New Zealand Curriculum: Level 3 to early Level 4
Learning Progression Frameworks: Measurement Sense, Signpost 5 to Signpost 7
Target students
These activities are intended for students who understand how to use units of measure to find lengths. They should understand the following characteristics of units:
- Relate to the attribute being measured (length is measures with a part of length)
- Identical
- Tiled with no gaps or overlaps to create a measure
- Sub-dividable (equally partitioned when greater accuracy is needed)
Students should also know how to use a measurement scale, such as a ruler or tape measure. They should be familiar with the most common metric units of length, metres, centimetres, and possibly millimetres, though they may not be able to convert measures (e.g., 45cm = 450mm). Students should have a partial or full grasp of their basic multiplication facts and the division equivalents.
The following diagnostic questions indicate students’ understanding of, and ability to find the areas and perimeters of flat spaces, particularly rectangles. Allow access to pencil and paper and to a calculator if students need it. (show diagnostic questions)
The questions should be presented orally and in a written form so that the student can refer to them. The questions have been posed using a money context but can be changed to other contexts that are engaging to your students.
- Here are three different floors. The numbers show how many tiles are on each side.
Which floor has the biggest number of square tiles?
(Use Copymaster page 1)
Signs of fluency and understanding:
Using multiplication to count the number of tiles, i.e., 3 x 6 = 18, 4 x 4 = 16, and 2 x 9 = 18. Recognises that the product of numbers of rows and columns gives the number of tiles.
What to notice if they don’t solve the problem fluently:
Counting, in ones, or by skip counting in twos, to find the total number of tiles in each floor. This indicates an inability to structure the rectangle in rows and columns and/or no recognition that multiplication might be used to count the number of tiles.
Supporting activity:
Finding areas of rectangles
- Measure the sides of each rectangle in centimetres. Write the measurements down.
Use the measurements to work out the area of each rectangle in square centimetres.
Write the areas using appropriate numbers and units.
(Use Copymaster page 2 with calculator available).
Signs of fluency and understanding:
Measure the side lengths accurately to the nearest centimetre and write the measures. Apply multiplication to find and record the areas, i.e., 2 x 18 = 36cm2, 3 x 12 = 36cm2, 6 x 6 = 36cm2, and 4 x 9= 3636cm2.
What to notice if they don’t solve the problem fluently:
Draw in lines to partition each rectangle into square units of 1cm2 then find a way to count all the squares. This indicates an inability to anticipate the structure of the rectangle in rows and columns.
Apply additive strategies to find the number of square units, such as 6 + 6 = 12, 12 = 12 = 24 and 24 + 12 = 36 to find the area of the square. This might indicate that the student has yet to establish multiplication as a binary (two numbers at a time) operation.
Supporting activity:
Finding the areas of rectangles using side lengths
- These rectangles have the same area. The numbers show the length of each side.
Do the rectangles have the same perimeter? Please explain.
(You may need to explain that perimeter is the distance around the outside of each rectangle)
(Use Copymaster page 3 with calculator available).
Signs of fluency and understanding:
Calculate the perimeter of each rectangle fluently, possibly using a calculator. Recognise that each side measure occurs twice in the perimeter. For example, may use 2 x 12 + 2 x 4 or 2 x (12 + 4) to work out the perimeter of the 12 x 4 rectangle (example).
Might know that regular rectangles tend to have lesser perimeter for a fixed area. Regularity means that the sides are equal so the rectangle closest to a square has the least perimeter.
What to notice if they don’t solve the problem fluently:
May use counting or additive strategies to find the perimeters, rather than a combination of addition and multiplication. This suggests the student needs to work on modelling situations that combine operations, like addition and multiplication.
Supporting activity:
Finding perimeters
- Which shape has the greatest area? Which shape has the greatest perimeter?
(Use Copymaster page 4).
Signs of fluency and understanding:
Calculate the area and perimeter of each rectangle including allowing for the half measures on side lengths. May use mental and paper calculation or a calculator using multiplicative reasoning. For example, the area of the 5½ x 5 rectangle is 5.5 x 5 = 27.5 square units, and the perimeter is 2 x (5.5 + 5) = 2 x 10.5 = 21 units.
Answers are:
Area: 5.5 x 5 = 27.5 square units for top left, 4 x 6.5 = 26 square units for top right, and 2.5 x 10 = 25 square units for bottom left. So the top left rectangle has the biggest area.
Perimeter: 2 x (5.5 + 5) = 2 x 10.5 = 21 units for top left, 2 x (4 + 6.5) = 2 x 10.5 = 21 units for top right, and 2 x (2.5 + 10) = 2 x 12.5 = 25 units for bottom left. So the bottom left rectangle has the biggest perimeter.
Recognises that length (perimeter) and area are different attributes, so the measures are expressed using different units, length units for perimeter and square units for area.
What to notice if they don’t solve the problem fluently:
May confuse the two attributes, area, and perimeter, when providing answers.
May have difficulty with partial units and either ignore them or be unsure about how to allow for half units in calculation. This might indicate that the student needs experience in finding areas of shapes that include partial units and in calculating areas and perimeters where partial units are involved.
Supporting activity:
Working with partial units
- Find the area and perimeter of this rectangle as accurately as you can.
Use a ruler to measure the side lengths first then find the answers using a calculator.
(Use Copymaster page 5).
Signs of fluency and understanding:
Measure the side lengths accurately to express those lengths as 18.5cm or 185mm and 11.3cm or 113mm.
Correctly enter an appropriate algorithm for find both area and perimeter. For area work out 18.5 x 11.3 = 209.05cm2 or 185 x 113 = 20 905mm2. For perimeter work out 2 x (18.5 + 11.3) = 59.6cm or 2 x (185 + 113) 596mm. Answers must include the number and unit of measure.
What to notice if they don’t solve the problem fluently:
May be unsure about how to accommodate fractional numbers of measurement units. This is likely to show in being unsure of how to express length and width as 18.5cm and 11.3cm respectively. This may indicate that the student needs experience with measuring lengths with greater accuracy using smaller units in the metric system.
May measure the side lengths accurately but either be unclear about how to find the area and perimeter by calculation and/or express the resulting measure using appropriate units. This indicates the student needs more experience with calculating areas and perimeters with partial units, as provided above in Working with partial units.
Supporting activity:
Finding areas and perimeters from decimal side lengths
Teaching activities
- Finding areas of rectangles
- Finding the areas of rectangles using side lengths
- Finding perimeters
- Working with partial unit
- Finding areas and perimeters from decimal side lengths
