Short Blocks

Maths Year 6 Spring Multiplication and Division (A)

Each unit has everything you need to teach a set of related skills and concepts. 'Teaching for Understanding' provides whole-class teaching and fully differentiated adult-led group activities. ‘Problem-solving and Reasoning’ develops these skills, and includes questions to enable you to assess mastery. Practice sheets ensure procedural fluency. Extra support activities enable targeted work with children who are well below ARE.

‘UNIT PLAN’ gives you a text version of all parts of the unit to use in your school planning documentation. ‘DOWNLOAD ALL FILES’ gives you that unit plan plus all of associated documents. These bulk downloads are added value for Hamilton Friends and School Subscribers.

Unit 1 Scale factor problems concerning area (suggested as 2 days)

Objectives

Solve problems using a scale factor in areas of shapes
Unit 1: ID# 6481

National Curriculum
Mult/div (iv) (viii)
Rat/Pr (iii)

Hamilton Objectives
34. Solve problems involving similar shapes where the scale factor is known or can be found.
13. Scale up or down by a factor of 2, 4, 5 or 10; solve scaling problems and those involving rates.
35. Solve problems involving simple ratios, using tables facts and knowledge of fractions and multiples.

Teaching and Group Activities for Understanding

Day 1 Teaching
Children sketch a right-angle triangle where the sides along the right angle are 4cm and 7cm long. They measure the longest side, compare, and find the area. Draw a similar triangle with each side next to the right angle twice the length. Compare the longest sides. Define similar triangles and model the scaling effect. Remind children how to find the area of a parallelogram (see below plan), multiplying the base by the height.
Group Activities
-- Scale up rectangles, triangles and parallelograms by a factor of 2 and 3 and see what happens to the area.

Day 2 Teaching
Define similar shapes as identical in shape, but not in size. Show ‘Similar shapes’ (see resources). Ask children to find 2 rectangles which look similar. Measure the sides of each. Ask children to work out the scale factor (1½). Repeat for triangles (scale factor is 3).
Group Activities
Use the ‘Geometry genius’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Identify pairs of similar rectangles and triangles. Use the scale factor to work out side lengths.
-- Draw similar shapes, using a scale factor of 2 or 3 or 1.5. Identify pairs of similar shapes drawn by others and find the scale factor?

You Will Need

  • Mini-whiteboards and pens
  • Rulers
  • ‘Scaling up’ activity sheet (see resources)
  • cm2 paper
  • Similar shapes (see resources)
  • Similar shapes - triangles and rectangles sheets (see resources)
  • ITP: Function Blocks

Procedural Fluency

Day 1
Calculate the dimensions of toys, given scale factors.

Day 2
Identify pairs of similar rectangles and triangles. Use a scale factor to calculate the lengths of sides.

Mastery: Reasoning and Problem-Solving

  • True or false?
    If one triangle is scaled up to have sides 3 times as long as another, the area is also 3 times as large.
    If two rectangles are similar and the scale factor is 4, then the area of the larger rectangle is 16 times that of the smaller rectangle.
  • Calculate the area of the triangle whose sides are half the length of this one (see download). Compare the two areas. What do you notice?
    Explain why the area of the smaller has this relation to the area of the larger.

In-depth Investigation: Geometry Genius
Children use what they know about how to find the areas of triangles and parallelograms to find the areas of rhombi, kites and trapezia.

Extra Support

Function Blocks
Tables knowledge is an important pre-requisite for scaling. Use Function Blocks ITP to generate multiplication by 3. Can children work out the function? Repeat with other single-step x and ÷ functions.

