MTH1W

Diagnostic Thinking Task

Integers

Teacher Notes

FACILITATING THE DIAGNOSTIC:

Slides with a grey background should not be shown to students.

The purpose of this diagnostic is to learn more about what students recall and what their comfort level is with various concepts. This is NOT a test.

It is HIGHLY recommended that this tool be used in the following ways:

• Provide 1-3 problems to students at a time for a short part of the period
• Emphasize that you want to learn about what they remember - this is NOT a reflection on them
• Emphasize that you want to learn how they think, so the visuals and explanations they give for their thinking will be really helpful to you

Paper Approach: Provide each student with a paper copy of the selected questions (e.g., slides 3-5 on the first day and slides 6 and 7 on the next). Provide two-sided counters or algebra tiles (for integer pieces) to students if needed.

Digital Approach: Provide students with a copy of the slides (only the problem slides) via Google Classroom or your VLE. Have students show their thinking digitally or by inserting pictures onto the slides. If necessary, describe how to use the arrows and counters digitally if you are doing the diagnostic that way. Note: You can stretch the arrows to the appropriate length or you can make copies of the arrows and keep them unstretched. It is better if students do not use calculators and instead show other ways of thinking about number.

For either approach, circulate / give students the explicit option of verbally explaining/showing their thinking to you instead of recording it on paper. Avoid confusion based on running the technology. Sometimes paper / writing by hand on a whiteboard can give you more information than trying to figure out how to do something with the technology. Make notes about observations to use for future record-keeping.

ASSESSING THE DIAGNOSTIC:

Observe where students have difficulty and make notes on the MTH1W Developmental Continuum (Integers tab) for each student. Use the look fors to help guide your observations. These notes will help you identify what you can reteach to close gaps in skills and concepts, differentiate instruction as needed, and provide intervention.

FOLLOW-UP STRATEGIES AND RESOURCES

Suggestions for how to differentiate instruction, close gaps and skills and provide intervention are provided here.

SOME BACKGROUND:

WHY MIGHT STUDENTS STRUGGLE WITH INTEGERS

They might...

• be confused with comparing integers (e.g., students might think -40 > -3 since 40 > 3)
• have difficulties with whole number operations that affect their ability to operate with integers
• struggle with the use of the - sign for both negative integers and subtraction, and the use of the + sign for both positive integers and addition
• consider subtraction only in terms of take-away, making it difficult to model the subtraction of a negative number on a number line
• confuse how to handle the signs for subtracting than for adding because the “rules” are different (e.g., for 5 + (-3), you use the sign of the integer farthest from 0 for the answer, but for 3 - (-5), you don’t)
• over-generalize some relationships with signs (e.g., since 4 - (-3) is the same as the positive amount 4 + 3, students might think that 2 negatives make a positive when adding -4 and -3)

Here is a numberline.

If the black circle is -12, label the other shapes with their possible values on each line:

0

-12

0

-12

Look at your number(s) for the triangle on each line.

Give reasons for the number(s) you chose.

NAME:

The black point below has coordinates (1, 1).

Choose TWO points.

What could their coordinates be?

NAME:

Determine the value of each expression.

Choose an expression. Use a model to show why your answer makes sense.

Explain the model.

NAME:

Choose THREE problems below.

What are the products?

You multiply two integers and the product (or answer) is (–36)

List two possible pairs of integers:

Choose a pair of numbers. Use a model to show why your answer makes sense.

Explain the model.

NAME:

You divide two integers and the quotient (or answer) is –12.

List two possible pairs of integers.

Choose a pair of numbers. Use a model to show why your answer makes sense.

Explain the model.

Choose THREE problems below.

What are the quotients?

NAME:

Look Fors

Prior to Grade 8 Expectation / Grade 8 Expectation / MTH1W Expectation

FOLLOW-UP TEACHING STRATEGIES

Instruction that follows the diagnostic task should be responsive to observations made about students’ current understanding on the continuum. The following are possible teaching strategies you might use to help students develop their understanding.

A word of caution: avoid telling students “rules” to use. Instead, work with students to develop understanding first based on visual representations.

MATERIALS / MANIPULATIVES

Students who struggle with understanding operations with integers will benefit greatly from using visual representations and concrete materials to represent quantities and operations.

Here are some materials / manipulatives that might be beneficial in supporting this development:

ADDITIONAL RESOURCES

These resources provide some sample problems and planning tools that can be used with students to develop their understanding.

Students will benefit from purposeful activities as well as opportunities to discuss and compare their thinking. Practices that encourage students to think - rather than mimic - should be emphasized when using these or any resources.