Measurement records and constraints

pre1. A parcel has mass 1 kg 8 g. Using 1 kg = 1000 g, enter its mass entirely in grams. Show the contribution from the kilogram and the leftover grams.

Answer: 1008

Look for:

• Labels 1000 g and 8 g as separate contributions, adds the remainder once and states that the measurement stays unchanged when renamed in grams.

Teacher response: One kilogram contributes 1000 g and the leftover mass contributes 8 g, giving 1008 g.

pre2. Strip A is 2 m 9 cm long and strip B is 190 cm long. Using 1 m = 100 cm, enter how many centimetres longer A is. Record both lengths in centimetres before comparing.

Answer: 19

Look for:

• Writes 209 cm and 190 cm before subtraction, preserves the 9 cm remainder and identifies A as the longer strip.

Teacher response: A is 209 cm and B is 190 cm; the common-unit difference is 19 cm.

pre3. A rectangle has area 35 square cm and width 5 cm. Enter its missing length in cm, and write a multiplication equation that checks the area.

Answer: 7

Look for:

• Uses 5 × unknown length = 35, solves by division and labels the side in cm rather than square cm.

Teacher response: The length is 35 ÷ 5 = 7 cm, and 5 × 7 reconstructs the required 35 square cm.

pre4. A rectangular plan requires area 40 square cm and width 5 cm. Only the rectangle's outside boundary receives edging, and at most 24 cm is available. Enter the extra centimetres of edging needed. Show the missing length, all four side lengths and whether the plan meets the limit.

Answer: 2

Look for:

• Recovers 8 cm from the area, counts both pairs of opposite edges and explains that a correct area can accompany a boundary that exceeds its limit.

Teacher response: The required length is 8 cm. Its full perimeter is 8 + 5 + 8 + 5 = 26 cm, which exceeds the 24 cm allowance by 2 cm.

pre5. A 2 cm × 2 cm × 1 cm prism and a 2 cm × 2 cm × 3 cm prism are stacked along one 2 cm × 2 cm face. Their interiors do not overlap. Enter their combined volume in cubic cm, give each part's volume and explain how the shared face is treated.

Answer: 16

Look for:

• Adds both non-overlapping interiors once, uses cubic units for volume and explains why subtracting the contact-face area would mix different quantities.

Teacher response: The separate volumes are 4 and 12 cubic cm, so the total is 16 cubic cm. The contact face has area 4 square cm and no thickness; it does not remove a filled region from either part.

pre6. A base prism is 5 cm long, 2 cm wide and 1 cm high. A top prism is 2 cm long, 2 cm wide and 2 cm high; it sits above the leftmost 2 cm of the base, sharing a 2 cm × 2 cm face with no overlapping interiors. Enter the stepped solid's volume in cubic cm. Explain why the outer 5 cm × 2 cm × 3 cm dimensions do not describe a completely filled prism.

Answer: 18

Look for:

• Uses each part's own height, identifies the empty upper-right region and explains why adding the two occupied parts counts the solid once.

Teacher response: The base contributes 10 cubic cm and the top contributes 8, giving 18. Above the rightmost 3 cm of the base is empty space 3 cm × 2 cm × 2 cm, with volume 12 cubic cm; the containing box would include it.

post1. A parcel has mass 2 kg 5 g. Using 1 kg = 1000 g, enter its mass entirely in grams. Show the contribution from the kilograms and the leftover grams.

Answer: 2005

Look for:

• Shows 2 × 1000 g and 5 g separately, preserves the small remainder and reconstructs 2 kg 5 g from 2005 g.

Teacher response: The two kilograms contribute 2000 g and the leftover mass contributes 5 g, giving 2005 g.

post2. Strip C is 3 m 6 cm long and strip D is 289 cm long. Using 1 m = 100 cm, enter how many centimetres longer C is. Record both lengths in centimetres before comparing.

Answer: 17

Look for:

• Writes 306 cm and 289 cm before subtraction and explains that the original numerals describe different-sized units.

Teacher response: C is 306 cm and D is 289 cm; subtracting matching units gives 17 cm.

post3. A rectangle has area 56 square cm and width 7 cm. Enter its missing length in cm, and write a multiplication equation that checks the area.

Answer: 8

Look for:

• Represents the unknown as a factor, uses division to recover it and distinguishes the length unit from the area unit.

Teacher response: The length is 56 ÷ 7 = 8 cm, and 7 × 8 reconstructs the required 56 square cm.

post4. A rectangular plan requires area 54 square cm and width 6 cm. Only its outside boundary receives edging, and at most 27 cm is available. Enter the extra centimetres of edging needed. Show the missing length, all four side lengths and whether the plan meets the limit.

Answer: 3

Look for:

• Verifies the 54-square-cm area, counts all four edges and states that the area and width can be correct while the edging constraint is unmet.

Teacher response: The required length is 9 cm. The complete perimeter is 9 + 6 + 9 + 6 = 30 cm, exceeding the 27 cm allowance by 3 cm.

post5. A 3 cm × 2 cm × 2 cm prism and a 3 cm × 2 cm × 3 cm prism are stacked along one 3 cm × 2 cm face. Their interiors do not overlap. Enter their combined volume in cubic cm, give each part's volume and explain how the shared face is treated.

