**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.

The following diagnostic questions indicate students’ understanding of, and ability to apply, multiplication and division to situations that involve scaling. 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. In the assessment 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 a cubes context but can be changed to other contexts that are engaging to your students, such as toy animals or vehicles.

**Here is a stack of 4 cubes. I want you to make a stack that is three times higher. How many cubes will you need?**

**(Ask for a prediction then let the student make their stack of 12 cubes)**

**What fraction of your stack, is my stack?**

__Signs of fluency and understanding:__

Anticipates that 12 cubes will be needed, preferably using 3 x 4 = 12 or by repeated addition, 4 + 4 + 4 = 12. Knows that the stack of 4 cubes fits into (measures) the stack of 12 cubes. Uses the fraction “one third.”

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

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

Adding on 8 cubes, by trial and error, rather than anticipation may indicate a lack of addition and basic fact knowledge.

Creating stacks of 4 cubes and ‘iterating’ the stacks to form a stack of 12 cubes indicates the student understands the meaning of “three times” but may lack fact knowledge to anticipate the result.

Inability to make fractional comparison between 4 and 12 may indicate that the student needs exposure to ‘sets’ models of fractions, both part to whole, and whole to whole. It may also suggest that the student lacks fraction names to describe what they notice.

__Supporting activity:__

Times as many

**I have 5 cubes and you have 20 cubes (model with linking cubes). Tell me about the number of cubes you have, and I have.**

**Can you use addition and subtraction words like “more” or “less” or “difference”?**

**Can you use multiplication and division words like “times as many” or fraction words?**

__Signs of fluency and understanding:__

Understands that comparison might be additive, such as “I have 15 more cubes than you” or “You have 15 less cubes than me.” Can also make multiplicative statements such as “I have four times as many cubes as you” or “You have one quarter the number of cubes I have.”

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

Counting up, or down, (e.g. 4, 5, 6, 7, 8, 9, …, 16, 17, 18, 19, 20) to establish the additive difference of 15 is a sign of the need to develop addition knowledge and strategies.

Ability to make additive comparison but not multiplicative comparison. This observation may indicate that student has yet to encounter both types of comparison.

Inability to make fractional comparison between 5 and 20 may indicate that the student needs exposure to ‘sets’ models of fractions, both part to whole, and whole to whole.

__Supporting activity:__

Comparing sets using addition and multiplication

**Here is a stack of 15 cubes. I want you to make a stack that is five times less than this stack.**

**How many cubes should be in the stack?**

**(Ask for a prediction then let the student make their stack of 3 cubes in their own way)**

**What fraction of my stack, is your stack?**

__Signs of fluency and understanding:__

Anticipates that 3 cubes will be needed, using 5 x □ = 15, or preferably 15 ÷ 5 =3. Uses the fraction “one fifth” and understands that “five times less” signals one fifth.

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

Inability to interpret ‘five time less” as an action suggests the student is unfamiliar with the associated mathematical language and needs more exposure to scaling situations that involve ‘shrinking’.

Trial and error strategies, such as breaking the 15-cube stack into five equal parts, suggests that the student understands the meaning of “five times less” but lacks multiplication or division fact knowledge.

__Supporting activity:__

Times less

**I have 21 cubes. You have 6 times as many cubes as me. How many cubes do you have?**

__Signs of fluency and understanding:__

Using a place value-based strategy that uses the distributive property of multiplication, such as 6 x 20 = 120 and 6 x 1 = 6, the product equals 120 + 6 = 126.

If your student uses a written algorithm question them about the meaning of their working to check that they are applying place value knowledge. Lack of understanding shows when students think that all the digits refer to ones, e.g. “I carried the 1” but is unable to explain that the ‘1’ represents 100.

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

Inability to turn the ‘six times as many” condition into action suggests that the students is unfamiliar with the mathematical language and needs more experience with scaling situations.

Improvised additive strategies can also cause problems for students where there is extra load on working memory. Look for signs like, “21 + 21 = 42, that’s twice as many. Another 21 equals 42 + 21 = 63 ...” If the student uses repeated addition, interrupt them, and ask, “Can you find a more efficient way to work this out?”

__Supporting activity:__

**I have 8 cubes and you have 48 cubes. How many times more cubes do you have than me?**

__Signs of fluency and understanding:__

Using an efficient multiplicative strategy, such considering □ x 8 = 48 (scanning the x 8 tables), or preferably using 48 ÷ 8 = 6.

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

Drawing pictures of the cubes, possibly as bars of 8, may indicate that your student can action the problem but either lacks addition or multiplication facts to solve it, or does not see the potential (affordance) to use those facts.

Building up to 48 additively, such as 8, 16, 24, … shows that your student creates an appropriate action, but lacks the connection between multiplication and repeated addition. Build up strategies are also associated with greater load on working memory.

Inability to create an appropriate action may indicate that your student is unfamiliar with the mathematical language and needs more exposure to scaling situations. Lack of action may also be due to the ‘multiplier unknown’ nature of the problem. The student may have only encountered ‘result unknown’ problems.

__Supporting activity:__

Multiplier-unknown scaling problems

**You have four times as many cubes as me. If you have 36 cubes, how many do I have?**

__Signs of fluency and understanding:__

Using a division-based strategy such as, 36 ÷ 4 = 9.

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

Applying trial and error with multiplication facts, such as 4 x 6 = 24, 4 x 7 = 28, 4 x 8 = 32, 4 x 9 = 36. This strategy may indicate that your student needs to work on division as the inverse operation to multiplication.

Applying trial and error by repeatedly adding one to four stacks. This strategy is likely to be supported by recording of diagrams and/or symbols, and may begin from a known fact, such as four stacks of five (4 x 5) equals 20. This strategy may indicate that the student lacks multiplication fact knowledge or does not see the potential to use facts.

Inability to create an action to solve the problem may be due to your student’s lack of familiarity with the mathematical language, and lack of exposure to ‘multiplicand unknown’ problems. Such problems require understanding of division as the inverse operation to multiplication.

__Supporting activity:__

Multiplicand-unknown scaling problems