Cambridge Lower Secondary Maths Teacher Guide 7

Unit 1A • Chapter 1

Place value and rounding Learning objectives Learning objectives covered in this chapter: 7Np1, 7Np2 and 7Np3 

Interpret decimal notation and place value.



Multiply and divide whole numbers and decimals by 10, 100 or 1000.



Order decimals including measurements, changing these to the same units.



Round whole numbers to the nearest 10, 100 or 1000 and

Key terms

• • • • • • • •

digit place value grid state round; rounding; rounded; to the nearest ... whole number decimal multiply divide

decimals, including measurements, to the nearest whole number or one decimal place.

Prior knowledge assumptions

• • • • •

Students know what each digit represents in whole numbers up to 1 million.

Students know what each digit represents in decimals of up to 2 decimal places.

Students can multiply and divide whole numbers by 10, 100 and 1000 where the answer is a whole number.

Students can round numbers to the nearest 10, 100 or 1000.

Students know the relationships between simple units of measurement, for example centimetres and metres.

Online resources

•

End of chapter test

Guidance on the hook Purpose: this hook encourages students to consider the place value of different digits in decimals with up to 2 decimal places, and to reason about how to maximise or minimise the value of such a number.

Use of the hook: Start by rolling a dice yourself and asking students to position the resulting numbers in any position on the grid, repeating until it is full. You should find that different students have produced different final numbers.

Ask if anyone thinks they have the biggest number and, if so, how they know that no one else can have a bigger number.

Then ask if anyone thinks they have the smallest number and, if so, how they know that no one else can have a smaller number.

Then ask students to explore the activity with their own dice in pairs. They should begin to think about the best strategy to make the largest (or smallest) possible number each time.

Bring the whole class back together for a discussion of their strategies.

BEWARE: Some students may perceive that numbers with more high-value digits are greater than those with fewer high-value digits, without realising that the first digit matters the most when deciding the size of the number and so on. © HarperCollinsPublishers 2018

Chapter 1: Place value and rounding

1

Adaptation: You could use an 8-sided dice or pack of number cards to change the possible digits being produced (and alter whether or not they can be repeated).

Extension: You could give students two empty grids to fill up at the same time (thus rolling 8 numbers) and ask them to position numbers to produce the largest and smallest possible pair of numbers.

Starter ideas Mental maths starter Write this number in numerals: Forty-three thousand and sixteen

Write this number in numerals: Eight thousand, two hundred and ninety

Write this number in numerals: Ten thousand and four

Write this number in numerals: Fourteen thousand, nine hundred and seven

Calculate 47 × 100

Calculate 290 × 10

Calculate 402 × 100

Calculate 450 × 10

Calculate 5000 ÷ 10

Calculate 4000 ÷ 100

Calculate 8900 ÷ 10

Calculate 8760 ÷ 10

Write the number that is one more than 8979

Write the number that is one less than 4100

Write the number that is one more than 12 699

Write the number that is one less than 11 000

Round 875 to the nearest 100

Round 878 to the nearest 10

Round 48 to the nearest 100

Round 192 to the nearest 10

Start point check: Where did it come from?

Give each student a mini-whiteboard and ask them to try to answer these questions with a unique answer.

After each question, get students to hold up their boards and discuss the range of answers provided. Then seek out any unique answers, awarding these students with one point each.

1. Write a number that has 4 tens.

2. Write a number that has 4 tens and 9 thousands.

3. Write a number that has 3 tenths and 7 hundredths.

4. Write a number that has 3 tenths and 7 hundredths.

5. Write a number that lies between 389 and 393.

6. Write a number that lies between 24 and 25.

7. A whole number has been rounded to 140 to the nearest 10. What could it be?

8. A whole number has been rounded to 3000 to the nearest 1000. What could it be?

After all the questions, see which students have the most points.

Ask the students which of the questions were easiest to obtain a unique answer on and why this might be.

Human number line You will need a space where students can stand in a line and the remaining students can view what they are holding.

Give each student a number (on a piece of card or on a mini-whiteboard) and ask them to stand in order.

2 • Unit 1A: Number and calculation

© HarperCollinsPublishers 2018

For example, use a subset of the decimal cards shown in the photocopiable resources. For example, 12.9, 13, 12.6, 12.96 and 13.9 Then give some additional students numbers and ask them to go to stand in the correct place between the students already in the line. For example, add in the numbers 12.69 and 12.962 Discuss how each student can find their position with the whole class.

Continue with additional numbers as required.

