × (height) × 1/2
The box is the one solid whose volume rule is already known, so every new rule in this unit is built by relating a new solid back to a box - here by cutting the box in two.
5.MD.C.5Solve An Easier Related Problem
4STEP 4
Slide the 1/2 next to the base area
Slide the half next to the base area.
(area of the base of the box) × (height) × 1/2 = (area of the base of the box) × 1/2 × (height)
Regrouping a product is free, and it is done here for a reason: the derivation wants the first two factors to fuse into a single quantity that has a name of its own.
5.NF.B.4Organize Information In More Ways
5STEP 5
Name what (area of the base of the box) x 1/2 really is
That is exactly the triangle's base area.
(area of the base of the box) × 1/2 = (area of the triangle) = (the triangular prism's base area)
This is the pay-off of the whole derivation: the 1/2 does not stay in the formula at all - it gets absorbed into the base, and a triangular prism ends up obeying the very same (base area) x (height) rule as a box.
6.G.A.1Organize Information In More Ways
6STEP 6
Part (2): cut the cube from its centre and count the pieces
Cutting a cube from its centre gives 6 pieces.
6 faces → 6 pyramids
Think of the cube as a room and the centre as a lamp: each of the six walls, floor and ceiling has its own pyramid of space in front of it, and the six pyramids together are the whole room.
5.MD.C.5Create A Physical Representation
7STEP 7
Why the six pyramids are congruent
The six pyramids are congruent.
(square pyramid) = (cube) × 1/6
A cube looks exactly the same no matter which face is turned to the floor, so nothing distinguishes one of the six pyramids from another - and identical pieces must hold identical amounts.
8.G.A.2Visualize Spatial Relationships
8STEP 8
The prism with the same base and the same height
The matching prism is half the cube.
e × e × e/2 = e³/2 = (cube) × 1/2
The apex sits at the centre of the cube, so it is only half an edge above the bottom face - the pyramid is a short one, and the matching prism is the bottom half of the cube.
The cube is the go-between: both the pyramid and the prism are measured against it, so comparing them is just comparing 1/6 with 1/2, and 1/6 is one third of 1/2.
5.NF.B.4Solve An Easier Related Problem
The pyramid turns out to be exactly one third of the prism on the same base with the same height.
Why?
Pouring a cone or pyramid full into a matching prism takes exactly three fills, which is the one-third rule you can test by hand.
🧱A cone is one third of its cylinderFill a cone-shaped cup and pour it into a cylinder of the same base and height: it takes exactly three cupfuls.
Why?
The six pyramids cut from the cube are identical, because each can be laid exactly onto any other, so each holds the same share.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Answer
2, 2, 2, 2, base area / 6, 6, 2, 3, 3
1/6 = 1/2 × 1/3
Put numbers in and check both rules. Take a box 7 cm by 5 cm by 3 cm: its volume is 7 x 5 x 3 = 105 cubic centimetres, and the triangular prism the derivation describes has base area 7 x 5 / 2 = 17.5 square centimetres and height 3 cm, giving 17.5 x 3 = 52.5 cubic centimetres - exactly half of 105, as the count of 2 pieces demands. Take a cube of edge 6 cm: its volume is 216 cubic centimetres, so each of the 6 pyramids should be 36 cubic centimetres; the derived rule gives (6 x 6) x 3 x 1/3 = 36, and 6 x 36 = 216 puts the cube back together exactly. The units also behave: a base area in square centimetres times a height in centimetres gives cubic centimetres, correct for volume, and multiplying by 1/2 or 1/3 does not change the unit. The sizes are sensible too - the pyramid sits inside the prism that has the same base and height, so its volume must be less than that prism's, and 1/3 is indeed less than 1.
Takeaway
Cut a solid into identical copies and just count them: 2 copies means each is 1/2, 6 copies means each is 1/6 - the whole formula falls out of the count.
Part (1): make the cut and count the pieces
Why the two pieces are congruent, not just 'about the same'
Turn 'half the box' into a formula
Slide the 1/2 next to the base area
Name what (area of the base of the box) x 1/2 really is
Part (2): cut the cube from its centre and count the pieces