**New Zealand Curriculum:** Level 3

**Learning Progression Frameworks: **Multiplicative thinking, Signpost 4 to Signpost 5

**Target students**

These activities are intended for students who understand multiplication as the repeated addition of equals sets, and who know some of the basic multiplication facts. It is also expected that students have an existing understanding of whole number place value, preferably to six places.

The following diagnostic questions indicate students’ understanding of, and ability to apply, multiplication and division by powers of ten to whole numbers. The questions are in order of complexity. If the student answers a question confidently and with understanding proceed to the next question. If not, then use the supporting activities to build and strengthen their fluency and understanding. Allow access to pencil and paper but not to a calculator unless stated. (show diagnostic questions)

The questions should be presented orally and in a written form so that the student can refer to them. The questions have been posed using various contexts but can be changed to other contexts that are engaging to your students. **Ensure students can record on a piece of paper.**

**The recipe calls for 6 ripe bananas. To make the same cake ten times the size, how many bananas do you need?**

__Signs of fluency and understanding:__

Anticipates that 60 bananas are needed using 10 x 6 = 60.

__What to notice if your student does not solve the problem fluently:__

Inability to turn “ten times bigger” into an action may indicate that the student has yet to be exposed to scaling situations and to the associated mathematical language.

Build-up of sixes by addition or skip counting is unusual but may occur. This indicates that student understands the meaning of “ten times bigger” but does not see an opportunity to use multiplication, or that 10 x 6 has the same answer as 6 x 10 = 60 (6 tens).

__Supporting activity:__

Multiplying by ten

*The recipe calls for 24 tomatoes. To make ten times as much soup, how many tomatoes do you need?*

__Signs of fluency and understanding:__

Anticipates that 240 tomatoes are needed using 10 x 24 = 240.

__What to notice if your student does not solve the problem fluently:__

Inability to turn “ten times bigger” into an action may indicate that the student has yet to be exposed to scaling situations and to the associated mathematical language.

Build-up of 24 by addition is unusual but may occur. This indicates that student understands the meaning of “ten times bigger” but does not see an opportunity to use multiplication, or that 10 x 24 has the same answer as 24 x 10 = 240 (24 tens). The later may indicate that the student does not understand the nested nature of whole number place value, such as ten tens make 100.

__Supporting activity:__

More multiplying by ten

*The phone costs $360. You get the money out of the ATM in $10 notes. How many ten-dollar notes do you have?*

*Explain how you worked out your answer.*

__Signs of fluency and understanding:__

Calculates the number of $10 notes using 36 x 10 = 360 or knows that $360 contains 36 tens (Place value knowledge). Explains answer using language of multiplication or division, e.g., “I know 36 times ten equals 360.”

__What to notice if your student does not solve the problem fluently:__

Build-up by tens and double tracking (possibly as tallies on paper), i.e., “10, 20, 30, …, 360” indicates a lack of place values knowledge, such as ten tens make 100. The tens will be counted to get an answer of 36.

Partial build up and tracking (possibly using pencil and paper), such as 10 tens equal 100, 20 tens equal 200, 3 tens equal 300, 310, 320, 330, …, 360. The number of tens will be tracked somehow, e.g. 10, 20 , 30, 31, 32, …, 36. The student knows that 10 tens make 100 but has not yet learnt to combine the tens in hundreds and decades in a single calculation.

Explanations of “subtracting a zero” may result in a correct answer but further questioning is needed to find out if the student understands the effect on quantities of dividing by powers of ten.

__Supporting activity:__

Dividing by 10

**The recipe calls for:****5 kilograms of oranges****750 grams of sugar****500 millilitres of water****20 grams of butter**

**To make a batch of marmalade that is 100 times bigger, how much of each ingredient do you need?**

**Explain how you worked out each amount.**

__Signs of fluency and understanding:__

Anticipates the amounts correctly using multiplication by 100, i.e. 100 x 1.5 = 150, 100 x 750 = 75 000, 100 x 500 = 50 000, 100 x 20 = 2 000

__What to notice if your student does not solve the problem fluently:__

Inability to turn “100 times bigger” into an action may indicate that the student has yet to be exposed to scaling situations and to the associated mathematical language.

Use of multiplication to correctly find the amounts but difficulty reading the numbers correctly indicates more learning is needed about using place value periods.

Explanations of “adding zeros” may result in correct answers but further questioning is needed to find out if the student understands the effect on quantities of multiplying by powers of ten.

Correct whole number answers but incorrect multiplication of 100 x 1.5 indicates the student needs to work on decimal place value. Common incorrect answers are 100.500, 1.500, and 100.5.

__Supporting activity:__

Multiplying by 100

*A team of 100 people win a prize of $27,080. If the prize is shared equally, how much money does each person get?*

Explain how you got your answer.

__Signs of fluency and understanding:__

Using an efficient division strategy such as 27 080 ÷ 100 = 270.8 . Explain that the answer means $270.80 in context. Use the language of division, such as “I divided because I needed to share the money equally.”

__What to notice if your student does not solve the problem fluently:__

Partial build up and tracking (Using pencil and paper), such as 10 hundreds make 1000, so 270 hundred make 27 000 indicates the student has strong place value knowledge and recognises that the problem involves equal sharing. The student needs to develop efficient calculation strategies for division by powers of ten.

Explanations of “subtracting two zeros” or a written algorithm may result in a correct answer, but further questioning is needed to find out if the student understands the effect on quantities of dividing by powers of ten.

__Supporting activity:__

Dividing by 100

*Here is the number 7 094. Multiply the number by 1000, make it one thousand times larger.*

Explain how you worked out your answer.

Now divide 7 094 by 1000, make it one thousandth of what it was.

Explain how you worked out your answer.

__Signs of fluency and understanding:__

Using multiplication and division-based strategies, such as, 1000 x 7 094 = 7 094 000 and

7 094 ÷ 1000 = 7.094. Recording should show attention to place values up to millions and down to thousandths. Explain that multiplying by 1000 shifts the digits three places to the left and dividing by 1000 shifts the digits three places to the right. Correctly states the answers.

__What to notice if your student does not solve the problem fluently:__

Adding zeros to 7 094 may mean the student has an algorithm for multiplying by 10, 100, 1000, etc. Further inquiry is needed to establish if the algorithm is supported by place value understanding.

Writing the correct answer, 7 094 000 but not being able to say the numbers indicates more fluency is needed with connecting numerals to words.

Inability to conduct the operation 7 094 ÷ 1000 = 7.094 or say the answer as “Seven point zero, nine, four” indicates the student need support with place value understanding of decimals.

__Supporting activity:__

Multiplying and dividing by up to 1000