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 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 given in order of complexity. If the student answers a question confidently and with understanding proceed to the next question. If not, use the supporting activities to build and strengthen their fluency and understanding. When presenting the questions, 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 with 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:
Unable to turn “three times higher” into an action. This may indicate that the student has yet been exposed to scaling situations and to the associated mathematical language.
Adds on 8 cubes, by trial and error, rather than anticipation. This may indicate a lack of addition and basic fact knowledge.
Creates stacks of 4 cubes and ‘iterates’ the stacks to form a stack of 12 cubes. This indicates that the student understands the meaning of “three times” but may lack fact knowledge to anticipate the result.
Unable to make fractional comparison between 4 and 12. This 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:
Makes additive comparisons, 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:
Counts up, or down, (e.g. 4, 5, 6, 7, 8, 9, …, 16, 17, 18, 19, 20) to establish the additive difference of 15. This may indicate the need to develop addition knowledge and strategies.
Able to make additive comparison but not multiplicative comparison. This may indicate that the student has yet to encounter both types of comparison.
Unable to make fractional comparison between 5 and 20. This 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 the need to create one fifth or divide by five.
What to notice if your student does not solve the problem fluently:
Unable to interpret ‘five time less” as an action. This indicates that the student is unfamiliar with the associated mathematical language and needs more exposure to scaling situations that involve ‘shrinking’.
Uses trial and error strategies, such as breaking the 15-cube stack into five equal parts. This indicates 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:
Uses a place value-based strategy, applying the distributive property of multiplication, such as 6 x 20 = 120 and 6 x 1 = 6, and explains that the product equals 120 + 6 = 126.
If your student uses a written algorithm, question them to check their understanding of place value. 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:
Unable to turn the ‘six times as many” condition into action. This indicates that the student is unfamiliar with the mathematical language and needs more experience with scaling situations.
Use of improvised additive strategies can also cause problems for students by creating additional 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?”
- I have 8 cubes and you have 48 cubes. How many times more cubes do you have than me?
Signs of fluency and understanding:
Uses an efficient multiplicative strategy, such as 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:
Draws pictures of the cubes, possibly as bars of 8. This may indicate that the 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.
Builds up to 48 additively, such as 8, 16, 24, … This may indicate that the student creates an appropriate action, but has not yet made the connection between multiplication and repeated addition. Use of repeated addition also adds greater load toworking memory.
Unable to identify what mathematical operation they should use. This may indicate that the student is unfamiliar with the mathematical language and needs more exposure to scaling situations. This 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:
Uses a division-based strategy such as, 36 ÷ 4 = 9.
What to notice if your student does not solve the problem fluently:
Applies 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 may indicate that your student needs to develop their recognition and use of division as the inverse operation to multiplication.
Applies trial and error by repeatedly adding one to four stacks. This strategy may indicate that the student lacks multiplication fact knowledge or does not see the potential to use facts. 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.
Unable to identify what mathematical operation they should use. This 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