Answer: 30

Look for:

• Adds 12 and 18 cubic cm, distinguishes the 6-square-cm boundary from a volume and explains the non-overlapping-interior condition.

Teacher response: The separate volumes are 12 and 18 cubic cm, giving 30. Their contact face has area 6 square cm and no thickness, so no duplicated filled volume is subtracted.

post6. A base prism is 6 cm long, 2 cm wide and 1 cm high. A top prism is 2 cm long, 2 cm wide and 2 cm high; it sits above the leftmost 2 cm of the base, sharing a 2 cm × 2 cm face with no overlapping interiors. Enter the stepped solid's volume in cubic cm. Explain why the outer 6 cm × 2 cm × 3 cm dimensions do not describe a completely filled prism.

Answer: 20

Look for:

• Labels the two occupied parts and their own heights, excludes the empty upper-right prism and explains why a containing box includes extra space.

Teacher response: The base contributes 12 cubic cm and the top contributes 8, giving 20. The empty region above the rightmost 4 cm of the base is 4 cm × 2 cm × 2 cm, with volume 16 cubic cm; the containing box's 36 cubic cm includes it.

transfer. Create one complete record for a stepped display model. Its rectangular base has footprint area 900 square cm, width 15 cm and height 2 cm. Only the base's rectangular footprint receives edging; 1 m 55 cm is available, with 1 m = 100 cm. A top prism is 20 cm long, 15 cm wide and 1 cm high. It sits above the leftmost 20 cm of the base, sharing a 20 cm × 15 cm face, with no overlapping interiors. Find the missing base length, the complete base boundary, whether the edging is sufficient and by how much, and the total volume. Explain the units and how the parts cover the solid once.

Answer: The base length is 900 ÷ 15 = 60 cm. Its full boundary is 60 + 15 + 60 + 15 = 150 cm. The available edging is 155 cm, so it is sufficient with 5 cm left. The base volume is 60 × 15 × 2 = 1800 cubic cm and the top volume is 20 × 15 × 1 = 300 cubic cm. Their non-overlapping interiors give a total of 2100 cubic cm. The 300-square-cm shared face is a boundary, not a volume to subtract.

Teacher response: The decision connects the three lesson records: convert the allowance to the boundary's unit, use area to recover a side and check all four edges, then add occupied prism volumes in cubic units. A second valid partition is a 20 cm × 15 cm × 3 cm left prism and a 40 cm × 15 cm × 2 cm right prism, giving 900 + 1200 = 2100 cubic cm.

lesson-1. Compare metric records written in different units · 40 minutes. Before: Prepare the conversion key and an object/original/common-unit record. Read pre1 and pre2 separately: pre1 tests preservation of the smaller-unit remainder, while pre2 tests comparison after conversion. A correct number without a unit-based explanation still needs a brief follow-up. Bring learners' records, rather than a combined assessment score, into the investigation's diagnostic. Focus: Run the existing investigation's 5-minute diagnostic, 10-minute model, 15-minute differentiated practice and 10-minute independent/transfer routine. Learners replace each larger unit using the exact key, add the smaller-unit remainder once and compare matching units. Have partners reverse a conversion to reconstruct the original record. Connect this to the next lesson: a boundary and its edging allowance can only be compared after their units match.

Teacher response: Evidence: Collect the investigation's independent record and the learner's reverse check. Look for both measurements in the same unit, preservation of the leftover smaller units and a difference labelled with the requested unit. Record the numerical answer separately from these explanations; a plausible difference alone does not show which units were compared. If secure: Move to the rectangle investigation. Ask the learner to label which numbers describe length, area and an edging allowance before calculating; the common-unit record becomes part of the final feasibility check. If not secure: Use a larger-unit box labelled with its smaller-unit count and write leftover units outside it. Complete the common-unit routing recheck and ask the learner to reverse it. Keep the conversion key visible during Lesson 2; postpone an independent mixed-unit comparison until the learner can explain the conversion.

lesson-2. Recover a missing rectangle side and check the boundary · 40 minutes. Before: Prepare a labelled rectangle with one missing side and distinct spaces for area, side lengths, perimeter and the edging limit. Use pre3 to check the missing-factor relationship and pre4 to check all four outside edges and the limit. If multiplication and division are not connected, prepare rows of unit squares before the investigation's diagnostic. Focus: Run the linked 5/10/15/10-minute routine. Learners recover the missing side from area ÷ known side, multiply to verify the area and then add both pairs of opposite sides. They compare the complete boundary with the edging allowance. Discuss a plan that meets one requirement but fails another. Connect to Lesson 3 by distinguishing a filled two-dimensional region from its boundary; a solid likewise needs a record of occupied space rather than its outline alone.

