**New Zealand Curriculum: **Level 2 to Level 3

**Learning Progression Frameworks:** Additive Thinking, Signpost 6 to Signpost 7

#### Target students

These activities are intended for students who understand place value with two- and three-digit numbers but are not yet able to use this knowledge to solve addition and subtraction problems. While these activities assume students know most of their addition and subtraction basic facts (number bonds to 20), some students will still need to work on these. For students who are not yet able to use place value to solve two-digit addition and subtraction problems see Using whole number place value for two-digit addition and subtraction.

The following diagnostic problems indicate students’ understanding of, and ability to use, place value to add and subtract whole numbers to three places. The numbering indicates a progression in complexity. Please do not ask a problem if the student has found the previous problem difficult. (show diagnostic questions)

Present the problems orally and in written form, so the student can refer to the context. You might change the objects in the context to suit the interests of your students. Allow access to pencil and paper but not to a calculator.

**There are 500 people in the town. 460 more people come to the town for the holidays.**

**How many people are in the town now?**

__Signs of fluency and understanding:__

Recognises that the hundred units 500 and 400 combine to form 900, then 60 adds to get a total of 960. Shows clear evidence to understanding the place value of each digit.

If the student uses a written algorithm that does not necessarily mean they understand place value unless they can explain the meaning of each digit.

__What to notice if they don’t solve the problem fluently:__

Counts on in hundreds, 500, 600, 700, 800, 900, then in units of ten, 900, 910, 920, 930, 940, 950, 960. This shows the student has control of some counting sequences, but they do not understand that units of ten and hundred can be added in the same way as ones can be added.

Unable to obtain an answer or may confuse place values to get answers like 546. This shows the student needs to work on the place value of three-digit whole numbers.

__Supporting activity:__

Addition with hundreds and tens

**There are 620 people at the netball game. 130 people leave early.**

How many people are left at the game?

__Signs of fluency and understanding:__

Recognises that the subtracting 120 people from 620 leaves 500 people. They subtract ten to get 490 people left at the game. This indicates the student has good understanding the ten tens make 100.

If the student uses a written algorithm that does not necessarily mean they understand place value unless they can explain the meaning of each digit. The algorithm involves renaming so ask the student about their working to see they know that 100 decomposes into ten tens.

__What to notice if they don’t solve the problem fluently:__

Counts backward in hundreds first, 520, 420, then tens, 420, 410,400, 390. This shows the student has control of some backward counting sequences, but they do not understand that units of ten and hundred can be subtracted in the same way as ones can be. They may also not realise that 100 can decompose into ten tens.

Uses lower from higher in each place value column, often associated with a written algorithm. For example, in the tens place subtract 30 – 20 = 10 rather than 20 – 30.

This suggests that the student needs experience with connecting their calculations to changes in quantities, modelled with place value materials.

__Supporting activity:__

Subtraction with hundreds and tens

**497 people are at the concert. 213 more people come along.**

How many people are at the concert now?

__Signs of fluency and understanding:__

Uses an efficient place value-based mental calculation that commonly involves renaming the calculation as 500 + 210 by moving three from the 213 onto the 497.

Records working as an algorithm or an empty number line.

Neither strategy is efficient but may show understanding of place value if the student can explain the renaming of ten tens as one hundred.

__What to notice if they don’t solve the problem fluently:__

Gets lost in the working due to excessive load on working memory. This frequently happens when the student attempts a mental strategy that combines place value units inefficiently (see number line above). This suggests that the student needs support with recording their working to reduce the memory load.

Performs an algorithm with errors and does not recognise the unreasonableness of the answer. For example:

This suggests that the student needs to work with materials and the written algorithm simultaneously, so they understand what the manipulation of numbers means for the quantities involved.

__Supporting activity:__

Addition with renaming

**There are usually 705 students at the school. 215 students are away today with ill-health.**

How many students are at the school today?

__Signs of fluency and understanding:__

Uses an efficient place value-based mental calculation that commonly involves either:
- Subtracting 200 to get 505 then subtracting 15 to get 490.
- Subtracting 5 from both numbers first to get 700 – 210 then subtracting 200 then subtracting 10.

Records working as an algorithm or an empty number line.

Both strategies are moderately efficient but ask the student to explain the meaning of the digits in the algorithm or to explain the jumps on the empty number line.

__What to notice if they don’t solve the problem fluently:__

Gets lost in the working due to excessive load on working memory. This frequently happens when the student attempts a mental strategy that combines place value units inefficiently (see number line above). This suggests the student needs support to use recording to ease mental load, such as using empty number lines or algorithms.

Performs an algorithm with errors and does not recognise the unreasonableness of the answer. The most common error is to take smaller from bigger. For example:

This suggests that the student needs to work with materials and the written algorithm simultaneously, so they understand what the manipulation of numbers means for the quantities involved.

__Supporting activity:__

Subtraction with renaming in one place

**There are 943 students at Rere School and 479 students at Kopu School.**

How many more students are at Rere School than Kopu School?

__Signs of fluency and understanding:__

Recognises that the problem can be solved by either subtracting 479 from 943 or adding onto 479 to get 943.

Uses some form of recording to manage the memory load of the renaming. Recording might include an empty number line or algorithm and obtains a correct answer of 464.

The student could create a manageable mental strategy by rounding and adjusting the answer, such as 943 – 500 + 21, or 950 – 480 – 7 + 1. Such strategies demonstrate strong understanding of place value and difference and indicate excellent working memory.

__What to notice if they don’t solve the problem fluently:__

Unable to create an operation to solve the problem. This indicates the student needs more experience with difference problems.

Gets lost in the working due to excessive load on working memory. This frequently happens when the student attempts a mental strategy that combines place value units inefficiently (see number line above). This suggests the student needs support to use recording, as a way of easing mental load, such as empty number lines or algorithms.

Performs an algorithm with errors and does not recognise the unreasonableness of the answer. The common error is to take smaller from bigger. For example:

This suggests that the student needs to work with materials and the written algorithm simultaneously, so they understand what the manipulation of numbers means for the quantities involved.

__Supporting activity:__

Difference with renaming in two places

#### Teaching activities