Unit 2 Solve rate and scaling problems (suggested as 3 days)

Objectives

Use mental strategies, factors and multiples to solve problems of rate and scaling
Unit 2: ID# 6491

National Curriculum
Mult/div (iv) (viii)
Rat/Pr (iii)

Hamilton Objectives
9. Know all multiplication and division facts up to 12 × 12; identify common factors, common multiples.
10. Multiply/divide whole numbers mentally, using facts to 12 × 12 and place value; use facts to work with larger numbers.
13. Scale up or down by a factor of 2, 4, 5 or 10; solve scaling problems and those involving rates.
14. Perform divisions mentally within the range of tables facts; divide multiples of 10 and 100 (4500 ÷ 9) and use mental strategies such as halving (450 ÷20).

Teaching and Group Activities for Understanding

Day 1 Teaching
A blue whale is typically 24–30m in length, 4 times as long as the class. It has the fastest growth rate: calves grow 90kg/day, 2.5cm in length per day. So how much in a week? Explain that these are rates: It eats 3 tonnes per day. It grows 90Kg per day. The rate is weight (kg/tonnes) per day (time).
Group Activities
-- Use internet research to find real life rates (e.g. speed of a cheetah, the amount a young lion eats per day ….) and then invent similar problems.
-- Work out how much a blue whale calf will grow in given times and compare to rate of growth for a human child.

Day 2 Teaching
Recap mental strategies to multiply by 5, by 20, by 6, by 8. Show a table of dimensions of scale models of dinosaurs. Each model is 1/20 of the actual height and length. Ask how we work out full size, drawing out multiplying by 20.
Group Activities
-- Work out the scaled-up dimensions of children and classroom furniture: scale factors 5, 6 and 8.

Day 3 Teaching
Recap mental strategies to divide by 5, by 20, by 6, by 8. Say that a toy shop is selling dinosaurs where each measurement is 1/200 of the size of the real dinosaurs. Ask children in pairs to find 1/200 of the actual measurements.
Group Activities
Use the ‘Get to the root’ in-depth problem-solving investigation below as today’s group activity.
Or, use these activities:
-- Scale down dimensions of dinosaurs by factors of 10 or 100 to sketch on paper. Look at relative sizes.

You Will Need

  • Tape measure or trundle wheel to measure the length of the classroom
  • ‘Solve problems involving rate’ sheet (see resources)
  • Access to the internet
  • ‘Scaling up’ sheet (see resources)
  • Mini-whiteboards and pens
  • ‘Dimensions for dinosaur toys’ sheet (see resources)
  • A3 paper
  • ‘Dinosaur measurements’ sheet (see resources)

Mental/Oral Maths Starters

Day 1
Double and halve 3-digit numbers (pre-requisite skills)

Suggested for Day 2
Multiplication and division facts (simmering skills)

Suggested for Day 3
Order of operations and brackets (simmering skills)

Procedural Fluency

Day 1
Answer problems about growth rates for blue whale calves and humans.

Day 2
Work out actual dinosaur sizes, given model dimensions.

Day 3
Work out the dimensions of dinosaur toys, given full size measurements.

Mastery: Reasoning and Problem-Solving

  • Charlie’s uncle drinks chocolate milk at a rate of 3 pints a day. How much milk does he drink in a week? In a year?
  • Find these products using mental strategies:
    (i) 234 × 5
    (ii) 450 × 20
    (iii) 1270 ÷ 8
    (iv) 253 × 6
    (v) 732 ÷ 5
    Explain which you found the most tricky and why.
  • A doll's house is made so that all the furniture is 1/12 size of actual real furniture. Write the dimensions of these items in the doll's house:
    -- Bed (actual size 2.4m by 1.2m)
    -- Sofa (actual size 1.8m by 120cm)
    -- Wardrobe (actual size 2.4m by 1.8m by 60cm)


In-depth Investigation: Get to the Root
Children use their fluency in mental multiplication to explore the patterns of digital roots in multiplication.

Extra Support

Factors and Multiples Game
Use this game where players take it in turns to cross out numbers, at each stage choosing a number that is a factor or multiple of the number just crossed out by the other player. Factors and Multiples Game from nrich.maths.org.