# Mathematics K–12

This is a thinking mathematically targeted teaching opportunity focussed on reasoning to compare which problems you would solve using a mental, written or digital strategy.

Inspired by McIntosh, Reys, Reys and Hope, Number SENSE: Simple Effective Number Sense Experiences, Grades 4-6 (1997)

## Syllabus

Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2023

• MAO-WM-01
• MA2-AR-01
• MA2-AR-02

## Collect resources

You will need:

• pencils or markers

• something to write on.

## Watch

Watch Which would you work out in your head part 1 video (1:05).

Which of these problems would you solve mentally?

### Transcript of Which would you do in your head? Stage 2 part 1 video

[A title over a navy-blue background: Which would you work out in your head 2? Part 1. Below the title is text in slightly smaller font: (Inspired by McIntosh Reys, Reys and Hope). Small font text in the upper left-hand corner reads: NSW Department of Education. In the lower left-hand corner is the white waratah of the NSW Government logo.]

#### Speaker

Hello there mathematicians.

[Text over a navy-blue background: We’re curious…]

So we're a little bit curious.

[A title in a blue text box: Which of these problems would you solve using a mental strategy? Which ones would you solve using a written or digital strategy?

Under the text box are 3 columns of math problems from a to f:

• 4.4 + (blank) = 13.4
• 0.980 – 0.5
• 7.5 + 0.15 + 6.5
• 1001 – 3
• 235 – 44
• 98 + (blank) = 266.]

When you look at these six problems here, which ones of these would you solve using a mental strategy, if any, and which ones would you prefer to solve using a written or digital strategy? Mm hmm. So in your notebook or with your classmates and your teacher, can you record which ones of those would you feel comfortable solving mentally, and which ones would you prefer to use a written or digital strategy? Yeah, and something that you could do, actually…

[Under the math problems, a speech bubble appears which says: You could survey your classmates!]

…is survey your classmates. Uh huh.

[Another speech bubble appears, it reads: Is there a problem that everyone would use mental strategies for?]

And find out, is there a problem or are there problems that everyone would use a mental strategy for?

[Another speech bubble appears, it reads: Is there a problem that everyone would use a written or digital strategy for?]

And also, is there a problem or problems that everyone would use a written or digital strategy for? Over to you, mathematicians and we'll be back together soon.

[Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]

[End of transcript]

## Reflection

• Which of these problems would you solve using a mental strategy?

• Which of these problems would you solve using a written or digital strategy?

## Instructi﻿ons

• Is there a problem that everyone would use mental strategies for?
• Is there a problem that everyone would use a written or digital strategy for?

## Watch

Watch Which would you work out in your head part 2 video (1:05).

Explore student responses to problems.

### Transcript of Which would you do in your head? Stage 2 part 2 video

[A title over a navy-blue background: Which would you work out in your head 2? Part 2. Below the title is text in slightly smaller font: (Inspired by McIntosh Reys, Reys and Hope). Small font text in the upper left-hand corner reads: NSW Department of Education. In the lower left-hand corner is the white waratah of the NSW Government logo.]

### Speaker

Welcome back, mathematicians.

[Text over a navy-blue background: We’re curious…]

Let's have a look, mm-hmm.

[A title in a blue text box: Which of these problems would you solve using a mental strategy? Which ones would you solve using a written or digital strategy?

Under the text box are 3 columns of math problems from a to f:

a. 4.4 + (blank) = 13.4

b. 0.980 – 0.5

c. 7.5 + 0.15 + 6.5

d. 1001 – 3

e. 235 – 44

f. 98 + (blank) = 266

Under the math problems are 3 speech bubbles that says:

· You could survey your classmates!

· Is there a problem that everyone would use mental strategies for?

· Is there a problem that everyone would use a written or digital strategy for?]

### Speaker

So we collected your thoughts, and you might have collected some ideas from your classmates or family and friends too.

[The second speech bubble with text: Is there a problem that everyone would use mental strategies for? turns pink. The third speech bubble with text: Is there a problem that everyone would use a written or digital strategy for? turns green.]

