Getting confident with place value

I thought I would share a bunch of questions I have been using with my low attaining Year 7 class. These guys are pretty good at the process of column addition and subtraction but were not confident with place value such that they could immediately answer a question like this one:

What number is a thousand more than 17407?

I have been dropping one of these into our starters every lesson for the last couple of weeks and they now “get it”.  The next step is to carry over to the next place value, for example:

What number is eighty more than 1843?

We have also tried subtraction, for example:

What number is 500 less than 1935?

Moving on from this I have been using questions like these:

These step up in difficulty quite considerably I think.  I will be using ones like the first few and see how they get on before looking at Qu 7 onwards.

You can access these on this Powerpoint slide. Clicking on the boxes reveals the answer beneath.

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A Decimals starter

Some simple questions to move on from ordering:

What other questions could be asked here?

Powerpoint file, with solutions here.

Zoomable Number Line

This little gadget on Mathisfun.com is so illuminating for teaching decimals. I always get a positive response from students when I show it.

I think it can be used it in a number of ways:

1. Predict what happens when I zoom in. As you zoom in, first the tick marks appear and then at the next level of zoom, numbers start appearing.  It’s a great way of “bridging” from the familiar to the new.  A number line is likely to be a familiar concept.  However, what is “in between” 0 and 1? An intentionally ambiguous question. Students are likely to say “half” or 0.5.  How else can we show half? What other numbers are in between 0 and 1?
2. Why are there 10 ticks between 0 and 1? We have divided 1 into 10 equal parts, what is each part.  How else can we represent 1/10?

From there we “zoom” into the next level of hundredths

This table might be useful as some practice to relate fractions to decimals.  I would love to hear some comments on this.

 

 

 

 

Palindromes

No 2 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey

This is the sort of exercise I can envisage taking a number of different paths depending on what my students do with it which is exciting. The book walks through a generalisation by looking for the lowest 4-digit palindromic number, 1001 and then noting that subsequent palindromic numbers can be found by adding 110. Since 1001 and 110 are multiples of 11, then all numbers in this series are multiples of 11. However, this series misses out other palindromic numbers, e.g. 7557 so we need to refine it further.

I am intrigued to see if this is indeed a path my students would follow or if we would discover something else in these numbers. Depending on the class, I might start by asking “how many 4 digit palindromic numbers are there?”  Before getting into the general, I would see this as an opportunity for purposeful practice of long division if that was something that my students require.  Some students might need a fair amount of direction to reach a proof, but I would aim to make sure that all students left this lesson with an appreciation of that proof even if I had to lead them through it.

 

Place value game

I often play this simple game with younger pupils to help them build a stronger understanding of place value. It’s simple and requires no resources.

Start off by drawing a place value chart on the board.  Depending on where you are at, you could do just hundreds, tens, ones or you could include digits either side of the decimal point, e.g. tens, ones, tenths and hundredths.  Use this as an opportunity to target questions at students as you are drawing the table “if we are doing place value, which column is this?”

Each group then has a row on the place value chart.  You want to limit it to about 6-7 groups otherwise it takes too long to get round to your go again.  It works really well with small classes working in pairs or threes.

Then we start randomly generating digits.  You could just use a dice (it doesn’t matter if you only have digits up to six), or using something like this from Classtools.net

There are a couple of twists that I have added to this over the years.

1, You don’t just get to place the digits in your own row, you can also place them in someone else’s row. So if you get a low digit, you can scupper someone else’s chances by putting it in their thousands column. This depends very much on the culture within the classroom. If you think there might be existing friendship issues amongst the group then it may be best to avoid this twist.

2, You could add some extra options as shown above, e.g. multiply or divide by ten.  This can make it a bit more interesting.  Other options might be to be able to erase a digit.

It’s good fun, but to get the most out of it, it is good to discuss at various stages who will definitely have the highest number / lowest number, etc.  You don’t always need to fill the grid completely to determine the order of the numbers. Why is this?

If you have used this or have any ideas for other “twists”, please drop me a line in the comments.

How many zeros?