Note: You could also introduce the idea of scale by marking the end points of the number line and asking students to estimate where they should stand. (For example, for the numbers above you could mark the ends of the number line at 10 and 15 or at 12 and 14).

Discussion ideas Probing questions

Teacher prompts

Convince me/a friend that 0.378 < 0.39

You could encourage students to use place value equipment to represent these numbers to show why one is larger than the other. They may wish to exchange a hundredth for ten thousandths, for example.

What’s the same and what’s different?

Again, you could get students to use place value equipment to represent these numbers to discover that they are all equivalent.

200 thousandths, 0.2, 20 hundredths, 2 tenths, 1 tenth and 10 hundredths

Then, challenge them to come up with another list of 5 numbers that are equivalent!

True or False? “When you divide a number by 1000, you get a decimal answer.”

Encourage students to try out different numbers first and then to give a structured answer that explains that numbers that end in 3 or more zeroes will give a whole number when divided by 1000, but all other numbers with give a decimal.

Convince me/a friend that 17 567 is closer to 18 000 than 17 000

You may wish to encourage students to use a number line to represent the numbers – suggest that they include the halfway value of 17 500 to help them.

Alice says, “The population of Romania is 22 million to the nearest million, and the population of Sri Lanka is 21.9 million to the nearest hundred thousand so Romania must have a higher population than Sri Lanka”. Do you agree with Alice?

Encourage students to explain that Romania’s population can be anything from 21 500 000 to 22 499 999, and Sri Lanka’s population can be anything from 21 850 000 to 21 950 000 and hence it is possible for Sri Lanka to have a higher population than Romania.

Common errors/misconceptions Misconception

Strategies to address

Misreading or misrepresenting numbers with 0s in, for example, 47 008

Ensure you include examples where some digits are 0

Reducing multiplying by 10 to simply ‘adding a 0’ and hence making errors when multiplying a decimal by 10

Start by showing students that this rule cannot work.

For example, can 4.5 ×10 – 4.50? Why not?

© HarperCollinsPublishers 2018

Instead of using such a rule, encourage students to represent numbers with place value equipment to see why the tens become hundreds, ones become tens, tenths become ones etc. when you multiply by 10.

Chapter 1: Place value and rounding

3

Misconception

Strategies to address

Dividing by 10 by simply ‘removing a 0’ and hence making errors when there is no 0 to remove

Start by showing students that this rule cannot work.

For example, how do you remove a 0 when calculating 73 ÷ 10?

Instead of using such a rule, encourage students to represent numbers with place value equipment to see why the hundreds become tens, tens become ones, ones become tenths etc, when you divide by 10.

Making errors when crossing from a whole number to a decimal or vice versa

Start by checking whether the answer seems reasonable using estimation.

For example, students may calculate 2.3 × 100 and wrongly obtain 2300, 2.300 or even 23.0.

For example, how do we know that 2.3 × 100 cannot be 2300? (What is 2 × 100 and so what kind of answer are we expecting?)

Try using the place value grid will help to eliminate these errors.

Finding it difficult to identify the near multiples of 10, 100, 1000 etc. when rounding

This error suggests students have a weaker understanding of the place value of the numbers.

Again, using the place value grid and a number line will help them to position the number mentally.

Struggling to round up when there is a 9-digit that requires rounding to 10 (and hence an effect on the next column)

Start by asking students to consider the nearest multiples.

For example, to round 4798 to the nearest 10, we are choosing between 4790 and 4800.

The key to eliminating this error is ensuring enough exposure to examples where the digit to be rounded up is a 9.

Incorrectly rounding when the answer should be 0

Exposure to examples where the answer could be 0.

For example, round 42 to the nearest 100

Developing conceptual understanding 7Np1 Interpret decimal notation and place value; multiply and divide whole numbers and decimals by 10, 100 or 1000.

•

FIX: Encourage students to represent numbers using a place value grid (Resource 2) so that they can easily read and compare numbers.

1 000 000

100 000

10 000

1 000

100

10

1

millions

hundred thousands

ten thousands

thousandths

hundreds

tens

units

·

0.1

0.01

0.001

tenths

hundredths

thousandths

· •

When multiplying and dividing numbers by 10 (and then 100 and 1000), encourage students to see that each element of the number becomes 10 times bigger or smaller. This will help them understand why the shortcut of moving the digits to the left or the right works and make it more memorable.

TIP: Encourage students to estimate the size of the answer they are expecting to check whether their answer is reasonable.

4 • Unit 1A: Number and calculation

© HarperCollinsPublishers 2018