### Speaker

We also asked some students, and we colour-coded their responses. So the ones highlighted in pink are problems that everybody agreed that would use a mental strategy for. So let's have a look at those first.

[The first and last speech bubble are cleared.]

### Speaker

These were the ones that they said they confidently could use a mental strategy.

[A pink text box appears around math problems d, e and f.]

### Speaker

And in this case, they think it would be the most efficient way for them to solve the problem. Yeah, so let's have a look at what they were thinking. They said for 1001 minus 3…

[Under d, the text ‘1001 – 1 – 2 = 998’ appears.]

…if you subtract one that gets you to 1,000, and then subtract two gets you to 998. Mm-hmm.

[Under e, the text ‘235 – 35 = 200’ and ‘200 – 9 = 191’ appears.]

### Speaker

And for 235 minus 44, some of the students were saying, "well, inside of 44 is 35 and nine. "So subtract 35 from 235 to get to the landmark of 200. "And then 200 minus 9 is 191." Mm-hmm.
And then for 98 plus something is 266, they said they would build up.

[Under f, the text ’98 + 2 = 100’, ‘100 + 166 = 266’ and ‘The missing addend is 168’ appears.]

So from 98, if you had two more you get to the nearest hundred. Then from 100, they know they just need 166 more. So that means that the missing addend is 168. Mm-hmm. But there were also some problems where they were like, "oh my gosh…

[the pink text box around d is cleared. A green text box appear around b and c.]

…"We think for this we would like to use a calculator or a written strategy."

[The third speech bubble with text: Is there a problem that everyone would use a written or digital strategy for? appears]

### Speaker

So we thought we'd have a look at these two in particular and see whether there are some mental strategies we could also apply if we were more confident in working flexibly with numbers with decimals. Mm-hmm.

[All other texts and text boxes is cleared except for:

• Is there a problem that everyone would use a written or digital strategy for?

b. 0.980 – 0.5

c. 7.5 + 0.15 + 6.5]

### Speaker

So let's have a look at seven and five tenths plus 15 hundredths plus six and five tenths.

[On a white background, the text box with 7.5 + 0.15 + 6.5] is at the top centre of the screen. Underneath, on the left, is a row of 7 blue squares and 1 rectangle (half the size of the squares). On the right side is a row of 6 blue squares and 1 rectangle (half the size of the squares). In between the two is a thin blue rectangle.]

### Speaker

And the first thing that we noticed actually was about the five tenths…

[The end rectangles move a row down from the squares. In the text box, the .5 in 7.5 and 6.5 of the problem turn white.]

…that seven and five tenths and six and five tenths both had a five tenths.

[The left rectangle moves across to join the right rectangle.]

And we could join that together, mm-hmm, to make another whole which would mean that we're rethinking of the number…

[The two rectangles join to form a square and moves to the right of the 6 squares. In the text box, both the 7.5 and 6.5 becomes 7.]

…. as seven and 15 tenths and seven whole more.

[The right row of 7 squares move under the left row of 7 squares.]

Yeah, and then you could join the two sevens together to get 14 and then add back on…

[The thin rectangle moves down next to the lower row of squares.]

…the 15 hundredths, and you'd get 14 and 15 hundredths.

[The text box text changes to 14.15]

Aha!

[All other texts and text boxes is cleared except for:

b. 0.980 – 0.5

c. 7.5 + 0.15 + 6.5]

So what looked like a problem where we thought, "ooh, I would use a written strategy "or a digital strategy like a calculator,"

[The text box for c turns pink.]

### Speaker

…when actually we could rethink about this by using strategies that we know, mm-hmm, like partitioning numbers and using known facts and renaming, mm-hmm.

So, mathematicians, over to you now to see how you went with your classmates. Are there any of those problems that you could now rethink? And go, "actually let's have a look at, are there any of these "that we could solve mentally that we thought before we wouldn't?"

[Text over a navy-blue background: Over to you, mathematicians.]

So over to you, mathematicians. Have a lovely day.

[Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]

[End of transcript]

## Share and submit

How can you help someone develop confidence in solving 17 + __= 34 using a flexible mental strategy?

• Work together with a classmate to develop your ideas together.