A conversation in the Maths department this week about how to extend a particularly strong Year 7 student whilst teaching multiplying and dividing by 10, 100, 1000 led to this idea:

If you multiply all the numbers from 1 to 100 how many zeros are at the end?

I like this question.  It’s actually really quite hard. Have a go. I had previously used is as, “How many zeros are the end of 100! ?” But phrasing it this way gets around the need to explain factorial notation in the middle of a lesson that is not about factorial notation and therefore is can be used as an extension challenge.  Another less challenging, but similar(ish) question:

How many numbers with repeated digits (e.g. 55, 101, 155) are there between 1 and 201?

It is sometimes hard when teaching “basic” topics like decimal place value to find suitable challenge.

Before thinking about finding hard questions, I think it’s worth putting ourselves in the shoes of some of our higher attaining Year 7s and their experience of our mixed attainment classes in the first few months of secondary school.

Primary school experiences vary a lot. Many are taught in sets, especially in larger 2 and 3 form entry schools which are common in London.  My hunch is that the propensity to set children for Maths increases the closer they get to Year 6 although I don’t have any data to back that up (if you know of such data, please share!)  Most of the primaries that I am working with this year that are implementing mastery approaches and so are moving away from setting to mixed attainment teaching, but starting in the lower years.

In many cases, children who felt they were “good at maths” in Primary, developed that self-perception by completing calculations and getting correct answers. After all, that’s what maths is, right?  In some cases they will have established rules in their own minds schemas, etc. An example might be “to multiply by 10, simply add a zero on the end”. And, of course, for integers this works.  Which is why it is so important to highlight misconceptions e.g. 3.4 x 10 = 3.40 as soon as possible before these “false friends” become established.  This is why Year 7 can be such a challenging year for pupils and their teachers.  There is often a need to undo and deconstruct some false ideas that have become well-established before building back up the correct concepts.

So, back to the problem.  Maybe it’s worth saying that there are 24 zeros at the end of that huge number.  That’s the answer.  But as we all know, the answer is just the beginning.

Decimal Place Value Charts

I use these a lot at the early stages of understanding of place value.  In my opinion, place value is the single most important mathematical concept that children need to master in upper KS2 to prepare them for secondary school maths. Many don’t, so it is incumbent on secondary maths teachers to ensure that any gaps are filled in Year 7.

As a pdf.  Or if you want to change things, as a Google Sheets or MS Excel file.

If you print them out on card, you can put them inside transparent A4 plastic wallets and then use mini-whiteboard pens to write on them.  There are different types of plastic wallet – you want the ones that have relatively thick plastic which is smooth, not textured.  Otherwise you will have a hard time rubbing off the pen!

Initial Activities

1. Show me 3 tens.  Now show me Thirty. Why do we need to put the zero in there?
2. Show me 5 tenths. Show me 5 hundreths.  Why do we need the zeros?
3. Show me 34.  What is 34 made up of?
4. Show me 6.75.  What is 6.75 made up of?

From here, of course, you might want to look at multiplying and dividing by powers of 10 and then eventually 4 operations involving decimals.  But don’t rush into that until you are confident they have secured a depth of understanding.  These questions are good for really testing that:

For me these are a crucial AfL tool to ensure the building blocks are in place before doing more complicated things with decimals.

 

 

 

Reinforcing Place Value at every opportunity

So, I learnt something today from a kid in Year 3.  He gave a perfect explanation of a column addition that made me stop and think.  How would you explain this?

His explanation:

“4 plus 1  is 5, so put a 5 in the ones column. Then twenty plus ninety is one hundred and ten, so we put a 1 in the hundreds column and a 1 in the tens column.  Then one hundred plus three hundred is four hundred so put a 4 in the hundreds column”

This has got me thinking about where else I can reinforce place value when discussing procedures.

 

 

Place Value – what is 34 tens and 15 ones?

I’ve been doing a Lesson Study this week with 2 colleagues on Place Value with a lower attaining Year 7 group. I might write about that later, but in the meantime, here is one of the resources that we have developed. I think it’s hugely differentiated, fairly low threshold high ceiling task.

In the lesson, I just wrote a few of these on the board.  It was something I hadn’t used before so I went away thinking about the range of difficulty in these types of questions and wrote a series of questions here.