**New Zealand Curriculum:** Level 3

**Learning Progression Frameworks: **Multiplicative thinking, Signpost 6 to Signpost 7

**Target students**

These activities are intended for students who have some previous experience with equal partitioning, such as finding lines of symmetry in shapes. It is preferable that they have knowledge of addition and multiplication facts though that is helpful but not essential.

The following diagnostic questions indicate students’ understanding of, and ability to apply fractions as multipliers operating on other numbers. At this level the multiplicand (second factor) is restricted 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. (show diagnostic questions)

The questions should be presented orally and in a written form so that the student can refer to them.

**Here are 18 counters. You can have one half of them. How many counters will you get?**

Write an equation for this problem.

__Signs of fluency and understanding:__

Identifies nine as one half of 18, using either 2 x 9 = 18 or 9 + 9 = 18.

Records one of these equations: 1/2 x 18 = 9 or 2 x 9 = 18 or 18 ÷ 2 = 9.

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

Tries various numbers to see what works, such as 6 + 6 = 12 and builds on that result.

Relies on physically dealing the 18 counters between two people, then counts the result.

__Supporting activity:__

Half as an operator

**Here are 24 counters. You can have three quarters of them. How many counters will you get?**

Write an equation for this problem.

__Signs of fluency and understanding:__

Understands that three quarters means three of four quarters that make the whole set of 24. Finds 1/4 of 24 using division by four, or a corresponding multiplication fact, 4 x 6 = 24. Multiplies 3 x 6 = 18 to find three quarters.

Records 3/4 x 4 = 18.

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

Inability to operationalize finding three quarters which may be due to lack of understanding of 3/4 = 1/4 + 1/4 + 1/4.

Additive methods to find one quarter of 24, such as 5 + 5 + 5 + 5 = 20 so 6 + 6 + 6 + 6 = 24. Additive methods used to find 6 + 6 + 6 =18. Such strategies place considerable demand on working memory.

Errors in calculations that are based on multiplication and division may be due to lack of basic fact knowledge.

Inability to record the equation as 3/4 x 24 = 18 that is due to lack of exposure to the multiplication symbol as ‘of’, and limited transfer between whole numbers as multipliers and fractions.

__Supporting activity:__

Non unit fractions as operators

*Here are 42 counters. You can have five sevenths of them. How many counters will you get?*

Write an equation for this problem.

__Signs of fluency and understanding:__

Understands that five sevenths as five sevenths that make the whole set of 42. Finds 1/7 of 42 using division by seven, or a corresponding multiplication fact, 7 x 6 = 42. Multiplies 5 x 6 = 30 to find five sevenths.

Records 5/7 x 42 = 30.

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

Inability to operationalize finding five sevenths which may be due to lack of understanding of 5/7 = 1/7 + 1/7 + 1/7 + 1/7 + 1/7

Additive methods to find one seventh, such as 5 + 5 + 5 + 5 + 5 + 5 + 5 = 35 so 6 + 6 + 6 + 6 + 6 + 6 + 6 = 42 are likely to place considerable demand on working memory. Often students will be unable to complete the strategy even with support of written recording.

Errors in calculations that are based on multiplication and division may be due to lack of basic fact knowledge.

Inability to record the equation as ¾ x 24 = 18 that is due to lack of exposure to the multiplication symbol as ‘of’, and limited transfer between whole numbers as multipliers and fractions.

__Supporting activity:__

Complex non-unit fractions as operators

**Here are 36 counters. You can have two-thirds of the counters or six-ninths of the counters. Which fraction gives you the most counters? Explain why.**

Write equations for this problem.

__Signs of fluency and understanding:__

Understands that two thirds and six ninths are equivalent. Therefore the numbers of counters for both fractions are the same. 2/3 x 36 = 6/9 x 36.

Calculates both amounts using multiplication and division, 2/3 x 36 = 24 and 6/9 x 36 = 24, then realizes that this is due to use of equivalent fractions.

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

Inability to operationalize finding two thirds of 36 or six ninths of 36 which may be due to lack of understanding of 2/3 = 1/3 + 1/3 and 6/9 = 1/9 + 1/9 + 1/9 + 1/9 + 1/9 + 1/9. This issue should show in earlier questions. In ability to write equations for the problem should also have occurred in previous problems.

Attempts additive methods. The demands on working memory will make such strategies difficult even with support of written recording.

Errors in calculations that are based on multiplication and division may be due to lack of basic fact knowledge.

Inability to recognise 2/3 and 6/9 as equivalent fractions is due to limited understanding of fractions as numbers.

__Supporting activity:__

Equivalent fractions as operators

*Here is a rectangle of paper. If you get one quarter of one third of the paper, how much of the whole rectangle do you get? Explain how you know.*

Write an equation for this problem.

__Signs of fluency and understanding:__

Recognises that the answer is one twelfth, using 3 x 4 = 12, and records ¼ x 1/3 = 1/12 or 1/3 x 1/4 = 1/12.

Explains that making quarters creates four equal parts and dividing each part into thirds creates 4 x 3 = 12 equal parts, twelfths.

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

Draws a diagram to model the problem. This shows the student has yet to develop anticipation of the result (a scheme). Similarly they may fold the paper to get an answer.

Inability to write an equation may indicate limited exposure to using x to represent ‘of’ when both factors are fractions.

__Supporting activity:__

Unit fraction of a unit fraction

**Here is a rectangle of paper. Imagine I fold the paper into fifths width-ways and shade four fifths. ****(Demonstrate if necessary and write down the fractions). **

**Then I fold the paper into thirds length-ways. I shade two thirds of four fifths. What fraction of the rectangle is double shaded?**

Write an equation for this problem.

__Signs of fluency and understanding:__

Recognises the problem as multiplication and records 2/3 x 4/5 = 8/15. Explains that the answer, 8/15, is two thirds or four fifths of the rectangle.

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

Recognises the problem as multiplication and records 2/3 x 4/5 = [ ] but is unable to complete the procedure. That indicates the need for fluency with fraction multiplication algorithms.

Unsure about the operation to perform on 2/3 and 4/5, and most commonly chooses addition or subtraction. This indicates a lack of connection between ‘of’ and multiplication as factors change from whole numbers to fractions.

Draws a diagram to model the problem. This shows the student has yet to develop anticipation of the result (a scheme). Similarly they may fold the paper and shade it to get an answer. May or may not be able to recognise the double shaded area as 8/15 of the rectangle.

__Supporting activity:__

Non-unit fraction of a non-unit fraction

#### Teaching activities