Teacher response: Evidence: Collect a labelled rectangle, its missing-factor equation, the reconstructed area, the four-edge perimeter and the feasibility decision. Distinguish a correct side with an incomplete boundary from an incorrect side. Check whether the learner states exactly how much edging is left or missing, including the possibility of meeting a limit exactly. If secure: Move to the joined-solid investigation once the single-prism volume prerequisite is available. Ask the learner to name the area units used for a face and the cubic units used for filled space before adding any prism volumes. If not secure: For a missing-factor error, rebuild complete unit-square rows and connect multiplication to division. For a boundary error, mark and trace all four edges before comparing with the limit. Use the rectangle-constraint routing recheck and retain separate evidence for the recovered side, perimeter and decision.

lesson-3. Account for every part of a joined solid · 40 minutes. Before: Confirm the learner can explain a cubic unit and calculate one prism's volume, using volume-as-equal-layers first if needed. Prepare two colours and a base/top record. Use pre5 to identify subtraction of a contact-face area and pre6 to identify multiplication of outer dimensions despite empty space. Grade 4 learners join this lesson as a supported extension after the prerequisite, while it is the Grade 5 additive-volume lesson in this plan. Focus: Run the existing investigation's 5/10/15/10-minute routine. Colour two parts with non-overlapping interiors, record each part's own dimensions and add their volumes. Contrast a shared face with a genuinely overlapping filled region. Use the challenge to repair overlapping region labels into a disjoint partition. Connect back to the earlier lessons: label the quantity and unit first, then ensure that the record describes the complete object exactly once.

Teacher response: Evidence: Collect a coloured partition, both part-volume calculations and the combined volume in cubic units. Ask the learner to point to the contact face and explain why it has area but contributes no duplicate filled volume. Use a containing-box check to see whether an empty region has been included. Record calculation success and coverage explanations separately. If secure: Administer the post-assessment and the integrated transfer. Ask the learner to compare two valid disjoint partitions or use a containing box with its empty prism identified. Describe readiness only for the skills shown by the current work. If not secure: If the learner cannot calculate a single prism, return to the prerequisite before repeating joined-solid work. If the problem is double coverage or empty space, colour occupied regions and create one record row per disjoint part. Use the solid-accounting routing recheck and require the shared-face explanation alongside the number.

route-1. common-unit: If the smaller-unit remainder disappears or changes size, use a labelled larger-unit box plus separate remainder marks. If conversion is correct but the comparison is wrong, place both common-unit records side by side before subtracting. Ask for the unit explanation even when the numerical response is correct. Recheck: A parcel has mass 3 kg 6 g. Using 1 kg = 1000 g, enter the mass in grams and explain each contribution. Reverse your conversion to recover the original record.

Answer: 3006

Teacher response: Three kilograms contribute 3000 g and the remainder contributes 6 g. Splitting 3006 g into those parts reconstructs 3 kg 6 g.

route-2. rectangle-constraint: For an incorrect missing side, tile equal rows and connect the area equation to its inverse division. For a correct side with an incorrect perimeter, label both pairs of opposite sides and trace the complete boundary. For an incorrect decision, compare that perimeter with the stated allowance and name any surplus or shortage. Recheck: A rectangle requires area 18 square cm and width 3 cm. Exactly 18 cm of edging is available for its complete outside boundary. Enter the perimeter in cm, show the missing side and explain whether the edging is enough.

Answer: 18

Teacher response: The missing side is 6 cm because 18 ÷ 3 = 6. The boundary is 6 + 3 + 6 + 3 = 18 cm, so the edging fits exactly with none left.

route-3. solid-accounting: First distinguish difficulty with one prism from difficulty partitioning a joined solid. Use volume-as-equal-layers for the single-prism prerequisite. For contact-face subtraction, compare square and cubic units. For an outer-dimensions product, colour the occupied regions and identify the empty part of the containing box. Resume the joined-solid lesson with one record row per disjoint interior. Recheck: A base prism is 3 cm × 2 cm × 1 cm. A 1 cm × 2 cm × 2 cm top prism sits above the leftmost 1 cm of the base, sharing a 1 cm × 2 cm face with no overlapping interiors. Enter the total volume in cubic cm and explain why the shared face is not subtracted.

Answer: 10

Teacher response: The base occupies 6 cubic cm and the top occupies 4, so together they occupy 10. The shared face has area 2 square cm and no thickness; each occupied interior is already counted once.

ready. Review the post-assessment and relevant rechecks by skill. A learner is ready to use that skill in the transfer when the current work includes both a correct calculation and its requested explanation: equivalent common-unit records, an area check plus all-edge boundary check, or a disjoint volume partition with cubic units. Record the evidence actually observed; this is a decision about the next task rather than a claim of general mastery.

support. Route each unmet explanation or calculation to its matching lesson and recheck. A numerical error with a sound representation calls for checking the calculation; a correct number without a quantity, unit or coverage explanation calls for a follow-up representation. Keep solid-accounting work marked not yet introduced for learners awaiting the volume prerequisite. Retain item-level records and use the two parallel forms to compare the calculations and explanations observed at each point.

extension. Have learners produce a second valid record for the transfer: reverse the metric conversion, verify the rectangle's area and boundary, then repartition the solid vertically. Compare the two volume sums and explain why equal totals are supported by coverage of the same occupied region. Change one constraint and state which conclusion must be recalculated.